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

Published online by Cambridge University Press:  03 June 2015

Marie-France Vigneras*
Affiliation:
UMR 7586, Institut de Mathematiques de Jussieu, 4 place Jussieu, Paris 75005, France ([email protected])

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.

Type
Research Article
Copyright
© Cambridge University Press 2015 

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