Skip Navigation

International Mathematics Research Papers (2006) Vol. 2006 : article ID 26439, 54 pages, doi:10.1155/IMRP/2006/26439
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Gan, W. L.
Right arrow Articles by Ginzburg, V.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Almost-commuting variety, D-modules, and Cherednik algebras

Wee Liang Gan and Victor Ginzburg

Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA E-mail address: wlgan{at}math.mit.edu
Department of Mathematics, University of Chicago Chicago, IL 60637, USA E-mail address: ginzburg{at}math.uchicago.edu (with an Appendix by Victor Ginzburg)

We study a scheme M closely related to the set of pairs of nxn matrices with rank-1 commutator. We show that M is a reduced complete intersection with n + 1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Mnil sub M. We introduce a category C of D-modules whose characteristic variety is contained in Mnil. Simple objects of that category are analogous to Lusztig's character sheaves. We construct an exact functor of quantum Hamiltonian reduction from our category C to the category O for type-A rational Cherednik algebra.



References

  1. Beilinson A., Bernstein J. Localisation de g-modules. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique (1981) 292(1):15–18.
  2. Beilinson A., Bernstein J. A proof of Jantzen conjectures. In: I. M. Gel'fand Seminar (1993) 16, Part 1. Rhode Island: American Mathematical Society. 1–50. Advances in Soviet Mathematics.
  3. Berest Yu., Etingof P., Ginzburg V. Cherednik algebras and differential operators on quasi-invariants. Duke Mathematical Journal (2003) 118(2):279–337.[CrossRef][Web of Science]
  4. Bezrukavnikov R., Finkelberg M., Ginzburg V. Cherednik algebras and Hilbert schemes in characteristic p. with an appendix by P. Etingof, to appear in Repr. Theory, http://xxx.sf.nchc.gov.tw/abs/math.RT/0312474.
  5. Broer A. Lectures on decomposition classes. In: NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences (1998) 514. Representation Theories and Algebraic Geometry, 1997: Montreal, PQ. Dordrecht: Kluwer Academic. 39–83.
  6. Brylinski J.-L. Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques [Canonical transformations, projective duality, Lefschetz theory, Fourier transforms and trigonometric sums]. Astérisque (1986) (140–141):3–134, 251.
  7. Chriss N., Ginzburg V. Representation Theory and Complex Geometry (1997) Massachusetts: Birkhäuser Boston. x+495.
  8. Crawley-Boevey W. Geometry of the moment map for representations of quivers. Compositio Mathematica (2001) 126(3):257–293.[CrossRef][Web of Science]
  9. Eisenbud D. Commutative Algebra. with a View Toward Algebraic Geometry. In: Graduate Texts in Mathematics (1995) 150. New York: Springer. xvi+785.
  10. Etingof P., Gan W. L., Ginzburg V., Oblomkov A. Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products. preprint, 2005, http://xxx.sf.nchc.gov.tw/abs/math.RT/0511489.
  11. Etingof P., Ginzburg V. Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Inventiones Mathematicae (2002) 147(2):243–348.[CrossRef][Web of Science]
  12. Finkelberg M., Ginzburg V. Character sheaves for Cherednik algebras. in preparation.
  13. Ginzburg V. Intégrales sur les orbites nilpotentes et représentations des groupes de Weyl [Nilpotent orbital integrals and representations of Weyl groups]. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique (1983) 296(5):249–252.
  14. Ginzburg V. Admissible modules on a symmetric space. Astérisque (1989) (173–174):9–10, 199–255.
  15. Gordon I., Stafford J. T. Rational Cherednik algebras and Hilbert schemes. to appear in Advances in Mathematics, http://xxx.sf.nchc.gov.tw/abs/math.RA/0407516.
