ENDOMORPHISMS OF DELIGNE-LUSZTIG VARIETIES

F. DIGNE AND J. MICHEL

Abstract. We study some conjectures on the endomorphism algebras of the cohomology of Deligne-Lusztig varieties which are a refinement of those of [BMi].

Contents

1. Introduction 1 2. Conjectures 2 3. Regular elements in braid groups 6 4. The case of Coxeter elements 9 5. Regular elements in type A 11 6. Regular elements in type B 16 7. The elements π and w_{0} 19 8. Pieces of the Deligne-Lusztig varieties 25 9. The n-th roots of π in type A_{n} 38 10. Conjecture 2.4 in type A 42 11. Conjecture 2.4 in type B 45 12. The 4-th roots of π in D_{4} 47 References 55

1. Introduction

Let G be a connected reductive algebraic group, defined over an algebraic closure 𝔽 of a finite field of characteristic p. Let F be an isogeny on G such that some power F^{δ} is the Frobenius endomorphism attached to a split 𝔽_{q^{δ}}-structure on G (where q is a real number such that q^{δ} is a power of p). The finite group G^{F} of fixed points under F is called a finite group of Lie type. When considering a simple group which is not a Ree or Suzuki group we may take F to be already a Frobenius endomorphism.

Let W be the Weyl group of G and let B (resp. B^{+}) be the corresponding braid group (resp. monoid). The canonical morphism of monoids β : B^{+} → W has a section that we denote by w → w: it sends an element of W to the only positive braid w such that β(w) = w and such that the length of w in B^{+} is the same as the Coxeter length of w; we write W = {w | w ∈ W } and S = {s | s ∈ S} where S is a set of Coxeter generators for W.

Let us recall how in [BMi] a “Deligne-Lusztig variety” is attached to each element of B^{+}. Let B be the variety of Borel subgroups of G. The orbits of B × B are in natural bijection with W. Let O(w) be the orbit corresponding to w ∈ W. Let b ∈ B^{+} and let b = w_{1} … w_{n} be a decomposition of b as a product of elements of W. To such a decomposition we attach the variety {(B_{1}, …, B_{n+1}) | (B_{i}, B_{i+1}) ∈ O(w_{i}) and B_{n+1} = F (B_{1})}. It is shown in [De, p. 163] that the varieties attached to two such decompositions of b are canonically isomorphic. The projective limit of this system of isomorphisms defines what we call the Deligne-Lusztig variety X(b) attached to b; it is the “usual” Deligne-Lusztig variety X(w) when we take b = w ∈ W.

When W is an irreducible Coxeter group, the center of the pure braid group is cyclic. We denote by π its positive generator; we define π in general as the product of the corresponding elements for the irreducible components of W. Another way of constructing π is as w_{0}^{2} where w_{0} is the longest element of W. The Frobenius endomorphism acts naturally as a diagram automorphism, i.e., an automorphism which preserves S (resp. S), on W (resp. B^{+}); we still denote by F these diagram automorphisms. We call “F -root of order d of π”, an element b ∈ B^{+} such that (bF)^{d} = πF^{d}; in [BMi] it is proved that β(b) is then a regular element of the coset W F (in the sense of [Sp]) for the eigenvalue e^{2iπ/d}; when F acts trivially we just have a root of order d, i.e., b^{d} = π. It is conjectured in loc. cit. that the G^{F} -endomorphisms of the ℓ-adic cohomology complex of X(b) form a “cyclotomic Hecke algebra” attached to the complex reflection group C_{W} (β(b)F). We will show below that for any F -root b of π except for π itself, there is another F -root w ∈ W of π of the same order and an equivalence of étale sites X(b) ≃ X(w), thus the conjecture is about the variety X(π) and some “ordinary” Deligne-Lusztig varieties.

We shall make more specific this conjecture by replacing it by a set of conjectures that we shall study, and prove in some specific examples. In [DMR], of which this paper is a continuation, we already obtained some general results on some of the conjectures. We will get here further results for the element π, and for all roots of π in the case of split groups of type A. We also study powers of Coxeter elements in type B and fourth roots of π in split type D_{4}.

2. Conjectures

First, we should state that a guide for the following conjectures is that, using Lusztig’s results in [Lu2], we will show in section 4 that they all hold in the case of Coxeter elements.

We recall from [DMR, 2.1.1] that a possible presentation of B is

⟨w ∈ W | w_{1}w_{2} = w_{3} when w_{1}w_{2} = w_{3} and l(w_{1}) + l(w_{2}) = l(w_{3})⟩.

We recall the action defined in [DMR, 5.1] of a submonoid of B^{+} on X(w), and of the group it generates (which will be equal to C_{B}(wF) in every case we study) on H_{c}^{∗}(X(w)) = ⊕_{i}H_{c}^{i}(X(w)).

First, we recall the definition of the morphism D_{t} : X(b) → X(t^{−1}bF (t)) defined when t is a left divisor of b: if b = tt′, and if t = w_{1} … w_{n} and t′ = w′_{1} … w′_{n′} are decompositions as products of elements of W, it sends the element (B_{1}, …, B_{n+n′+1}) ∈ X(b) attached to the decomposition w_{1} … w_{n}w′_{1} … w′_{n′} to the element (B_{n+1}, …, B_{n+n′}, F (B_{1}), …, F (B_{n+1})) attached to the decomposition w′_{1} … w′_{n′}F (w_{1}) … F (w_{n}).

Then we introduce categories as in [DMR, 5.1]: B is a category with objects the elements of B, and such that Hom_{B}(b, b′) = {y ∈ B|b′ = y^{−1}bF (y)}; composition of maps corresponds to the product in B; one has End_{B}(b) = C_{B}(bF).

D^{+} is the smallest subcategory of B which contains the objects in B^{+} and such that

{y ∈ B^{+} | y ≼ b, y^{−1}bF (y) = b′} ⊂ Hom_{D^{+}}(b, b′),

where ≼ denotes left divisibility in the braid monoid, and D is the smallest subcategory of B containing D^{+} and where all maps are invertible.

C^{+} is the category of quasi-projective varieties on 𝔽, together with proper morphisms. C is the localized category by morphisms inducing equivalences of étale sites. An isomorphism in C induces a linear isomorphism in l-adic cohomology.

It is shown in [DMR, 5.2.1] that the map which sends the object b to X(b) and the map t to D_{t} extends to a functor D^{+} → C^{+}, which itself extends to a functor D → C.

In the following we denote by H_{c}^{∗}(X) the graded vector space ⊕_{i}H_{c}^{i}(X, \overline{ℚ}_{ℓ}), i.e., the ℓ-adic cohomology with compact support of the quasi-projective variety X. With this notation, the monoid End_{D^{+}}(b) acts on X(b) as a monoid of endomorphisms, and the group End_{D}(b) acts linearly on H_{c}^{∗}(X(b)).

Conjecture 2.1. When b is an F -root of π we have End_{D}(b) = End_{B}(b) = C_{B}(bF).

We will show this conjecture for π, w_{0}, Coxeter elements, all roots of π in types A and B, and 4-th roots of π in type D_{4}. We should note that in [DMR, 5.2.5] we have defined an action of C_{B^{+}(bF)} on X(b) which extends the action of End_{D^{+}}(b), but we are not able to determine its image in H_{c}^{∗}(X(b)) (except via conjecture 2.1).

It is proved in [BMi, 6.8] that, except when b = π, of course, there is a morphism in D^{+} between any d-th F -root b of π and a “good” d-th F -root w, “good” meaning that (wF)^{i} ∈ W.F^{i} for i ≤ d/2. Thus the variety X(b) is isomorphic in C to X(w), as was asserted above. We thus need only to consider a variety associated to a good F -root. We actually need only to consider one of them, according to the

Conjecture 2.2. There is always a morphism in D^{+} between any two F -roots of π of the same order.

This says in particular that two such roots are F -conjugate in B. The result of [BMi, 6.8] shows that it is sufficient to consider “good” F -roots of π in the above conjecture.

We will show this conjecture for w_{0}, for Coxeter elements in split groups and n-th roots of π in split type A_{n}. It has now been proved in split type A as a consequence of a recent result of Birman, Gebhardt and Gonzales-Meneses (personal communication) which states that in general there is a morphism in D^{+} between two conjugate roots, and of the theorem of Eilenberg [E] stating that in type A two roots of the same order are conjugate.

Since, when w is an F -root of π, wF = β(w)F is a regular element of W F (cf. [BMi, 6.6]), the group C_{W} (wF) is naturally a complex reflection group. We will denote by B(w) the corresponding braid group; it is shown in e.g., [BDM] in the case F = Id and in section 3 below in the general case that there is a natural map γ : B(w) → C_{B}(wF) such that the image of β∘γ is C_{W} (wF). We recall the following conjecture from [BDM, 0.1]

Conjecture 2.3. γ is an isomorphism.

The above conjecture is easy when w = π or w = w_{0}. It has been proved in [BDM] for split types A and B. We prove it for Coxeter elements in split groups, and for 4th roots of π in type D_{4}. For this last case we use programs of N. Franco and J. Gonzales-Meneses which compute centralizers in Garside groups.

Assuming conjecture 2.1, and since the operators D_{t} commute with the action of G^{F}, we get an action of B(w) as G^{F} -endomorphisms of H_{c}^{∗}(X(w)). The next conjecture states that this action factors through a cyclotomic Hecke algebra for B(w). Let us recall their definition; the braid group B(w) is generated by so-called “braid reflections” (see definition 3.1) which form conjugacy classes in bijection with conjugacy classes of distinguished reflections in C_{W} (wF) (see again definition 3.1) (cf. [BMR, 2.15 and Appendix 1]). Let C be a set of representatives of conjugacy classes of distinguished reflections in C_{W} (wF); for s ∈ C we choose a representative s of the corresponding class of braid reflections, and we denote by e_{s} the order of s. Let A = \overline{ℚ}_{ℓ}[u_{s,j}]_{s,j} where \{u_{s,j}\}_{\{s∈C,j=0...e_{s}−1\}} are indeterminates. The generic Hecke algebra of C_{W} (wF) over the ring A is defined as the quotient of A[B(w)] by the ideal generated by (s − u_{s,0}) … (s − u_{s,e_{s−1}}). A d-cyclotomic Hecke algebra for C_{W} (wF) is a “one-variable specialization” of the generic algebra which specializes to \overline{ℚ}_{ℓ}[C_{W} (wF)] by the further specialization of the variable to e^{2iπ/d}. To make this precise, we need some definitions: we choose an integer a and we denote by e^{\frac{2iπ}{a|W |}} a primitive a|W |-th root of unity in \overline{ℚ}_{ℓ}; we choose an indeterminate denoted by x^{1/a}. Then a d-cyclotomic Hecke algebra is a specialization of the generic Hecke algebra of the form u_{s,j} ↦ e^{2iπj/e_{s}}(e^{−2iπ/ad}x^{1/a})^{n_{s,j}} for some integers n_{s,j} (it is defined over \overline{ℚ}_{ℓ}[x^{1/a}]; it specializes to \overline{ℚ}_{ℓ}[C_{W} (wF)] by the further specialization x^{1/a} ↦ e^{2iπ/ad}).

Conjecture 2.4. The action of B(w) on H_{c}^{∗}(X(w)) factors through a specialization x ↦ q of a cyclotomic Hecke algebra H(w) for C_{W} (wF).

More precisely we have to state the specialization as x^{1/a} ↦ q^{1/a}. This conjecture is proved for w = w_{0} and w = π in [DMR, 5.4.1] and [BMi, 2.7] respectively (see also [DMR, 5.3.4]). We will prove this conjecture for all roots of π in split type A for roots of even order in type B and for 4-th roots of π in type D_{4}.

Assuming conjecture 2.4, let H_{q}(w) be the above specialization (the specialization for x^{1/a} ↦ q^{1/a} of H(w)). We thus have a virtual representation ρ_{w} of H_{q}(w) on ∑_{i}(−1)^{i}H_{c}^{i}(X(w)). If we decompose ∑_{i}(−1)^{i}H_{c}^{i}(X(w)) = ∑_{λ∈Irr(G^{F})} a_{λ}λ in the Grothendieck group of G^{F}, we get thus for each λ a virtual character χ_{λ} of H_{q}(w) of dimension a_{λ}.

We call a representation special if its trace defines up to a scalar the canonical symmetrizing trace form on H_{q}(w) (see [BMM, 2.1] for the definition of the canonical trace form). The canonical trace form has not been proved to exist for all complex reflection groups; however it is known to exist for those groups that we will encounter in the present paper.

Conjecture 2.5. (i) The χ_{λ} generate the Grothendieck group of H_{q}(w) and

are irreducible up to sign. (ii) The representation ρ_{w} is special.

(i) above means that the image of H_{q}(w) by ρ_{w} is the “full G^{F} -endomorphism algebra of ∑_{i}(−1)^{i}H_{c}^{i}(X(w))”. We will be able to prove conjecture 2.5 when w = π and G is split of type A_{n}, G_{2}, E_{6} and some small rank cases and also for the cases when we can prove the next conjecture.

Conjecture 2.6. The groups H_{c}^{i}(X(w)) are disjoint from each other as G^{F} -modules.

Conjectures 2.5 and 2.6 thus imply that H_{q}(w) ≃ End_{G^{F}} (H_{c}^{∗}(X(w))). Conjecture 2.6 is the hardest in some sense, since it is very difficult to determine individually the cohomology groups of a Deligne-Lusztig variety, except when w is rather short. In addition to the known case of Coxeter elements, we will show conjecture 2.6 for n-th roots of π in type A_{n} and 4-th roots of π in type D_{4}. Also, when G is of rank 2, conjecture 2.6 follows from [DMR, 4.2.4, 4.2.9, 4.3.4, 4.3.5, 4.4.3 and 4.4.4] (with some indeterminacy left in type split G_{2}).

It should be pointed out that conjectures 2.4 to 2.6 are consequences of a special case of the version for reductive groups of the Broué conjectures on blocks with abelian defect. They already have been formulated in a very similar form by Broué, see [Br], [BMa] and [BMi]. In particular, compared to [BMi, 5.7], we have only inverted the order of the assertions, in order to present them by order of increasing difficulty, and made the connection with the braid group a little bit more specific via conjectures 2.1 to 2.3.

3. Regular elements in braid groups

In this section we provide the needed background on braid monoids and groups. We fix a complex vector space V of finite dimension. A complex reflection is

an element of finite order of GL(V) whose fixed point space is a hyperplane. Let W ∈ GL(V) be a finite group generated by complex reflections, let A be the set of reflecting hyperplanes for reflections of W, and let V^{reg} = V − ⋃_{H∈A} H. Choose x ∈ V^{reg} and let \overline{x} ∈ V^{reg}/W be its image. The group Π_{1}(V^{reg}, x) is called the pure braid group of W and the group B = Π_{1}(V^{reg}/W, \overline{x}) is called the braid group of W. The map V^{reg} → V^{reg}/W is an étale covering, so that it gives rise to the exact sequence

1 → Π_{1}(V^{reg}, x) → B ⟶^{β} W → 1.

We denote by N_{W} (E) (resp. C_{W} (E)) the stabilizer (resp. pointwise stabilizer) of a subset E ⊂ V in W.

We choose distinguished generators of W and B as follows (see [BMR, 2.15]):

Definition 3.1. (i) A reflection s ∈ W of hyperplane H is distinguished if its

only non trivial eigenvalue is e^{2iπ/e_{H}} where e_{H} = |C_{W} (H)|. (ii) Let H ∈ A and let s_{H} be a distinguished reflection of hyperplane H. We call “braid reflection” associated to s_{H} an element of B of the form γ ∘ \overline{λ} ∘ γ^{−1}, where: γ is a path from \overline{x} to a point \overline{x_{H}} which is the image of a point x_{H} ∈ V^{reg} “close to H” in the sense that there is a ball around x_{H} which meets H and no other hyperplane, and contains the path λ : t ↦ proj_{H}(x_{H})+ e^{2iπt/e_{H}} proj_{H^{⊥}}(x_{H}) where proj means orthogonal projection (with respect to a chosen W -invariant inner product); and where \overline{λ} is the image of λ.

It is clear from the definition that if s_{H} is a braid reflection associated to s_{H}, then β(s_{H}) = s_{H}. Moreover it is proved in [BMR, 2.8] that braid reflections generate B and in [BMR, 2.14] that β induces a bijection from the conjugacy classes of braid reflections in B to the conjugacy classes of distinguished reflections in W.

We assume now given ϕ ∈ GL(V) normalizing W; thus V^{reg} is ϕ-stable. We fix a d-regular element wϕ ∈ W.ϕ i.e., an element which has an eigenvector in V^{reg} for the eigenvalue ζ = e^{2iπ/d}. We refer to [Sp] and [BMi, §B] for the properties of such elements. If V_{ζ} is the ζ-eigenspace of wϕ, then C_{W} (wϕ) = N_{W} (V_{ζ}); we denote this group by W_{ζ}. Its representation on V_{ζ} is faithful and makes it into a complex reflection group, with reflecting hyperplanes the traces on V_{ζ} of the reflecting hyperplanes of W.

To construct the braid group B(w) of W_{ζ} we choose a base point x_{ζ} in V_{ζ}^{reg} = V^{reg} ∩ V_{ζ}. Then B(w) = Π_{1}(V_{ζ}^{reg}/W_{ζ}, \overline{x}_{ζ}), where \overline{x}_{ζ} is the image of x_{ζ} in V_{ζ}^{reg}/W_{ζ}. Let γ : B(w) → B be the morphism induced by the injection V_{ζ}^{reg}/W_{ζ} ↪ V^{reg}/W_{ζ} composed with the quotient V^{reg}/W_{ζ} → V^{reg}/W. It is shown in [Be1, 1.2 (ii)] that γ factorizes through a (unique) homeomorphism V_{ζ}^{reg}/W_{ζ} ≃ (V^{reg}/W)^{<ζ>}, where for a group G acting on a vector space V we denote by V^{G} the the set of G-fixed points in V.

Let δ be the path t ↦ e^{2iπt/d}x_{ζ} from x_{ζ} to ζx_{ζ} in V_{ζ}^{reg} and let \overline{δ} be its image in V^{reg}/W. Then the map wϕ : λ ↦ \overline{δ} ∘ (wϕ)(λ) ∘ \overline{δ}^{−1} is a lift to B of the action of wϕ on W.

Remark 3.2. Note that we have not defined independently w and ϕ; if ϕ is 1-regular this can be done in the following way: let x be a fixed point of ϕ, and choose a path η_{0} from x to x_{ζ}. Let η_{1} be the path ϕ(η_{0})^{−1} ∘ η_{0}, from ϕ(x_{ζ}) to x_{ζ}. The image in V^{reg}/W of the path δ ∘ w(η_{1}) from x_{ζ} to w(x_{ζ}) is an element w of B which by definition lifts w ∈ W.

Then if \overline{η}_{1} is the image of η_{1} in V^{reg}/W, the map ϕ : λ ↦ \overline{η}_{1}^{−1} ∘ ϕ(λ) ∘ \overline{η}_{1} is an automorphism of B which lifts the action of ϕ on W.

Note also the following:

Remark 3.3. Let ϕ and w be as in remark 3.2, and let π be the loop t ↦ e^{2iπt}x_{ζ} in V^{reg}; it is a generator of the center of the pure braid group. Then (wϕ)^{d} = πϕ^{d} in the semi-direct product B⋊ < ϕ >. Indeed the element (wϕ)^{d}ϕ^{−d} is represented by the path δ ∘ wϕ(δ) ∘ (wϕ)^{2}(δ) ∘ ⋯ ∘ (wϕ)^{d−1}(δ) which is equal to π.

Lemma 3.4. We have γ(B(w)) ⊂ C_{B}(wϕ).

Proof. It is easily checked that for any λ ∈ B the path \overline{δ} ∘ λ ∘ \overline{δ}^{−1} is homotopic to ζ^{−1}λ, so (wϕ)(λ) is homotopic to ζ^{−1}(wϕ)(λ). If λ is the image of a path in V_{ζ}^{reg}, we have ζ^{−1}(wϕ)(λ) = λ; so the action of wϕ on the image of λ in B is trivial, as claimed. □

We now assume that W is a Coxeter group. The space V is the complexification of the real reflection representation of W and the real hyperplanes define chambers. We choose a fundamental chamber, which defines a Coxeter generating set S for W.

When ϕ is trivial, and W not of type F_{4} or E_{n}, the morphism γ has been proved injective [Be1, 4.1]; if moreover W is of type A(split) or B it is proved to be an isomorphism onto C_{B}(wϕ) in [BDM].

We assume now that ϕ induces a diagram automorphism of W, i.e., that it stabilizes the fundamental chamber. Then ϕ is 1-regular: in fact it has a fixed point in the fundamental chamber. We recall the following result from [VdL]:

Proposition 3.5. Assume that we have fixed a base point whose real part is in the fundamental chamber. Let W be the set of elements of B which can be represented by paths λ in V^{reg}, starting from the base point and satisfying the two following properties:

(i) The real part of λ meets each element of A at most once.

(ii) When the real part of λ meets H ∈ A, the imaginary part of λ is on the same side of H as the fundamental chamber; then W is in bijection with W via the map B ⟶^{β} W. Moreover if S ⊂ W is such that β(S) = S then B has a presentation with generators S and relations the braid relations given by the Coxeter diagram of W.

The elements of S are braid reflections.

Corollary 3.6. Let x and y be two points with real parts in the fundamental chamber. If γ is a path from x to y with real part in the fundamental chamber, the isomorphism Π_{1}(V^{reg}/W, x) → Π_{1}(V^{reg}/W, y) which it defines is independent of γ.

Proof. Two such isomorphisms differ by the inner automorphism of Π_{1}(V^{reg}/W, x) defined by a loop with real part in the fundamental chamber. An element defined by a loop is in the pure braid group. But by the proposition 3.5 this element is also in W, so it is trivial. □

Definition 3.7. The braid monoid B^{+} is defined to be the submonoid of B generated by W. Its elements are called positive braids.

Recall that a d-regular element in W.ϕ is called a Springer element if it has a ζeigenvector with real part in the fundamental chamber (cf. [BMi, 3.10]). We choose as base point a fixed point of ϕ in the fundamental chamber. If wϕ is a Springer element we can choose η_{0} in remark 3.2 such that Re(η_{0}) is in the fundamental chamber.

Proposition 3.8. Assume that d > 1 and that wϕ is a Springer element. If η_{0} in remark 3.2 is chosen with real part in the fundamental chamber then the element w is independent of the choice of η_{0} and is in W.

Proof. The element w is represented by δ ∘ w(η_{1}). As Re(w(η_{1})) does not meet any reflecting hyperplane, the only possibility for the real part of this path to meet H ∈ A is that for some t ∈]0, 1[ we have l_{H}(Re(δ(t))) = 0, where l_{H} is a real linear form defining H. Write x_{ζ} = a + ib with a and b real and let θ = 2πt/d; since d > 1 we have θ ∈]0, π[. We have l_{H}(Re(δ(t))) = (cos θ)l_{H}(a) − (sin θ)l_{H}(b) = 0. As this equation has only one solution in ]0, π[, property 3.5 (i) above is satisfied. Moreover

if θ is such a solution we have l_{H}(Im(δ(t))) = l_{H}(a)(sin θ + \frac{cos^{2} θ}{sin θ}) = \frac{l_{H}(a)}{sin θ} which has the same sign as l_{H}(a) since sin θ > 0. So property 3.5 (ii) is also satisfied. □

When d = 1 we still have that similarly π is independent of the choice of η_{0} and is in B^{+}; a way to see this is to use that π = w_{0}^{2} where w_{0} is the element w obtained for d = 2.

We show now how in some cases we can lift to B a distinguished reflection of C_{W} (wϕ). We still assume that wϕ is a Springer element. Recall that a parabolic subgroup of W is the stabilizer of a subset of V; and a standard parabolic subgroup of W is the subgroup generated by a subset of S (it is a parabolic subgroup).

First, to H ∈ A we associate the hyperplane H ∩ V_{ζ} of V_{ζ} and the distinguished reflection t_{H} of W_{ζ} with reflecting hyperplane H ∩ V_{ζ}. Let W_{H} = C_{W} (H ∩ V_{ζ}); it is a parabolic subgroup of W, thus a reflection subgroup. The element wϕ normalizes W_{H} as it acts by ζ on H ∩ V_{ζ}; it is a regular element of W_{H}.wϕ and we have C_{W_{H}}(wϕ) =< t_{H} >. We can apply the constructions of this section to W_{H}: let V^{reg^{H}} = V − ⋃_{{H′∈A|H′⊃H∩V_{ζ}}} H′ and let V_{ζ}^{reg_{H}} = V_{ζ} ∩ V^{reg^{H}}. We get a morphism Π_{1}(V_{ζ}^{reg_{H}}/C_{W_{H}}(wϕ), \overline{x}_{ζ}) → Π_{1}(V^{reg^{H}}/W_{H}, \overline{x}_{ζ}), whose image centralizes wϕ as in lemma 3.4. As x_{ζ} and x are both in the fundamental chamber of W_{H}, by corollary 3.6 any path from x to x_{ζ} whose real part stays in this fundamental chamber defines a canonical isomorphism between Π_{1}(V^{reg^{H}}/W_{H}, \overline{x}_{ζ}) and Π_{1}(V^{reg^{H}}/W_{H}, \overline{x}) which commutes with wϕ. Let us denote by B_{H} this group. By composition with this isomorphism we get a morphism Π_{1}(V_{ζ}^{reg_{H}}/C_{W_{H}}(wϕ), \overline{x}_{ζ}) → B_{H}, whose image centralizes wϕ. Let t_{H} be the generator of the infinite cyclic group Π_{1}(V_{ζ}^{reg_{H}}/C_{W_{H}}(wϕ)) such that its image in C_{W} (wϕ) is t_{H}; then t_{H} is a braid reflection in B(w) and, if e_{H} is the order of t_{H}, then t_{H}^{e_{H}} is the loop t ↦ e^{2iπt}x_{ζ} i.e., the element π_{H} of B_{H}. If H is such that W_{H} is a standard parabolic subgroup generated by a subset I of S then B_{H} = Π_{1}(V^{reg^{H}}/W_{H}, \overline{x}) is canonically embedded in B as the subgroup of B generated by the lift I ⊂ S of I and if s_{H} ∈ B_{H} is the image of t_{H} by the above morphism we have s_{H} ∈ B_{H} ∩ C_{B}(wϕ).

4. The case of Coxeter elements

We prove here, using the results of [BDM], [Be2] and [Lu2] that all our conjectures hold in the case of Coxeter elements for an untwisted quasi-simple reductive group G (the assumption of G being quasi-simple is equivalent to W being irreducible). Even though the results of Lusztig cover them, we are unable to handle twisted groups because the construction of a dual braid monoid when F is not trivial has not yet been carried out.

Let h be the Coxeter number of G and denote by n the semisimple rank of G, which is also the Coxeter rank of W. We begin with

Proposition 4.1. The h-th roots of π are the lift to W of Coxeter elements of W. Conjecture 2.2 holds for h-th roots of π, that is, there is always a morphism in D^{+} between two such roots.

Proof. By e.g., [BMi, 3.11] h-th roots of π exist. Such a root is an element of B^{+} of length n, whose image in W is in the conjugacy class of Coxeter elements by [BMi, 3.12]. Since the minimal length of an element in this conjugacy class is n, which is attained exactly for Coxeter elements, we conclude that an h-th root of π is in W, and its image is a Coxeter element.

