Compatibility between Satake and Bernstein isomorphisms in characteristic p

Ollivier, Rachel

doi:10.2140/ant.2014.8.1071

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Compatibility between Satake and Bernstein isomorphisms in characteristic $p$

Rachel Ollivier

Vol. 8 (2014), No. 5, 1071–1111

DOI: 10.2140/ant.2014.8.1071

Abstract

We study the center of the pro- $p$ Iwahori–Hecke ring ${\tilde{H}}_{ℤ}$ of a connected split $p$ -adic reductive group $G$ . For $k$ an algebraically closed field of characteristic $p$ , we prove that the center of the $k$ -algebra ${\tilde{H}}_{ℤ} \otimes_{ℤ} k$ contains an affine semigroup algebra which is naturally isomorphic to the Hecke $k$ -algebra $ℋ (G, ρ)$ attached to an irreducible smooth $k$ -representation $ρ$ of a given hyperspecial maximal compact subgroup of $G$ . This isomorphism is obtained using the inverse Satake isomorphism defined in our previous work.

We apply this to classify the simple supersingular ${\tilde{H}}_{ℤ} \otimes_{ℤ} k$ -modules, study the supersingular block in the category of finite-length ${\tilde{H}}_{ℤ} \otimes_{ℤ} k$ -modules, and relate the latter to supersingular representations of $G$ .

Keywords

Hecke algebras, characteristic $p$, Satake isomorphism, supersingularity

Mathematical Subject Classification 2010

Primary: 20C08

Secondary: 22E50

Milestones

Received: 25 April 2013

Revised: 27 December 2013

Accepted: 27 February 2014

Published: 16 September 2014

Authors

Rachel Ollivier
Department of Mathematics Columbia University 2990 Broadway New York, NY 10027 United States

Vol. 8, No. 5, 2014