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Hecke algebras for inner forms of \(p\)-adic special linear groups. (English) Zbl 1436.20090
Summary: 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 \(\mathrm{GL}_{n}(F)\).
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.

20G25 Linear algebraic groups over local fields and their integers
20C08 Hecke algebras and their representations
22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups
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