Now, by [Bou, chap. V §6, Proposition 1], any two such elements are connected by a morphism in D^{+} (it is easy to identify the conjugating process used in loc. cit. to morphisms in D^{+}). □

We now show

Proposition 4.2. Let c be the lift in W of a Coxeter element. Then C_{B}(c) is the cyclic group generated by c.

Proof. Here we use the results of [BDM] and [Be2]. By [Be2, 2.3.2], B admits a Garside structure where c is a Garside element. By [BDM, 2.26], the centralizer of c is generated by the lcm of orbits of atoms under the action of c. Such an element is a c-stable simple element of the dual braid monoid. By [Be2, 1.4.3], it is the lift to W of an element c_{1} in the Coxeter class of a parabolic subgroup of W. But the centralizer of c in W is the cyclic subgroup generated by c, in particular a simple element centralizes c only if its image in W is a power of c. Thus, we have to show that no power c^{k} of c with 1 < k < h is the image of a simple element, or equivalently that no such power divides c for the reflection length. By [Be2, 1.2.1] this is equivalent to showing that the equality dim ker(c^{k} −1)+dim ker(c^{1−k} −1) = n cannot hold for such k. But, by [Sp, 4.2], the eigenvalues of c are ζ^{1−d_{i}} where ζ = e^{2iπ/h} and where d_{i} are the reflection degrees of W. Thus the equality to study becomes |{i | (1 − d_{i})k ≡ 0 (mod h)}| + |{i | (1 − d_{i})(1 − k) ≡ 0 (mod h)}| = n. Both conditions cannot occur simultaneously since this would imply d_{i} ≡ 1 (mod h), which is impossible since the irreducibility of W implies that 1 < d_{i} ≤ h. Thus it is sufficient to exhibit a d_{i} which satisfies neither condition. But d_{i} = h itself is such a d_{i}, whence the result. □

It follows immediately from proposition 4.2 that conjecture 2.1 holds for Coxeter elements, that is End_{D}(c) = C_{B}(c) = <c>, since by definition c ∈ End_{D}(c).

Let us now prove that conjecture 2.3 holds. The space V_{ζ} is one-dimensional, and for any x_{ζ} ∈ V_{ζ}^{reg} = V_{ζ} − {0} the group Π_{1}(V_{ζ}/C_{W} (c), x_{ζ}) is cyclic, generated by the loop b = t ↦ e^{2iπt/h}. Doing if necessary a conjugation in B, we may take any h-th root of π to prove conjecture 2.3. We will choose a Springer element, so we may assume that Re(x_{ζ}) is in the real fundamental chamber of W. Then, by proposition 3.8, the image γ(b) is in W, and since its image in W is c it is equal to c. We have shown that γ is an isomorphism.

Conjectures 2.5 and 2.6 will follow from the following proposition from [Lu2]:

Proposition 4.3. Let c be a Coxeter element. Then

(i) F is a semisimple automorphism of ⊕_{i}H_{c}^{i}(X(c)); it has h distinct eigenvalues; the corresponding eigenspaces are mutually non-isomorphic irreducible G^{F} -modules.

(ii) For s = 1, …, h − 1, the endomorphism F^{s} has no fixed points on X(c). (iii) The eigenvalues of F are monomials in q which, under the specialization

q ↦ e^{2iπ/h}, specialize to 1, ζ, ζ^{2}, …, ζ^{h−1} where ζ = e^{2iπ/h}.

Proof. (i) is [Lu2, 6.1 (i)]; (ii) is 6.1.2 of loc. cit.; (iii) results from the tables pages 146–147 of loc. cit. □

We show now how this implies conjectures 2.5 and 2.6. Conjecture 2.6 is immediate from proposition 4.3(i). The generic Hecke algebra H(c) of the cyclic group C_{W} (c) of order h is generated by one element T with the relation (T − u_{0}) … (T − u_{h−1}) = 0. The map which sends c to D_{c} = F is thus a representation of this algebra, specialized to u_{i} ↦ λ_{i} where λ_{0}, …, λ_{h−1} are the eigenvalues of F on ⊕_{i}H_{c}^{i}(X(c)). By proposition 4.3(iii) this is indeed an h-cyclotomic algebra for C_{W} (c). It remains to see that the virtual representation ∑_{i}(−1)^{i}H_{c}^{i}(X(c)) of H(c) is special. But, by e.g., [BMa, 2.2], the symmetrizing trace on H(c) is characterized by its vanishing on T^{i} for i = 1, …, h − 1. By the Lefschetz trace formula, one has ∑_{i}(−1)^{i} Trace(F^{s}|H_{c}^{i}(X(c), \overline{ℚ}_{ℓ})) = |X(c)^{F^{s}}|, so this vanishing is a consequence of proposition 4.3(ii).

5. Regular elements in type A

We prove conjecture 2.1 for roots of π when W is of type A and F acts trivially on W. Here we assume that d > 1. The case of π will be treated in section 7.

Let W be a Coxeter group of type A_{n−1} and let B be the associated braid group. There are d-regular elements in W for any d dividing n or n − 1. To handle the case d|n − 1, it will be simpler to embed W as a standard parabolic subgroup of a group W′ of type A_{n}, to embed B as the corresponding subgroup of the associated braid group B′ and to consider d-regular elements for d|n in W′.

We denote by S = {σ_{1}, …, σ_{n}} the set of generators of B′, the generators of B being σ_{1}, …, σ_{n−1}, where the Coxeter diagram of W′ is ◯_{σ_{1}} ◯_{σ_{2}}⋯◯_{σ_{n−1}} ◯_{σ_{n}}. We

denote by π and π′ respectively the positive generators of the centers of the pure braid groups respectively associated to W and W′.

Let c = σ_{1}σ_{2} … σ_{n−1}, the lift in W of a Coxeter element. Let r and d be two integers such that rd = n and let w = c^{r}; it is a d-th root of π (cf. proposition 4.1). The image w of w in W is a regular element of order d and its centralizer C_{W} (w) is isomorphic to the complex reflection group G(d, 1, r) which has a presentation given

by the diagram: ◯_{t}^{d} ◯_{s_{r−1}}⋯◯_{s_{2}} ◯_{s_{1}} .

We denote by B(d, 1, r) the braid group associated to G(d, 1, r); it has a presentation given by the same diagram (deleting the relations giving the orders of the generators).

Conjecture 2.3 holds in our case. Indeed, Bessis [Be1, 4.1] has proved that the morphism γ : B(w) → C_{B}(w) is injective and in [BDM], this morphism is proved to be bijective. More precisely Bessis proves that γ(s_{i}) = ∏_{j=0}^{d−1} σ_{i+rj} and γ(∏_{i=1}^{r−1} s_{i}t) = c, where t, s_{1}, …, s_{r−1} are the generators of B(w) given in [BMR, 3.6]: they are braid reflections which satisfy the braid relations given by the above diagram for G(d, 1, r). We shall identify B(w) with its image, so that we shall identify s_{i} and t with the elements given by the above formulas.

Let now c′ = σ_{1} … σ_{n}σ_{n}, a n-th root of π′ in B′, cf. [BMi, A1.1]. Let w′ = c^{′r}; it is a d-th root of π′ and if w′ is its image in W′, the centralizer C_{W′}(w′) is also isomorphic to G(d, 1, r). It has been proved in [BDM, 5.2] that C_{B}(w) ≃ C_{B′}(w′). More precisely, let X_{n} be the configuration space of n distinct points in ℂ, let μ_{n} be the set of n-th roots of 1, and let ν_{n+1} = μ_{n}∪{0}. We have B′ = Π_{1}(X_{n+1}, μ_{n+1}). Let X_{n}^{∗} be the configuration space of n non-zero distinct points in ℂ; we have morphisms Π_{1}(X_{n+1}, ν_{n+1}) ⟵^{Ψ} Π_{1}(X_{n}^{∗}, μ_{n}) ⟶^{Θ} Π_{1}(X_{n}, μ_{n}), (these morphisms are called A and B in loc. cit.). One gets the map Ψ by adjoining to a braid a constant string at 0.

If we choose an isotopy from ν_{n+1} to μ_{n+1} we get an isomorphism of Π_{1}(X_{n+1}, ν_{n+1}) with B′; this can be done by bringing 0 along a path ending at e^{2iπ \frac{n}{n+1}} .

Similarly, if we choose an isotopy mapping the n first n + 1-th roots of 1 to μ_{n}, we get an isomorphism α : Π_{1}(X_{n}, μ_{n}) ⟶^{∼} B.

It is shown in loc. cit. that the map α ∘ Θ has a section Θ′ above C_{B}(w).

Let ψ be the restriction to C_{B}(w) of Ψ ∘ Θ′; it is an isomorphism from C_{B}(w) to C_{B′}(w′). We have ψ(c) = c′ and ψ(s_{i}) = s_{i}. Let t′ = ψ(t); it satisfies ∏_{i=1}^{r−1} s_{i}t′ = c′.

The following theorem proves conjecture 2.1 in type A. Note that if we have a standard parabolic subgroup W_{1} of a Coxeter group W_{2}, and if B_{1} and B_{2} are the corresponding braid groups, the category D_{1} (resp. D_{1}^{+}) associated to B_{1} as in section 1 is a full subcategory of the category D_{2} (resp. D_{2}^{+}) associated to B_{2}. This allows us in the following theorem to state the results in term of the categories associated to B′. We will denote these categories by D and D^{+}.

Theorem 5.1. One has s_{i} ∈ End_{D^{+}}(w) ∩ End_{D^{+}}(w′), t ∈ End_{D}(w) and t′ ∈ End_{D}(w′), so that End_{D}(w) = C_{B}(w) ≃ B(d, 1, r) and End_{D}(w′) = C_{B′}(w′) ≃ B(d, 1, r).

The end of this section is devoted to the proof of that theorem.

In the next lemma ≼ (resp. ≽) denotes left divisibility (resp. right divisibility) in the braid monoid.

Lemma 5.2. Let B be the braid group of an arbitrary finite Coxeter group; let x, y, z ∈ B^{+} be such that xy^{−1}z ∈ B^{+}. Then there exist x_{1}, z_{1} ∈ B^{+} such that x ≽ x_{1}, z_{1} ≼ z and y = z_{1}x_{1}.

Proof. Let b = xy^{−1}z, so that y^{−1}z = x^{−1}b. By [Mi, 3.2], if we denote by z_{1} the left gcd of y and z, and by x_{2} the left gcd of x and b, we have z_{1}^{−1}y = x_{2}^{−1}x, whence the result, putting x_{1} = x_{2}^{−1}x. □

Lemma 5.3. For i = 1, …, n, let c_{i} = σ_{1} … σ_{i}.

(i) We have c_{k}σ_{i} = σ_{i+1}c_{k} for i < k. (i’) We have c′σ_{i} = σ_{i+1}c′ for i < n − 1. (ii) We have ^{c^{2}}σ_{n−1} = σ_{1}. (ii’) We have ^{c^{′2}}σ_{n−1} = σ_{1}. (iii) For x ∈ B^{+}, one has σ_{i+1} ≼ c_{k}x ⇔ σ_{i} ≼ x for i < k.

(iv) For j ≤ k, we have {i | σ_{i} ≼ c_{k}^{j}} = {1, …, j}.

(iv’) For j ≤ n we have {i | σ_{i} ≼ c^{′j}} = {1, …, j}.

(v) c^{′j} = c_{n}^{j}σ_{n−j+1} … σ_{n} for 1 ≤ j ≤ n.

Proof. Let us prove (i) and (i’). We get c_{k}σ_{i} = σ_{i+1}c_{k} by commuting σ_{i} (resp. σ_{i+1}) with the factors of c_{k} and applying once the braid relation between σ_{i} and σ_{i+1}. Moreover, as c′ = c_{n}σ_{n} and σ_{n} commutes with σ_{i} for i < n − 1, we get also (i’).

From (i), by induction on j we get (v).

Let us prove (ii). For proving ^{c^{2}}σ_{n−1} = σ_{1}, we use (i) to get cσ_{1} … σ_{n−2} = σ_{2} … σ_{n−1}c, then we multiply both sides on the right by σ_{n−1} and on the left by σ_{1} to get σ_{1}c^{2} = c^{2}σ_{n−1}.

We deduce (ii’): we have ^{c′}σ_{n−1} = ^{cσ_{n}}σ_{n−1} = ^{cσ_{n−1}^{−1}}σ_{n} = ^{σ_{n}^{−1}c}σ_{n}, so that ^{c^{′2}}σ_{n−1} = cσ_{n}σ_{n}^{−1}cσ_{n} = σ_{1} by (ii).

Let us prove (iii). We have σ_{i+1} ≼ c_{k}x ⇔ σ_{i+1}^{−1}c_{k}x ∈ B^{+}; by (i) σ_{i+1}^{−1}c_{k}x = c_{k}σ_{i}^{−1}x and by lemma 5.2 this implies that either σ_{i} ≼ x or c_{k} ≽ σ_{i}. But, as no braid relation can be applied in c_{k}, the only j such that c_{k} ≽ σ_{j} is k. So we are in the case σ_{i} ≼ x, whence the implication from left to right. The converse implication comes from the fact that σ_{i} ≼ x ⇒ c_{k}σ_{i}^{−1}x ∈ B^{+}.

Let us prove (iv). If σ_{i} ≼ c_{k}^{j} we cannot have i > j otherwise applying j times (iii), we get σ_{i−j} ≼ 1 which is false. Conversely, if i ≤ j, by applying i − 1 times (iii) we get σ_{i} ≼ c_{k}^{j} ⇔ σ_{1} ≼ c_{k}^{j−i+1} which is true.

(v) is a direct computation using (i).

Let us prove (iv’). By (iv) and (v), we have σ_{i} ≼ c^{′j} for i ≤ j. On the other hand by (iii) and (v) we see that if σ_{i} ≼ c^{′j} with i > j, then σ_{i−j} ≼ σ_{n−j+1} … σ_{n} which is impossible. □

In the next lemma, as in [DMR, 5.2.7], for I ⊂ S we denote by B_{I}^{+} the submonoid of B^{+} generated by I = {s ∈ S | s ∈ I}, and by D_{I}^{+} the “parabolic” subcategory of D^{+} where we keep only the maps coming from elements of B_{I}^{+}.

Lemma 5.4. Assume 1 ≤ i < r and let I_{i} = {σ_{i}, σ_{i+r}, …, σ_{i+(d−1)r}}. Then (i) s_{i} ∈ End_{D_{I_{i}}^{+}}(w) ∩ End_{D_{I_{i}}^{+}}(w′). In particular s_{i} ∈ End_{D^{+}}(w) ∩ End_{D^{+}}(w′). (ii) The conjugation by either w or w′ stabilizes I_{i} and induces on this set the

cyclic permutation σ_{i+jr} ↦ σ_{i+(j+1)r (mod n)}.

Proof. We can assume r ≥ 2 otherwise there is nothing to prove. Let y_{j} = σ_{i+r(j−1)} for j = 1, …, d, and, following the notation of [DMR, 5.1.1 (i)] let w_{1} = w and w_{j+1} = y_{1}^{−1}wy_{j} for j = 1, …, d. We have w_{j} ∈ B^{+} as y_{1} = σ_{i} divides w = c_{n−1}^{r} by lemma 5.3(iv). We have w_{j+1} = y_{j}^{−1}w_{j}y_{j} by using ^{w}y_{j−1} = y_{j}, which is a consequence of lemma 5.3(i), and the fact that y_{j} and y_{1} commute as r ≥ 2; from lemma 5.3(ii) we have ^{w}y_{d} = y_{1}, whence w_{d+1} = w so that (i) is proved for w. We get (i) for w′ by the same computation, replacing w by w′ and using lemma 5.3 (i’), (ii’) and (iv’) instead of lemma 5.3 (i), (ii) and (iv). We have also got (ii) along the way. □

To prove that t ∈ End_{D}(w), we shall find y ∈ B^{+} such that ywy^{−1} ∈ B^{+}, y ∈ Hom_{D^{+}}(ywy^{−1}, w) and yty^{−1} ∈ End_{D^{+}}(ywy^{−1}). Then, as D is a groupoid, we will get t ∈ End_{D}(w).

We will follow the same lines to prove that t′ ∈ End_{D}(w′).

Lemma 5.5. When j ≤ k, let σ_{j,k} = σ_{j}σ_{j+1} … σ_{k} and let x_{i} = σ_{i,i+r−2}. With this notation, let y = ∏_{i=1}^{d−1} y_{i} where y_{i} = ∏_{j=d}^{i+1} x_{i(r−1)+j}. Then ywy^{−1} ∈ B^{+} and y ∈ Hom_{D^{+}}(ywy^{−1}, w).

Proof. We set w_{d} = w, and then by decreasing induction on i we define y_{i}^{−1}w_{i}y_{i} = w_{i+1}. It is enough to show that y_{i}^{−1}w_{i} ∈ B^{+}; we will get this by proving by induction that w_{i} = y_{i}c^{r−1}c_{i(r−1)+d−1}. This formula is true for i = d (here y_{d} = 1). Let us assume it true for i + 1 and let us prove it for i. We have

w_{i+1} = y_{i+1}c^{r−1}c_{(i+1)(r−1)+d−1}
= x_{(i+1)(r−1)+d} … x_{(i+1)(r−1)+i+1}c^{r−1}c_{(i+1)(r−1)+d−1}
= c^{r−1}x_{i(r−1)+d} … x_{i(r−1)+i+1}c_{(i+1)(r−1)+d−1} (by 5.3(i))
= c^{r−1}c_{(i+1)(r−1)+d−1}x_{i(r−1)+d−1} … x_{i(r−1)+i} (by 5.3(i))
= c^{r−1}c_{i(r−1)+d−1}x_{i(r−1)+d}x_{i(r−1)+d−1} … x_{i(r−1)+i}
= c^{r−1}c_{i(r−1)+d−1}y_{i}

which, conjugating by y_{i}, gives the stated value for w_{i} . □

We note that in particular we have ywy^{−1} = w_{1} = c^{r−1}c_{d−1}y_{0} = σ_{r,r+d−2}c^{r−1}x_{d} … x_{1}. We prove now the analogous lemma for w′. Let c′_{i} = c_{i}σ_{i}.

Lemma 5.6. Let y′ = ∏_{i=1}^{d−1} y′_{i} where y′_{i} = ∏_{j=d+1}^{i+1} x_{i(r−1)+j}. Then y′w′y^{′−1} ∈ B^{+} and y′ ∈ Hom_{D^{+}}(y′w′y^{′−1}, w′).

Proof. We can assume r ≥ 2 since for r = 1 we have y′ = 1. We set w′_{d} = w′, and then by decreasing induction on i, we define w′_{i} = y′_{i}w′_{i+1}y_{i}^{′−1}. We will show that y_{i}^{′−1}w′_{i} ∈ B^{+} by proving by decreasing induction on i that w′_{i} = y′_{i}c_{rd}^{r−1}c′_{(r−1)i+d}. This equality has no meaning for i = d, as x_{dr+1} does not make sense. So we rewrite it as w′_{i} = c_{rd}^{r−1}δ_{r−1}(y′_{i})c′_{(r−1)i+d}, where we define formally δ_{i}(x_{j}) = x_{j−i} which makes sense when j − i ≤ n. Now the formula for i = d becomes w′ = c_{rd}^{r}σ_{rd−r+1,rd}, which holds by lemma 5.3 (v). This is the starting point of the induction. Let us assume the formula true for i and let us prove it for i − 1. We have

w′_{i} = c_{rd}^{r−1}δ_{r−1}(y′_{i})c′_{(r−1)i+d}
= c_{rd}^{r−1}δ_{r−1}(y′_{i})c_{(r−1)i+d}σ_{(r−1)i+d}
= c_{rd}^{r−1}c_{(r−1)i+d}δ_{r−2}(y′_{i})σ_{(r−1)i+d}

the last equality as r ≥ 2 implies that the largest j such that σ_{j} occurs in δ_{r−1}(y′_{i}) is at most (r − 1)i + d,

= c_{rd}^{r−1}c_{(r−1)(i−1)+d}y′_{i−1}σ_{(r−1)i+d}
= c_{rd}^{r−1}c′_{(r−1)(i−1)+d}y′_{i−1},

where we get the last line by commuting σ_{(r−1)i+d} and x_{(i−1)(r−1)+j} for j < d, and applying formula x_{a+1}x_{a}σ_{a+r−1} = σ_{a}x_{a+1}x_{a} to the two first terms of y′_{i−1}. □

From lemma 5.6 we get y′w′y^{′−1} = w′_{1} = c_{rd}^{r−1}c′_{d}y′_{0} = σ_{r,r+d−1}σ_{r+d−1}c_{rd}^{r−1}x_{d+1} … x_{1}. Let us now prove

Lemma 5.7. (i) We have yty^{−1} = σ_{r,r+d−2} (ii) We have y′t′y^{′−1} = σ_{r,r+d−1}σ_{r+d−1}.

Proof. Let us prove (i). We show that t = ∏_{j=1}^{d−1} a_{jr,(j+1)r−1} where a_{k,l} = σ_{k,l}σ_{k,l−1}^{−1}. For this, replace each a_{k,l} by its value to get ∏_{i=1}^{d−1} σ_{ir,(i+1)r−1} ∏_{i=1}^{d−1} σ_{ir,(i+1)r−2}^{−1}, which is equal to σ_{r,dr−1} ∏_{i=1}^{d−1} σ_{ir,(i+1)r−2}^{−1}, in turn equal to c_{r−1}^{−1}c ∏_{i=1}^{d−1} σ_{ir,(i+1)r−2}^{−1}. We can put c on the right of this product if we replace σ_{k,l} by σ_{k+1,l+1} for all k, l. We get ∏_{i=0}^{d−1} σ_{ir+1,(i+1)r−1}^{−1}c, which is equal to t = (∏_{i=1}^{r−1} s_{i})^{−1}c, as wanted, since ∏_{i=1}^{r−1} s_{i} = ∏_{j=0}^{d−1} x_{jr+1} where all factors commute.

By decreasing induction on i, we find ∏_{j=i}^{d−1} y_{j} = ∏_{j=d}^{i+1} σ_{ir+j−i,jr−1}, so, setting z_{i} = σ_{i+r,(i+1)r−1}, we have y = ∏_{i=d−1}^{1} z_{i}. But z_{j} conjugates a_{jr,(j+1)r−1} into a_{j+r,(j+1)r−1} and a_{j+r−1,(j+1)r−1} into σ_{j+r−1}. It commutes with σ_{i} for i < j + r − 1 and with a_{ir,(i+1)r−1} for i > j. So by induction on j we see that ∏_{i=j}^{1} z_{i} conjugates t into (∏_{i=1}^{j} σ_{i+r−1})a_{j+r,(j+1)r−1}(∏_{i=j+1}^{d−1} a_{ir,(i+1)r−1}), whence (i).

We now prove (ii). We claim that y′ = σ_{d+r,dr}y: indeed y′_{i} = x_{i(r−1)+d+1}y_{i} and x_{i(r−1)+d+1} commutes with y_{k} for k < i; this gives the claim as ∏_{i=1}^{d−1} x_{i(r−1)+d+1} = σ_{d+r,dr}.

Let us now conjugate t′ = tσ_{dr}^{2} by y′ = σ_{d+r,dr}y. By (i), y′ conjugates t into σ_{r,r+d−2} as σ_{d+r,dr} commutes with σ_{r,r+d−2}. On the other hand conjugation by y′ has the same effect on σ_{dr} as conjugation by σ_{d+r,dr}z_{d−1} = σ_{d+r,dr}σ_{d+r−1,dr−1} as z_{i} for i < d−1 commutes with σ_{dr}. It remains to see that σ_{d+r,dr}σ_{d+r−1,dr−1} conjugates σ_{dr} into σ_{d+r−1} i.e., σ_{d+r,dr}σ_{d+r−1,dr−1}σ_{dr} = σ_{d+r−1}σ_{d+r,dr}σ_{d+r−1,dr−1}; this can be written σ_{d+r,dr}σ_{d+r−1,dr} = σ_{d+r−1,dr}σ_{d+r−1,dr−1}, which is true. □

Corollary 5.8. We have t ∈ End_{D}(w) and t′ ∈ End_{D}(w′).

Proof. If I = {σ_{r}, …, σ_{r+d−2}}, then σ_{r,r+d−2} ∈ End_{D_{I}^{+}}(ywy^{−1}), as σ_{r,r+d−2} ≼ ywy^{−1}. By lemma 5.5 and the remarks made above that lemma, we get the first assertion.

Similarly, if I′ = {σ_{r}, …, σ_{r+d−1}}, then σ_{r,r+d−1}σ_{r+d−1} ∈ End_{D_{I′}^{+}}(y′w′y^{′−1}) as σ_{r,r+d−1}σ_{r+d−1} ≺ ywy^{−1}. By lemma 5.6, we get the second assertion. □

6. Regular elements in type B

We now prove conjecture 2.1 for roots of π when W is of type B_{n}. We will see W as the centralizer in a Coxeter group W′ of type A_{2n−1} of the longest element w_{0}. Let V′ be the vector space which affords the reflection representation of W′. As w_{0} is a 2-regular element, the eigenspace V′_{−1} of w_{0} affords the reflection representation of W, and if we choose a base point in V_{−1}^{′reg} we get an embedding Π_{1}(V^{reg}/W) ↪ Π_{1}(V^{′reg}/W′) of the braid group B of type B_{n} into the braid group B′ of type A_{2n−1}. If S′ = {σ′_{1}, … σ′_{2n−1}} is the generating set of B′ then it is known (see [Mi, 4.4]) that σ_{1} = σ′_{n} and σ_{n+1−i} = σ′_{i}σ′_{2n−i} for 1 ≤ i < n are generators of B such that the relations are given by the Coxeter diagram ◯_{σ_{1}} ◯_{σ_{2}} ◯_{σ_{3}}⋯◯_{σ_{n}}.

Let w be a d-regular element of W for some d, and let ζ = e^{2iπ/d}. With notation as above 3.2, we have V_{ζ}^{reg}/C_{W} (w) ≃ (V^{reg}/W)^{<ζ>} and V^{reg}/W = V_{−1}^{′reg}/C_{W′}(w_{0}) ≃ (V^{′reg}/W′)^{<−1>}, so that V_{ζ}^{reg}/C_{W} (w) ≃ (V^{′reg}/W′)^{<−1,ζ>}. This is equal to (V^{′reg}/W′)^{<ζ>} if d is even and to (V^{′reg}/W′)^{<ζ′>} with ζ′ = e^{iπ/d} if d is odd. As C_{W} (w) = C_{W′}(w, w_{0}), we see that if w_{0} is a power of w, which implies that d is even, then C_{W} (w) = C_{W′}(w). In this case, as V_{ζ}^{reg} = V_{ζ}^{′reg} we see that the map γ from B(w) to C_{B′}(w) is the composition of the map which we still denote by γ from B(w) to C_{B}(w) and of the embedding B → B′.

We make a specific choice of a regular element. Let c = σ_{1} … σ_{n}. It is a 2n-th root of π (cf. proposition 4.1). Let r and d be two integers such that rd = 2n and let w = c^{r}; it is a d-th root of π and its image w ∈ W is a good regular element.