  16. Gordon I., Stafford J. T. Rational Cherednik algebras and Hilbert schemes II: representations and sheaves. to appear in Duke Mathematical Journal, http://xxx.sf.nchc.gov.tw/abs/math.RT/0410293.
  17. Grothendieck A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Institut des Hautes Études Scientifiques. Publications Mathématiques (1965) (24):5–231.
  18. Guralnick R. M. A note on pairs of matrices with rank one commutator. Linear and Multilinear Algebra (1979/1980) 8(2):97–99.[CrossRef]
  19. Haiman M. t,q-Catalan numbers and the Hilbert scheme. Selected papers in honor of Adriano Garsia (Taormina, 1994). Discrete Mathematics (1998) 193(1–3):201–224.[CrossRef][Web of Science]
  20. Haiman M. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. Journal of the American Mathematical Society (2001) 14(4):941–1006.[CrossRef][Web of Science]
  21. Hotta R., Kashiwara M. The invariant holonomic system on a semisimple Lie algebra. Inventiones Mathematicae (1984) 75(2):327–358.[CrossRef][Web of Science]
  22. Kac V. G. Root systems, representations of quivers and invariant theory. In: Lecture Notes in Mathematics (1983) 996. Invariant Theory, 1982: Montecatini. Berlin: Springer. 74–108.[Web of Science]
  23. Kazhdan D., Kostant B., Sternberg S. Hamiltonian group actions and dynamical systems of Calogero type. Communications on Pure and Applied Mathematics (1978) 31(4):481–507.[CrossRef][Web of Science]
  24. Kraft H. Geometrische Methoden in der Invariantentheorie [Geometrical Methods in Invariant Theory]. In: Aspects of Mathematics (1984) D1. Braunschweig: Friedr. Vieweg & Sohn. x+308.
  25. Kraft H., Riedtmann Ch. Geometry of representations of quivers. In: London Mathematical Society Lecture Note Series (1986) 116. Representations of Algebras, 1985: Durham. Cambridge: Cambridge University Press. 109–145.
  26. Lusztig G. Intersection cohomology complexes on a reductive group. Inventiones Mathematicae (1984) 75(2):205–272.[CrossRef][Web of Science]
  27. Lusztig G. Character sheaves. I. Advances in Mathematics (1985) 56(3):193–237.[CrossRef][Web of Science]
  28. Lusztig G. Character sheaves. II. Advances in Mathematics (1985) 57(3):226–265.[CrossRef][Web of Science]
  29. Lusztig G. Character sheaves. III. Advances in Mathematics (1985) 57(3):266–315.[CrossRef][Web of Science]
  30. Lusztig G. Character sheaves. IV. Advances in Mathematics (1986) 59(1):1–63.[CrossRef][Web of Science]
  31. Lusztig G. Character sheaves. V. Advances in Mathematics (1986) 61(2):103–155.[CrossRef][Web of Science]
  32. Lusztig G. Quivers, perverse sheaves, and quantized enveloping algebras. Journal of the American Mathematical Society (1991) 4(2):365–421.[CrossRef]
  33. Nakajima H. Quiver varieties and Kac-Moody algebras. Duke Mathematical Journal (1998) 91(3):515–560.[CrossRef][Web of Science]
  34. Nakajima H. Lectures on Hilbert Schemes of Points on Surfaces. In: University Lecture Series (1999) 18. Rhode Island: American Mathematical Society. xii+132.
  35. Neubauer M. G. The variety of pairs of matrices with rank (AB – BA) ≤ 1. Proceedings of the American Mathematical Society (1989) 105(4):787–792.[CrossRef][Web of Science]
  36. Richardson R. W. Commuting varieties of semisimple Lie algebras and algebraic groups. Compositio Mathematica (1979) 38(3):311–327.[Web of Science]
  37. Schofield A. General representations of quivers. Proceedings of the London Mathematical Society. Third Series (1992) 65(1):46–64.
  38. Wilson G. Collisions of Calogero-Moser particles and an adelic Grassmannian. Inventiones Mathematicae (1998) 133(1):1–41. with an appendix by I. G. Macdonald.[CrossRef][Web of Science]

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Gan, W. L.
Right arrow Articles by Ginzburg, V.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?