Inverse Satake isomorphism and change of weight

Abstract

Let $G$ be any connected reductive $p$-adic group. Let $K\subset G$ be any special parahoric subgroup and $V,V^{\prime }$ 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 ${\mathrm{End}}_{{\overline{\mathbb{F}}}_p[K]}({c-Ind}_K^{G}V,{c-Ind}_K^G{V}^{\prime })$ 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 $K$).

arXiv comment60 pages

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Cited by in corpus (1)

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