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HECKE ALGEBRAS FOR INNER FORMS OF $p$-ADIC SPECIAL LINEAR GROUPS

Part of: Linear algebraic groups and related topics Lie groups

Published online by Cambridge University Press:  05 May 2015

Anne-Marie Aubert ,
Paul Baum ,
Roger Plymen  and
Maarten Solleveld
Anne-Marie Aubert
Affiliation:
Institut de Mathématiques de Jussieu – Paris Rive Gauche, U.M.R. 7586 du C.N.R.S., U.P.M.C., 4 place Jussieu 75005 Paris, France (anne-marie.aubert@imj-prg.fr)
Paul Baum
Affiliation:
Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA (baum@math.psu.edu)
Roger Plymen
Affiliation:
School of Mathematics, Southampton University, Southampton SO17 1BJ, England (r.j.plymen@soton.ac.uk) School of Mathematics, Manchester University, Manchester M13 9PL, England (plymen@manchester.ac.uk)
Maarten Solleveld
Affiliation:
Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands (m.solleveld@science.ru.nl)
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Abstract

Let $F$ be a non-Archimedean local field, and let $G^{\sharp }$ be the group of $F$-rational points of an inner form of $\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of $G^{\sharp }$, via restriction from an inner form $G$ of $\text{GL}_{n}(F)$.

For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth $G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of $G^{\sharp }$, and the idempotent is derived from a type for $G$. We show that the Hecke algebras for Bernstein components of $G^{\sharp }$ are similar to affine Hecke algebras of type $A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

Keywords

representation theorydivision algebraHecke algebratypes

MSC classification

Primary: 20G25: Linear algebraic groups over local fields and their integers
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 2 , April 2017 , pp. 351 - 419
Copyright
© Cambridge University Press 2015 

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