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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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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