Parabolic induction and the Harish-Chandra $\boldsymbol {\mathcal {D}}$-module
HTML articles powered by AMS MathViewer
- by Victor Ginzburg PDF
- Represent. Theory 26 (2022), 388-401 Request permission
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
- D. Ben-Zvi and S. Gunningham, Symmetries of categorical representations and the quantum Ngo action, arXiv:1712.01963, 2017.
- Roman Bezrukavnikov and Alexander Yom Din, On parabolic restriction of perverse sheaves, Publ. Res. Inst. Math. Sci. 57 (2021), no. [3-4], 1089–1107. MR 4322008, DOI 10.4171/prims/57-3-12
- T.-H. Chen, On the conjectures of Braverman-Kazhdan, arXiv:1909.05467. J. Amer. Math. Soc., https://doi.org/10.1090/jams/992.
- V. Drinfeld and D. Gaitsgory, On a theorem of Braden, Transform. Groups 19 (2014), no. 2, 313–358. MR 3200429, DOI 10.1007/s00031-014-9267-8
- Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243–348. MR 1881922, DOI 10.1007/s002220100171
- Victor Ginzburg, Isospectral commuting variety, the Harish-Chandra $\scr D$-module, and principal nilpotent pairs, Duke Math. J. 161 (2012), no. 11, 2023–2111. MR 2957698, DOI 10.1215/00127094-1699392
- Sam Gunningham, Generalized Springer theory for $D$-modules on a reductive Lie algebra, Selecta Math. (N.S.) 24 (2018), no. 5, 4223–4277. MR 3874694, DOI 10.1007/s00029-018-0443-x
- Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc. 119 (1965), 457–508. MR 180631, DOI 10.1090/S0002-9947-1965-0180631-0
- R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), no. 2, 327–358. MR 732550, DOI 10.1007/BF01388568
- R. Hotta and M. Kashiwara, Quotients of the Harish-Chandra system by primitive ideals, Geometry today (Rome, 1984) Progr. Math., vol. 60, Birkhäuser Boston, Boston, MA, 1985, pp. 185–205. MR 895154
- Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361, DOI 10.1007/978-0-8176-4523-6
- Masaki Kashiwara, The invariant holonomic system on a semisimple Lie group, Algebraic analysis, Vol. I, Academic Press, Boston, MA, 1988, pp. 277–286. MR 992461
- T. Levasseur and J. T. Stafford, Invariant differential operators and an homomorphism of Harish-Chandra, J. Amer. Math. Soc. 8 (1995), no. 2, 365–372. MR 1284849, DOI 10.1090/S0894-0347-1995-1284849-8
- T. Levasseur and J. T. Stafford, The kernel of an homomorphism of Harish-Chandra, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 3, 385–397. MR 1386924
- Sam Raskin, A generalization of the $b$-function lemma, Compos. Math. 157 (2021), no. 10, 2199–2214. MR 4311556, DOI 10.1112/S0010437X21007491
- R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math. 38 (1979), no. 3, 311–327. MR 535074
- Nolan R. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc. 6 (1993), no. 4, 779–816. MR 1212243, DOI 10.1090/S0894-0347-1993-1212243-2
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