If d is odd, we have C_{B}(w) = C_{B}(c^{r}) = C_{B}(c^{pgcd(n,r)}) = C_{B}(c^{r/2}) and C_{W} (w) = C_{W} (c^{r/2}) since w_{0} = c^{n} is central. So we are reduced to study C_{B}(w) when d is even and C_{B}(w^{2}) when d is even and d/2 odd (see theorem 6.3 (ii), below).

When d is even we have seen above that C_{W} (w) = C_{W′}(w), and this group is a complex reflection group of type G(d, 1, r) (see also [BMi, A.1.2]).

We have c = σ′_{n}σ′_{n−1}σ′_{n+1} … σ′_{1}σ′_{2n−1}. In order to apply the results of section 5, we use a conjugation by v^{−1} where v is the canonical lift of the longest element of the standard parabolic subgroup of W′ generated by σ′_{1}, …, σ′_{n−1}: indeed we have c = v^{−1}σ′_{1} … σ′_{2n−1}v.

This proves that the generators of C_{B}(w) are the elements s_{i} = v^{−1}(∏_{j=0}^{d−1} σ′_{i−1+rj})v = ∏_{k=0}^{d/2−1} σ_{i+kr} for i = 2, …, r and the element t such that t ∏_{i=2}^{r} s_{i} = c; note that we have not chosen for the element t the conjugate by v^{−1} of the element t of section 5 but we have applied a further conjugation by c in order to simplify the computations. We have t = (σ̂_{2}σ_{3} … σ_{r+1}σ̂_{r+2} … σ_{(d/2−1)r+1})^{−1}σ_{1}σ_{2} … σ_{(d/2−1)r+1}, where σ̂_{i} means deleting σ_{i} from the product: we have deleted all σ_{i} such that i ≡ 2 (mod r).

We first prove a lemma analogous to lemma 5.3.

Lemma 6.1. (i) We have cσ_{i} = σ_{i+1}c for 2 ≤ i < n − 1.

(ii) We have ^{c^{2}}σ_{n} = σ_{2}.

(iii) For x ∈ B^{+} and 2 ≤ i < n we have σ_{i} ≺ x ⇔ σ_{i+1} ≺ cx.

(iv) We have {i | σ_{i} ≺ c^{j}} = {1, …, j} for j ≤ n.

Proof. Statements (i) and (iii) have the same proof as the corresponding statements in lemma 5.3.

Let us prove (ii). We have

c^{2} = σ_{1}σ_{2}(σ_{3} … σ_{n})c = σ_{1}σ_{2}c(σ_{2} … σ_{n−1}) = σ_{1}σ_{2}σ_{1}σ_{2}(σ_{3} … σ_{n})(σ_{2} … σ_{n−1}) = σ_{2}σ_{1}σ_{2}σ_{1}(σ_{3} … σ_{n})(σ_{2} … σ_{n−1}) = σ_{2}σ_{1}σ_{2}(σ_{3} … σ_{n})σ_{1}(σ_{2} … σ_{n−1}) = σ_{2}cσ_{1} … σ_{n−1},

whence c^{2}σ_{n} = σ_{2}c^{2}.

The proof of (iv) is similar to that of lemma 5.3 (iv): it uses similarly the fact that σ_{i} ≺ c if and only if i = 1, but it also needs the fact that σ_{2} ≺ c^{2}, which we have seen in the proof of (ii). □

Lemma 6.2. The group C_{B}(w) has a presentation with generators t, s_{2}, …, s_{r}, the relations being the braid relations given by the diagram ◯_{t} ◯_{s_{2}} ◯_{s_{3}}⋯◯_{s_{r}}.

Proof. We already know that C_{B}(w) has a presentation with generators s_{2}, …, s_{r}, t′ and relations the braid relations given by ◯_{s_{2}} ◯_{s_{3}}⋯◯_{s_{r}} ◯_{t′}, where t′ is the

element conjugate by v^{−1} of the element t of section 5. We have t = ct′c^{−1} = (s_{2} … s_{r})t′(s_{2} … s_{r})^{−1}. The commutation relation t′ with s_{i} for i < r is equivalent by 6.1 (i) to the commutation relation of t with s_{i} for i > 2. It remains to see that the braid relation between t′ and s_{r} is equivalent to the braid relation between t and s_{2}. This is proved by decreasing induction on i using the following fact that is the result of a simple computation: if s_{i−1}, s_{i} and u′ are elements of a group and if u = s_{i}u′s_{i}^{−1} then the braid relations given by ◯_{s_{i−1}} ◯_{s_{i}} ◯_{u′} imply s_{i−1}us_{i−1}u = us_{i−1}us_{i−1}.

Conjugating by s_{2} the relation s_{2}^{s_{3}...s_{r}}t′s_{2}^{s_{3}...s_{r}}t′ = ^{s_{3}...s_{r}}t′s_{2}^{s_{3}...s_{r}}t′s_{2}, which we get at the end of the induction, gives the braid relation which we want. The converse is similar. □

Theorem 6.3. Assume d even; then

(i) We have s_{i} ∈ End_{D^{+}}(w) (i = 1 …, r − 1) and t ∈ End_{D}(w), so that

End_{D}(w) = C_{B}(w) ≃ B(d, 1, r).

(ii) If d/2 odd, we have s_{i} ∈ End_{D^{+}}(w^{2}) (i = 1 …, r − 1) and t ∈ End_{D}(w^{2}),

so that End_{D}(w^{2}) = C_{B}(w^{2}) ≃ B(d, 1, r).

Proof. In the following lemma, the statements about w^{2} assume that d/2 is odd. The proof follows the same lines as that of lemma 5.4, using lemma 6.1 instead of lemma 5.3.

Lemma 6.4. Assume 2 ≤ i ≤ r and let I_{i} = {σ_{i}, σ_{i+r}, …, σ_{i+(d/2−1)r}}. Then

(i) s_{i} ∈ End_{D_{I_{i}}^{+}}(w) and s_{i} ∈ End_{D_{I_{i}}^{+}}(w^{2}). In particular s_{i} ∈ End_{D^{+}}(w) and s_{i} ∈ End_{D^{+}}(w^{2}).

(ii) The conjugation by w (resp. w^{2}) stabilizes I_{i} and induces the cyclic permutation σ_{i+jr} ↦ σ_{i+(j+1)r (mod n)} (resp. σ_{i+jr} ↦ σ_{i+(j+2)r (mod n)}).

Lemma 6.5. For i ≤ j we set σ_{i,j} = σ_{i}σ_{i+1} … σ_{j}, and we set x_{i} = σ_{i+1,i+r−1}. Let y = ∏_{i=1}^{d/2−1} y_{i} where y_{i} = ∏_{k=1}^{d/2−i} x_{(i−1)(r−1)+d/2−k+1}. Then ywy^{−1} ∈ B^{+} and y ∈ Hom_{D^{+}}(ywy^{−1}, w).

Proof. We set c_{i} = σ_{1,i}. Let w_{d/2} = w, and by decreasing induction on i define y_{i}^{−1}w_{i}y_{i} = w_{i+1}. We claim that y_{i}^{−1}w_{i} ∈ B^{+}, which implies the result: in fact we prove by induction that w_{i} = y_{i}c_{i(r−1)+d/2}c^{r−1}. This equality is clearly true for i = d/2 (with y_{d/2} = 1). Let us assume it to be true for i + 1 and let us prove it for i. We have

w_{i+1} = y_{i+1}c_{(i+1)(r−1)+d/2}c^{r−1}

= ∏_{k=1}^{d/2−i−1} x_{i(r−1)+d/2−k+1}c_{(i+1)(r−1)+d/2}c^{r−1}

= c_{(i+1)(r−1)+d/2} ∏_{k=1}^{d/2−i−1} x_{i(r−1)+d/2−k}c^{r−1} (by 6.1(i))

= c_{i(r−1)+d/2} ∏_{k=0}^{d/2−i−1} x_{i(r−1)+d/2−k}c^{r−1}

= c_{i(r−1)+d/2}c^{r−1} ∏_{k=0}^{d/2−i−1} x_{(i−1)(r−1)+d/2−k} (by 6.1(i))

= c_{i(r−1)+d/2}c^{r−1}y_{i}

which, conjugating by y_{i} gives the equality for w_{i}. □

We note that the above proof shows that

ywy^{−1} = y_{1}c_{r−1+d/2}c^{r−1} = ∏_{i=d/2}^{2} x_{i} ∏_{i=1}^{r−1+d/2} σ_{i}c^{r−1} = σ_{1,r−1+d/2} ∏_{i=d/2−1}^{1} x_{i}c^{r−1}

the last equality by lemma 6.1 (i), so ywy^{−1} = σ_{1,d/2} ∏_{i=d/2}^{1} x_{i}c^{r−1}.

Lemma 6.6. We have yty^{−1} = σ_{1,d/2}.

Proof. For i = 1, …, d/2 − 1, let t_{i} = (σ̂_{2}σ_{3} … σ_{r+1}σ̂_{r+2} … σ_{ir+1})^{−1}σ_{1,i(r−1)+d/2}. We have t = t_{d/2−1} and t_{0} = σ_{1,d/2}. We prove by induction that y_{i} (cf. lemma 6.5) conjugates t_{i} into t_{i−1}, which proves the lemma. Keeping the notation of lemma 6.5, we have t_{i} = (x_{2}x_{r+2} … x_{(i−1)r+2})^{−1}σ_{1,i(r−1)+d/2}. By definition y_{i} = ∏_{k=1}^{d/2−i} x_{(i−1)(r−1)+d/2−k+1}. This product commutes with x_{2}x_{r+2} … x_{(i−2)r+2}. As the factor indexed by k = d/2 − i in y_{i} is equal to x_{(i−1)r+2}, we get

y_{i}t_{i}y_{i}^{−1} = (x_{2}x_{r+2} … x_{(i−2)r+2})^{−1} ∏_{k=1}^{d/2−i−1} x_{(i−1)(r−1)+d/2−k+1}σ_{1,i(r−1)+d/2}y_{i}^{−1}.

We use the fact that conjugation by σ_{1,i(r−1)+d/2} of the factor x_{(i−1)(r−1)+d/2−k+1} in y_{i}^{−1} for k > 1 changes k into k − 1: this allows to simplify the product and we get

y_{i}t_{i}y_{i}^{−1} = (x_{2}x_{r+2} … x_{(i−2)r+2})^{−1}σ_{1,i(r−1)+d/2}x_{(i−1)(r−1)+d/2}^{−1}
= (x_{2}x_{r+2} … x_{(i−2)r+2})^{−1}σ_{1,(i−1)(r−1)+d/2} = t_{i−1}.

□

Let I = {σ_{1}, …, σ_{d/2}}; we have σ_{i,d/2} ∈ End_{D_{I}^{+}}(ywy^{−1}), by the remark following the proof of lemma 6.5. This, together with lemmas 6.5 and 6.6 proves the statements about t in the theorem. □

7. The elements π and w_{0}

We consider here the order one F -root of π, which is y = π, and the order two F -root y = w_{0}. For w_{0}, we will just show how conjectures 2.1 to 2.4 follow from known results. For π, we will in addition prove conjecture 2.5 in a certain number of cases, including split type A in general.

We recall that (cf. [DMR, proposition 2.1.6]) the group C_{W} (F) (resp. C_{W} (w_{0}F)) is a Coxeter group with Coxeter generators the elements w_{0}^{I} for I an element of the set of orbits S/F (resp. I ∈ S/w_{0}F). The corresponding braid groups C_{B}(πF) (resp. C_{B}(w_{0}F)) have as generators the corresponding elements w_{0}^{I}.

Since the generators w_{0}^{I} divide π (resp. w_{0}) in B^{+}, conjecture 2.1 is trivial. For π, conjecture 2.2 is also trivial. For w_{0} it results from the remark below 2.2 and the fact that w_{0} is the only “good” square F -root of π since it is the only element of W of its length.

Since πF (resp. w_{0}F) acts as a diagram automorphism, conjecture 2.3 is [Mi, corollary 4.4].

Conjecture 2.4 holds for the cases y = π and y = w_{0} by the following results:

Proposition 7.1. [DMR, 5.3.4] The map t ↦ D_{t} from C_{B}(F) to the G^{F} -endo-morphisms of H_{c}^{∗}(X(π)) factors through the specialization x ↦ q of a 1-cyclotomic Hecke algebra for C_{W} (F) which is the specialization u_{w_{0}^{I},0} ↦ x^{l(w_{0}^{I})}, u_{w_{0}^{I},1} ↦ −1 of the generic Hecke algebra of C_{W} (F).

Proposition 7.2. [DMR, 5.4.1] The map t ↦ D_{t} from C_{B}(w_{0}F) to the G^{F} - endomorphisms of H_{c}^{∗}(X(w_{0})) factors through the specialization x ↦ q of a 2cyclotomic Hecke algebra for C_{W} (w_{0}F) which is the specialization u_{w_{0}^{I},ε} ↦ x^{l(w_{0}^{I})}, u_{w_{0}^{I},1−ε} ↦ (−1)^{1+l(w_{0}^{I})}, where ε = 0 if l(w_{0}^{I}) is even and 1 otherwise, of the generic Hecke algebra of C_{W} (w_{0}F) (here the names u_{w_{0}^{I},0} and u_{w_{0}^{I},1} are given in accordance with the definition of a cyclotomic Hecke algebra in section 2).

We will now consider conjecture 2.5 for y = π. Let H be the cyclotomic algebra of proposition 7.1. Since some characters of H are only defined over \overline{ℚ}_{ℓ}[x^{1/2}] (for W irreducible, this happens only for the characters of degree 512 of W (E_{7}) and those of degree 4096 of W (E_{8})), we need to take the integer a defined above 2.4 equal to 2 and thus consider the specialization f : x^{1/2} ↦ q^{1/2} of the algebra with parameters u_{w_{0}^{I},0} ↦ ((−x^{1/2})^{2l(w_{0}^{I})}) and u_{w_{0}^{I},1} ↦ −1. In the terms of [DMR, 5.3.1] this corresponds to the specialization x^{1/2} ↦ −q^{1/2} of H_{x}(W, F). We recall from [DMR, 5.3.2] that if f′ is the specialization x^{1/2} ↦ −q^{1/2} (which corresponds to the specialization x^{1/2} ↦ q^{1/2} of H_{x}(W, F)), and if we fix an F -stable Borel subgroup B, then H ⊗_{f′} \overline{ℚ}_{ℓ} ≃ End_{_{\overline{ℚ}_{ℓ}}G_{F}} (Ind_{B^{F}}^{G^{F}} Id).

Let σ be the semi-linear automorphism, coming from Gal(\overline{ℚ}_{ℓ}(x^{1/2})/\overline{ℚ}_{ℓ}(x)), given by x^{1/2} ↦ −x^{1/2} of H; thus f ∘ σ = f′. Let H_{q} = H ⊗_{f′} \overline{ℚ}_{ℓ} and let χ ↦ χ_{q} be the bijection between characters of H and H_{q} obtained via f′; we then denote by χ_{q} ↦ ρ_{χ} the bijection between characters of H_{q} and characters of G^{F} occurring in Ind_{B^{F}}^{G^{F}} Id coming from [DMR, 5.3.2].

Let us recall that the representation Ind_{B^{F}}^{G^{F}} Id of H ⊗_{f′} \overline{ℚ}_{ℓ} is special (cf. 2.5 (ii)); this follows from the fact that the image of any non-trivial w ∈ W has zero trace in this representation, which characterizes the canonical trace form for Hecke algebras of Coxeter groups.

It follows that conjecture 2.5 is implied by the

Conjecture 7.3. We have an equality of virtual G^{F} × H_{q}-modules

∑_{i}(−1)^{i}H_{c}^{i}(X(π), \overline{ℚ}_{ℓ}) = ∑_{χ∈Irr(W^{F})} ρ_{χ} ⊗ σ(χ)_{q}.

In the remaining part of this section we will prove the following theorem:

Theorem 7.4. Conjecture 7.3 holds if the characteristic is almost good for G and if (W, F) is irreducible of type untwisted A_{n}, B_{2}, B_{3}, B_{4}, D_{4}, D_{5}, D_{6}, D_{7}, G_{2} or E_{6}.

Recall that the characteristic is almost good for G if it is good for each simple component of exceptional type of G.

Proof. We have to prove that the virtual character of H_{q} appearing in the ρ_{χ}-isotypic component of ∑_{i}(−1)^{i}H_{c}^{i}(X(π), \overline{ℚ}_{ℓ}) is equal to σ(χ)_{q}. This is equivalent to proving that for any x ∈ C_{B^{+}}(F) and any χ ∈ Irr(W^{F}), we have:

(1)  ⟨ g ↦ ∑_{i}(−1)^{i} Trace(gD_{x}|H_{c}^{i}(X(π), \overline{ℚ}_{ℓ})), ρ_{χ} ⟩_{G^{F}} = σ(χ)_{q}(T_{x}),

where T_{x} denotes the image of x in H_{q}. We will prove this equality for sufficiently many elements of C_{B^{+}}(F) to deduce it for all elements for groups in the list of theorem 7.4.

The lemmas used in the proof have a larger scope of validity than the theorem. The proof proceeds by induction on the semi-simple rank of G because of the next lemma.

Lemma 7.5. Let G be an arbitrary reductive group as in section 1. If conjecture 7.3 holds for any reductive group with semi-simple rank less than that of G then (1) holds for any x ∈ C_{B_{I}^{+}}(F) for any F -stable proper subset I of S.

Proof. If L_{I} is the standard Levi subgroup of G corresponding to I, then by [DMR, théorème 5.2.10], for x ∈ C_{B_{I}^{+}}(F) we have:

⟨ g ↦ ∑_{i}(−1)^{i} Trace(gD_{x}|H_{c}^{i}(X(π), \overline{ℚ}_{ℓ})), ρ_{χ} ⟩_{G^{F}} =

⟨ l ↦ ∑_{i}(−1)^{i} Trace(lD_{x}|H_{c}^{i}(X_{L_{I}}(π_{I}), \overline{ℚ}_{ℓ})),^{∗}R_{L_{I}^{F}}^{G^{F}} ρ_{χ} ⟩_{L_{I}^{F}} ,

which by assumption is equal to

∑_{φ∈Irr(W_{I}^{F})} σ(φ)_{q}(T_{x})⟨ ρ_{φ},^{∗}R_{L_{I}^{F}}^{G^{F}} ρ_{χ} ⟩_{L_{I}^{F}} .

As ⟨ ρ_{φ},^{∗}R_{L_{I}^{F}}^{G^{F}} ρ_{χ} ⟩_{L_{I}^{F}} = ⟨ φ, Res_{W_{I}^{F}}^{W^{F}} χ ⟩_{W_{I}^{F}}, cf. [CR, Theorem 70.24], we get (1) for x. □

Lemma 7.6. Here again G is arbitrary. Equality (1) holds if x = π^{n}, with n multiple of δ.

Proof. As π^{n} acts by F^{n} on X_{π}, we have ∑_{i}(−1)^{i} Trace(gD_{π}^{n}|H_{c}^{i}(X(π), \overline{ℚ}_{ℓ})) = |X(π)^{gF^{n}}| by the Lefschetz trace formula. We shall use the same methods and notation as in [BMi, §2.B and §6.D]. Proposition [DMR, 3.3.7] shows that

|X(π)^{gF^{n}}| = ∑_{ρ∈Irr(G^{F})} ρ(g) ∑_{χ∈Irr(W)^{F}}⟨ρ, Rχ̃⟩χ̃_{q}(T_{π}^{n}F).

We have χ̃_{q}(T_{π}^{n}F) = q^{n(2N−a_{χ}−A_{χ})}χ̃(F) [BMi, proposition 6.11] whence we get as in the proof of [BMi, Proposition 2.5]

|X(π)^{gF^{n}}| = ∑_{ρ∈Irr(G^{F})} ρ(g)⟨ρ, Ind_{B^{F}}^{G^{F}} Id⟩q^{n(2N−a_{ρ}−A_{ρ})} = ∑_{χ∈Irr(W^{F})} χ(1)q^{n(2N−a_{χ}−A_{χ})}ρ_{χ}(g).

We now use χ_{q}(T_{π}^{n}) = σ(χ)_{q}(T_{π}^{n}) = χ(1)q^{n(2N−a_{χ}−A_{χ})} [BMi, corollaire 4.21], which gives the result. □

Lemma 7.7. If the characteristic is almost good for the split irreducible group G, equality (1) holds when x is a root of π.

Proof. Lemma 7.6 shows the result for x = π; we thus assume that x is a d-th root of π with d ≥ 2. By [DMR, 5.2.2 (i)] the endomorphism D_{w} of X(π) satisfies the trace formula so that

∑_{i}(−1)^{i} Trace(gD_{w}|H_{c}^{i}(X(π), \overline{ℚ}_{ℓ})) = |X(π)^{gD_{w}}|.

Moreover, by [DMR, 5.2.2 (ii)], we have (g ↦ |X(π)^{gD_{w}}|) = Sh^{d}(g ↦ Trace(gT_{w} | Ind_{B^{F}}^{G^{F}} Id)) where Sh is the twisting operator on G^{F} -class functions. So we have to prove

Sh^{d}(g ↦ Trace(gT_{w} | Ind_{B^{F}}^{G^{F}} Id)) = ∑_{χ_{q}∈Irr(H_{q}(W,F))} σ(χ)_{q}(T_{w})ρ_{χ},

which is equivalent to

(1′)  ∑_{χ_{q}∈Irr(H_{q}(W,F))} χ_{q}(T_{w}) Sh^{d} ρ_{χ} = ∑_{χ_{q}∈Irr(H_{q}(W,F))} σ(χ)_{q}(T_{w})ρ_{χ}.

To prove this, we may replace w by a conjugate in B so we may assume that w is a “good” root, in particular that w ∈ W. As usual we set w = β(w).

We have χ_{q}(T_{w}) = χ(w)q^{\frac{2N−aρ_{χ} −Aρ_{χ}}{d}}: one gets this by applying [BMM, 6.15(2)] to H; with the notation of loc. cit., it is a principal algebra (see loc. cit. 6.3), with θ_{0}(w_{0}^{I}) = x^{l(w_{0}^{I})}; we have D_{0} = 2N and if we take P (q) = |(G/B)^{F} | the degree Deg_{χ}^{(P)} identifies with the generic degree of ρ_{χ}.

As H is split over ℤ[x^{1/2}, x^{−1/2}], there exists a sign ε_{d,χ} depending only on (a_{ρ_{χ}} + A_{ρ_{χ}})/d such that σ(χ)_{q}(T_{w}) = ε_{d,χ}χ_{q}(T_{w}). This sign is equal to −1 if and only if (a_{ρ_{χ}} + A_{ρ_{χ}})/d ∈ ℤ + 1/2 and χ(w) ≠ 0. Equation (1′) becomes then

(1′′) ∑_{χ_{q}∈Irr(H_{q}(W,F))} q^{\frac{2N−aρ_{χ} −Aρ_{χ}}{d}} χ(w) Sh^{d} ρ_{χ} = ∑_{χ_{q}∈Irr(H_{q}(W,F))} ε_{d,χ}q^{\frac{2N−aρ_{χ} −Aρ_{χ}}{d}} χ(w)ρ_{χ}.

For computing Sh^{d}, we shall use Shoji’s results on the identification of character sheaves with almost characters. Here we need the assumption that the characteristic is almost good. We recall these results: unipotent characters of G^{F} have been divided by Lusztig into families. Unipotent character sheaves have also been divided into families which are in one-to-one correspondence with the families of unipotent characters. In [Sh2, 3.2 and 4.1] Shoji proves that the transition matrix from the unipotent characters to the characteristic functions of the unipotent character sheaves is block diagonal according to the families, and in [Sh1, 3.3] he proves that the characteristic functions of the character sheaves are eigenvectors of Sh. From this we see that (1′′) is equivalent to the set of its projections on each family. Moreover a_{ρ} and A_{ρ} are constant when ρ runs over a family of unipotent characters. So (1′′) is equivalent to the set of equations

(1′′′)  ∑_{ρ_{χ}∈F} χ(w) Sh^{d} ρ_{χ} = ε_{d,F} ∑_{ρ_{χ}∈F} χ(w)ρ_{χ},

where F runs over the families. We have written ε_{d,F} instead of ε_{d,χ} because this sign depends only on the family of ρ_{χ}.

We can also assume that G is adjoint as the unipotent characters factorize through the adjoint group and Sh is compatible with this factorization.

Lusztig defined in [Lu3, 4.24.1] almost characters R_{ρ} indexed by unipotent characters. If R_{w} is the Deligne-Lusztig character given by the virtual representation ∑_{i≥0}(−1)^{i}H_{c}^{i}(X(w)), we have R_{w} = ∑_{χ∈Irr(W)} χ(w)R_{ρ_{χ}}; for any unipotent character ρ we also have ⟨ ρ, R_{ρ_{χ}} ⟩_{G^{F}} = Δ_{ρ}⟨ R_{ρ}, ρ_{χ} ⟩_{G^{F}} for a sign Δ_{ρ} defined in [Lu3]. Almost characters being an orthonormal basis of the space of unipotent class functions, we get ∑_{ρ_{χ}∈F} χ(w)ρ_{χ} = ∑_{ρ∈F} ⟨ R_{w}, ρ ⟩_{G^{F}} Δ_{ρ}R_{ρ}.

In [Lu4, 23.1] Lusztig has defined a bijection ρ ↦ A_{ρ} from the set of unipotent characters to the set of unipotent character sheaves, compatible with the partition into families. Shoji, in ([Sh1] and [Sh2]) proved that the almost character R_{ρ} is a multiple of the characteristic function of χ_{A_{ρ}} relative to the Frobenius endomorphism F of the character sheaf A_{ρ} and that (cf. [Sh1, 3.6 and 3.8]) Sh(χ_{A_{ρ}}) = λ_{ρ}χ_{A_{ρ}} where λ_{ρ} is as in [DMR, 3.3.4].

Using this, we see that (1′′′) is equivalent to:

(1′′′′)  if ⟨ R_{w}, ρ ⟩_{G^{F}} ≠ 0 then λ_{ρ}^{d} = ε_{d,F}

This would be a consequence of conjecture [BMi, 5.13]. We prove it by a case by case analysis.

If G is classical, we have always ε_{d,F} = 1 and λ_{ρ} = ±1, so (1′′′′) holds if d is even. Assume d odd; one checks that in a Coxeter group of type A_{n}, B_{n} or D_{n}, any odd order element lies in a parabolic subgroup of type A. Let us denote by L the corresponding Levi subgroup of G, which is an F -stable Levi subgroup of an F - stable parabolic subgroup. We have R_{w} = R_{L}^{G}(R_{w}^{L}) where R_{w}^{L} is the Deligne-Lusztig character of L^{F} associated to w. As λ_{ρ} is constant in a Harish-Chandra series and is equal to 1 for a group of type A, we get the result in this case.

