Inverse Satake isomorphism and change of weight

Abstract

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