Vol. 8, No. 5, 2014

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Compatibility between Satake and Bernstein isomorphisms in characteristic $p$

Rachel Ollivier

Vol. 8 (2014), No. 5, 1071–1111
Abstract

We study the center of the pro-p Iwahori–Hecke ring H̃ 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 H̃ k contains an affine semigroup algebra which is naturally isomorphic to the Hecke k-algebra (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 H̃ k-modules, study the supersingular block in the category of finite-length H̃ 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