## Parabolic induction and the Harish-Chandra $\boldsymbol {\mathcal {D}}$-module

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- by Victor Ginzburg PDF
- Represent. Theory
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## Abstract:

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.## References

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## Additional Information

**Victor Ginzburg**- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 190015
- Email: ginzburg@math.uchicago.edu
- Received by editor(s): April 16, 2021
- Received by editor(s) in revised form: December 16, 2021
- Published electronically: March 24, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 388-401 - MSC (2020): Primary 22E47; Secondary 14F10
- DOI: https://doi.org/10.1090/ert/603
- MathSciNet review: 4399287

Dedicated: To the memory of Tom Nevins