If G is of exceptional type we can check the result, using the explicit description of the coefficients ⟨ R_{w}, ρ ⟩_{G^{F′}} and of λ_{ρ} in [Lu3]. The most complicated case to check is when for some d we have ε_{d,F} = −1. In type E_{7} there is exactly one such family; it contains 4 unipotent characters. Two of them are some ρ_{χ} for a χ such that a_{χ} + A_{χ} = 63. In type E_{8} there are two such families, each with 4 unipotent characters. In each of these families there are two ρ_{χ} with respectively a_{χ}+A_{χ} = 105 and a_{χ} + A_{χ} = 135. So in all cases we have ε_{d,F} = −1 if and only if d ≡ 2 (mod 4). In each case for the two other unipotent characters of the family one has λ_{ρ} = ±i. One checks that if ρ ∈ F and ⟨ ρ, R_{w} ⟩_{G^{F}} ≠ 0 then if d ≢ 2 (mod 4) one has λ_{ρ}^{d} ≠ 1 whence the result in this case; and if d ≡ 2 (mod 4) we have λ_{ρ} = ±i, thus λ_{ρ}^{d} = −1 and we also get the result in that case. □

Let Φ be the class function on H_{q} with values in the Grothendieck group of G^{F} given by

Φ(T_{x}) = (g ↦ ∑_{i}(−1)^{i} Trace(gD_{x}|H_{c}^{i}(X(π), \overline{ℚ}_{ℓ}))) − ∑_{χ} σ(χ)_{q}(T_{x})ρ_{χ}.

To prove theorem 7.4 we have to prove that Φ = 0. By lemmas 7.5, 7.6 and 7.7 respectively we know that

(a) Φ(T_{x}) = 0 for x ∈ B_{I} for any proper subset I of S.

(b) Φ(T_{π}^{n}) = 0 for n > 0.

(c) We have Φ(T_{x}) = 0 if x is a root of π and the characteristic is almost good. We shall prove that in any of the cases considered in theorem 7.4 a class function

on H_{q} which satisfies these three properties is zero. Such a class function, can be written ∑_{χ} λ_{χ}χ_{q}. We show that the three above properties imply λ_{χ} = 0 for all χ. Let us translate each of these properties into a property of (λ_{χ})_{χ}.

Lemma 7.8. Property (a) means that (λ_{χ})_{χ} is linearly spanned by vectors (χ(w))_{χ} with w ∈ W cuspidal (i.e., the conjugacy class of w has no representative in a proper parabolic subgroup of W).

Proof. Consider the scalar product on the Grothendieck group of H_{q} such that the χ_{q} form an orthonormal basis (which corresponds to the usual scalar product on the vectors (λ_{χ})_{χ}); then the χ_{c,q} = ∑_{χ∈Irr(W^{F})} χ(c)χ are pairwise orthogonal when c runs over a set of representatives of the conjugacy classes in W. The statement to prove is that a class function satisfies property (a) if and only if it is orthogonal to the χ_{c,q} with c non cuspidal.

With our choice of scalar product, restriction and induction satisfy Frobenius reciprocity, as the scalar product is compatible with the specialization to W, as are restriction and induction. So for I ⊂ S, a class function is zero on H_{q}(W_{I}) if and only if it is orthogonal to any Ind_{H_{q}(W_{I})}^{H_{q}} ϕ; but the χ_{c,q} with c non-cuspidal span the same subspace as the Ind_{H_{I}}^{H} ϕ with I ⊊ S, so we get the result. □

Lemma 7.9. If x is a d-th root of π, property (c) is equivalent to ∑_{χ} λ_{χ}χ(x)q^{\frac{2N−aρ_{χ} −Aρ_{χ}}{d}} = 0.

Proof. This is a simple translation of (c), using the value of χ_{q}(T_{x}). □

We now prove the theorem when G is split of type A_{n}. The only cuspidal class is the class of a Coxeter element c. So by lemma 7.8 (λ_{χ})_{χ} has to be equal to a(χ(c))_{χ} for some a ∈ \overline{ℚ}_{ℓ}. Lemma 7.9 then gives a ∑_{χ} χ(c)^{2}q^{\frac{2N−aρ_{χ} −Aρ_{χ}}{d}} = 0, so that a = 0, as all summands are non-negative and at least one is non-zero.

For the other types we need property (b).

Lemma 7.10. Property (b) is equivalent to: for all i, we have ∑_{{χ|a_{ρχ}+A_{ρχ}=i}} λ_{χ}χ(1) = 0.

Proof. Using the value of χ_{q}(T_{π}^{n}) property (b) is equivalent to the fact that for all n we have

∑_{i} q^{n(2N−i)} ∑_{{χ|a_{ρχ}+A_{ρχ}=i}} λ_{χ}χ(1) = 0.

We get the result using the linear independence of the characters of ℤ. □

The proof of theorem 7.4 in the remaining types is obtained by a computer calculation which shows that the vectors given by lemmas 7.10 and 7.9 span for any q the space given by lemma 7.8 (note that only the vectors given by lemma 7.9 depend on q). □

8. Pieces of the Deligne-Lusztig varieties

In this section, we introduce a technique inspired by [Lu2], which will allow us to compute Harish-Chandra restrictions of the cohomology of some Deligne-Lusztig varieties; we will also find a criterion for irreducibility of a generalized Deligne-Lusztig variety (see proposition 8.4).

The technique is intersecting with Bruhat cells. Let B be the variety of Borel subgroups of G. We recall from [DMR, 2.2.18] that, given a decomposition t = w_{1} … w_{k} of t ∈ B^{+} as a product of elements of W, and denoting by O(w) the G-orbit in B × B indexed by w ∈ W, the variety O(w_{1}, …, w_{k}) = {(B_{1}, …, B_{r+1}) | (B_{i}, B_{i+1}) ∈ O(w_{i})} depends only on t and not on the chosen decomposition; it affords two canonical projections p′(B_{1}, …, B_{r+1}) = B_{1} and p′′(B_{1}, …, B_{r+1}) = B_{r+1} and the Deligne-Lusztig variety is X(t) = {x ∈ O(t) | p′′(x) = F (p′(x))}. We fix an F -stable Borel subgroup B = TU ⊂ G, where U is the unipotent radical of B and T is an F -stable maximal torus. We identify W with N_{G}(T)/T. We put B^{−} = ^{w_{0}}B = TU^{−}. For v ∈ W, we define the piece X^{v}(t) = {x ∈ X(t) | (B, p′(x)) ∈ O(v)}. We have X(t) = ∐_{v∈W} X^{v}(t), and the action of G^{F} on X(t) restricts to an action of B^{F} on each piece.

Remark 8.1. If t = w_{1} … w_{k} is a decomposition of t ∈ B as a product of elements of W, we recall from [DMR, 2.2.12 et 2.3.2] that

X(t) = {(g_{1}B, g_{2}B, …, g_{k}B)|g_{i}^{−1}g_{i+1} ∈ Bw_{i}B, for i = 1, …, k−1 and g_{k}^{−1}^{F}g_{1} ∈ Bw_{k}B}.

In this model we get X^{v}(t) by adding the condition g_{1} ∈ BvB.

Let H(W) be the generic Hecke algebra of W over ℂ[x]. This is the quotient of the group algebra ℂ[x]B by the relations (s + 1)(s − x) = 0 for s ∈ S. We denote by T_{b} the image of b ∈ B^{+} in H(W). The algebra H(W) has a basis {T_{w} | w ∈ W}. We will also sometimes denote by T_{w} the elements of this basis. We will denote by H_{q}(W) the specialized algebra by the specialization x ↦ q and keep the notation T_{w} for the basis of this algebra (trying to make clear by the context which algebra is meant). Finally we note A|T_{v} the coefficient of the element A ∈ H(W) on the basis element T_{v}. We recall that the canonical symmetrizing form is T_{v} ↦ T_{v}|1. Since T_{v} and q^{−l(v)}T_{v^{−1}} are dual bases for this form we have A|T_{v} = q^{−l(v)}AT_{v^{−1}}|1. With this notation, we have

Proposition 8.2. Let t ∈ B^{+} and v ∈ W; for any m multiple of δ we have |(U^{F} \X^{v}(t))^{F^{m}} | = T_{v}T_{t}|T_{^{F}v}, where the elements on the right-hand side are taken in the Hecke algebra H_{q^{m}}(W).

Proof. We may assume t ∈ W. Indeed, by [DMR, 2.3.3], if t = w_{1} … w_{k} is a decomposition as a product of elements of W, and if F_{1} is the isogeny on G^{k} defined by F_{1}(g_{1}, …, g_{k}) = (g_{2}, …, g_{k}, F (g_{1})), then X(t) ≃ X_{G^{k}}((w_{1}, …, w_{k}), F_{1}) and this isomorphism restricts to

X^{v}(t) ≃ ∐_{v_{2},...,v_{k}∈W}(X_{G^{k}}^{(v,v_{2},...,v_{k})}((w_{1}, …, w_{k}), F_{1})).

Thus (U^{F} \X^{v}(t))^{F^{m}} ≃ ∐_{v_{2},...,v_{k}} ((U^{k})^{F_{1}}\(X_{G^{k}}((w_{1}, …, w_{k}), F_{1}))^{(v,v_{2},...,v_{k})})^{F_{1}^{km}}, and it is also clear that:

∑_{v_{2},...,v_{k}} T_{(v,v_{2},...,v_{k})}T_{(w_{1},...,w_{k})}|T_{^{F_{1}}(v,v_{2},...,v_{k})} =

∑_{v_{2},...,v_{k}}(T_{v}T_{w_{1}}|T_{v_{2}})(T_{v_{2}}T_{w_{2}}|T_{v_{3}}) … (T_{v_{k}}T_{w_{k}}|T_{^{F}v}) = T_{v}T_{t}|T_{^{F}v}.

As F_{1}^{kδ} is the smallest power of F_{1} which is a split Frobenius, we are indeed reduced to the same statement for G^{k}, F_{1}, (w_{1}, …, w_{k}), (v, v_{2}, …, v_{k}).

We then assume t ∈ W. Thus X^{v}(t) = {gB | g ∈ BvB, g^{−1}^{F}g ∈ BtB}. The map u ↦ uvB then fibers {u ∈ U | (uv)^{−1}^{F}(uv) ∈ BtB} on X^{v}(t) with fibers isomorphic to U ∩^{v}U, since BvB = U_{v}vB, where for v ∈ W we set U_{v} = U ∩^{v}U^{−}. This fibration is U^{F} -equivariant for the action of U^{F} by left multiplication on both spaces. The quotient by U^{F} is thus obtained by u ↦ u^{−1}.^{F}u which maps the above variety to U ∩ vBtB^{F}v^{−1}. As the fibers U ∩ ^{v}U are connected and have q^{ml(w_{0}v)} fixed points under F^{m}, the cardinality we seek is thus q^{−ml(w_{0}v)}|U^{F^{m}} ∩ vBtB^{F}v^{−1}|. Thus the proposition results from the following lemma, applied with F replaced by F^{m} and w by ^{F}v:

Lemma 8.3. Assume F split. For v, t, w ∈ W and T_{v}, T_{t}, T_{w} ∈ H_{q}(W) we have

T_{v}T_{t}|T_{w} = q^{−l(w_{0}v)}|(U ∩ vBtBw^{−1})^{F} |.

Proof. We have T_{v}T_{t}|T_{w} = T_{t^{−1}}T_{v^{−1}}|T_{w^{−1}} = q^{−l(w)}T_{t^{−1}}T_{v^{−1}}T_{w}|1 = q^{l(t)−l(w)}T_{v^{−1}}T_{w}|T_{t}. We recall that H_{q}(W) may be realized as a subalgebra of ℂ[G^{F}] via the isomorphism H_{q}(W) ≃ End_{G^{F}} Ind_{B^{F}}^{G^{F}} ℂ. By this isomorphism, T_{w} corresponds to q^{l(w)}e_{B}ẇe_{B} where ẇ is a representative of w in N(T)^{F} and where e_{B} is the idempotent |B^{F} |^{−1} ∑_{b∈B^{F}} b. Thus:

T_{v^{−1}}T_{w}|T_{t} = q^{l(v)+l(w)−l(t)}e_{B}v^{−1}e_{B}we_{B} | e_{B}te_{B}
= |B^{F} |^{−1}q^{l(v)+l(w)−l(t)}|v^{−1}B^{F} w ∩ B^{F} tB^{F} |
= q^{−l(w_{0})+l(v)+l(w)−l(t)}|U^{F} ∩ vB^{F} tB^{F} w^{−1}|.

The lemma follows, since (U ∩ vBtBw^{−1})^{F} = U^{F} ∩ vB^{F} tB^{F} w^{−1} which may be seen by using the uniqueness properties of the Bruhat decomposition. □

□

For t ∈ B^{+}, we call support of t the set of s ∈ S which appear in a decomposition of t as a product of elements of S: this set does not depend on the decomposition as it is not changed by a braid relation. With this notation, we have

Proposition 8.4. Let t ∈ B^{+}. The variety X(t) is irreducible (in particular, with the convention of [DMR, 3.3.5], H_{c}^{2l(t)}(X(t)) is given by h^{2l(t)}t^{l(t)} Id) if and only if the support of t meets every orbit of F on S (i.e., if the group L_{I} of [DMR, 2.3.8] cannot be taken different from G).

Proof. We adapt the proof of [Lu1, 3.10(d)] to our case. From proposition 8.2 we get that if m is a multiple of δ, |(U^{F} \X^{v}(t))^{F^{m}}| is the coefficient T_{v}T_{t}|T_{^{F}v} in H_{q^{m}}(W); this coefficient is equal to T_{t}T_{^{F}v}|T_{v} thus |(U^{F} \X(t))^{F^{m}}| is the trace of the endomorphism x ↦ T_{t}^{F}x of H_{q^{m}}(W).

The next lemma generalizes [Lu5, lemma 10.4(c)].

Lemma 8.5. Let x_{1}, …, x_{n} ∈ W; then the coefficient T_{x_{1}}T_{x_{2}} … T_{x_{n}}|1 is a polynomial in q^{m} of degree less than or equal to ∑_{i=1}^{n} l(x_{i}) − sup{l(x_{1}), …, l(x_{n})}.

Proof of the lemma. As the coefficient T_{x_{1}}T_{x_{2}} … T_{x_{n}}|1 is invariant under cyclic permutation of the factors, it is sufficient to prove that its degree is at most l(x_{1})+⋯+ l(x_{n−1}). We prove it by induction on n and for a fixed n by induction on l(x_{1}). If n = 0 then 1|1 = 1 has indeed degree 0. Assume now n > 0. If l(x_{1}) = 0 we get the result by induction on n. Assume now l(x_{1}) > 0. If there exists x′′_{1} ≠ 1 such that x_{1} = x′_{1}x′′_{1} with l(x_{1}) = l(x′_{1}) + l(x′′_{1}) and l(x′′_{1}) + l(x_{2}) = l(x′′_{1}x_{2}), then we get the result by induction on l(x_{1}), as l(x_{1})+l(x_{2})+⋯+l(x_{n−1}) = l(x′_{1})+l(x′′_{1}x_{2})+l(x_{3})+⋯+l(x_{n−1}) and T_{x_{1}}T_{x_{2}} … T_{x_{n}}|1 = T_{x′_{1}}T_{x′′_{1}x_{2}} … T_{x_{n}}|1. Otherwise there exists s ∈ S, y_{1}, y_{2} ∈ W such that x_{1} = y_{1}s, x_{2} = sy_{2} and l(y_{i}) = l(x_{i}) − 1 for i = 1, 2. We then have T_{x_{1}}T_{x_{2}} … T_{x_{n}}|1 = q^{m}(T_{y_{1}}T_{y_{2}} … T_{x_{n}}|1) + (q^{m} − 1)(T_{y_{1}}T_{x_{2}} … T_{x_{n}}|1), and we get the result by induction on l(x_{1}) as 1+l(y_{1})+l(x_{2})+⋯+l(x_{n−1}) = l(x_{1})+⋯+l(x_{n−1}). □

Lemma 8.6. Let t ∈ B^{+}; then Trace(x ↦ T_{t}^{F}x | H_{q^{m}}(W)) is a polynomial in q^{m} of degree l(t) and the coefficient of q^{ml(t)} in this polynomial is the number of v ∈ W^{F} who are right multiples of all elements of the support of t.

Proof. As we have T_{t}T_{^{F}v}|T_{v} = q^{−ml(v)}(T_{t}T_{^{F}v}T_{v^{−1}}|1), to show the lemma, it is enough to show that T_{t}T_{^{F}v}T_{v^{−1}}|1 is a polynomial in q^{m} of degree < l(t) + l(v) except if v = ^{F}v and all s in the support of t divide v on the left, and that in this last case it is a unitary polynomial of degree l(t) + l(v). Let us write t = t′s where s ∈ S. If l(s^{F}v) > l(^{F}v), then by lemma 8.5 the degree of T_{t}T_{^{F}v}T_{v^{−1}}|1 = T_{t′}T_{s^{F}v}T_{v^{−1}}|1 is less than l(t)+l(v). Otherwise, T_{t}T_{^{F}v}T_{v^{−1}}|1 = q^{m}(T_{t′}T_{^{F}v}T_{v^{−1}}|1)+(q^{m}−1)(T_{t′}T_{s^{F}v}T_{v^{−1}}|1). By lemma 8.5, we see that only the first summand can contribute to q^{m(l(t)+l(v))}; and by induction on l(t′), we see that the contribution to q^{m(l(t)+l(v))} of T_{t}T_{^{F}v}T_{v^{−1}}|1 is T_{^{F}v}T_{v^{−1}}|1 if all s in the support of t divide on the left ^{F}v, and is 0 otherwise. The result follows. □

From the last lemma, |(U^{F} \X(t))^{F^{m}}| is a polynomial of degree l(t) in q^{m}, unitary if and only if the support of t meets every F -orbit in S. As all irreducible components of U^{F} \X(t) have the same dimension, since X(t) is the transverse intersection of the graph of F with the smooth irreducible variety O(t), this variety is irreducible if and only if |(U^{F} \X(t))^{F^{m}}| is a unitary polynomial in q^{m}.

To prove the proposition it remains to check that X(t) is irreducible if and only if U^{F} \X(t) is. The“only if” part is clear; to see the “only” part, we may follow the arguments of [Lu2, 4.8]: if U^{F} \X(t) is irreducible, the set π of irreducible components of X(t) is a single orbit under U^{F}, so its cardinality is a power of p. The set π is in bijection with the set of irreducible components of the (smooth) compactification X(\underline{s}_{1}, …, \underline{s}_{r}) (see [DMR, 2.3.4]). But the G^{F} -stabilizer of (B, …, B) ∈ X(\underline{s}_{1}, …, \underline{s}_{r}) is B^{F}, thus the orbit of (B, …, B) (and a fortiori the number of irreducible components of X(\underline{s}_{1}, …, \underline{s}_{r})) has cardinality a divisor of |G^{F} /B^{F} |, which is prime to p, whence the result. □

By proposition 8.2, we see that the variety X^{v}(w) is non-empty if and only if T_{v}T_{w}|T_{^{F}v} ≠ 0. We shall study this condition, especially when v is F -stable. In what follows, we will denote by ≤ the Bruhat order on W.

Proposition 8.7. Assume that w ∈ B^{+} is of the form w = w_{1} … w_{k} where w_{i} ∈ W have mutually disjoint support. Then T_{v}T_{w}|T_{v} ≠ 0 is equivalent to T_{v}T_{w_{i}}|T_{v} ≠ 0 for all i.

Proof. By induction on k, it is enough to show the case k = 2 of the proposition. By the isomorphism of the Hecke algebra with a subalgebra of the group algebra of G^{F}, we have T_{v}T_{w}|T_{v} ≠ 0 if and only if BvBw_{1}Bw_{2}B ⊃ BvB. We then use the following lemma:

Lemma 8.8. For w, w′ ∈ W we have: BwBw′B ⊂ (∐_{v′≤w′} Bwv′B)∩(∐_{v≤w} Bvw′B).

Proof. The inclusion in e.g., the left union is an easy induction on l(w′), using the exchange lemma. □

Thus BvBw_{1}Bw_{2}B = ∐_{v_{1}} Bvv_{1}Bw_{2}B for some v_{1} ≤ w_{1}, and in turn this last union is a union of double cosets of the form Bvv_{1}v_{2}B, where v_{2} ≤ w_{2}; now the assumption on supports implies that vv_{1}v_{2} = v if and only if v_{1} = v_{2} = 1. Since v_{1} = 1 occurring is equivalent to BvBw_{1}B ⊃ BvB and then in turn v_{2} = 1 occurring is equivalent to BvBw_{2}B ⊃ BvB, we get the proposition. □

Lemma 8.9. If v, w, x ∈ W and if T_{v}T_{w}|T_{x} ≠ 0, then x ≥ vw.

Proof. The condition is equivalent to T_{w}T_{x^{−1}}|T_{v^{−1}} ≠ 0. Applying lemma 8.8, this implies v^{−1} = wx′ with x′ ≤ x^{−1}, i.e., vw = x^{′−1} with x^{′−1} ≤ x. □

Lemma 8.10. Let v, t ∈ W where t is a reflection of root α > 0. Then T_{v}T_{t}|T_{v} ≠ 0 if and only if vα < 0.

Proof. By [Dy, 1.2 and 1.12], vα < 0 if and only if vt < v. If T_{v}T_{t}|T_{v} ≠ 0, then by lemma 8.9 we have v > vt thus vα < 0. Conversely, if vα < 0, we will show by induction on l(t) that T_{v}T_{t}|T_{v} ≠ 0. If l(t) = 1, then v = v′t with l(v) = l(v′) + 1. We have then T_{v}T_{t} = (q − 1)T_{v} + qT_{vt}, thus T_{v}T_{t}|T_{v} = q − 1 ≠ 0. Otherwise, by [Dy, 1.4], we may write t = at′a where a ∈ S and l(t) = l(t′) + 2. We have: T_{v}T_{at′a}|T_{v} = q^{−l(v)}T_{v}T_{a}T_{t′}T_{a}T_{v^{−1}}|1; when va > v this is equal to qT_{va}T_{t′}|T_{va}. Otherwise, it is equal to q^{−l(v)}(q^{2}(T_{va}T_{t′}T_{av^{−1}}|1) + (q − 1)^{2}(T_{v}T_{t′}T_{v^{−1}}|1) + q(q − 1)(T_{va}T_{t′}T_{v^{−1}}|1) + q(q − 1)(T_{v}T_{t′}T_{av^{−1}}|1)) whose first term is equal to q^{3}T_{va}T_{t′}|T_{va}. Since the structure constants of the Hecke algebra are polynomials which positive highest coefficient, we see in both cases that T_{v}T_{t}|T_{v} will be non-zero if T_{va}T_{t′}|T_{va} ≠ 0 is non-zero. Since t′ is a reflection of root aα, we see by induction, that this coefficient is non-zero if va(aα) < 0, i.e., vα < 0, whence the result. □

Recall that an element w ∈ W is reduced-I, with I ⊂ S, if it is of minimal length in its coset wW_{I}. To continue our study, we define E_{W} (w) = {w_{0}v ∈ W | T_{v}T_{w}|T_{v} ≠ 0}. With this notation, we have

Lemma 8.11. Let I ⊂ S be F -stable. Assume that w ∈ W is of the form w = sw′ with s ∈ S − I and and w′ ∈ W_{I}. Then E_{W} (w) consists of the products v_{1}v_{2} where v_{2} ∈ E_{W_{I}}(w′) and where v_{1} is a reduced-I element such that l(v_{1}v_{2}s) > l(v_{1}v_{2}).

Proof. By proposition 8.7, T_{v}T_{w}|T_{v} ≠ 0 if and only if T_{v}T_{s}|T_{v} ≠ 0 and T_{v}T_{w′}|T_{v} ≠ 0. Let w_{0}v = v_{1}v_{2} where v_{2} ∈ W_{I} and v_{1} is reduced-I. Then we have v = (w_{0}v_{1}w_{0}^{I}).(w_{0}^{I}v_{2}). Note that w_{0}v_{1}w_{0}^{I} is still reduced-I; it follows that T_{v} = T_{w_{0}v_{1}w_{0}^{I}}T_{w_{0}^{I}v_{2}} and that the condition T_{w_{0}v_{1}w_{0}^{I}}T_{w_{0}^{I}v_{2}}T_{w′}|T_{w_{0}v_{1}w_{0}^{I}}T_{w_{0}^{I}v_{2}} ≠ 0 is equivalent to T_{w_{0}^{I}v_{2}}T_{w′}|T_{w_{0}^{I}v_{2}} ≠ 0. It remains to express the condition T_{w_{0}v_{1}v_{2}}T_{s}|T_{w_{0}v_{1}v_{2}} ≠ 0; this condition is equivalent to l(w_{0}v_{1}v_{2}s) < l(w_{0}v_{1}v_{2}), which in turn is equivalent to l(v_{1}v_{2}s) > l(v_{1}v_{2}). □

Note that v_{1} = 1 and v_{2} arbitrary in E_{W_{I}}(w′) satisfy the above condition, so that E_{W} (w) ⊃ E_{W_{I}}(w′).

We will apply lemma 8.11 in a more specific situation where the following holds:

Proposition 8.12. Under the assumptions of lemma 8.11, assume in addition that S = I ∪ {s}, that there is a unique s′ ∈ I which does not commute with s and that ss′s = s′ss′. Assume also that any v ∈ E_{W_{I}}(w′) whose support contains s′ is such that s′ is not in the support of s′v. Then E_{W} (w) = E_{W_{I}}(w′) ∪ { sv | v ∈ E_{W_{I}}(w′) and s′v < v }.

Proof. We take v_{1}v_{2} ∈ E_{W} (w) as in lemma 8.11; if v_{1} ≠ 1, then l(v_{1}s) < l(v_{1}) since v_{1} is reduced-I. It follows that v_{2} does not commute with s, thus the support of v_{2} must contain s′. We claim that v_{1} = s; otherwise, as v_{1} is reduced-I, it would end with s′s since s′′s = ss′′ for s′′ ≠ s′; but then s′sv_{2}s would not be reduced since by the assumption of the proposition v_{2}s has a reduced expression starting with s′s as s commutes with all terms of a reduced expression for v_{2} excepted s′. □

For I ⊂ S, we denote by Φ_{I} the corresponding parabolic root subsystem, and we denote by L_{I} the Levi subgroup generated by T and { U_{α} }_{α∈Φ_{I}}. We denote B_{I} (resp. B_{I}^{−}, W_{I}, U_{I}, U_{I}^{−}) the intersection with L_{I} of B (resp. B^{−}, W, U, U^{−}), by P_{I} the parabolic subgroup L_{I}B and by U_{P_{I}} its unipotent radical. We will use the following proposition in the proof of proposition 8.17:

Proposition 8.13. Let I_{1}, …, I_{k} be mutually disjoint subsets of S and let x_{i} ∈ L_{I_{i}}. Then the condition x_{1} … x_{k} ∈ U^{−}B is equivalent to x_{i} ∈ U_{I_{i}}^{−}B_{I_{i}} for all i.

Proof. If k = 1, let us write x_{1} = uv̇b with u ∈ U_{I_{1}}^{−}, with v̇ a representative of v ∈ W_{I_{1}} and b ∈ B_{I_{1}}. As U_{I_{1}}^{−} ⊂ U^{−} and B_{I_{1}}^{−} ⊂ B, the existence of the Bruhat decomposition with respect to the pair of Borel subgroups (B^{−}, B), which is obtained by multiplying on the left by w_{0} the classical Bruhat decomposition, implies that v = 1.

