Wall-crossing functors and $\mathcal {D}$-modules
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- by Alexander Beilinson and Victor Ginzburg
- Represent. Theory 3 (1999), 1-31
- DOI: https://doi.org/10.1090/S1088-4165-99-00063-1
- Published electronically: January 11, 1999
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Abstract:
We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using ${\mathcal {D}}$-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem; two important results were established earlier by W. Soergel in a totally different way. Other applications to the category ${\mathcal {O}}$ of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.References
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
- Alexander Beĭlinson and Joseph Bernstein, A generalization of Casselman’s submodule theorem, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 35–52. MR 733805, DOI 10.1007/978-1-4684-6730-7_{3}
- A. Beĭlinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1–50. MR 1237825, DOI 10.1090/advsov/016.1/01
- A. Beilinson and V. Ginzburg, Mixed categories, Ext-duality and representations (results and conjectures), Preprint, Moscow (1986).
- Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
- Joseph Bernstein, Trace in categories, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 417–423. MR 1103598
- I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, A certain category of ${\mathfrak {g}}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097, DOI 10.1007/BF01077933
- J. N. Bernstein and S. I. Gel′fand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245–285. MR 581584
- Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527, DOI 10.1007/BFb0073549
- W. Borho, J.-L. Brylinski, Differential operators on flag manifolds II, MPI Preprint (1989).
- Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
- M. Finkelberg, I. Mirkovic, Semi-infinite flags I. Case of global curve ${\mathbb {P}}^{1}$, Preprint 1997, alg-geom/9707010.
- I. Frenkel, F. Malikov, Kazhdan-Lusztig tensoring and Harish-Chandra categories, Preprint 1997, q-alg/9703010.
- V. Ginzburg, $\mathfrak {g}$-modules, Springer representations, and bivariant Chern classes, Adv. Math. 61 (1986), 1–48.
- Victor Ginsburg, Perverse sheaves and $\textbf {C}^*$-actions, J. Amer. Math. Soc. 4 (1991), no. 3, 483–490. MR 1091465, DOI 10.1090/S0894-0347-1991-1091465-6
- Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943, DOI 10.1007/BFb0069521
- Masaki Kashiwara, Representation theory and $D$-modules on flag varieties, Astérisque 173-174 (1989), 9, 55–109. Orbites unipotentes et représentations, III. MR 1021510
- Masaki Kashiwara, Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 407–433. MR 1106905, DOI 10.1007/978-0-8176-4575-5_{1}0
- Masaki Kashiwara and Toshiyuki Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J. 77 (1995), no. 1, 21–62. MR 1317626, DOI 10.1215/S0012-7094-95-07702-3
- Bertram Kostant, On the tensor product of a finite and an infinite dimensional representation, J. Functional Analysis 20 (1975), no. 4, 257–285. MR 0414796, DOI 10.1016/0022-1236(75)90035-x
- James Lepowsky and Nolan R. Wallach, Finite- and infinite-dimensional representation of linear semisimple groups, Trans. Amer. Math. Soc. 184 (1973), 223–246. MR 327978, DOI 10.1090/S0002-9947-1973-0327978-7
- Wolfgang Soergel, Kategorie $\scr O$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445 (German, with English summary). MR 1029692, DOI 10.1090/S0894-0347-1990-1029692-5
- Wolfgang Soergel, Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren, Represent. Theory 1 (1997), 115–132. MR 1445716, DOI 10.1090/S1088-4165-97-00017-4
- Wolfgang Soergel, The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math. 429 (1992), 49–74. MR 1173115, DOI 10.1515/crll.1992.429.49
- Nolan R. Wallach, Cyclic vectors and irreducibility for principal series representations, Trans. Amer. Math. Soc. 158 (1971), 107–113. MR 281844, DOI 10.1090/S0002-9947-1971-0281844-2
- Gregg Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. of Math. (2) 106 (1977), no. 2, 295–308. MR 457636, DOI 10.2307/1971097
Bibliographic Information
- Alexander Beilinson
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 33735
- Email: sasha@math.uchicago.edu
- Victor Ginzburg
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- Email: ginzburg@math.uchicago.edu
- Published electronically: January 11, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Represent. Theory 3 (1999), 1-31
- MSC (1991): Primary 05E99, 17B37
- DOI: https://doi.org/10.1090/S1088-4165-99-00063-1
- MathSciNet review: 1659527