Central elements in affine modpHecke algebras via perverse -sheaves

Abstract

AbstractLet$G$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.