By induction on k it is enough to prove the statement for k = 2. Let I = I_{1} ∪ I_{2}; we have U_{I}^{−} = U_{I_{1}}^{−}U_{I_{2}}^{−} and B_{I} = B_{I_{1}}B_{I_{2}}. From the case k = 1 we get that x_{1}x_{2} ∈ U^{−}B is equivalent to x_{1}x_{2} ∈ U_{I_{1}}^{−}U_{I_{2}}^{−}B_{I_{1}}B_{I_{2}}. As B_{I_{1}} normalizes U_{I_{2}}^{−}, this is equivalent to x_{1}x_{2} ∈ (U_{I_{1}}^{−}B_{I_{1}}).(U_{I_{2}}^{−}B_{I_{2}}). Let us write x_{1}x_{2} = y_{1}y_{2} according to this decomposition; as L_{I_{1}} ∩ L_{I_{2}} = T we get x_{i} ∈ y_{i}T ⊂ U_{I_{i}}^{−}B_{I_{i}} for i = 1, 2. □

Lemma 8.14. Let w = v_{1} … v_{k} ∈ W be such that l(v_{1}) + … + l(v_{k}) = l(w) and let ẇ, v̇_{1}, …, v̇_{k} be representatives in N_{G}(T) such that ẇ = v̇_{1} … v̇_{k}. Then U_{w}ẇ = U_{v_{1}} v̇_{1} … U_{v_{k}} v̇_{k}.

Proof. We have U_{v_{1}} v̇_{1} … U_{v_{k}} v̇_{k} = U_{v_{1}}^{v_{1}}U_{v_{2}} …^{v_{1}...v_{k−1}}U_{v_{k}} ẇ, and we have U_{w} = ∏_{α∈N(w^{−1})} U_{α}, where N(w) = { α > 0 | ^{w}α < 0}. Let ∐ represent disjoint union. The lemma is thus a consequence of N(w^{−1}) = ∐_{i} v_{1} … v_{i−1}(N(v_{i}^{−1})), which itself is obtained by iterating the well known formula: l(x) + l(y) = l(xy) ⇔ N(xy) = y^{−1}(N(x)) ∐ N(y). □

Corollary 8.15. Let I_{1}, …, I_{k} be disjoint parts of S, and let v_{i} ∈ W_{I_{i}}. Then Bv_{1} … v_{k}B ∩ U^{−} = ∏_{i}((B_{I_{i}}v_{i}B_{I_{i}}) ∩ U^{−}).

Proof. As in proposition 8.13, it is enough to prove the result for k = 2. By lemma 8.14 we have Bv_{1}v_{2}B = U_{v_{1}v_{2}}v_{1}v_{2}B = U_{v_{1}} v̇_{1}U_{v_{2}} v̇_{2}B. Let x ∈ Bv_{1}v_{2}B ∩ U^{−} and write accordingly x = x_{1}x_{2}b where x_{1} ∈ U_{v_{1}} v̇_{1}, x_{2} ∈ U_{v_{2}} v̇_{2} and b ∈ B. We have x_{1}x_{2} ∈ U^{−}B thus by proposition 8.13 x_{1} = u_{1}b_{1} ∈ U_{I_{1}}^{−}B. We have u_{1} = x_{1}b_{1}^{−1} ∈ U_{v_{1}} v̇_{1}B ∩ U^{−} = Bv_{1}B ∩ U^{−}. As x ∈ U^{−} we have also b_{1}x_{2}b ∈ U^{−}, thus b_{1}x_{2}b ∈ Bv_{2}B ∩ U^{−}. □

Proposition 8.16. Let w ∈ B^{+}, and let I be an F -stable subset of S. Then for any v ∈ W, the left multiplication action of P_{I}^{F} on X(w) stabilizes ∐_{v′∈W_{I}v} X^{v′}(w).

Proof. An element (g_{1}B, …, g_{r}B) ∈ X(w) is in X^{v}(w) if and only if g_{1} ∈ BvB. If p ∈ Bw′B with w′ ∈ W_{I}, then by lemma 8.8 we have pg_{1} ∈ Bw′′vB with w′′ ≤ w′, thus w′′ ∈ W_{I}, whence the result. □

In the next proposition, for I ⊂ S we denote by B_{I}^{+} the submonoid of B^{+} generated by I = {s ∈ S | s ∈ I}.

Proposition 8.17. Under the assumptions of proposition 8.16, assume in addition that ^{w_{0}}w = sw′ where s ∉ I and where w′ ∈ B_{I}^{+}. Let U_{P_{I}} be the unipotent radical of P_{I}. Then, if w_{0}^{I} ∈ W lifts w_{0}^{I}, for any I we have an isomorphism of L_{I}^{F} ×<F^{δ}>- modules

H_{c}^{i}(( ∐_{v∈W_{I}w_{0}} X^{v}(w)) /U_{P_{I}}^{F}) ⟶^{∼} H_{c}^{i−2}(X_{L_{I}}(^{w_{0}^{I}}w′))(−1) ⊕ H_{c}^{i−1}(X_{L_{I}}(^{w_{0}^{I}}w′))

Proof. To simplify the notation we write just L, P for L_{I}, P_{I}, and we set Y = ∐_{v∈W_{I}w_{0}} X^{v}(w). Let us see first that the proposition follows from its special case where w ∈ W. If w = w_{1} … w_{k} is a decomposition of w as a product of elements of W, we have w′ = w′_{1}^{w_{0}}w_{2} …^{w_{0}}w_{k}, where ^{w_{0}}w_{1} = sw′_{1}. Using [DMR, 2.3.3] as in the beginning of the proof of proposition 8.2, we apply the proposition with G^{k}, F_{1}, L^{k}, (w_{1}, …, w_{k}) and (s, 1, …, 1) replacing respectively G, F, L, w and s. If we set Y′ = ∐_{v_{1},...v_{k}∈W_{I}w_{0}} (X_{G^{k}}^{(v_{1},...,v_{k})}((w_{1}, …, w_{k}), F_{1})) we obtain that the cohomology groups

of Y′/(U_{P}^{k})^{F_{1}} are sums of those of X_{L^{k}}((^{w_{0}^{I}}w′_{1}, w_{2} …, w_{k}), F_{1}). This last variety is isomorphic to X_{L}(^{w_{0}^{I}}w′). On the other hand (U_{P}^{k})^{F_{1}} ≃ U_{P}^{F} and Y′ is formed from pieces from Y. Indeed, by the beginning of the proof of proposition 8.2, we have Y = ∐_{v∈W_{I},v_{2},...,v_{k}∈W} (X_{G^{k}}^{(v,v_{2},...,v_{k})}((w_{1}, …, w_{k}), F_{1})). We show that the only non-empty pieces of Y are those such that v_{i} ∈ W_{I} for all i, i.e., those of Y′: the (v, v_{2}, …, v_{k}) piece is non-empty if and only if T_{(w_{1},...,w_{k})}T_{(v_{2}^{−1},v_{3}^{−1},...,^{F}v^{−1})}|T_{(v^{−1},v_{2}^{−1},...,v_{k}^{−1})} ≠ 0. As w_{2}, …, w_{k} ∈ ^{w_{0}}W_{I} and v ∈ W_{I}w_{0} the non-vanishing of this coefficient implies that v_{2}, …, v_{k} ∈ W_{I}w_{0}; indeed, proceeding by induction on i, if v_{i+1} ∈ W_{I}w_{0} with i ≥ 2, then the product T_{w_{i}}T_{v_{i+1}^{−1}} involves only T_{y} for y ∈ W_{I}w_{0}, thus v_{i} ∈ W_{I}w_{0}.

We thus assume now w ∈ W. We use then the model 8.1 of X^{v}(w) taking k = 2, w_{1} = ^{w_{0}}s and w_{2} = ^{w_{0}}w′:

X^{v}(w) = {(g_{1}B, g_{2}B)|g_{1}^{−1}g_{2} ∈ B^{w_{0}}sB, g_{2}^{−1}^{F}g_{1} ∈ B^{w_{0}}w′B, and g_{1} ∈ BvB}.

Let ẇ_{0} be a rational representative of w_{0}. Taking g_{1}ẇ_{0} and g_{2}ẇ_{0} as variables we get

X^{v}(w) = {(g_{1}B^{−}, g_{2}B^{−}) | g_{1} ∈ Bvw_{0}B^{−}, g_{1}^{−1}.g_{2} ∈ B^{−}sB^{−}, g_{2}^{−1}.^{F}g_{1} ∈ B^{−}w′B^{−}}.

By arguing as in the proof of proposition 8.16 we see that ∪_{v∈W_{I}w_{0}}Bvw_{0}B^{−} = PB^{−}. Thus (1) Y = {(g_{1}B^{−}, g_{2}B^{−}) | g_{1} ∈ PB^{−}, g_{2} ∈ G, g_{1}^{−1}.g_{2} ∈ B^{−}sB^{−}, g_{2}^{−1}.^{F}g_{1} ∈ B^{−}w′B^{−}}.

The action of P^{F} is by left multiplication. In the following, we fix F^{δ}-stable representatives, denoted ṡ and ẇ′ of s and w′. For v ∈ W, let U_{v}^{−} = U^{−} ∩ ^{v}U.

Lemma 8.18. The variety X = {p ∈ P | p^{−1}^{F}p ∈ B^{−}sw′B^{−}} admits natural actions of B^{−} by right multiplication and of P^{F} by left multiplication. The map p ↦ (pB^{−}, pu_{p}ṡB^{−}) where u_{p} is the unique element of U_{s}^{−} such that p^{−1}^{F}p ∈ u_{p}ṡB^{−}w′B^{−} defines a P^{F} and F^{δ} equivariant isomorphism of varieties X/B_{I}^{−} ⟶^{∼} Y.

Proof. The existence and uniqueness of u_{p} come from lemma 8.14: we have

B^{−}sw′B^{−} = U_{sw′}^{−}ṡẇ′B^{−} = U_{s}^{−}ṡU_{w′}^{−}ẇ′B^{−} = U_{s}^{−}ṡB^{−}w′B^{−}

where the U_{s}^{−} part is unique. The image of the map p ↦ (pB^{−}, pu_{p}ṡB^{−}) is easily checked to be in the model (1) of Y. If b ∈ B_{I}^{−}, the element u_{pb} is determined by b^{−1}u_{p}ṡB^{−} = u_{pb}ṡB^{−}. We have thus pu_{p}ṡB^{−} = pbu_{pb}ṡB^{−}, which shows that p and pb have the same image in Y.

Conversely, given (g_{1}B^{−}, g_{2}B^{−}) ∈ Y, the equality g_{1}B^{−} = pB^{−} defines p ∈ X up to right translation by B_{I}^{−}. We must check that g_{2}B^{−} = pu_{p}ṡB^{−}. By definition of Y, we have g_{2} ∈ ^{F}g_{1}B^{−}w^{′−1}B^{−} ∩ g_{1}B^{−}sB^{−} = ^{F}pB^{−}w^{′−1}B^{−} ∩ pB^{−}sB^{−} whence p^{−1}g_{2} ∈ p^{−1}^{F}pB^{−}w^{′−1}B^{−} ∩ B^{−}sB^{−} ⊂ u_{p}ṡB^{−}w′B^{−}w^{′−1}B^{−} ∩ B^{−}sB^{−}; but B^{−}w′B^{−}w^{′−1}B^{−} is a union of double cosets of the form B^{−}vB^{−}, where v ∈ W_{I}. Thus u_{p}ṡB^{−}ẇ′B^{−}w^{′−1}B^{−} ∩ B^{−}sB^{−} = u_{p}ṡB^{−}, whence the result. □

Let us decompose p ∈ X as ul, with u ∈ U_{P} and l ∈ L. The action of B_{I}^{−} does not change the component u thus the quotient of X/B_{I}^{−} by U_{P^{F}} is realized by the Lang map (u, l) ↦ (u^{−1}.^{F}u, l). If we take ^{l^{−1}}(u^{−1}.^{F}u) and l as variables we get

Y/U_{P}^{F} ≃ {u ∈ U_{P}, l ∈ L | ul^{−1}.^{F}l ∈ B^{−}sw′B^{−}}/B_{I}^{−},

where the action of b ∈ B_{I}^{−} is by conjugation by b^{−1} on u and by right multiplication on l, and where the action of L^{F} is by left multiplication on l.

Lemma 8.19. For u ∈ U_{P}, l ∈ L, the condition ul ∈ B^{−}sw′B^{−} is equivalent to u ∈ B_{{s}}^{−}sB_{{s}}^{−} and l ∈ B_{I}^{−}w′B_{I}^{−}.

Proof. We have B^{−}sw′B^{−} = U_{s}^{−}ṡU_{w}^{−}ẇB_{I}^{−}U_{P}^{−} = U_{s}^{−}ṡB_{I}^{−}wB_{I}^{−}U_{P}^{−}. Thus using that l normalizes U_{P}^{−}, we see that there exists u′ ∈ U_{P}^{−} such that uu′ ∈ U_{s}^{−}ṡB_{I}^{−}wB_{I}^{−}l^{−1}. Thus there exists l_{s} ∈ U_{s}^{−}ṡ ⊂ L_{{s}}, l′ ∈ B_{I}^{−}wB_{I}^{−}l^{−1} ⊂ L such that uu′ = l_{s}l′. We may then apply proposition 8.13 with k = 2, I_{1} = {s} and I_{2} = I (exchanging the roles of B and of B^{−}), and we get l_{s} ∈ U_{s}B_{{s}}^{−}. Thus there exists u_{s} ∈ U_{s} such that l_{s} ∈ u_{s}B_{{s}}^{−}. We have u_{s}^{−1}u ∈ B_{{s}}^{−}Lu^{′−1} ⊂ LU_{P}^{−} thus u_{s}^{−1}u ∈ U_{P} ∩ LU_{P}^{−} = {1} thus u = u_{s} ∈ U_{s}, and since l_{s} ∈ U_{s}^{−}ṡ we even have u ∈ U_{s}^{−}ṡB_{{s}}^{−} = B_{{s}}^{−}sB_{{s}}^{−}. The condition uu′ = l_{s}l′ becomes thus u′ = b_{s}l′ for some b_{s} ∈ B_{{s}}^{−}; as TU_{P}^{−} ∩ L = T, this implies l′ ∈ T thus l ∈ B_{I}^{−}w′B_{I}^{−} q.e.d. □

As we have U ∩ B_{{s}}^{−}sB_{{s}}^{−} = U_{s}^{∗} ⊂ U_{P}, we get thus Y/U_{P}^{F} ≃ {u ∈ U_{s}^{∗}, l ∈ L | l^{−1}^{F}l ∈ B_{I}^{−}w′B_{I}^{−}}/B_{I}^{−} where the action of L^{F} is by left multiplication on l and the action of b ∈ B_{I}^{−} is by right multiplication on l and conjugation by b^{−1} on u. Note that, as U_{I}^{−} centralizes U_{s} since no root in Φ_{I}^{−} can add to the simple root corresponding to s, the action of B_{I}^{−} on u is through T.

We have X_{L_{I}}(^{w_{0}^{I}}w′) = {l ∈ L | l^{−1}^{F}l ∈ B_{I}^{−}w′B_{I}^{−}}/B_{I}^{−}. We conclude arguing as in [DMR, 3.2.10]. To simplify the notation we write w′′ for ^{w_{0}^{I}}w′. Let Ỹ be the variety {u ∈ U_{s}, l ∈ L | l^{−1}^{F}l ∈ B_{I}^{−}w′B_{I}^{−}}/B_{I}^{−}; the projection π : Ỹ → X_{L_{I}}(w′′) defined by (u, l) ↦ l is a fibration by affine lines, and π restricted to Ỹ − Y/U_{P}^{F} is an isomorphism. Let i be the closed inclusion Ỹ − Y/U_{P}^{F} ↪ Ỹ and j the open inclusion Y/U_{P}^{F} ↪ Ỹ. If we make L^{F} act by left multiplication on Ỹ then i, j and π are L^{F} -equivariant. Let Λ_{X} be the constant sheaf \overline{ℚ}_{ℓ} on a variety X; we have an exact sequence 0 → j_{!}Λ_{Y/U_{P}^{F}} → ΛỸ → i_{!}Λ_{Ỹ−Y/U_{P}^{F}} → 0. Using that Rπ_{!}ΛỸ ≃ Λ_{X_{L_{I}} (w′′)}[−2](−1) and that Rπ_{!}i_{!}Λ_{Ỹ−Y/U_{P}^{F}} ≃ Λ_{X_{L_{I}} (w′′)}, we deduce a distinguished triangle Rπ_{!}j_{!}Λ_{Y/U_{P}^{F}} → Λ_{X_{L_{I}} (w′′)}[−2](−1) ⟶^{∂} Λ_{X_{L_{I}} (w′′)} ⇝ where ∂ ∈ Ext^{2}(Λ_{X_{L_{I}} (w′′)}, Λ_{X_{L_{I}} (w′′)}) = H^{2}(X_{L_{I}}(w′′), \overline{ℚ}_{ℓ}). All maps being G^{F} -equivariant, we even have ∂ ∈ H^{2}(X_{L_{I}}(w′′), \overline{ℚ}_{ℓ})^{G^{F}} ≃ H_{c}^{2l−2}(X_{L_{I}}(w′′), \overline{ℚ}_{ℓ})^{G^{F}} where the isomorphism comes from the smoothness of the variety X_{L_{I}}(w′′). But by [DMR, 3.3.14] we have H_{c}^{2l−2}(X_{L_{I}}(w′′), \overline{ℚ}_{ℓ})^{G^{F}} = 0, thus ∂ = 0 and the distinguished triangle gives an isomorphism Rπ_{!}j_{!}Λ_{Y/U_{P}^{F}} ≃ Λ_{X_{L_{I}} (w′′)}[−2](−1)⊕Λ_{X_{(L_{I})}(w′′)}[−1] whence H_{c}^{i}(Y/U_{P}^{F}) ≃ H_{c}^{i−2}(X_{L_{I}}(w′′))(−1) ⊕ H_{c}^{i−1}(X_{L_{I}}(w′′)) as wanted. □

In the proof of proposition 8.21 we will use the next lemma.

Lemma 8.20. Let ẇ_{1}, …, ẇ_{k} be representatives in N_{G}(T) of w_{1}, …, w_{k} ∈ W; for any u_{1}, …, u_{k} ∈ U, there exist unique u′_{i} ∈ U_{w_{i}} such that for all i we have u_{1}ẇ_{1} … u_{i}ẇ_{i} ∈ u′_{1}ẇ_{1} … u′_{i}ẇ_{i}U. This defines a morphism U^{k} → ∏_{i=1}^{k} U_{w_{i}}.

Proof. It is known that for v ∈ W, the equality uv̇ = u′v̇u′′ with u ∈ U, u′ ∈ U_{v} and u′′ ∈ U ∩ ^{v}U defines an isomorphism U ⟶^{∼} U_{v} × U ∩ ^{v}U. The lemma is a consequence of this fact by induction on k. □

The elements w that we will handle in this paper will have E_{W} (w) of the form W_{I}w_{0} ∪ {v}, where v satisfies the assumptions of the next proposition.

Proposition 8.21. Let w = w_{1} … w_{k} be a decomposition of w with w_{i} ∈ W; let v ∈ W^{F}, and let I be an F -stable subset of S; we write P for P_{I}. Then

(i) For all i we have an isomorphism of <F >-modules:

H_{c}^{i}(X^{v}(w))^{U_{P}^{F}} ≃ H_{c}^{i+2l(w_{0}v)}(Z_{w}^{v})(l(w_{0}v)),

where (l(w_{0}v)) is a “Tate twist” and

Z_{w}^{v} = {(y, x, u_{1}, …, u_{k}) ∈ U_{P} × U_{I} × ∏_{i} U_{w_{i}} | yx^{−1}^{F}x ∈ v̇u_{1}ẇ_{1} … u_{k}ẇ_{k}Bv^{−1}}.

(ii) Let \overline{Z}_{w}^{v} be the variety \{(y, x, u_{1}, …, u_{k}) ∈ Z_{w}^{v} | y ∈ U_{P} ∩ ^{v}U^{−}\}. The map Z_{w}^{v} → \overline{Z}_{w}^{v} given by (y, x, u_{1}, …, u_{k}) ↦ (y_{2}, x, u′_{1}, …, u′_{k}) where y_{2} is defined by y = y_{1}y_{2} with y_{1} ∈ U_{P} ∩ ^{v}U and y_{2} ∈ U_{P} ∩ ^{v}U^{−} and where the u′_{i} are defined by ^{v̇^{−1}}y_{1}^{−1}u_{1}ẇ_{1} … u_{i}ẇ_{i} ∈ u′_{1}ẇ_{1} … u′_{i}ẇ_{i}U for any i (cf. lemma 8.20), is a fibration whose fibers are all isomorphic to ^{v^{−1}}U_{P} ∩ U.

(iii) If in addition v is the unique element of W_{I}v such that X^{v}(w) is nonempty, there is an action of L_{I}^{F} on Z_{w}^{v} such that the isomorphism of 8.21(i) is L_{I}^{F} -equivariant (for the action of L_{I}^{F} on X^{v}(w) given by proposition 8.16).

(iv) If in addition ^{v^{−1}}U_{I} ⊂ U^{−} and proj_{U_{I}}(U ∩ vBw_{1}B … w_{k}Bv^{−1}) = ∏_{s∈I} U_{s}^{∗} (where we have denoted proj_{U_{I}} the natural projection U → U_{I}), then the projection π : (y, x, u_{1}, …, u_{k}) ↦ x is an epimorphism Z_{w}^{v} → X_{L_{I}}(c), where c is a Coxeter element of W_{I}, which is L_{I}^{F} -equivariant (for the action of L_{I}^{F} on Z_{w}^{v} given in 8.21(iii)).

Proof.

Lemma 8.22. Let v̇, ẇ_{1}, …, ẇ_{k} be representatives in N_{G}(T) of v, w_{1}, …, w_{k} ∈ W and let

O^{v}(w_{1}, …, w_{k}) = {(x_{1}B, …, x_{k}B) ∈ (G/B)^{k} | x_{i}^{−1}x_{i+1} ∈ Bw_{i}B and x_{1} ∈ BvB};

there exist unique u ∈ U_{v} and u_{i} ∈ U_{w_{i}} such that x_{1} ∈ uvB and that for all i we have x_{i+1} ∈ uv̇u_{1}ẇ_{1} … u_{i}ẇ_{i}B. This defines a morphism O^{v}(w_{1}, …, w_{k}) → U_{v} × (∏_{i=1}^{k} U_{w_{i}}).

Proof. The proof is similar to that of lemma 8.20. □

Consider the map Ψ : (u, g_{2}B, …, g_{k}B) ↦ (uvB, g_{2}B, …, g_{k}B) from the variety

Z = {(u, g_{2}B, …, g_{k}B) | u ∈ U, (uv)^{−1}g_{2} ∈ Bw_{1}B, g_{i}^{−1}g_{i+1} ∈ Bw_{i}B, g_{k}^{−1}^{F}(uv) ∈ Bw_{k}B}

to X^{v}(w). As BvB = U_{v}vB, it is a fibration whose fibers are isomorphic to U ∩ ^{v}U, an affine space of dimension l(w_{0}v). This fibration is U^{F} (thus U_{P}^{F})- equivariant, for the action by left multiplication of all components. Applying lemma 8.22 with x_{i} = (uv)^{−1}g_{i+1} we get Z ≃ {(u, u_{1}, …, u_{k}) | u ∈ U, u_{i} ∈ U_{w_{i}}, u^{−1}^{F}u ∈ v̇u_{1}ẇ_{1} … u_{k}ẇ_{k}Bv^{−1}}. The map u = x_{P}x ↦ (y = ^{x^{−1}}(x_{P}^{−1}^{F}x_{P}), x), where x_{P} ∈ U_{P} and x ∈ U_{I}, defines thus an isomorphism between Z/U_{P}^{F} and Z_{w}^{v}; 8.21(i) results immediately from this isomorphism and the isomorphism of cohomology implied by Ψ.

Let us prove 8.21(ii). It is clear that the image of the map is in \overline{Z}_{w}^{v}. Consider now the fiber of (y_{2}, x, u′_{1}, …, u′_{k}). Let y_{1} in U_{P} ∩ ^{v}U be arbitrary; the formulas v̇^{−1}y_{1}^{−1}u_{1}ẇ_{1} … u_{i}ẇ_{i} ∈ u′_{1}ẇ_{1} … u′_{i}ẇ_{i}B for any i define unique u_{i} (cf. lemma 8.20), and the element thus obtained (y_{1}y_{2}, x, u_{1}, …, u_{k}) is in the fiber and we get thus all the fiber.

We prove now 8.21(iii). If v is the unique element of W_{I}v such that X^{v}(w) is non-empty, we will define an action of P^{F} on Z such that Ψ is P^{F} -equivariant. Under this assumption, by proposition 8.16 X^{v}(w) is P^{F} -stable, i.e., if p ∈ P^{F} and (uvB, g_{2}B, …, g_{k}B) ∈ X^{v}(w) where u ∈ U_{v}, then pu ∈ U_{v}vB. The action of p ∈ P^{F} on X^{v}(w) is thus given by

(uvB, g_{2}B, …, g_{k}B) ↦ (\overline{u}vB, pg_{2}B, …, pg_{k}B)

where \overline{u} ∈ U_{v} is defined by pu = \overline{u}b with b ∈ ^{v}B. Let z ∈ Z have image (uvB, g_{2}B, …, g_{k}B) in X^{v}(w), i.e., z = (uu′, g_{2}B, …, g_{k}B) where u′ ∈ U ∩ ^{v}U. We want to define the action of p by the map

z ↦ (\overline{u}. proj_{U∩^{v}U}(bu′), pg_{2}B, …, pg_{k}B)

where pu = \overline{u}b as above and where proj_{U∩^{v}U} is the projection of ^{v}B on U ∩ ^{v}U according to the decomposition ^{v}B = (U ∩ ^{v}U).(U^{−} ∩ ^{v}U).T. This map clearly commutes with Ψ, but it is not obvious that it defines an action. Let p′ ∈ P^{F}; let us write p′\overline{u} = \overline{\overline{u}}b′ where \overline{\overline{u}} ∈ U_{v} and b′ ∈ ^{v}B. We must check that the action of pp′ is the composition of that of p and that of p′; this is equivalent to proj_{U∩^{v}U}(b′bu′) = proj_{U∩^{v}U}(b′ proj_{U∩^{v}U}(bu′)), and this last equality is easy to check. This action of P^{F} gives after quotienting Z by U_{P}^{F} an action of L_{I}^{F} on Z_{w}^{v}, for which 8.21(iii) holds.

Let us now prove 8.21(iv). By [Lu2, 2.5] the variety X_{L_{I}}(c) has a single piece X_{L_{I}}^{w_{0}^{I}}(c). As c ∈ W, the model 8.1 gives X_{L_{I}}^{w_{0}^{I}}(c) ≃ {gB_{I} ∈ L_{I}/B_{I} | g^{−1}^{F}g ∈

B_{I}cB_{I} and g ∈ B_{I}w_{0}^{I}B_{I}}. Defining u ∈ U_{I} by g ∈ uw_{0}^{I}B_{I} we get X_{L_{I}}^{w_{0}^{I}}(c) ≃ {u ∈ U_{I} | ^{w_{0}^{I}}(u^{−1}^{F}u) ∈ B_{I}cB_{I}}. As B_{I}cB_{I} ∩ U_{I}^{−} = ∏_{s∈I} U_{s}^{−∗} (cf. [Lu2, 2.6]) and w_{0}^{I}(∏_{s∈I} U_{s}^{∗}) = ∏_{s∈I} U_{s}^{−∗}, we get X_{L_{I}}^{w_{0}^{I}}(c) ≃ {u ∈ U_{I} | u^{−1}^{F}u ∈ ∏_{s∈I} U_{s}^{∗}}, on which t ∈ T^{F} acts by u ↦ ^{t}u and u_{+} ∈ U_{I}^{F} acts by u ↦ u_{+}u; the action of u_{−} ∈ U_{I}^{−F} maps u on the element u′ of U_{I} such that u_{−}u ∈ u′B_{I}^{−}: such an element exists by uniqueness of the piece X_{L_{I}}^{w_{0}^{I}}(c). On the other hand the projection on the first two components of Z^{v}(w) is surjective on couples (y, x) ∈ U_{P} × U_{I} which satisfy yx^{−1}.^{F}x ∈ U ∩ vBw_{1}B … w_{k}Bv^{−1} and the projection of U ∩ vBw_{1}B … w_{k}Bv^{−1} on U_{I} is equal to ∏_{s∈I} U_{s}^{∗} by assumption. Thus we see that π indeed defines an epimorphism Z_{w}^{v} → X_{L_{I}}(c).

