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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 and the Harish-Chandra $\boldsymbol {\mathcal {D}}$-module
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by Victor Ginzburg PDF
Represent. Theory 26 (2022), 388-401 Request permission


Let $G$ be a reductive group and $L$ a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between $\operatorname {Ad}$-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) ${\mathscr {D}}$-modules on $G$ and $L$, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where $L=T$ is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra ${\mathscr {D}}$-module on ${G\times T}$. We show that this module is flat over ${\mathscr {D}}(T)$, which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of ${\mathscr {D}}$-modules.
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Additional Information
  • Victor Ginzburg
  • Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
  • MR Author ID: 190015
  • Email:
  • Received by editor(s): April 16, 2021
  • Received by editor(s) in revised form: December 16, 2021
  • Published electronically: March 24, 2022

  • Dedicated: To the memory of Tom Nevins
  • © Copyright 2022 American Mathematical Society
  • Journal: Represent. Theory 26 (2022), 388-401
  • MSC (2020): Primary 22E47; Secondary 14F10
  • DOI:
  • MathSciNet review: 4399287