Cohomology of standard modules on partial flag varieties

Kitchen, S.

doi:10.1090/S1088-4165-2012-00419-8

Cohomology of standard modules on partial flag varieties
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by S. N. Kitchen PDF

Represent. Theory 16 (2012), 317-344 Request permission

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.

References

Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
A. Beĭlinson and J. Bernstein, 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
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
Frédéric V. Bien, ${\scr D}$-modules and spherical representations, Mathematical Notes, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1082342, DOI 10.1515/9781400862078
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
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
Jing-Song Huang and Pavle Pandžić, Dirac operators in representation theory, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2244116
Sarah Noelle Kitchen, Localization of cohomologically induced modules to partial flag varieties, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–The University of Utah. MR 2736734
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
Dragan Miličić, Lectures on algebraic theory of $\mathscr {D}$-modules, Course lecture notes, University of Utah, http://www.math.utah.edu/$\sim$milicic.
——–, Localization and representation theory of reductive Lie groups, in progress, http://www.math.utah.edu/$\sim$milicic.
Dragan Miličić, Algebraic ${\scr D}$-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, DOI 10.1090/conm/154/01361
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
Pavle Pandžić, Zuckerman functors between equivariant derived categories, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2191–2220. MR 2276617, DOI 10.1090/S0002-9947-06-04013-X
N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640