It remains to check that this epimorphism is L_{I}^{F} -equivariant. For this, it is enough to check that the above action and that on the component x of (y, x, u_{1}, …, u_{k}) ∈ Z_{w}^{v}, which results itself of the action on the projection on U_{I} of the first component of an element of Z, coincide. For this it is enough to check separately that the actions of B_{I}^{F} and U_{I}^{−F} coincide.

Let u ∈ U_{v} and u′ ∈ U∩^{v}U; the first component of the image of z = (uu′, g_{2}B, …) ∈ Z by the action of xt ∈ B_{I}^{F}, with t ∈ T^{F} and x ∈ U_{I}^{F}, is x^{t}u proj_{U∩^{v}U}(tu′) = x^{t}u^{t}u′, since x ∈ U_{v} and u′ ∈ U ∩ ^{v}U. Since U_{I} ⊂ U_{v}, the action of xt on the projection u_{I} of uu′ is thus u_{I} ↦ x^{t}u_{I}, which coincides with the action on X_{L_{I}}(c).

Similarly, the action of y ∈ U_{I}^{−F} maps u_{I} on proj_{U_{I}}(\overline{u}) where yu ∈ \overline{u}^{v}B with \overline{u} ∈ U_{v}. On the other hand, the image by y of u_{I} ∈ X_{L_{I}}^{w_{0}^{I}}(c) is u′_{I} such that yu_{I} ∈ u′_{I}B_{I}^{−} ⊂ u′_{I}^{v}B. Let us write u = u_{P}u_{I} with u_{P} ∈ U_{P}; thus yu_{P}u_{I} = ^{y}u_{P}yu_{I} ∈ yu_{P}u′_{I}^{v}B = u′_{I}(^{u_{I}^{′−1}y}u_{P})^{v}B = u′_{I}(^{u_{I}^{′−1}y}u_{P})_{−}^{v}B = ^{u′_{I}}(^{u_{I}^{′−1}y}u_{P})_{−}u′_{I}^{v}B, where we have denoted by x_{−} the projection on U_{v} of an element x ∈ U = U_{v}.(U ∩ ^{v}U). But we have ^{u′_{I}}(^{u_{I}^{′−1}y}u_{P})_{−} ∈ U_{P} ∩ ^{v}U^{−} since u′_{I} ∈ U_{I} ∩ ^{v}U^{−}, thus \overline{u} = ^{u′_{I}}(^{u_{I}^{′−1}y}u_{P})_{−}u′_{I} and its component in U_{I} is u′_{I} as required. □

The map π of 8.21(iv) is the composition of the fibration of 8.21(ii) and the projection

\overline{π} : (y, x, u_{1}, …, u_{k}) ↦ x

which is an epimorphism \overline{Z}_{w}^{v} → X_{L_{I}}(c). All fibers of Z_{w}^{v} → \overline{Z}_{w}^{v} are affine spaces of dimension l(w_{0}v). It will be easier to compute the fibers of \overline{π} than those of π. In the case where we will apply proposition 8.21, we will be in the situation of one of the next two propositions:

Proposition 8.23. Assume under the assumptions of 8.21(iv) that the fibers of \overline{π} are affine spaces of dimension d. Then for every i we have an isomorphism of L_{I}^{F} × <F >-modules: H_{c}^{i}(X^{v}(w))^{U_{P_{I}}^{F}} ≃ H_{c}^{i−2d}(X_{L_{I}}(c))(−d).

Proof.

Lemma 8.24. With the same notation as above proposition 8.13, let V be a variety given with an action of P_{I}^{F}, let V′ be a variety given with an action of L_{I}^{F} and let π : V/U_{P_{I}}^{F} → V′ be an L_{I}^{F} -equivariant epimorphism whose fibers are all affine spaces of dimension d. Then for all i we have an isomorphism H_{c}^{i}(V)^{U_{P_{I}}^{F}} ≃ H_{c}^{i−2d}(V′)(−d) of L_{I}^{F} × <F >-modules.

Proof. This lemma results from standard properties of ℓ-adic cohomology , see e.g., [DM, 10.10 and 10.12]. □

The proposition results immediately from proposition 8.21 and lemma 8.24, taking in account the “Tate twist” induced by the quotient Z_{w}^{v} → \overline{Z}_{w}^{v}. □

In the next proposition we assume w ∈ W and write X(w) for X(w).

Proposition 8.25. Assume that in addition to w, there is another element w′ < w satisfying the assumptions of 8.21(iv) with the same v; let \overline{π}′ : \overline{Z}_{w′}^{v} → X_{L_{I}}(c) be the epimorphism analogous to \overline{π} and assume that the fibers of \overline{π}′ are affine lines and that the fibers of

\overline{π} ∐ \overline{π}′ : \overline{Z}_{w}^{v} ∐ \overline{Z}_{w′}^{v} → X_{L_{I}}(c)

are affine planes, the above union being taken in (U_{P_{I}} ∩^{v}U^{−}) × U_{I} × U. Then for any i we have an isomorphism of L_{I}^{F} × <F >-modules:

H_{c}^{i}(X^{v}(w))^{U_{P_{I}}^{F}} ≃ H_{c}^{i−3}(X_{L_{I}}(c))(−1) ⊕ H_{c}^{i−4}(X_{L_{I}}(c))(−2).

Proof. Proposition 8.23 and lemma 8.24 give respectively the isomorphisms of L_{I}^{F} × <F >-modules H_{c}^{i}(X^{v}(w′))^{U_{P_{I}}^{F}} ≃ H_{c}^{i−2}(X_{L_{I}}(c))(−1) and H_{c}^{i}(X^{v}(w) ∐ X^{v}(w′))^{U_{P_{I}}^{F}} ≃ H_{c}^{i−4}(X_{L_{I}}(c))(−2). The assumption w′ < w implies that X^{v}(w) is open in X^{v}(w) ∐ X^{v}(w′). We deduce a long exact sequence

… → H_{c}^{i−3}(X_{L_{I}}(c))(−1) → H_{c}^{i}(X^{v}(w))^{U_{P_{I}}^{F}} → H_{c}^{i−4}(X_{L_{I}}(c))(−2)
→ H_{c}^{i−2}(X_{L_{I}}(c))(−1) → …

We deduce the proposition by observing that a morphism of L_{I}^{F} -modules from H_{c}^{i−4}(X_{L_{I}}(c))(−2) to H_{c}^{i−2}(X_{L_{I}}(c))(−1) must be 0. Indeed cohomology groups of different degrees of X_{L_{I}}(c) are disjoint as L_{I}^{F} -modules. □

9. The n-th roots of π in type A_{n}

In this section we compute the cohomology groups H_{c}^{i}(X(w)) as G^{F} × <F >- modules when G is a split group of type A_{n} (n ≥ 1) and w an n-th root of π and we show conjectures 2.1 to 2.6 for this case. The Coxeter presentation of W is given by the diagram ◯_{s_{1}} ◯_{s_{2}}⋯◯_{s_{n}}, and we denote S = {s_{1}, …, s_{n}} the corresponding

generating set of B.

Conjecture 2.1 holds by theorem 5.1, and conjecture 2.3 holds as remarked in section 5. As noticed in section 2, conjecture 2.2 follows from a result of Eilenberg and a recent result of Birman, Gebhardt and Gonzales-Meneses. However, we will give a simple proof of it in our case.

Proposition 9.1. There is a morphism in D^{+} between any two of n-th roots of π in B^{+}.

Proof. We will show the proposition by showing that any two roots b and b′ are equivalent by the transitive closure of the relation on B^{+} given by xy ∼ yx.

By [BMi, 3.12 and A.1.1] the image in w, identified with the symmetric group 𝔖_{n+1}, of any n-th root of π is an n-cycle. So it is enough to see that if b ∈ B^{+} has length n + 1 and support S, and is such that β(b) is an n-cycle then there exists a morphism from b to s_{1} … s_{n−1}s_{n}s_{n} in D^{+}.

We first show that there exists a morphism in D^{+} from b to an element of the form xs_{n}s_{n}y. By assumption, in the decomposition of b as a product of elements of S, exactly one, say s_{i} is present twice. We write b = xs_{i}ys_{i}z where x, y, z ∈ B^{+} do not have s_{i} in their support. There are three cases:

(a) The support of y contains neither s_{i+1} nor s_{i−1}. Then b = xys_{i}s_{i}z, so that β(b) = β(xyz) is an element of length n − 1 of support S − {s_{i}}. Such an element can be an n-cycle only if i = 1 or i = n. In the latter case b is as desired. If i = 1, as s_{1} commutes with all elements of S − {s_{2}}, we see that b has the form xs_{1}s_{1}s_{2}y or xs_{2}s_{1}s_{1}y. In both cases b is equivalent to xs_{1}s_{2}s_{1}y = xs_{2}s_{1}s_{2}y and we are reduced to case (c) below with i = 2.

(b) The support of y contains s_{i+1} and s_{i−1}. Then we use that b is equivalent to ys_{i}zxs_{i}, and the support of zx contains neither s_{i+1} nor s_{i−1}. We are back to case (a).

(c) The support of y contains one of s_{i+1} and s_{i−1}. Then s_{i} commutes with either x or z and if i = n then b is equivalent to an element of the form we want. Otherwise replacing if needed b by the equivalent element ys_{i}zxs_{i}, we may assume that y involves s_{i+1}. Then b can be written xs_{i}s_{i+1}s_{i}y′z = xs_{i+1}s_{i}s_{i+1}y′z. By induction on i we are reduced to i = n whence the result.

We prove now that any pair of elements xs_{n}s_{n}y of length n + 1 with support S are connected by a morphism in D^{+}. We use [Bou, chap. V §6, lemma 1] which says “If X is a finite forest and if x ↦ g_{x} is a mapping from X to a group Γ such that g_{x} and g_{y} commute if x and y are not connected in X, then the elements of Γ which are the products of all the g_{x} in some order are conjugate by cyclic permutation”, where conjugation by cyclic permutation is the transitive closure of g_{x_{1}} … g_{x_{k}} ↦ g_{x_{2}} … g_{x_{k}}g_{x_{1}} ([Bou] does not state that the conjugation is by cyclic permutation but it is established in the proof). We apply this result to the map from the Coxeter diagram of type A_{n} which maps the i-th vertex to s_{i} ∈ B, with the exception of the n-th vertex which is mapped onto s_{n}s_{n}. This gives the result. □

We denote by ρ_{b}^{(n)} the unipotent representation of G^{F} which corresponds to the partition 1, …, 1, 2, b of n + 1. Let St^{(n)} be the Steinberg representation and Id^{(n)} be the identity representation of G^{F}. We will deduce conjectures 2.4 to 2.6 from the following theorem. In this theorem, we adopt the conventions of [DMR, 3.3.5] to describe the cohomology of a variety X(w) as a G^{F} ×<F >-module; we describe the cohomology as a 2-variable polynomial with coefficients in the Grothendieck group of G^{F}, where the degree in the variable h represents the degree of the cohomology group, and where the degree in t encodes the eigenvalues of F: by a theorem of Lusztig, given a unipotent character ρ, the eigenvalues of F on the ρ-isotypic part of a cohomology group H_{c}^{j}(X(w)) are of the form q^{i}λ_{ρ} where λ_{ρ} is a complex number of module 1 or q^{1/2} which depends only on ρ and neither on j nor on w. We encode such an eigenvalue by t^{i}.

Theorem 9.2. Let w ∈ B^{+} be an n-th root of π; then we have as G^{F} × <F >- modules:

∑_{i} h^{i}H_{c}^{i}(X(w)) = St^{(n)} h^{n+1} + ∑_{b=2}^{n−1} ρ_{b}^{(n)}t^{b}h^{n+b} + Id^{(n)} t^{n+1}h^{2(n+1)}.

Proof. We prove the theorem by induction on n. If n = 1 we have w = π. Then the only unipotent representations of G^{F} are St^{(1)} and Id^{(1)} and the result is given by [DMR, 3.3.14] and [DMR, 3.3.15]. If n ≥ 2, by proposition 9.1 and [DMR, 3.1.6], it is sufficient to prove the result for a fixed root of π.

We choose w = s_{1} … s_{n−1}s_{n}s_{n}. We shall prove the theorem using the results of section 8. Let I = {s_{1}, …, s_{n−1}}.

Lemma 9.3. The variety X^{v}(w) is not empty if and only if v is the longest element in its coset vW_{I}.

Proof. We apply proposition 8.2: X^{v}(w) is not empty if and only if T_{v}T_{w}|T_{v} is not equal to zero. In the Hecke algebra T_{w} = (q −1)T_{s_{1}...s_{n}} +qT_{s_{1}...s_{n−1}}, so if T_{v}T_{w}|T_{v} ≠ 0 then T_{v}T_{s_{1}...s_{n}}|T_{v} or T_{v}T_{s_{1}...s_{n−1}}|T_{v} is not zero. By [Lu2, 2.5] or proposition 8.7, the only v such that T_{v}T_{s_{1}...s_{n}}|T_{v} ≠ 0 is w_{0}. Let us write v = xy with x reduced-I and y ∈ W_{I}; then T_{v}T_{s_{1}...s_{n−1}}|T_{v} ≠ 0 if and only if T_{y}T_{s_{1}...s_{n−1}}|T_{y} ≠ 0; by the same result as above, applied in W_{I}, this coefficient is not zero if and only if y = w_{0}^{I}. So v has to be the longest element in its coset, and we have shown that this is equivalent to the non-vanishing of T_{v}T_{w}|T_{v} except possibly for v = w_{0}. In this last case T_{w_{0}}T_{s_{1}...s_{n}}|T_{w_{0}} = (q −1)^{n} and T_{w_{0}}T_{s_{1}...s_{n−1}}|T_{w_{0}} = (q −1)^{n−1} and the sum of these two coefficients is again non-zero. □

The reduced-I elements are s_{i}s_{i+1} … s_{n}, for i ≤ n; their number is n+1 = |W/W_{I}|. The elements v of maximal length in their cosets vW_{I} are then s_{i}s_{i+1} … s_{n}w_{0}^{I} = s_{1} … s_{i−1}w_{0}. They are in the coset W_{I}w_{0}, except s_{1} … s_{n}w_{0} = w_{0}^{I}. Let P denote the parabolic subgroup P_{I}; by proposition 8.16 X(w) is the union of two P^{F} -stable pieces: ⋃_{v∈W_{I}} X^{vw_{0}}(w), which is an open subvariety as ⋃_{v∈W_{I}} Bvw_{0}B is open in ⋃_{v∈W_{I}} Bvw_{0}B ⋃ Bw_{0}^{I}B, and the closed subvariety X^{w_{0}^{I}}(w). As H_{c}^{i}(X(w)/U_{P}^{F}) = ∗R_{L}^{G}(H_{c}^{i}(X(w))) (cf. e.g., [DM, 10.10]), we get, setting X_{1} = (⋃_{v∈W_{I}} X^{vw_{0}}(w))/U_{P}^{F} and X_{2} = X^{w_{0}^{I}}(w)/U_{P}^{F}, the following long exact sequence of L^{F} × <F >-modules, where L denotes L_{I}:

(1)  … → H_{c}^{i}(X_{1}) → ^{∗}R_{L}^{G}(H_{c}^{i}(X(w))) → H_{c}^{i}(X_{2}) → H_{c}^{i+1}(X_{1}) → …

We now apply proposition 8.17 with s = s_{1} (indeed ^{w_{0}}w = s_{n}w′ where w′ = s_{n−1} … s_{2}s_{1}s_{1} ∈ B_{I}^{+}), whence

H_{c}^{i}(X_{1}) ≃ H_{c}^{i−2}(X_{L}(^{w_{0}^{I}}w′))(−1) ⊕ H_{c}^{i−1}(X_{L}(^{w_{0}^{I}}w′)).

The element ^{w_{0}^{I}}w′ = s_{1} … s_{n−2}s_{n−1}s_{n−1} is the element of L analogous to w. So by induction on n we get the equality of L^{F} × <F >-modules:

(1′) ∑_{i} h^{i}H_{c}^{i}(X_{1}) = (th^{2} + h)(St^{(n−1)} h^{n} + ∑_{b=2}^{n−2} ρ_{b}^{(n−1)}t^{b}h^{n+b−1} + Id^{(n−1)} t^{n}h^{2n}).

To compute the L^{F} × <F >-module ∑_{i} h^{i}H_{c}^{i}(X_{2}), we apply propositions 8.21 and 8.23 with v = w_{0}^{I}, with k = n + 1 and with s_{1}, s_{2}, …, s_{n}, s_{n} for w_{1}, …, w_{k}. Let us check the assumptions: the assumption (iii) on v holds by lemma 9.3; assumption (iv) holds since ^{v^{−1}}U_{I} = U_{I}^{−} ⊂ U^{−}; the other assumptions to check are

(2)  proj_{U_{I}}(U ∩ vBs_{1}B … Bs_{n−1}Bs_{n}Bs_{n}Bv^{−1}) = ∏_{α∈I} U_{α}^{∗}

and that the fibers of \overline{π} are affine lines. We have

proj_{U_{I}}(U ∩ vBs_{1}B … Bs_{n−1}Bs_{n}Bs_{n}Bv^{−1}) =
v(proj_{U_{I}^{−}}(^{v^{−1}}U ∩ Bs_{1}B … Bs_{n}Bs_{n}B)) =
v(proj_{U_{I}^{−}}((U_{I}^{−} · U_{P}) ∩ Bs_{1}B … Bs_{n−1}Bs_{n}Bs_{n}B)) =
v(U_{I}^{−} ∩ Bs_{1}B … Bs_{n−1}Bs_{n}Bs_{n}B),

the last equality as, since U_{P} ⊂ B we have

(U_{I}^{−} · U_{P}) ∩ Bs_{1}B … Bs_{n−1}Bs_{n}Bs_{n}B = (U_{I}^{−} ∩ Bs_{1}B … Bs_{n−1}Bs_{n}Bs_{n}B) · U_{P}.

But we have Bs_{1}B … Bs_{n−1}Bs_{n}Bs_{n}B = Bs_{1} … s_{n−1}s_{n}B∪Bs_{1} … s_{n−1}B. By corollary 8.15 we have Bs_{1} … s_{n−1}B∩U^{−} = ∏_{i=1}^{i=n−1} U_{−α_{i}}^{∗} ⊂ U_{I}^{−}, and in the same way (or by [Lu2, 2.2]) we have Bs_{1} … s_{n}B ∩ U^{−} = ∏_{i=1}^{i=n} U_{−α_{i}}^{∗} which has empty intersection with U_{I}^{−}. So (2) is proved.

Let us compute the fibers of \overline{π}. We have U_{P} ∩ ^{v}U^{−} = U_{P} ∩ (U_{I}.U_{P}^{−}) = 1, so

\overline{Z_{w}^{v}} ≃ \{(x, u_{1}, …, u_{n+1}) | x ∈ U_{I}, u_{i} ∈ U_{s_{i}}(i = 1, …, n), u_{n+1} ∈ U_{s_{n}}, x^{−1} · ^{F}x ∈ v̇u_{1}ṡ_{1} … u_{n−1}ṡ_{n−1}u_{n}ṡ_{n}u_{n+1}ṡ_{n}Bv^{−1}}.

Using the above description of the projection we see that the fibers of \overline{π} are those of the map from

{(u_{1}, …, u_{n+1}) | u_{i} ∈ U_{s_{i}}(i = 1, …, n), u_{n+1} ∈ U_{s_{n}}, u_{1}ṡ_{1} … u_{n−1}ṡ_{n−1}u_{n}ṡ_{n}u_{n+1}ṡ_{n}B ∈ Bs_{1} … s_{n−1}B}

to G/B given by (u_{1}, …, u_{n+1}) ↦ u_{1}ṡ_{1} … u_{n}ṡ_{n}u_{n+1}ṡ_{n}B. The condition on the u_{i} implies u_{n+1} = 1 and that the image of an n-tuple (u_{1}, …, u_{n−1}, u_{n}) does not depend on u_{n} and is injective on (u_{1}, …, u_{n−1}). So the fibers are indeed affine lines.

So we may apply proposition 8.23: let us write γ_{b}^{(n)} for the unipotent character corresponding to the partition 1, …, 1, b of a split group of type A_{n}; multiplying by th^{2} the two variable polynomial which encodes the cohomology of the Coxeter variety L, we get that the cohomology of X_{2} is given by ∑_{b=1}^{n} t^{b}h^{n+b}γ_{b}^{(n−1)} as a L^{F} × <F >-module.

We now compute the t^{b}-isotypic part of the exact sequence (1). For 2 ≤ b ≤ n − 1 we get the exact sequence 0 → ρ_{b}^{(n−1)} + ρ_{b−1}^{(n−1)} → (^{∗}R_{L}^{G}H_{c}^{n+b}(X(w)))_{t^{b}} → γ_{b}^{(n−1)} → 0 which gives ^{∗}R_{L}^{G}H_{c}^{n+b}(X(w)) = ρ_{b}^{(n−1)} + ρ_{b−1}^{(n−1)} + γ_{b}^{(n−1)}. The Littlewood-Richardson formula allows to compute ^{∗}R_{L}^{G} of any unipotent character. In particular it shows that the only characters of G^{F} whose ^{∗}R_{L}^{G} contains γ_{b}^{(n−1)} are ρ_{b}^{(n)},γ_{b+1}^{(n)} and γ_{b}^{(n)}; so H_{c}^{n+b}(X(w)) contains one of these three characters. But ^{∗}R_{L}^{G}(γ_{b+1}^{(n)}) and ^{∗}R_{L}^{G}(γ_{b}^{(n)}) contain characters different from ρ_{b}^{(n−1)}, ρ_{b−1}^{(n−1)} and γ_{b}^{(n−1)}. So H_{c}^{n+b}(X(w)) contains ρ_{b}^{(n)}. But ^{∗}R_{L}^{G}(ρ_{b}^{(n)}) = ρ_{b}^{(n−1)} + ρ_{b−1}^{(n−1)} + γ_{b}^{(n−1)}. So H_{c}^{n+b}(X(w)) = ρ_{b}^{(n)} as a G^{F} - module.

For b = 0, 1, n, n + 1 we get respectively the exact sequences:

0 → St^{(n−1)} → (^{∗}R_{L}^{G}H_{c}^{n+1}(X(w)))_{t^{0}} → 0

0 → (^{∗}R_{L}^{G}H_{c}^{n+1}(X(w)))_{t} → St^{(n−1)} → St^{(n−1)} → (^{∗}R_{L}^{G}H_{c}^{n+2}(X(w)))_{t} → 0

0 → (^{∗}R_{L}^{G}H_{c}^{2n}(X(w)))_{t^{n}} → Id^{(n−1)} → Id^{(n−1)} → (^{∗}R_{L}^{G}H_{c}^{2n+1}(X(w)))_{t^{n}} → 0

0 → Id^{(n−1)} → (^{∗}R_{L}^{G}H_{c}^{2n+2}(X(w)))_{t^{n+1}} → 0

We know that the only character χ of G^{F} such that ^{∗}R_{L}^{G}χ is St^{(n−1)}-isotypic is St^{(n)}, and that the only character χ such that ^{∗}R_{L}^{G}χ is Id^{(n−1)}-isotypic is Id^{(n)}. So we see that the (H^{i}(X(w)))_{t^{b}} in the above exact sequences are Id^{(n)}-isotypic or St^{(n)}- isotypic. The exact sequence for b = 0 (resp. b = n + 1) gives H^{n+1}(X(w)) (resp. H^{2n+2}(X(w))). For b = 1 or b = n, applying propositions [DMR, 3.3.14] and [DMR, 3.3.15] we see that the (H^{i}(X(w)))_{t^{b}} in the above exact sequences must be zero and the arrows St^{(n−1)} → St^{(n−1)} and Id^{(n−1)} → Id^{(n−1)} must be isomorphisms. This completes the proof of the theorem. □

Let us explain now why theorem 9.2 implies conjectures 2.4 to 2.6; conjecture 2.6 is immediate. Let us show conjecture 2.4. By section 5, the centralizer C_{B}(w) is cyclic, generated by w, and C_{W} (w) = G(1, 1, n). The endomorphism D_{w} acts as F on X(w). Thus the value of the eigenvalues of F given in theorem 9.2 shows that the representation w ↦ D_{w} of B(1, 1, n) on End_{G^{F}} (⊕_{i}H_{c}^{i}(X(w))) factors through an n-cyclotomic Hecke algebra H(w) with parameters (1, x^{2}, x^{3}, …, x^{n−1}, x^{n}). To show conjecture 2.5, it remains to see that the virtual representation ∑_{i}(−1)^{i}H_{c}^{i}(X(w)) of H(w) is special. Proceeding as in section 4, it is enough to show that |X(w)^{F^{i}}| = 0 for i = 1, …, n − 1. But this is exactly the statement [BMi, 5.2].

10. Conjecture 2.4 in type A

We consider a group G of type A_{n−1} and a group G′ of type A_{n}. We keep the notation of section 5: W (resp. W′) is the Weyl group of G (resp. G′) and we consider w = c^{r} (resp. w′ = c^{′r}), a d-th root of π in the braid group B (resp. B′) of W (resp. W′). We have dr = n with d ≥ 2.

In section 5 we had two incarnations of the braid group B(d, 1, r): one as C_{B}(w), with generators t, s_{1}, …, s_{r−1} and one as C_{B′}(w′) with generators t′, s_{1}, …, s_{r−1}; in both cases these generators correspond to braid reflections of B(d, 1, r).

The group G(d, 1, r) has two orbits of reflecting hyperplanes corresponding to reflections of order d and 2 respectively, so that for a choice of indeterminates u = (u_{t,0}, …, u_{t,d−1}; u_{s_{1},0}, u_{s_{1},1}) the generic Hecke algebra H_{u} of G(d, 1, r) is the quotient of \overline{ℚ}_{ℓ}[u]B(d, 1, r) by the relations (t − u_{t,0}) … (t − u_{t,d−1}) = 0 and (s_{1} − u_{s_{1},0})(s_{1} − u_{s_{1},1}) = 0; we will write the first relation as (t′ − u_{t,0}) … (t′ − u_{t,d−1}) = 0 when considering the other incarnation of B(d, 1, r).

Next theorem proves that 2.4 holds.

Theorem 10.1. • The map x ↦ D_{x} from C_{B}(w) to End_{G^{F}} (⊕_{i}H_{c}^{i}(X(w))) factors through the specialization x ↦ q of a d-cyclotomic Hecke algebra H for B(d, 1, r) with parameters (1, x, x^{2}, …, x^{d−1}; x^{d}, −1).

