Zuckerman functors between equivariant derived categories

Pandžić, Pavle

doi:10.1090/S0002-9947-06-04013-X

Zuckerman functors between equivariant derived categories
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by Pavle Pandžić PDF

Trans. Amer. Math. Soc. 359 (2007), 2191-2220 Request permission

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

A. Beilinson, J. Bernstein, A proof of the Jantzen conjecture, (preprint), M.I.T. and Harvard University (1989).
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
Joseph Bernstein and Valery Lunts, Localization for derived categories of $({\mathfrak {g}},K)$-modules, J. Amer. Math. Soc. 8 (1995), no. 4, 819–856. MR 1317229, DOI 10.1090/S0894-0347-1995-1317229-7
Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 1 à 3, Hermann, Paris, 1970 (French). MR 0274237
Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. MR 0354654
Michel Duflo and Michèle Vergne, Sur le foncteur de Zuckerman, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 16, 467–469 (French, with English summary). MR 894570
T. J. Enright and N. R. Wallach, Notes on homological algebra and representations of Lie algebras, Duke Math. J. 47 (1980), no. 1, 1–15. MR 563362
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original. MR 1438306, DOI 10.1007/978-3-662-03220-6
V. A. Ginzburg, Equivariant cohomology and Kähler geometry, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 19–34, 96 (Russian). MR 925070
Henryk Hecht, Dragan Miličić, Wilfried Schmid, and Joseph A. Wolf, Localization and standard modules for real semisimple Lie groups. I. The duality theorem, Invent. Math. 90 (1987), no. 2, 297–332. MR 910203, DOI 10.1007/BF01388707
G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603. MR 54581, DOI 10.2307/1969740
Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006, DOI 10.1007/978-3-662-02661-8
Anthony W. Knapp and David A. Vogan Jr., Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR 1330919, DOI 10.1515/9781400883936
D. Miličić, Lectures on derived categories, http://www.math.utah.edu/$^\sim$milicic/ dercat.pdf.
Dragan Miličić and Pavle Pandžić, Equivariant derived categories, Zuckerman functors and localization, Geometry and representation theory of real and $p$-adic groups (Córdoba, 1995) Progr. Math., vol. 158, Birkhäuser Boston, Boston, MA, 1998, pp. 209–242. MR 1486143
D. Miličić, P. Pandžić, Cohomology of standard Harish-Chandra sheaves, (in preparation), University of Utah and University of Zagreb.
P. Pandžić, Equivariant analogues of Zuckerman functors, Ph.D. thesis, University of Utah, 1995.
Pavle Pandžić, A simple proof of Bernstein-Lunts equivalence, Manuscripta Math. 118 (2005), no. 1, 71–84. MR 2171292, DOI 10.1007/s00229-005-0580-3
Manfred Scheunert, The theory of Lie superalgebras, Lecture Notes in Mathematics, vol. 716, Springer, Berlin, 1979. An introduction. MR 537441
N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), xii+253 pp. (1997) (French, with French summary). With a preface by Luc Illusie; Edited and with a note by Georges Maltsiniotis. MR 1453167
David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51–90. MR 762307
Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
G. J. Zuckerman, Lecture Series “Construction of representations via derived functors", Institute for Advanced Study, Princeton, N.J., Jan.–Mar., 1978.