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

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.

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