Séminaires à venir

The representation theory of the finite group GLn(Fq) is well understood, thanks to its combinatorial description (given by Green) and its geometric interpretation due to the results of Deligne and Lusztig. Still, we have very little understanding of the multiplicities, i.e. of the decomposition of tensor product of irreducible representations. Hausel, Letellier and Rodriguez-Villegas gave a combinatorial description of the multiplicities in the generic case. This description relates these multiplicities to the cohomology of complex character varieties for GLn(C). In a joint work with Emmanuel Letellier, we try to generalize these results to other groups of type A. In particular, we study multiplicities for characteristic functions of character sheaves of SLn(Fq), rather than irreducible characters, and relate them to the cohomology of complex character varieties for PGLn(C). We expect more generally that , for a finite reductive G(Fq), multiplicities should be related to the character varieties for its complex Langlands dual.