Hecke algebras and affine flag varieties in characteristic p

Abstract

Let G be a split semi-simple p-adic group and let H be its Iwahori–Hecke algebra with coefficients in the algebraic closure F‾p of Fp. Let F be the affine flag variety associated with G. We show, in the simply connected simple case, that the K′-theory of F with coefficients in F‾p admits an action of H by Demazure operators and that this provides a model for the regular representation of H. A similar result holds for the affine Grassmannian and the mod p Satake–Hecke algebra of G.