Hecke algebras and affine flag varieties in characteristic p

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