Algebra Seminar

Let G(F_q) be a finite reductive group and H(F_q) the subgroup of fixed points of some involution. The set of cosets G(F_q)/H(F_q) is called a finite symmetric space The spherical functions on this space are the averages, over these cosets, of the irreducible characters of G(F_q). Computing the values of these spherical functions is a generalization of the much-studied problem of computing the character table of finite reductive groups. To solve the latter problem, Lusztig introduced the notion of character sheaves on groups. In this talk I will explain how the more general character sheaves on symmetric spaces (in the sense of Ginzburg and Grojnowski) help in the former problem, and examine some special cases where they lead to a solution.