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On typical representations for depth-zero components of split classical groups

Authors: Santosh Nadimpalli and Amiya Kumar Mondal
Journal: Represent. Theory 23 (2019), 249-277
MSC (2010): Primary 22E50; Secondary 11F70
Published electronically: September 12, 2019
MathSciNet review: 4007167
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Abstract: Let $\mathbf {G}$ be a split classical group over a non-Archimedean local field $F$ with the cardinality of the residue field $q_F>5$. Let $M$ be the group of $F$-points of a Levi factor of a proper $F$-parabolic subgroup of $\mathbf {G}$. Let $[M, \sigma _M]_M$ be an inertial class such that $\sigma _M$ contains a depth-zero Moy–Prasad type of the form $(K_M, \tau _M)$, where $K_M$ is a hyperspecial maximal compact subgroup of $M$. Let $K$ be a hyperspecial maximal compact subgroup of $\mathbf {G}(F)$ such that $K$ contains $K_M$. In this article, we classify $\mathfrak {s}$-typical representations of $K$. In particular, we show that the $\mathfrak {s}$-typical representations of $K$ are precisely the irreducible subrepresentations of $\operatorname {ind}_J^K\lambda$, where $(J, \lambda )$ is a level-zero $G$-cover of $(K\cap M, \tau _M)$.

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Additional Information

Santosh Nadimpalli
Affiliation: IMAPP, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, the Netherlands
MR Author ID: 1183126
ORCID: 0000-0002-2637-6159

Amiya Kumar Mondal
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan 529002, Israel
MR Author ID: 1115151

Keywords: Representation theory of $p$-adic groups, theory of types, inertial equivalence, Bernstein decomposition, typical representations, depth-zero inertial classes, classical groups
Received by editor(s): October 2, 2018
Received by editor(s) in revised form: August 7, 2019
Published electronically: September 12, 2019
Additional Notes: The first author was supported by the NWO Vidi grant “A Hecke algebra approach to the local Langlands correspondence” (nr. 639.032.528)
The second author was funded by Israel Science Foundation grant No. 421/17 (Mondal)
Article copyright: © Copyright 2019 American Mathematical Society