• The map x ↦ D_{x} from C_{B′}(w′) to End_{G^{F}} (⊕_{i}H_{c}^{i}(X(w′))) factors through the specialization x ↦ q of a d-cyclotomic Hecke algebra H′ for B(d, 1, r) with parameters (1, x^{2}, x^{3}, …, x^{d−1}, x^{d+1}; x^{d}, −1).

With the notation of section 5, we have to show that the operators induced on the cohomology by D_{s_{i}}, D_{t} and D_{t′} satisfy the expected polynomial relations. The end of this section is devoted to the proof of this theorem.

The next lemma will allow us to compute by induction the relations satisfied by the D_{s_{i}}, using [DMR, 5.2.9].

Lemma 10.2. For i = 1, …, r − 1 let I_{i} = {σ_{i}, σ_{i+r}, …, σ_{i+(d−1)r}} as in lemma 5.4; then we have α_{I_{i}}(w) = α_{I_{i}}(w′) = σ_{i}^{2}.

Proof. As the elements of I_{i} commute pairwise and as, by lemma 5.3 (iii), σ_{i} is the only divisor of w in I_{i}, we have α_{I_{i}}(w) = σ_{i}^{k} for some k. Now

σ_{i}^{k} ≼ w ⇔ σ_{i}^{−k}w ∈ B^{+} ⇔ c^{i−1}σ_{1}^{−k}c^{r−i+1} ∈ B^{+}.

But σ_{1} does not divide c^{i−1} on the right for i < n by the “right-side version” of lemma 5.3 (iv) so, by lemma 5.2, σ_{i}^{k} ≼ w if and only if σ_{1}^{k} ≼ c^{r−i+1}. To prove the assertion about α_{I_{i}}(w) it is sufficient to show that σ_{1}^{2} ≼ c^{2} and σ_{1}^{3} ⋠ c^{r}. The former statement follows from lemma 5.3 (ii) which gives σ_{1}^{−2}c^{2} = σ_{1}^{−1}c^{2}σ_{n−1}^{−1} ∈ B^{+}. To get the latter we write σ_{1}^{−3}c^{r} = (σ_{1}^{−2}c^{2})σ_{n−1}^{−1}c^{r−2} and, as σ_{n−1} ⋠ c^{r−2}, we have to see by lemma 5.2 that σ_{1}^{−2}c^{2} ⋡ σ_{n−1}, which is equivalent to σ_{1}^{−2}c^{2}σ_{n−1} = σ_{1}^{−3}c^{2} ∉ B^{+}. But σ_{1}^{3} ≼ c^{2} is impossible by [Mi, 4.8] as ν(σ_{1}^{3}) = 3 and ν(c^{2}) ≤ 2 (recall that ν(b) = inf{k ∈ ℕ|b ≼ w_{0}^{k}} for b ∈ B^{+}).

The proof of the assertion about α_{I_{i}}(w′) follows the same lines, using lemma 5.3 (iv’) instead of lemma 5.3 (iv) and lemma 5.3 (ii’) instead of lemma 5.3 (ii). At the end we have to see that σ_{1}^{3} ⋠ c^{′2}. But by lemma 5.3 (v), we have c^{′2} = c^{2}σ_{n−1}σ_{n}, which can be written c^{′2} = (cσ_{1} … σ_{n−1})(σ_{n}σ_{n−1}σ_{n}). The two factors are in W, so that ν(c^{′2}) = 2, and we conclude as in the w case. □

Proposition 10.3. For i ∈ {1, …, r−1}, the image T_{s_{i}} of D_{s_{i}} in either End_{G^{F}} (⊕_{j}H_{c}^{j}(X(w))) or End_{G^{F}} (⊕_{j}H_{c}^{j}(X(w′))) satisfies (T_{s_{i}} + 1)(T_{s_{i}} − q^{d}) = 0.

Proof. We prove the statement for w, the proof for w′ being exactly the same. By [DMR, 5.2.9], which can be applied by lemma 5.4 (i), the T_{s_{i}} satisfy the same relations as the D_{s_{i}} on ⊕_{j}H_{c}^{j}(X_{L_{I_{i}}}(α_{I_{i}}(w), ω_{I_{i}}(w)F)). By the above lemma we have α_{I_{i}}(w) = σ_{i}^{2}. The group L_{I_{i}} has type A_{1}^{d} where the components are permuted cyclically by ω_{I_{i}}(w)F. If s denotes the positive generator of the braid group of type A_{1}, through the isomorphism with A_{1}^{d} we have σ_{i}^{2} ↔ (s^{2}, 1, …, 1), s_{i} ↔ (s, s, …, s) and ω_{I_{i}}(w)F corresponds to (x_{1}, …, x_{d}) ↦ (^{F}x_{2}, …,^{F}x_{d},^{F}x_{1}). The variety we have to study is X_{L_{I_{i}}}(s^{2}, 1, …, 1), which can be identified with the three-term sequences of d-tuples of Borel subgroups of L_{I_{i}} of the form

(B_{1}, …, B_{d}) ⟶^{(s,1,...,1)} (B′_{1}, …, B′_{d}) ⟶^{(s,1,...,1)} (^{F}B_{2}, …,^{F}B_{d},^{F}B_{1}),

where for two Borel subgroups B and B′, we write B ⟶^{v} B′ to say that (B, B′) ∈ O(v) (we say that B and B′ are in relative position v). The conditions on the relative positions imply B′_{2} = B_{2},…, B′_{d} = B_{d} and ^{F}B_{3} = B_{2},…, ^{F}B_{d} = B_{d−1},^{F}B_{1} = B_{d}; so that X_{L_{I_{i}}}(s^{2}, 1, …, 1) identifies with the three-term sequences B_{1} ⟶^{s} B′_{1} ⟶^{s} FB_{2} = ^{F^{d}}B_{1}, i.e., to the variety X(s^{2}) of a group of type A_{1} with Frobenius endomorphism F^{d}. Let us put s_{(i)} = (s, 1, …, 1, s, 1, …, 1) where the second s is at the place i with i > 1. In the same way as above we identify the variety X_{L_{I_{i}}}(s_{(i)}) with

the variety X(s^{2}) by identifying a sequence (B_{1}, …, B_{d}) such that (B_{1}, …, B_{d}) ⟶^{s_{(i)}} (^{F}B_{2}, …,^{F}B_{d−1},^{F}B_{d}) with the three-term sequence B_{1} ⟶^{s} ^{F^{i−1}}B_{i} ⟶^{s} ^{F^{d}}B_{1}. We can decompose the morphism D_{s,...,s} as D_{s^{(2)}} ∘ … ∘ D_{s^{(d)}} ∘ D_{s^{(1)}}, where s^{(i)} = (1, …, 1, s, 1, …, 1), with s at the ith place. Then D_{s^{(1)}} sends X(s^{2}, 1, …, 1) to X(s_{(d)}) and D_{s^{(i)}} sends X(s_{(i)}) to X(s_{(i−1)}) for i > 1. With the above identifications, one checks that D_{s^{(1)}} sends B_{1} ⟶^{s} B′_{1} ⟶^{s} ^{F^{d}}B_{1} to B′_{1} ⟶^{s} ^{F^{d}}B_{1} ⟶^{s} ^{F^{d}}B′_{1} and

that D_{s^{(i)}} sends B_{1} ⟶^{s} ^{F^{i−1}}B_{i} ⟶^{s} ^{F^{d}}B_{1} to B_{1} ⟶^{s} ^{F^{i−2}}(^{F}B_{i}) ⟶^{s} ^{F^{d}}B_{1} for i > 1.

So that D_{(s,...,s)} identifies with the operator D_{s} on the variety X(s^{2}) of a group of type A_{1} with Frobenius endomorphism F^{d}. This is a particular case of [DMR, 5.3.4], where it is proved that the operator T_{s} induced by D_{s} on H_{c}^{∗}(X(s^{2}) satisfies (T_{s} − q^{d})(T_{s} + 1) = 0, whence the proposition. □

We will now prove that the operators induced on the cohomology of X(w) and X(w′) respectively by D_{t} and D_{t′} satisfy the claimed polynomial relation.

Lemma 10.4. Let I = {σ_{r}, …, σ_{r+d−2}} and I′ = {σ_{r}, …, σ_{r+d−1}}, as in corollary 5.8; we have

(i) α_{I}(ywy^{−1}) = yty^{−1}, (i’) α_{I′}(y′w′y^{′−1}) = y′t′y^{′−1}. (ii) ω_{I}(ywy^{−1}) commutes with σ_{r+i} for 0 ≤ i ≤ d − 2, (ii’) ω_{I′}(y′w′y^{′−1}) commutes with σ_{r+i} for 0 ≤ i ≤ d − 1.

Proof. Let us prove (i). As yty^{−1} = σ_{r,r+d−2} ∈ B_{I}^{+} and ywy^{−1} = yty^{−1}c^{r−1}x_{d} … x_{1} by the remark which follows the proof of lemma 5.5, it is enough to see that α_{I}(c^{r−1}x_{d} … x_{1}) = 1, i.e., that for i ∈ 0, …, d − 2 we have σ_{r+i} ⋠ c^{r−1}x_{d} … x_{1}. By lemma 5.2, this amounts to prove that for i ∈ 1, …, d−1 we have σ_{i} ⋠ x_{d} … x_{1}. But σ_{i} commutes with x_{j} for j > i + 1, so that σ_{i}^{−1}x_{d} … x_{1} = x_{d} … x_{i+2}σ_{i}^{−1}x_{i+1} … x_{1}. As σ_{i} is not in the support of x_{d} … x_{i+2}, we have x_{d} … x_{i+2} ⋡ σ_{i}, so by lemma 5.2 it is enough to see that σ_{i} ⋠ x_{i+1} … x_{1}. But σ_{i}^{−1}x_{i+1}x_{i} = x_{i+1}x_{i}σ_{i+r−1}^{−1}: indeed we can write this equality as σ_{i+1,i+r−1}σ_{i,i+r−1} = σ_{i,i+r−1}σ_{i,i+r−2}, using lemma 5.3(i) in the parabolic subgroup generated by σ_{i}, …, σ_{i+r−1}. So, as σ_{i+r−1} ⋠ x_{i−1} … x_{1} because σ_{i+r−1} is not in the support of this element, again by lemma 5.2 it is sufficient to see that x_{i+1}x_{i} ⋡ σ_{i+r−1}. But this is the exact analogue in the parabolic subgroup generated by σ_{i}, …, σ_{i+r−1} of what we have proved at the end of lemma 10.2, as σ_{i}^{−2}σ_{i,i+r−1}^{2} = x_{i+1}x_{i}.

The proof of (i’) is along the same lines: we start with y′w′y^{′−1} = y′t′y^{′−1}c_{rd}^{r−1}x_{d+1} … x_{1} as noticed in the remark following lemma 5.6. We have to see that α_{I′}(c_{rd}^{r−1}x_{d+1} … x_{1}) = 1, i.e., that σ_{r+i} ⋠ c_{rd}^{r−1}x_{d+1} … x_{1} for i ∈ 0, …, d − 1. By lemma 5.2, this amounts to prove that for 1, …, d we have σ_{i} ⋠ x_{d+1} … x_{1}. But σ_{i} commutes with x_{j} for j > i + 1, and as in the proof of (i) we are reduced to prove that σ_{i} ⋠ x_{i+1} … x_{1}. Then we finish exactly as in (i).

Let us prove (ii). We have to see that σ_{r+i}c^{r−1}x_{d} … x_{1} = c^{r−1}x_{d} … x_{1}σ_{r+i}. This can be written

c^{r−1}x_{d} … x_{i+3}σ_{i+1}x_{i+2}x_{i+1} … x_{1} = c^{r−1}x_{d} … x_{i+2}x_{i+1}σ_{r+i}x_{i} … x_{1}.

So we have to prove that σ_{i+1}x_{i+2}x_{i+1} = x_{i+2}x_{i+1}σ_{r+i}, i.e., σ_{i+1,i+r}σ_{i+1,i+r−1} = σ_{i+2,i+r}σ_{i+1,r+i}. This last equality is a consequence of lemma 5.3 (i) applied in the parabolic subgroup generated by {σ_{i+1}, …, σ_{i+r}}.

The proof of (ii’) is exactly the same, replacing I by I′, c by c_{rd} and x_{d} … x_{1} by x_{d+1} … x_{1}. □

Corollary 10.5. • The image of T_{t} of D_{t} in End_{G^{F}} (⊕_{i}H_{c}^{i}(X(w))) satisfies

(T_{t} − 1)(T_{t} − q) … (T_{t} − q^{d−1}) = 0.

• The image of T_{t′} of D_{t′} in End_{G^{F}} (⊕_{i}H_{c}^{i}(X(w′))) satisfies (T_{t′} − 1)(T_{t′} −

q^{2})(T_{t′} − q^{3}) … (T_{t′} − q^{d−1})(T_{t′} − q^{d+1}) = 0.

Proof. We prove first the result for t. Using conjugation by D_{y} we see that it is equivalent to prove that the image of D_{yty^{−1}} in End_{G^{F}} (⊕_{i}H_{c}^{i}(X(ywy^{−1}))) satisfies the same relation. By [DMR, 5.2.9] and lemma 10.4 this operator satisfies the same relations as D_{yty^{−1}} on ⊕_{i}H_{c}^{i}(X_{L_{I}}(α_{I}(ywy^{−1}), ω_{I}(ywy^{−1})F)) = ⊕_{i}H_{c}^{i}(X_{L_{I}}(σ_{r,r+d−2})). The element σ_{r,r+d−2} is the lift of a Coxeter element in the braid group of type I. The group L_{I} has type split A_{d−1}, and D_{yty^{−1}} acts as F on X_{L_{I}}(σ_{r,r+d−2}). The eigenvalues of F in the case of a Coxeter element are given in [Lu2] and gives the relation we want for T_{t}.

We follow the same lines for proving the assertion on t′. We are reduced to compute the relation satisfied by D_{y′t′y^{′−1}} on ⊕_{i}H_{c}^{i}(X_{L_{I′}}(α_{I′}(y′w′y^{′−1}), ω_{I′}(y′w′y^{′−1})F)) = ⊕_{i}H_{c}^{i}(X_{L_{I′}}(σ_{r,r+d−1}σ_{r+d−1})). The element σ_{r,r+d−1}σ_{r+d−1} relatively to the group L_{I′} is an element as studied in section 9. The group L_{I′} is split of type A_{d} and D_{y′t′y^{′−1}} acts as F on X_{L_{I′}}(σ_{r,r+d−1}σ_{r+d−1}). We end the proof by noticing, as in the end of section 9, that the eigenvalues of F given in theorem 9.2 give the relation we want for T_{t′}. □

11. Conjecture 2.4 in type B

In this section we keep the notation of section 6; we prove conjecture 2.4 in a group G of type B_{n} for a d-regular element with d even. So we have w = c^{r} with dr = 2n and d even.

Theorem 11.1. The map x ↦ D_{x} from C_{B}(w) to End_{G^{F}} (⊕_{i}H_{c}^{i}(X(w))) factors through the specialization x ↦ q of a d-cyclotomic Hecke algebra H for B(d, 1, r) with parameters (1, x, x^{2}, …, x^{d/2−1}, x^{d/2}, −x, −x^{2}, …, −x^{d/2−1}; x^{d/2}, −1).

With the notation of section 6, we have to show that the operators induced on the cohomology by D_{s_{i}} and D_{t} satisfy the expected polynomial relations.

The end of this section is devoted to the proof of this theorem. The next lemma is the analogue of lemma 10.2.

Lemma 11.2. For i = 2, …, r let I_{i} = {σ_{i}, σ_{i+r}, …, σ_{i+(d/2−1)r}} as in lemma 6.4; then we have α_{I_{i}}(w) = σ_{i}.

Proof. As the elements of I_{i} commute pairwise and as, by lemma 6.1 (iv), σ_{i} is the only divisor of w in I_{i}, we have α_{I_{i}}(w) = σ_{i}^{k} for some k. But c is a good root of π, so for any r ≤ n we have ν(c^{r}) = 1. As ν(σ_{i}^{2}) = 2 we cannot have σ_{i}^{2} ≼ w, whence the result. □

Corollary 11.3. For i ∈ {1, …, r−1} the image T_{s_{i}} of D_{s_{i}} in End_{G^{F}} (⊕_{j}H_{c}^{j}(X(w))) satisfies (T_{s_{i}} + 1)(T_{s_{i}} − q^{d/2}) = 0.

Proof. The proof is similar to that of proposition 10.3, except that here D_{s_{i}} eventually identifies with the operator D_{s} on the variety X(s) for a group of type A_{1} with Frobenius endomorphism F^{d/2}, whence the result. □

Lemma 11.4. Let I = {σ_{1}, …, σ_{d/2}} be as at the end of section 6; we have (i) α_{I}(ywy^{−1}) = yty^{−1}.

(ii) ω_{I}(ywy^{−1}) commutes with σ_{i} for 1 ≤ i ≤ d/2.

Proof. Let us prove (i). As yty^{−1} = σ_{1,d/2} and ywy^{−1} = yty^{−1} ∏_{i=d/2}^{1} x_{i}c^{r−1} by the remark which follows the proof of lemma 6.5, it is enough to see that σ_{i} ⋠ ∏_{i=d/2}^{1} x_{i}c^{r−1} for i = 0, …, d/2. We use the following lemma which is an immediate consequence of lemma 5.2:

Lemma 11.5. Assume that a, b ∈ B^{+}, that σ_{i} ≼ ab, that σ_{i}a = aσ_{j} and that σ_{i} ⋠ a; then σ_{j} ≼ b.

By this lemma, for i > 1 we have σ_{i} ≼ ∏_{j=d/2}^{1} x_{j}c^{r−1} ⇔ σ_{i} ≼ ∏_{j=i}^{1} x_{j}c^{r−1} ⇔ σ_{i+r−1} ≼ ∏_{j=i−2}^{1} x_{j}c^{r−1} ⇔ σ_{i+r−1} ≼ c^{r−1}, the second equivalence as, by the proof of lemma 10.4, we have σ_{i}^{−1}x_{i}x_{i−1} = x_{i}x_{i−1}σ_{i+r−1}^{−1}. But σ_{i+r−1} ≼ c^{r−1} is false by lemma 6.1(iv).

For i = 1 we get σ_{1} ≼ ∏_{j=d/2}^{1} x_{j}c^{r−1} ⇔ σ_{1} ≼ x_{1}c^{r−1} ⇔ σ_{1}^{2} ≼ σ_{1} … σ_{r}c^{r−1} ⇒ σ_{1}^{2} ≼ c^{r}, the last equivalence as σ_{r+1} … σ_{n}c^{r−1} = c^{r−1}σ_{2} … σ_{n−r+1}. But σ_{1}^{2} ≼ c^{r} is impossible as ν(c^{r}) = 1.

Let us prove (ii). By (i) we have ω_{I}(ywy^{−1}) = ∏_{i=d/2}^{1} x_{i}c^{r−1}. For i > 1, we have ∏_{j=d/2}^{1} x_{j}c^{r−1}σ_{i} = ∏_{j=d/2}^{1} x_{j}σ_{i+r−1}c^{r−1} = ∏_{j=d/2}^{i+1} x_{i}x_{i−1}σ_{i+r−1} ∏_{j=i−2}^{1} x_{j}c^{r−1} = ∏_{j=d/2}^{i+1} σ_{i}x_{i}x_{i−1} ∏_{j=i−2}^{1} x_{j}c^{r−1} = σ_{i} ∏_{j=d/2}^{1} x_{j}c^{r−1}.

For i = 1, we prove by induction on i that σ_{2} … σ_{i+1}c^{i}σ_{1} = σ_{1} … σ_{i+1}c^{i}; this equality for i = r − 1 gives x_{1}c^{r−1}σ_{1} = σ_{1}x_{1}c^{r−1}; then we use that σ_{1} commutes with x_{i} for i > 1. □

As for corollary 10.5, we deduce the following corollary which ends the proof of the theorem.

Corollary 11.6. Let T_{t} be the image of D_{t} in End_{G^{F}} (⊕_{i}H_{c}^{i}(X(w))); then we have (T_{t} − 1)(T_{t} − q) … (T_{t} − q^{d/2})(T_{t} + q)(T_{t} + q^{2}) … (T_{t} + q^{d/2−1}) = 0.

Proof. Using conjugation by D_{y}, we see that it is equivalent to prove that the image of D_{yty^{−1}} in End_{G^{F}} (⊕_{i}H_{c}^{i}(X(ywy^{−1}))) satisfies the same relation. By [DMR, 5.2.9] and lemma 11.4 this operator satisfies the same relation as D_{yty^{−1}} on

⊕_{i}H_{c}^{i}(X_{L_{I}}(α_{I}(ywy^{−1}), ω_{I}(ywy^{−1})F)) = ⊕_{i}H_{c}^{i}(X_{L_{I}}(∏_{i=1}^{d/2} σ_{i})).

The element ∏_{i=1}^{d/2} σ_{i} is the lift of a Coxeter element in the braid group of type I. The group L_{I} has type B_{d/2} and D_{yty^{−1}} acts as F on X_{L_{I}}(∏_{i=1}^{d/2} σ_{i}). The eigenvalues of F in the case of a Coxeter element are given in [Lu2] and give the relation which we want for T_{t}. □

12. The 4-th roots of π in D_{4}

We use the same numbering for the reflections of W = W (D_{4}) as in Bourbaki: we will study the 4-th root of π given by w = s_{2}s_{3}s_{1}s_{3}s_{4}s_{3}.

◯_{s_{1}} ◯ s_{2} ◯_{s_{3}} ◯_{s_{4}}

The centralizer of w is the complex reflection group G(4, 2, 2). We show conjectures 2.1 to 2.6 for w.

Proposition 12.1. The element w verifies conjectures 2.1 and 2.2. In particular, the map which sends the standard generators of the braid group B(4, 2, 2) to b_{1} = (s_{1}s_{2})^{s_{3}}, b_{2} = s_{1}s_{4} and b_{3} = (s_{2}s_{4})^{s_{3}s_{4}} is an isomorphism B(4, 2, 2) ≃ C_{B}(w).

Proof. A 4-th root of π is an element of length 6 whose image is a 4-regular element. As all twelve 4-regular elements of W are of length 6, all 4-th roots of π are in W. There are thus 12 good roots of π, and a direct check shows that there are morphisms in D^{+} between any two of them, thus proving conjecture 2.2 in this case. We may show conjecture 2.1 using the programs written by Franco and Gonzalez-Meneses to compute centralizers in braid groups. They give that b_{1}, b_{2} and b_{3} generate C_{B}(w). However, one sees that the elements wb_{1} = s_{1}s_{2}s_{3}s_{1}s_{2}s_{4}s_{1}s_{3}, b_{2} and wb_{3} = s_{2}s_{3}s_{1}s_{2}s_{3}s_{4}s_{3}s_{3} are in End_{D^{+}}(w) by using the decompositions wb_{1} = (s_{1}s_{2}s_{3}s_{1})(s_{2}s_{4})(s_{1}s_{3}), b_{2} = (s_{1})(s_{4}) and wb_{3} = (s_{2}s_{3}s_{1})(s_{4})(s_{2}s_{3}s_{4})(s_{3}). We get thus that b_{1}, b_{2} and b_{3} are in End_{D}(w) whence conjecture 2.1.

Let us show that the elements b_{1}, b_{2}, b_{3} are the image of braid reflections around hyperplanes of C_{W} (w). We start with b_{2}; let V be the reflection representation of W, and let V_{i} be the eigenspace of w in V for the eigenvalue i. This is a 2-dimensional space, and the set of fixed points of s_{1}s_{4} on this space has dimension 1 and is of the form H∩V_{i} for some reflecting hyperplane of W. We use now the constructions at the end of section 3: the group B_{H} is the parabolic subgroup of B generated by s_{1} and s_{4}, and the element s_{H} is a square root of the element π of this parabolic subgroup, so is equal to s_{1}s_{4}; thus this element is indeed the image in B of a the braid reflection t_{H} of B(w). To handle the other elements we use the triality automorphism of G. This automorphism corresponds to an automorphism of V which induces on B the automorphism given by s_{2} ↦ s_{1} ↦ s_{4} ↦ s_{2} and the corresponding automorphism on W. Let ψ be the automorphism of B which is the triality followed by conjugation by s_{2}s_{3} and let ψ be the corresponding automorphism of V. Then ψ fixes w and does the permutation b_{1} ↦ b_{2} ↦ b_{3} ↦ b_{1}. Thus ψ induces an automorphism of V_{i}, and the braid reflections which are images of t_{H} by the powers of this automorphism have images b_{1} and b_{3}.

The fact that the images of these braid reflections generate C_{B}(w) implies that they generate B(w); thus conjecture 2.3 holds. They satisfy the defining braid relations for the braid group of G(4, 2, 2), that is b_{1}b_{2}b_{3} = b_{2}b_{3}b_{1} = b_{3}b_{1}b_{2} (all three products are easily checked to be equal to w). □

The next proposition shows conjecture 2.4.

Proposition 12.2. The map b_{i} ↦ D_{b_{i}} factors through a representation H_{q}(w) → End(⊕_{i}H_{c}^{i}(X(w))) of the specialization x ↦ q of a 4-cyclotomic Hecke algebra H(w) for G(4, 2, 2) with parameters 1 and x^{2}.

We note that the above algebra is indeed a 4-cyclotomic algebra since the parameters specialize to 1 and −1 by the specialization x ↦ i.

Proof. It is enough to prove the quadratic relations (D_{b_{i}} − 1)(D_{b_{i}} − q^{2}) = 0.

For b_{2} = s_{1}s_{4}, we use proposition [DMR, 5.2.8] with x = y = s_{1} and I = {s_{1}, s_{4}}. The subgroup L_{I} is of type A_{1}×A_{1}; The element z = s_{1}^{−1}w normalises L_{I}, exchanging the two A_{1} components. The variety X_{L_{I}}(s_{1}, żF) is mapped by D_{s_{1}}, which acts as żF, to X_{L_{I}}(s_{4}, żF). Then D_{s_{4}} maps X_{L_{I}}(s_{4}, żF) to X_{L_{I}}(s_{1}, żF), and D_{s_{4}} ∘ D_{s_{1}} induces F^{2} on X_{L_{I}}(s_{1}, żF); by e.g., [DMR, 3.3.16] this variety has two non-zero cohomology groups with compact support: H_{c}^{1} on which F^{2} acts as 1 and H_{c}^{2} on which F^{2} acts as q^{2}. Thus D_{b_{2}} satisfies the quadratic relation (D_{b_{2}}−1)(D_{b_{2}}−q^{2}) = 0 on the cohomology of X_{L_{I}}(s_{1}, żF), and thus by [DMR, 2.3.13] and the Künneth formula, it also satisfies the same relation on the cohomology of X(w).

To show that D_{b_{3}} satisfies the same quadratic relation we use [DMR, 3.1.8] applied to the triality automorphism of G. The triality maps w to s_{1}s_{3}s_{4}s_{3}s_{2}s_{3} and D_{b_{2}} to D_{s_{2}s_{4}} acting on X(s_{1}s_{3}s_{4}s_{3}s_{2}s_{3}) which by the same argument as above, with x = y = s_{4} and I = {s_{2}, s_{4}}, satisfies the expected quadratic relation. The conjugation by s_{2}s_{3} ∈ Hom_{D}(w, s_{1}s_{3}s_{4}s_{3}s_{2}s_{3}) composed with the triality is equal to ψ; and the conjugation by s_{2}s_{3} maps back D_{s_{2}s_{4}} to D_{b_{3}} which thus satisfies the quadratic relation.

