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Cohomology of standard modules on partial flag varieties


Author: S. N. Kitchen
Journal: Represent. Theory 16 (2012), 317-344
MSC (2010): Primary 22-xx
DOI: https://doi.org/10.1090/S1088-4165-2012-00419-8
Published electronically: July 11, 2012
MathSciNet review: 2945222
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Abstract: Cohomological induction gives an algebraic method for constructing representations of a real reductive Lie group $ G$ from irreducible representations of reductive subgroups. Beilinson-Bernstein localization alternatively gives a geometric method for constructing Harish-Chandra modules for $ G$ from certain representations of a Cartan subgroup. The duality theorem of Hecht, Miličić, Schmid and Wolf establishes a relationship between modules cohomologically induced from minimal parabolics and the cohomology of the $ \mathscr {D}$-modules on the complex flag variety for $ G$ determined by the Beilinson-Bernstein construction. The main results of this paper give a generalization of the duality theorem to partial flag varieties, which recovers cohomologically induced modules arising from nonminimal parabolics.


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  • [BB1] Alexandre Beilinson and Joseph Bernstein, Localisation de g-modules, C. R. Acad. Sci.  Paris Ser.  I Math.  292 (1981), no. 1, 15-18. MR 610137 (82k:14015)
  • [BB2] ----, 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 (95a:22022)
  • [BL] 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 (95m:17004)
  • [Bi] Fredric Bien, $ D$-modules and spherical representations, Mathematical Notes, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1082342 (92f:22025)
  • [Bo] A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic D-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000 (89g:32014)
  • [HMSW] 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 (89e:22025)
  • [HP] Jing-Song Huang and Pavle Pandžić, Dirac operators in representation theory, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2244116 (2007j:22025)
  • [K] S. N. Kitchen, Localization of cohomologically induced modules to partial flag varieties, University of Utah thesis, University of Utah, Salt Lake City, UT, 2010. MR 2736734
  • [KV] Anthony W. Knapp and David A. Vogan, Jr., Cohomological induction and unitary representations, Princeton Matematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR 1330919 (96c:22023)
  • [M1] Dragan Miličić, Lectures on algebraic theory of $ \mathscr {D}$-modules, Course lecture notes, University of Utah, http://www.math.utah.edu/$ \sim $milicic.
  • [M2] ----, Localization and representation theory of reductive Lie groups, in progress, http://www.math.utah.edu/$ \sim $milicic.
  • [M3] ----, Algebraic $ \mathscr {D}$-modules and representation theory of semisimple Lie groups, The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992), Contemp. Math., vol. 154, Amer. Math. Soc., Providence, RI, 1993, pp. 133-168. MR 1246382 (94i:22035)
  • [MP] Dragan Miličić and Pavle Pandžić, Equivariant derived categories, Zuckerman functors and localization, Geometry and representation theory of real and $ p$-adic groups (Cordoba, 1995), Progr. Math., vol. 158, Birkhäuser Boston, Boston, MA, 1998, pp. 209-242. MR 1486143 (2000f:22018)
  • [P] Pavle Pandžić, Zuckerman functors between equivariant derived categories, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2191-2220 (electronic). MR 2276617 (2008a:22016)
  • [S] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121-154. MR 932640 (89m:18013)

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Additional Information

S. N. Kitchen
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, 79104 Freiburg im Breisgau, Germany
Email: sarah.kitchen@math.uni-freiburg.de

DOI: https://doi.org/10.1090/S1088-4165-2012-00419-8
Received by editor(s): February 7, 2011
Received by editor(s) in revised form: January 20, 2012, and February 24, 2012
Published electronically: July 11, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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