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Regular Bernstein blocks

  • Jeffrey D. Adler and Manish Mishra ORCID logo EMAIL logo

Abstract

For a connected reductive group G defined over a non-archimedean local field F, we consider the Bernstein blocks in the category of smooth representations of G(F). Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called regular Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of F is not too small. Under mild hypotheses on the residual characteristic, we show that the Bernstein center of a regular Bernstein block of G(F) is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of G0(F), where G0 is a certain twisted Levi subgroup of G. In some cases, we show that the blocks themselves are equivalent, and as a consequence we prove the ABPS Conjecture in some new cases.

Funding statement: The second named author was partially supported by SERB MATRICS and SERB ERCA grants.

Acknowledgements

It is a pleasure for the authors to thank the following. Maarten Solleveld clarified for us the definition of the twisted extended quotient and pointed out an error in a previous draft. Anne-Marie Aubert pointed out that the isomorphism of Hecke algebras in Remark 1.2 is false in general for tame Bernstein blocks. Adèle Bourgeois pointed out a typographical error in Definition 5.2. An anonymous referee provided comments that helped us to improve the exposition of this work.

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Received: 2019-11-15
Revised: 2020-12-08
Published Online: 2021-03-16
Published in Print: 2021-06-01

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