Electronic Only Electronic Research Announcements
Representation Theory
ISSN 1088-4165
Previous volume | Table of contents | Next volume
Articles in press | Most recent volume | All volumes
Next Article
     

Wall-crossing functors and ${\mathcal{D}}$-modules

Author(s): Alexander Beilinson; Victor Ginzburg
Journal: Represent. Theory 3 (1999), 1-31.
MSC (1991): Primary 05E99, 17B37
Posted: January 11, 1999
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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:

[BBD]
A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Astérisque 100 (1982). MR 86g:32015

[BB1]
A. Beilinson, J. Bernstein, Localisation de ${\mathfrak{g}}$-modules, C.R. Acad. Sci. Paris, Sér. 1 292 (1981), 15-18. MR 82k:14015

[BB2]
A. Beilinson, J. Bernstein, A generalization of Casselman's submodule theorem. Birkhäuser, Progress in Math. 40, Boston, 1983, pp. 35-52. MR 85e:22024

[BB3]
A. Beilinson, J. Bernstein, Proof of Jantzen conjectures, Advances in Soviet Math. 16, Gelfand Seminar, 1983, pp. 1-50. MR 95a:22022

[BG]
A. Beilinson and V. Ginzburg, Mixed categories, Ext-duality and representations (results and conjectures), Preprint, Moscow (1986).

[BGS]
A. Beilinson, V. Ginzburg, and V. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527. MR 96k:17010

[Be]
J. Bernstein, Trace in categories, in: Proceedings of the Dixmier Colloquium, Birkhäuser, Progress in Math. 92, Boston, 1990, pp. 417-423. MR 92d:17010

[BGG]
J. Bernstein, I. Gelfand, and S. Gelfand, A certain category of ${\mathfrak{g}}$-modules, Funct. Anal. and its Appl. 10 (1976), 1-8. MR 53:10880

[BeGe]
J. Bernstein, S. Gelfand, Tensor products of finite and infinite dimensional representations of semisimple Lie algebras, Comp. Math. 41 (1980), 245-285. MR 82c:17003

[BeLu]
J. Bernstein, V. Lunts, Equivariant sheaves and functors, Springer Lecture Notes in Math. 1578 (1994). MR 95k:55012

[BoBr]
W. Borho, J.-L. Brylinski, Differential operators on flag manifolds II, MPI Preprint (1989).

[CG]
N. Chriss, V. Ginzburg, Representation theory and complex geometry, Birkhäuser, Boston, 1997. MR 98i:22021

[FM]
M. Finkelberg, I. Mirkovic, Semi-infinite flags I. Case of global curve ${\mathbb{P}}^{1}$, Preprint 1997, alg-geom/9707010.

[FrMa]
I. Frenkel, F. Malikov, Kazhdan-Lusztig tensoring and Harish-Chandra categories, Preprint 1997, q-alg/9703010.

[Gi1]
V. Ginzburg, $\mathfrak{g}$-modules, Springer representations, and bivariant Chern classes, Adv. Math. 61 (1986), 1-48.

[Gi2]
V. Ginzburg, Perverse sheaves and ${\mathbb{C}}^{*}$-actions, J. Amer. Math. Soc. 4 (1991), 483-490. MR 92d:14013

[Ja]
J. C. Jantzen, Moduln mit Einem Höchsten Gewicht, Lecture Notes in Math. 750 (1979). MR 81m:17011

[Ka]
M. Kashiwara, Representation theory and $D$-modules, in: Orbites Unipotentes et Représentations, III, Astérisque 173-174 (1989), pp. 1-31. MR 90k:17029

[KT1]
M. Kashiwara, T. Tanisaki, The Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody algebras, in: The Grothendieck Festschrift, II, Birkhäuser, Progress in Mathem. 87, Boston, 1990, pp. 407-433. MR 93a:17026

[KT2]
M. Kashiwara, T. Tanisaki, The Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J. 77 (1995), 21-62. MR 96j:17016

[Ko]
B. Kostant, On the tensor product of a finite and an infinite dimensional representation, Journ. Funct. Anal. 20 (1975), 257-285. MR 54:2888

[LW]
J. Lepowsky, N. Wallach, Finite and infinite-dimensional representations of linear semisimple groups, Trans. Amer. Math. Soc. 184 (1973), 223-246. MR 48:6320

[S1]
W. Soergel, Kategorie ${\mathcal{O}}$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445. MR 91e:17007

[S2]
W. Soergel, Charakterformeln Für Kipp-Moduln Über Kac-Moody-Algebren, Representation Theory (Electronic Journal of the AMS) 1 (1997), 115-132. MR 98f:17016

[S3]
W. Soergel, The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math. 429 (1992), 49-74 (see also: Langlands' philosophy and Koszul duality, Preprint, 1992). MR 94b:17011

[W]
N. Wallach, Cyclic vectors and irreducibility for principal series representations, Trans. Amer. Math. Soc. 158 (1971), 107-113. MR 43:7558

[Z]
G. Zuckerman, Tensor products of finite and infinite-dimensional representations of semisimple Lie groups, Annals Math. 106 (1977), 295-308. MR 56:15841

Similar Articles:

Retrieve articles in Representation Theory with MSC (1991): 05E99, 17B37

Retrieve articles in all Journals with MSC (1991): 05E99, 17B37

Additional Information:

Alexander Beilinson
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: sasha@math.uchicago.edu

Victor Ginzburg
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: ginzburg@math.uchicago.edu

DOI: 10.1090/S1088-4165-99-00063-1
PII: S 1088-4165(99)00063-1
Posted: January 11, 1999
Copyright of article: Copyright 1999, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google