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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Parabolic induction via the parabolic pro-$p$ Iwahori–Hecke algebra
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by Claudius Heyer PDF
Represent. Theory 25 (2021), 807-843 Request permission

Abstract:

Let $\mathbf {G}$ be a connected reductive group defined over a locally compact non-archimedean field $F$, let $\mathbf {P}$ be a parabolic subgroup with Levi $\mathbf {M}$ and compatible with a pro-$p$ Iwahori subgroup of $G ≔\mathbf {G}(F)$. Let $R$ be a commutative unital ring.

We introduce the parabolic pro-$p$ Iwahori–Hecke $R$-algebra $\mathcal {H}_R(P)$ of $P ≔\mathbf {P}(F)$ and construct two $R$-algebra morphisms $\Theta ^P_M\colon \mathcal {H}_R(P)\to \mathcal {H}_R(M)$ and $\Xi ^P_G\colon \mathcal {H}_R(P) \to \mathcal {H}_R(G)$ into the pro-$p$ Iwahori–Hecke $R$-algebra of $M ≔\mathbf {M}(F)$ and $G$, respectively. We prove that the resulting functor $\operatorname {Mod}\text {-}\mathcal {H}_R(M) \to \operatorname {Mod}\text {-}\mathcal {H}_R(G)$ from the category of right $\mathcal {H}_R(M)$-modules to the category of right $\mathcal {H}_R(G)$-modules (obtained by pulling back via $\Theta ^P_M$ and extension of scalars along $\Xi ^P_G$) coincides with the parabolic induction due to Ollivier–Vignéras.

The maps $\Theta ^P_M$ and $\Xi ^P_G$ factor through a common subalgebra $\mathcal {H}_R(M,G)$ of $\mathcal {H}_R(G)$ which is very similar to $\mathcal {H}_R(M)$. Studying these algebras $\mathcal {H}_R(M,G)$ for varying $(M,G)$ we prove a transitivity property for tensor products. As an application we give a new proof of the transitivity of parabolic induction.

References
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Additional Information
  • Claudius Heyer
  • Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstraße 62, D-48149 Münster, Germany
  • ORCID: 0000-0001-6794-7870
  • Email: cheyer@uni-muenster.de
  • Received by editor(s): October 29, 2020
  • Received by editor(s) in revised form: March 31, 2021
  • Published electronically: October 5, 2021
  • Additional Notes: The author was funded by the University of Münster and Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics-Geometry-Structure.
  • © Copyright 2021 American Mathematical Society
  • Journal: Represent. Theory 25 (2021), 807-843
  • MSC (2020): Primary 11E95, 20C08, 20G25
  • DOI: https://doi.org/10.1090/ert/585
  • MathSciNet review: 4321333