George Lusztig (MIT): Unipotent representations: changing q to -q.

Let G be a connected reductive group defined and split over a finite field F_q and such that the longest element in the Weyl group W of G is central in W. Let U be the set of isomorphism classes of unipotent representations of the finite group G(F_q). We define an involution ξ ↦ ξ^! of U such that, for any ξ ∈ U, the dimension of ξ^! (a polynomial in q with rational coefficients) is obtained (up to sign) by changing q to −q in the dimension of ξ (also a polynomial in q with rational coefficients). This is a recent joint work with P. Deligne.