Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Zuckerman functors between equivariant derived categories


Author: Pavle Pandzic
Journal: Trans. Amer. Math. Soc. 359 (2007), 2191-2220
MSC (2000): Primary 22E46
DOI: https://doi.org/10.1090/S0002-9947-06-04013-X
Published electronically: December 19, 2006
MathSciNet review: 2276617
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We review the Beilinson-Ginzburg construction of equivariant derived categories of Harish-Chandra modules, and introduce analogues of Zuckerman functors in this setting. They are given by an explicit formula, which works equally well in the case of modules with a given infinitesimal character. This is important if one wants to apply Beilinson-Bernstein localization. We also show how to recover the usual Zuckerman functors from the equivariant ones by passing to cohomology.


References [Enhancements On Off] (What's this?)

  • [BB] A. Beilinson, J. Bernstein, A proof of the Jantzen conjecture, (preprint), M.I.T. and Harvard University (1989).
  • [BL1] J. Bernstein, V. Lunts, Equivariant sheaves and functors, Lecture Notes in Math., vol. 1578, Springer-Verlag, 1994. MR 1299527 (95k:55012)
  • [BL2] J. Bernstein, V. Lunts, Localization for derived categories of $ (\mathfrak{g},K)$-modules, J. Amer. Math. Soc. 8 No. 4 (1995), 819-856. MR 1317229 (95m:17004)
  • [BW] A. Borel, N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Math. Studies, vol. 94, Princeton University Press, Princeton, 1980. MR 0554917 (83c:22018)
  • [Bo] N. Bourbaki, Algèbre, Ch. 1 à 3, Bourbaki, Paris, 1970. MR 0274237 (43:2)
  • [De] P. Deligne, Cohomologie à supports propres, SGA 4, Lecture Notes in Math., vol. 305, Springer-Verlag, Berlin, Heidelberg, 1973. MR 0354654 (50:7132)
  • [DV] M. Duflo, M. Vergne, Sur les functeurs de Zuckerman, C. R. Acad. Sci. Paris 304 (1987), 467-469. MR 0894570 (89h:22025)
  • [EW] T.J. Enright, N.R. Wallach, Notes on homological algebra and representations of Lie algebras, Duke Math. J. 47 (1980), 1-15. MR 0563362 (81c:17013)
  • [GM] S. I. Gelfand, Yu.I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, Heidelberg, New York, 1996. MR 1438306 (97j:18001)
  • [G] V. A. Ginzburg, Equivariant cohomology and Kähler geometry, (Russian), Funktsional. Anal. i Prilozhen. 21 no. 4 (1987), 19-34. MR 0925070 (89b:58013)
  • [HMSW] H. Hecht, D. Milicic, W. Schmid, J.A. Wolf, Localization and standard modules for real semisimple Lie groups I: The duality theorem, Inventiones Math. 90 (1987), 297-332. MR 0910203 (89e:22025)
  • [HS] G. Hochschild, J.P. Serre, Cohomology of Lie algebras, Annals of Math. 57 (1953), 591-603. MR 0054581 (14:943c)
  • [Il] L. Illusie, Complexe cotangent et déformations I, Lecture Notes in Math., vol. 239, 1971; II, Lecture Notes in Math., vol. 283, Springer-Verlag, Berlin, Heidelberg, 1972. MR 0491680 (58:10886a); MR 0491681 (58:10886b)
  • [KS] M. Kashiwara, P. Schapira, Sheaves on manifolds, Springer-Verlag, Berlin, Heidelberg, 1990.MR 1074006 (92a:58132)
  • [KV] A.W. Knapp, D.A. Vogan, Cohomological induction and unitary representations, Princeton University Press, Princeton, 1995. MR 1330919 (96c:22023)
  • [M] D. Milicic, Lectures on derived categories, http://www.math.utah.edu/$ ^\sim$milicic/ dercat.pdf.
  • [MP1] D. Milicic, P. Pandzic, Equivariant derived categories, Zuckerman functors and localization, Geometry and representation theory of real and $ p$-adic Lie groups (J. Tirao, D. Vogan, J. A. Wolf, eds.), Progress in Mathematics 158, Birkhäuser, Boston, 1996, pp. 209-242.MR 1486143 (2000f:22018)
  • [MP2] D. Milicic, P. Pandzic, Cohomology of standard Harish-Chandra sheaves, (in preparation), University of Utah and University of Zagreb.
  • [P1] P. Pandzic, Equivariant analogues of Zuckerman functors, Ph.D. thesis, University of Utah, 1995.
  • [P2] P. Pandzic, A simple proof of Bernstein-Lunts equivalence, Manuscripta Math. 118 (2005), 71-84. MR 2171292
  • [Sch] M. Schenuert, The theory of Lie superalgebras, Lecture Notes in Math., vol. 716, Springer-Verlag, Berlin, Heidelberg, 1979. MR 0537441 (80i:17005)
  • [Sp] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), 121-154.MR 0932640 (89m:18013)
  • [Ve1] J.L. Verdier, Catégories dérivées, état 0, SGA 4 $ {\frac{1}{2}}$, Lecture Notes in Math., vol. 569, Springer-Verlag, 1977.MR 0463174 (57:3132)
  • [Ve2] J.L. Verdier, Des catégories dérivées des catégories abéliennes, with a preface by Luc Illusie; edited and with a note by Georges Maltsiniotis, Astérisque 239 (1996). MR 1453167 (98c:18007)
  • [Vo] D.A. Vogan, Representations of real reductive Lie groups, Birkhäuser, Boston, 1981.MR 0632407 (83c:22022)
  • [VZ] D.A. Vogan, G.J. Zuckerman, Unitary representations with non-zero cohomology, Compositio Math. 53 (1984), 51-90. MR 0762307 (86k:22040)
  • [W] N.R. Wallach, Real reductive Groups I, Academic Press, Boston, 1988.MR 0929683 (89i:22029)
  • [Z] G. J. Zuckerman, Lecture Series ``Construction of representations via derived functors", Institute for Advanced Study, Princeton, N.J., Jan.-Mar., 1978.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 22E46

Retrieve articles in all journals with MSC (2000): 22E46


Additional Information

Pavle Pandzic
Affiliation: Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia
Email: pandzic@math.hr

DOI: https://doi.org/10.1090/S0002-9947-06-04013-X
Received by editor(s): January 15, 2004
Received by editor(s) in revised form: March 9, 2005
Published electronically: December 19, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society