One proceeds similarly for D_{b_{1}}, using the square of the triality automorphism which maps D_{s_{1}s_{4}} to D_{s_{2}s_{1}} acting on X(s_{4}s_{3}s_{2}s_{3}s_{1}s_{3}). One has that ψ^{2} is the square of triality followed by conjugation by s_{2}s_{3}s_{1}s_{3}, and conjugating by s_{2}s_{3}s_{1}s_{3} ∈ Hom_{D}(w, s_{4}s_{3}s_{2}s_{3}s_{1}s_{3}) maps D_{s_{2}s_{1}} to D_{b_{1}}. □

If we denote by T_{1}, T_{2}, T_{3} the images in H_{q}(w) of b_{1}, b_{2}, b_{3}, we thus get a representation ρ of H_{q}(w) on ⊕H_{c}^{i}(X(w)) which maps T_{i} to D_{b_{i}}. We note that triality commutes with the automorphism of H_{q}(w) given by T_{1} ↦ T_{2} ↦ T_{3} ↦ T_{1}.

The next proposition and theorem 12.4 show that conjecture 2.5 holds.

Proposition 12.3. The representation induced by ρ on ∑(−1)^{i}H_{c}^{i}(X(w)) is special.

Proof. Let Φ = T_{1}T_{2}T_{3}. It follows from the computation of the generic degrees and the character table of H_{q}(w) in [BMa, 5.A] that a representation is special if (and only if) the elements {(T_{i})_{i=1,2,3}, (T_{i}T_{j})_{i=1,2,3,j=1+(i mod 3)}, (Φ^{i})_{i=1,2,3}} have trace zero on it. The quadratic relations show that B = {(T_{i})_{i=1,2,3}, (T_{i}Φ)_{i=1,2,3}, (Φ^{i})_{i=1,2,3}} generates the same subspace of H_{q}(w) as these elements. We shall show that the trace of any element in B vanishes on ∑(−1)^{i}H_{c}^{i}(X(w)). It is enough to check this on just one element in each orbit of B under triality, e.g., for T_{2}, T_{1}Φ, Φ^{i}. We have ρ(Φ) = D_{w} = F. Thus D_{w} as well as its square and cube verify the trace formula, and by [DMR, 5.2.3] they all have trace zero on ∑(−1)^{i}H_{c}^{i}(X(w)). Thus it only remains to compute the traces of D_{b_{2}} and D_{b_{1}w}.

To compute the trace of D_{b_{2}} = D_{s_{1}s_{4}}, we apply [DMR, 5.2.10] with g = 1, I = {s_{1}, s_{4}} and x = s_{1}s_{4}, which gives that the trace of D_{s_{1}s_{4}} on ∑(−1)^{i}H_{c}^{i}(X(w)) is the value at 1 of the class function

R_{L_{I}^{żF}}^{G^{F}} (l ↦ ∑_{i}(−1)^{i} Trace(lD_{s_{1}s_{4}}|H_{c}^{i}(X_{L_{I}}(s_{1}, żF), \overline{ℚ}_{ℓ}))) ,

with z as in 12.2. By a general result on Lusztig induction (see e.g., [DM, 12.17]), this value is a multiple of

∑_{i}(−1)^{i} Trace(D_{s_{1}s_{4}}|H_{c}^{i}(X_{L_{I}}(s_{1}, żF), \overline{ℚ}_{ℓ})).

But this last trace is zero since D_{s_{1}s_{4}} acts as F^{2} on X_{L_{I}}(s_{1}, żF), thus has no fixed points.

Using the square of the triality automorphism as in the proof of 12.2, we see that the trace of D_{b_{1}w} on ∑(−1)^{i}H_{c}^{i}(X(w)) is equal to the trace of D_{s_{2}s_{1}}F on ∑(−1)^{i}H_{c}^{i}(X(s_{3}s_{2}s_{3}s_{1}s_{3}s_{4})). By [DMR, 2.3.13] and [DMR, 5.2.9] applied with I = {s_{1}, s_{2}} this trace is equal to

|L_{I}^{żF} |^{−1} ∑_{l∈L_{I}^{żF}} ∑_{i}(−1)^{i} Trace(lD_{s_{2}s_{1}}F |H_{c}^{i}(X_{L_{I}}(s_{2}, żF), \overline{ℚ}_{ℓ}))

∑_{i}(−1)^{i} Trace(lF |H_{c}^{i}(X̃_{(I)}(ż), \overline{ℚ}_{ℓ}))

where z = s_{3}s_{2}s_{1}s_{3}s_{4} and where X̃_{(I)}(ż) = {gU_{I} | g^{−1}F (g) ∈ U_{I} żF (U_{I})} (the action of F decomposes as a product since U_{I} is F -stable). By the Lefschetz trace formula one has ∑_{i}(−1)^{i} Trace(lF |H_{c}^{i}(X̃_{(I)}(ż), \overline{ℚ}_{ℓ})) = |X̃_{(I)}(ż)^{lF} | = |\{gU_{I} | g^{−1}^{F}g ∈ (U_{I} żU_{I}) ∩ (lU_{I})}|. But this last intersection is empty since z ∉ W_{I} so that P_{I} does not intersect U_{I} żU_{I}. Thus the trace vanishes. □

We shall now prove conjecture 2.6 by giving a full description of H_{c}^{∗}(X(w)) as a G^{F} × <F >-module. To give the result, we first introduce a notation for the irreducible characters of W (D_{4}): they are parametrized by the pairs of partitions of total sum 4, except that a pair of equal partitions corresponds to 2 characters. We shall thus denote the characters by

1^{2}+, 1^{2}−, 1.1^{3}, 1^{4}, 1^{2}.2, 1.21, 21^{2}, 2+, 2−, 2^{2}, 1.3, 31, 4

where a missing “.” means that one of the partitions is empty. If λ is a parameter as above, we shall denote by γ_{λ} the corresponding unipotent character of the principal series, except that we denote respectively by St and Id the characters γ_{1^{4}} and γ_{4}. There is one more unipotent character of D_{4}, a cuspidal one, that we shall denote by θ. We will use the same convention as in theorem 9.2 to describe the cohomology by a two variable polynomial. With these notation, we have

Theorem 12.4. The cohomology of X(w) is given by

h^{6} St +t^{2}h^{7}(γ_{1^{2}+} +γ_{1^{2}−} +γ_{21^{2}})+2t^{3}h^{7}θ +2t^{3}h^{8}γ_{1.21} +t^{4}h^{9}(γ_{2+} +γ_{2−} +γ_{31})+t^{6}h^{12} Id .

Proof. We will use the parabolic subgroup of type A_{3} generated by I = {s_{1}, s_{3}, s_{4}}. Let w′ = s_{3}s_{1}s_{3}s_{4}s_{3}. We will need the value of the sets defined in lemma 8.11: they are E_{W_{I}}(w′) = {1, s_{1}, s_{3}, s_{4}} and E_{W} (w) = E_{W_{I}}(w′) ∪ {s_{2}s_{3}}. The value for w′ is obtained by a direct computation in the Hecke algebra of W_{I} and the value for w = s_{2}w′ comes from proposition 8.12 whose assumptions are easily checked.

We first compute the cohomology of X(w^{′w_{0}^{I}}) = X(s_{3}s_{4}s_{3}s_{1}s_{3}) in the Levi subgroup L_{I} corresponding to I. We use [DMR, 3.2.10]; with the notation of the proof of theorem 9.2 (ρ_{2} corresponds to the partition 2, 2, γ_{2} to 1, 1, 2, and γ_{3} to 1, 3) we know by [Lu2] the cohomology of the Coxeter variety X(s_{4}s_{3}s_{1}):

∑_{i} h^{i} · H_{c}^{i}(X(s_{4}s_{3}s_{1})) = h^{3} St +h^{4}tγ_{2} + h^{5}t^{2}γ_{3} + h^{6}t^{3} Id

and since s_{3}s_{4}s_{3}s_{1} is a 3-rd root of π_{I} we know its cohomology by theorem 9.2

∑_{i} h^{i} · H_{c}^{i}(X(s_{3}s_{4}s_{3}s_{1})) = h^{4} St +t^{2}h^{5}ρ_{2} + h^{8}t^{4} Id .

The exact sequence given by [DMR, 3.2.10] then completely determines H_{c}^{i}(X(s_{3}s_{4}s_{3}s_{1}s_{3})), except for the Id-isotypic and St-isotypic parts; we get these by [DMR, 3.3.14] and [DMR, 3.3.15], and we obtain

∑_{i} h^{i} · H_{c}^{i}(X(w′)) = h^{5} St +t^{2}h^{6}(γ_{2} + ρ_{2}) + t^{3}h^{7}(γ_{3} + ρ_{2}) + t^{5}h^{10} Id .

We now determine the principal series part of the cohomology of X(w) by the same method as in the proof of theorem 9.2. The value of E_{W} (w) shows that

X(w) = X^{w_{0}}(w) ∐ X^{w_{0}s_{1}}(w) ∐ X^{w_{0}s_{3}}(w) ∐ X^{w_{0}s_{4}}(w) ∐ X^{w_{0}s_{2}s_{3}}(w).

We may apply proposition 8.17 with s = s_{2}. If we set

Y = (X^{w_{0}}(w) ∐ X^{w_{0}s_{1}}(w) ∐ X^{w_{0}s_{3}}(w) ∐ X^{w_{0}s_{4}}(w)) /U_{P_{I}}^{F},

we have thus an equality of L_{I}^{F} -modules:

∑_{i} h^{i} · H_{c}^{i}(Y) = (th^{2} + h) × (∑_{i} h^{i} · H_{c}^{i}(X(w_{0}^{I}w′w_{0}^{I})))

= h^{6} St +h^{7}(t St +t^{2}(γ_{2}+ρ_{2}))+h^{8}t^{3}(γ_{2}+γ_{3}+2ρ_{2})+h^{9}t^{4}(γ_{3}+ρ_{2})+t^{5}h^{11} Id +t^{6}h^{12} Id .

To study the remaining piece X^{w_{0}s_{2}s_{3}}(w) we use proposition 8.25 with v = w_{0}s_{2}s_{3} and taking for w′ the element w′′ = s_{2}s_{3}s_{1}s_{4}s_{3}. We show that w and w′′ satisfy the assumptions of proposition 8.25.

We first check the assumptions of 8.21(iii) for w and w′′. We check on E_{W} (w) that v is the only element of W_{I}v such that X^{v}(w) ≠ ∅. On the other hand, one gets by proposition 8.12 whose assumptions are easily checked that E_{W} (w′′) = E_{W_{I}}(s_{3}s_{1}s_{4}s_{3}) ∪ {s_{2}s_{3}, s_{2}s_{3}s_{1}s_{4}}, where E_{W_{I}}(s_{3}s_{1}s_{4}s_{3}) = {1, s_{1}, s_{3}, s_{4}, s_{1}s_{4}, s_{3}s_{1}s_{4}}, which results from a direct computation in the Hecke algebra of W_{I}. Thus v is indeed the only element of W_{I}v such that X^{v}(w′′) ≠ ∅.

We check now the assumption 8.21(iv) for w and w′′. If α_{1}, α_{2}, α_{3}, α_{4} are the simple roots of D_{4}, we have v^{−1}({α_{1}, α_{3}, α_{4}}) = {−α_{1} − α_{3}, −α_{2}, −α_{3} − α_{4}}, thus v^{−1}U_{I} ⊂ U^{−} . The projection onto ^{v^{−1}}U_{I} of ^{v^{−1}}U ∩ (BwB ∐ Bw′′B) is the same as that of ^{v^{−1}}U ∩ U^{−} ∩ (BwB ∐ Bw′′B); indeed each double coset is invariant by right multiplication by ^{v^{−1}}U ∩ U, thus its intersection with ^{v^{−1}}U is the product of its respective intersections with ^{v^{−1}}U ∩ U^{−} and ^{v^{−1}}U ∩ U. As ^{v^{−1}}U_{I} ⊂ U^{−}, the part in ^{v^{−1}}U ∩ U has a trivial projection.

Let us compute ^{v^{−1}}U ∩ U^{−} ∩ (BwB ∐ Bw′′B). By proposition 8.13 we have BwB ∩ U^{−} = U_{−α_{2}}^{∗}.(B_{I}w′B_{I} ∩ U_{I}^{−}) and Bw′′B ∩ U^{−} = U_{−α_{2}}^{∗}.(B_{I}s_{3}s_{1}s_{4}s_{3}B_{I} ∩ U_{I}^{−}), whence, from the decomposition ^{v^{−1}}U ∩ U^{−} = (^{v^{−1}}U ∩ U_{P_{I}}^{−}).(^{v^{−1}}U ∩ U_{I}^{−}) we get

v^{−1}U ∩ U^{−} ∩ (BwB ∐ Bw′′B) = U_{−α_{2}}^{∗}.(^{v^{−1}}U ∩ U_{I}^{−} ∩ (B_{I}w′B_{I} ∐ B_{I}s_{3}s_{1}s_{4}s_{3}B_{I})).

We have ^{v^{−1}}U ∩ U_{I}^{−} = ∏_{α∈Φ_{I}^{−}−{−α_{3}}} U_{α}. In order to make explicit computations, we may replace L_{I} by the group GL_{4} as the Borel subgroup varieties, as well as the groups U_{I} and U_{α} depend only on the isogeny type together with the Frobenius action on the root system. The variety ^{v^{−1}}U ∩ U_{I}^{−} is thus isomorphic to the variety

of matrices ( \begin{matrix}1 & 0 & 0 & 0 \\ ∗ & 1 & 0 & 0 \\ ∗ & 0 & 1 & 0 \\ ∗ & ∗ & ∗ & 1 \end{matrix} ).

We now determine the variety of matrices in GL_{4} representing elements of ^{v^{−1}}U ∩ U_{I}^{−} ∩ (B_{I}w′B_{I} ∐ B_{I}s_{3}s_{1}s_{4}s_{3}B_{I}). For this we use the following:

Lemma 12.5. Let B be the Borel subgroup of GL_{n} of upper triangular matrices; let W be the Weyl group relative to the torus of diagonal matrices; let (w_{ij}) be the matrix for w ∈ W; then (a_{ij}) ∈ GL_{n} is in BwB if and only if the ranks of the submatrices (a_{ij})_{i=k,...,n,j=1,...,l} and (w_{ij})_{i=k,...,n,j=1,...,l} coincide for all k and l.

Proof. The condition on ranks is invariant by left multiplication (resp. right multiplication) by B since this replaces each line (resp. column) by a non-zero multiple of itself plus a linear combination of the following lines (resp. columns). This condition defines thus a union of double B-cosets. It remains to see that elements of W are determined by the rank conditions: but indeed, in line k, the position l of the non-zero coefficient is the smallest integer such that the rank of the matrices (w_{ij})_{i=k,...,n,j=1,...,l} and (w_{ij})_{i=k+1,...,n,j=1,...,l} differ. □

We thus obtain, characterizing w′ and s_{3}s_{1}s_{4}s_{3} by rank conditions, that ^{v^{−1}}U ∩

U_{I}^{−}∩(B_{I}w′B_{I} ∐ B_{I}s_{3}s_{1}s_{4}s_{3}B_{I}) is the variety of matrices ( \begin{matrix}1 & 0 & 0 & 0 \\ d & 1 & 0 & 0 \\ α & 0 & 1 & 0 \\ 0 & β & f & 1 \end{matrix} ), with α, β, d and

f in \overline{𝔽}_{q}, such that | \begin{matrix}α & 0 \\ 0 & β \end{matrix} | ≠ 0; the open subset corresponding to B_{I}w′B_{I} is given by the

condition | \begin{matrix}d & 1 & 0 \\ α & 0 & 1 \\ 0 & β & f \end{matrix} | ≠ 0. These matrices may be written as ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ α & 0 & 1 & 0 \\ 0 & β & 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 & 0 & 0 \\ d & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ −dβ & 0 & f & 1 \end{matrix} ) ∈

v^{−1}U_{I}^{v^{−1}}U_{P_{I}}. The projection on ^{v^{−1}}U_{I} of ^{v^{−1}}U ∩ U^{−} ∩ BwB as well as that of v^{−1}U ∩ U^{−} ∩ Bw′′B is thus U_{−α_{2}}^{∗}U_{−α_{1}−α_{3}}^{∗}U_{−α_{3}−α_{4}}^{∗}. The assumption 8.21(iv) thus holds for w and w′′.

To check the assumptions of proposition 8.25, we must compute the fibers of the maps \overline{π}′ and \overline{π} ∐ \overline{π}′ of proposition 8.25.

The above computations show that for y ∈ ^{v^{−1}}U_{P_{I}} ∩ U^{−} and x ∈ ^{v^{−1}}U_{I}, we have y.x^{−1}^{F}x ∈ BwB ∐ Bw′′B if and only if x^{−1}.^{F}x ∈ ∏_{α∈^{v^{−1}}U_{I}} U_{α}^{∗} and ^{^{F}x^{−1}x}y is in U_{−α_{1}}U_{−α_{4}}U_{−α_{1}−α_{4}} and are such that the latter element corresponds to the

matrix ( \begin{matrix}1 & 0 & 0 & 0 \\ d & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ −dβ & 0 & f & 1 \end{matrix} ) where the projection of x^{−1}^{F}x on U_{I}^{−} is given by the matrix ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ α & 0 & 1 & 0 \\ 0 & β & 0 & 1 \end{matrix} ). The closed subset corresponding to Bw′′B is given by dβ + fα = 0.

We see thus that the fibers of the map \overline{π} ∐ \overline{π}′ of proposition 8.25 are 2-dimensional affine spaces corresponding to the d and f coordinates of the matrix for y and that the fibers of the map \overline{π}′ are 1-dimensional affine subspaces corresponding to the equation dβ + fα = 0.

The assumptions of proposition 8.25 thus hold, and we get the cohomology of X^{v}(w)/U_{P_{I}}^{F} as an L^{F} × <F >-module by multiplying by th^{3} + t^{2}h^{4} the two-variable polynomial encoding the cohomology of the Coxeter variety for L_{I}. We get

∑_{i} h^{i}·H_{c}^{i}(X^{v}(w))^{U_{P_{I}}^{F}} = h^{6}t St +h^{7}t^{2}(γ_{2}+St)+h^{8}t^{3}(γ_{2}+γ_{3})+h^{9}t^{4}(γ_{3}+Id)+h^{10}t^{5} Id .

The long exact sequence of L_{I}^{F} × <F >-modules

… → H_{c}^{i}(Y) → H_{c}^{i}(X(w))^{U_{P_{I}}^{F}} → H_{c}^{i}(X^{v}(w))^{U_{P_{I}}^{F}} → H_{c}^{i+1}(Y) → …

gives

0 → St → H_{c}^{6}(X(w))^{U_{P_{I}}^{F}} → t St → t^{2}(γ_{2}+ρ_{2})+t St → H_{c}^{7}(X(w))^{U_{P_{I}}^{F}} → t^{2}(γ_{2}+St) → 0

H_{c}^{8}(X(w))^{U_{P_{I}}^{F}} = t^{3}(2γ_{2} + 2ρ_{2} + 2γ_{3})
H_{c}^{9}(X(w))^{U_{P_{I}}^{F}} = t^{4}(ρ_{2} + 2γ_{3} + Id)
0 → H_{c}^{10}(X(w))^{U_{P_{I}}^{F}} → t^{5} Id → t^{5} Id → H_{c}^{11}(X(w))^{U_{P_{I}}^{F}} → 0
H_{c}^{12}(X(w))^{U_{P_{I}}^{F}} = t^{6} Id

To obtain the non-cuspidal part of H_{c}^{∗}(X(w)), we first use [DMR, 3.3.14] and [DMR, 3.3.15] which give the Id and St isotypic parts. We then consider for each b the t^{b}- isotypic part of the above exact sequences arguing as the proof of theorem 9.2 and using the following table which describes ⟨ Ind_{A_{3}}^{D_{4}} χ, ψ ⟩_{D_{4}}:

1^{2}+ 1^{2}− 1.1^{3} St 1^{2}.2 1.21 21^{2} 2+ 2− 2^{2} 1.3 31 Id
Id 0  0  0  0 0  0  0 0 1 0 1 0 1
γ_{2} 1  0  1  0 1  1  1 0 0 0 0 0 0
ρ_{2} 0  1  0  0 0  1  0 0 1 1 0 0 0
γ_{3} 0  0  0  0 1  1  0 1 0 0 1 1 0
St 0  1  1  1 0  0  0 0 0 0 0 0 0

We get:

∑_{i} h^{i} · H_{c}^{i}(X(w))_{non-cuspidal} =

h^{6} St +t^{2}h^{7}(γ_{1^{2}+} + γ_{1^{2}−} + γ_{21^{2}}) + 2t^{3}h^{8}γ_{1.21} + t^{4}h^{9}(γ_{2+} + γ_{2−} + γ_{31}) + t^{6}h^{12} Id .

To study the θ-part of the cohomology of X(w), we will use the variety \overline{X}(w). One may check that all Kazhdan-Lusztig polynomials P_{y,w} for y ≤ w are 1, thus \overline{X}(w) is rationally smooth (cf. [DMR, 3.2.5]), which by [DMR, 3.3.8 (ii)] allows to compute ∑_{i} h^{i} · (H_{c}^{i}(\overline{X}(w)))_{θ} = t^{3}h^{6}θ.

By [DMR, 3.1.3], if y lies in a proper standard parabolic subgroup, H_{c}^{i}(X(y)) cannot have a cuspidal part. The only y < w for which this does not hold are the elements of

C = {s_{2}s_{3}s_{1}s_{4}s_{3}, s_{2}s_{1}s_{3}s_{4}s_{3}, s_{2}s_{1}s_{3}s_{1}s_{4}, s_{2}s_{3}s_{1}s_{4}, s_{2}s_{1}s_{4}s_{3}, s_{2}s_{1}s_{3}s_{4}}.

Thus \overline{X}(w) = X(w) ∐ X ∐ Y where X = ∐_{v∈C} X(v) and where Y is a union of Deligne-Lusztig varieties, closed in \overline{X}(w) and such that H_{c}^{i}(Y)_{θ} = 0 for any i. The long exact sequence corresponding to \overline{X}(w) = Y ∐(\overline{X}(w) − Y) shows thus that for any i we have H_{c}^{i}(\overline{X}(w))_{θ} = H_{c}^{i}(X(w) ∪ X)_{θ}.

The varieties X(s_{2}s_{3}s_{1}s_{4}), X(s_{2}s_{1}s_{4}s_{3}) and X(s_{2}s_{1}s_{3}s_{4}), corresponding to Coxeter elements, satisfy (cf. e.g., [Lu2]) ∑_{i} h^{i} · (H_{c}^{i})_{θ} = t^{2}h^{4}, and they are connected components of their union, thus ∑_{i} h^{i} · H_{c}^{i}(X(s_{2}s_{3}s_{1}s_{4}) ∪ X(s_{2}s_{1}s_{4}s_{3}) ∪ X(s_{2}s_{1}s_{3}s_{4}))_{θ} = 3t^{2}h^{4}. The elements s_{2}s_{3}s_{1}s_{4}s_{3}, s_{2}s_{1}s_{3}s_{4}s_{3} and s_{2}s_{1}s_{3}s_{1}s_{4} are conjugate by cyclic permutation respectively to s_{2}s_{3}s_{1}s_{4}s_{2}, s_{4}s_{2}s_{1}s_{3}s_{4} and s_{1}s_{2}s_{3}s_{4}s_{1} which, by [DMR, 3.1.6] and [DMR, 3.2.10], allows to compute the cuspidal part of their cohomology For each of them we get ∑_{i} h^{i} · (H_{c}^{i})_{θ} = t^{2}h^{5} + t^{3}h^{6}, thus for their union (of which they are connected components) ∑_{i} h^{i} · (H_{c}^{i})_{θ} = 3t^{2}h^{5} + 3t^{3}h^{6}. The union

X(s_{2}s_{3}s_{1}s_{4}s_{3}) ∐ X(s_{2}s_{1}s_{3}s_{4}s_{3}) ∐ X(s_{2}s_{1}s_{3}s_{1}s_{4})

is open in X; the corresponding long exact sequence gives

0 → H_{c}^{4}(X)_{θ} → 3t^{2}θ → 3t^{2}θ → H_{c}^{5}(X)_{θ} → 0,

and H_{c}^{6}(X)_{θ} = 3t^{3}θ. There exists thus an integer η ≤ 3 such that ∑_{i} h^{i} · H_{c}^{i}(X)_{θ} = ηt^{2}(h^{4} + h^{5})θ + 3t^{3}h^{6}θ. The long exact sequence corresponding to the union of X and X(w) shows then that only the characters t^{2} and t^{3} of <F > may occur in the cohomology of X(w), and gives for the corresponding isotypic parts of the cohomology: H_{c}^{5}(X(w))_{θ,t^{2}} = H_{c}^{6}(X(w))_{θ,t^{2}} = ηt^{2}θ, and

0 → H_{c}^{6}(X(w))_{θ,t^{3}} → t^{3}θ → 3t^{3}θ → H_{c}^{7}(X(w))_{θ,t^{3}} → 0.

By [DMR, 3.3.22] we have H_{c}^{5}(X(w)) = 0, thus also H_{c}^{6}(X(w))_{θ,t^{2}} = 0.

To lift the ambiguity on (θ, t^{3}), we now use [DMR, 3.3.21]; we take for w′ a Coxeter element, which is not conjugate to w; thus any eigenvalue of F on H_{c}^{6}(X(w))_{θ} must have a module less than q^{3}. This shows that H_{c}^{6}(X(w))_{θ} = 0, thus H_{c}^{7}(X(w))_{θ,t^{3}} = 2t^{3}θ. □

From the values of the generic degrees of H_{q}(w) (see [BMa, 5.A]), we see that if we denote by ρ(x_{1}, x_{2}, x_{3}) where x_{i} ∈ {1, q^{2}} the 1-dimensional representation of H_{q}(w) given by T_{i} ↦ x_{i}, and ρ^{+} (resp. ρ^{−}) the irreducible 2-dimensional representation where T_{1}T_{2}T_{3} acts as the scalar q^{3} (resp. −q^{3}), we have the following equalities, if we denote by m_{ρ} the multiplicity of ρ in ∑_{i}(−1)^{i}H_{c}^{i}(X(w)): m_{ρ(1,1,1)} = dim St, m_{ρ(q^{2},q^{2},q^{2})} = dim Id, m_{ρ(q^{2},q^{2},1)} = m_{ρ(1,q^{2},q^{2})} = m_{ρ(q^{2},1,q^{2})} = − dim γ_{1^{2}+} = − dim γ_{1^{2}−} = − dim γ_{21^{2}}, m_{ρ(q^{2},1,1)} = m_{ρ(1,q^{2},1)} = m_{ρ(1,1,q^{2})} = − dim γ_{2+} = − dim γ_{2−} = − dim γ_{31}, m_{ρ^{+}} = dim γ_{1.21} and m_{ρ^{−}} = − dim θ. Thus we can determine ⊕_{i}H_{c}^{i}(X(w)) as a G^{F} × H_{q}(w)-module up to an ambiguity on the correspondence between characters of G^{F} and of H_{q}(w) which appear in the same degree.

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