Inverse Satake isomorphism and change of weight

TL;DR

This paper computes the inverse of the Satake transform for mod p representations of p-adic groups, providing explicit formulas and a simpler proof of the change of weight theorem crucial for classifying such representations.

Contribution

It explicitly determines the inverse of the Satake transform for special parahoric subgroups, advancing the understanding of mod p representation theory of p-adic groups.

Findings

Explicit formula for the inverse Satake transform using pro-p Iwahori Hecke algebra.

Proof of the change of weight theorem in mod p representation classification.

Simplified proof of the change of weight theorem for split groups.

Abstract

Let $G$ be any connected reductive $p$-adic group. Let $K\subset G$ be any special parahoric subgroup and $V,V'$ be any two irreducible smooth $\overline {\mathbb F}_p[K]$-modules. The main goal of this article is to compute the image of the Hecke bi-module $\operatorname{End}_{\overline {\mathbb F}_p[K]}(\operatorname{c-Ind}_K^G V, \operatorname{c-Ind}_K^G V')$ by the generalized Satake transform and to give an explicit formula for its inverse, using the pro-$p$ Iwahori Hecke algebra of $G$. This immediately implies the "change of weight theorem" in the proof of the classification of mod $p$ irreducible admissible representations of $G$ in terms of supersingular ones. A simpler proof of the change of weight theorem, not using the pro-$p$ Iwahori Hecke algebra or the Lusztig-Kato formula, is given when $G$ is split (and in the appendix when $G$ is quasi-split, for almost all $K$).