Hecke algebras and affine flag varieties in characteristic p

TL;DR

This paper explores the relationship between the Iwahori-Hecke algebra of a split semi-simple p-adic group and the K'-theory of its affine flag variety over a field of characteristic p, revealing a model for the algebra's regular representation.

Contribution

It demonstrates that the torus-equivariant K'-theory of the affine flag variety admits an H-action via Demazure operators, providing a new geometric model for the regular representation of the Hecke algebra.

Findings

K'-theory of affine flag variety admits H-action

Provides a geometric model for the regular representation of H

Establishes Demazure operators as symmetries in K'-theory

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 k of the finite field with p elements. Let F be the affine flag variety over k associated with G. We show, in the simply connected simple case, that a torus-equivariant K'-theory of F (with coefficients in k) admits an H-action by Demazure operators and that this provides a model for the regular representation of H.