Central elements in affine mod $p$ Hecke algebras via perverse $\mathbb{F}_p$-sheaves

TL;DR

This paper constructs central perverse $F_p$-sheaves on the affine flag variety of a split reductive group, providing explicit formulas for the centers of mod $p$ Hecke algebras and analyzing related geometric properties.

Contribution

It introduces a geometric construction of central perverse $F_p$-sheaves and derives explicit formulas for the centers of mod $p$ Hecke algebras, advancing understanding of their structure.

Findings

Explicit isomorphism between spherical and Iwahori mod $p$ Hecke algebra centers.

Construction of a nearby cycles functor for perverse $F_p$-sheaves.

Proved certain local models are strongly $F$-regular with pseudo-rational singularities.

Abstract

Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text{der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb{F}_p$-sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse $\mathbb{F}_p$-sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly $F$-regular, and hence they are $F$-rational and have pseudo-rational singularities.