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THE PRO-p-IWAHORI HECKE ALGEBRA OF A REDUCTIVE p-ADIC GROUP III (SPHERICAL HECKE ALGEBRAS AND SUPERSINGULAR MODULES)

Part of: Representation theory of groups Lie groups

Published online by Cambridge University Press:  03 June 2015

Marie-France Vigneras
Marie-France Vigneras*
Affiliation:
UMR 7586, Institut de Mathematiques de Jussieu, 4 place Jussieu, Paris 75005, France (vigneras@math.jussieu.fr)
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Abstract

Let $R$ be a large field of characteristic $p$. We classify the supersingular simple modules of the pro-$p$-Iwahori Hecke $R$-algebra ${\mathcal{H}}$ of a general reductive $p$-adic group $G$. We show that the functor of pro-$p$-Iwahori invariants behaves well when restricted to the representations compactly induced from an irreducible smooth $R$-representation $\unicode[STIX]{x1D70C}$ of a special parahoric subgroup $K$ of $G$. We give an almost-isomorphism between the center of ${\mathcal{H}}$ and the center of the spherical Hecke algebra ${\mathcal{H}}(G,K,\unicode[STIX]{x1D70C})$, and a Satake-type isomorphism for ${\mathcal{H}}(G,K,\unicode[STIX]{x1D70C})$. This generalizes results obtained by Ollivier for $G$ split and $K$ hyperspecial to $G$ general and $K$ special.

Keywords

group theory and generalizationsnumber theory

MSC classification

Primary: 20C08: Hecke algebras and their representations
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 3 , June 2017 , pp. 571 - 608
Copyright
© Cambridge University Press 2015 

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