Produit scalaire elliptique

Abstract

Suppose that W is a Weyl group, let C ( W ) be a space of functions on W, with complex values, invariant under conjugation. We can define an “elliptic scalar product” on C ( W ) . It is a natural ingredient to the representation theory of p-adic reductive groups. Let G be a reductive group over the algebraic closure of a finite field. The generalized Springer correspondence gives a bijection between two sets: – the set of pairs ( U , E ) , where U is an unipotent orbit of G and E is a G-equivariant irreducible local system on U; – the disjoint union of the sets of irreducible representations of certain Weyl groups related to G. Using Kazhdan–Lusztig polynomials, we modify the generalized Springer correspondence. By the modified correspondence, a pair ( U , E ) as above maps to a representation of a certain Weyl group, and this representation is, in general, reducible. There is no simple formula that relates the elliptic scalar product and the generalized Springer correspondence. But a simple formula does exist, and we prove it, that relates the elliptic scalar product and the modified generalized Springer correspondence. Our result is, in fact, a corollary of a theorem of Lusztig on the restriction of character-sheaves to the unipotent variety.