Geometrization of the Satake transform for mod $p$ Hecke algebras

TL;DR

This paper geometrizes the mod p Satake isomorphism for reductive groups using Witt vector affine flag varieties, providing explicit formulas for convolution products and advancing towards a geometric understanding of the mod p Local Langlands Correspondence.

Contribution

It introduces a geometric framework for the mod p Satake transform using Witt vector affine flag varieties and derives explicit convolution formulas for parahoric Hecke algebras.

Findings

Derived a geometric Satake isomorphism for mod p Hecke algebras

Provided explicit convolution formulas for parahoric Hecke algebras

Extended methods to groups in equal characteristic

Abstract

We geometrize the mod $p$ Satake isomorphism of Herzig and Henniart-Vign\'eras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirkovi\'c-Vilonen cycles, for the Satake transform of an arbitrary pararhoric mod $p$ Hecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric mod $p$ Hecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step towards a geometrization of a mod $p$ Local Langlands Correspondence.