Abstract
Let G be a complex reductive linear algebraic group and \(G_0 \subseteq G\) a real form. Suppose P is a parabolic subgroup of G and assume that P has a Levi factor L such that \(G_0 \cap L = L_0 \) is a real form of L.Using the minimal globalization V min of a finite length admissible representation for L 0, one can define a homogeneous analytic vector bundle on the G 0 orbit S of P in the generalized flag manifold \(Y = G/P\). Let\(\mathcal{A}(P,V_{min} )\) denote the corresponding sheaf of polarized sections. In this article we analyze the G 0 representations obtained on the compactly supported sheaf cohomology groups\(H_c^p (S,\mathcal{A}(P,V_{\min } ))\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beilinson, A. and Bernstein, J.: A Proof of JantzenConjecturs, preprint,MathematicsDepartment, Harvard University.
Ibid., Localization de g modules, C. R. Acad. Sci. Paris. 292 (1981), 15–18.
Borel, A. et al.: Algebraic D-Modules, No. 2 in Perspectives in Mathematics, Academic Press, Inc., 1987.
Bott, R.: Homogeneous vector bundles, Ann. of Math.66 (1957), 203–248.
Casselman,W.: Jacuet modules for real reductive groups, International congress of Math.Helsinki, (1978), 557–563.
Ibid., Canonical extensions of Harish-Chandra modules to representations of G, Can. J. Math., XLI (1989).
Chang, J.: Special k-types, tempered characers and the Beilinson-Bernstein realization, Duke Math. J., 56 (1988), 345–383.
Ibid., Remarks on localization and standard modules: the duality theorem on a generalized flag manifold, Proc. Amer. Math. Soc. 117 (1993), 585–591.
Deligne, P.: Équations Differentielles á Points Singuliers Réguliers, Springer-Verlag, 1973.
Harish-Chandra: Representations of semisimple Lie groups VI, Amer. J Math.78 (1956), 564–628.
Ibid., Harmonic analysis on real reductive groups I, J. Func. Anal. 19 (1975), 104–204.
Hecht, H. and Taylor, J.: Some Remarks on Characters of Semisimple Lie Groups, preprint, Mathematics Department, University of Utah.
Ibid., Analytic localization of group representations, Advances in Math.79 (1990), 139–212.
Ibid., A comparison theorem for n homology, Composito Mathematica, 86 (1993), 189–207.
Hecht, H., Miličić, D., Schmid, W. and Wolf, J.: Localization and standard modules for semisimile Lie groups I: the duality theorem, Invent. Math.90 (1987), 297–332.
Kashiwara M. and Schmid, W.: Quasi-equivariantD-modules, equivariant derived category and representations of reductive Lie groups, Research announcement, Research Institute for Mathematical Sciences, Kyoto University, 1994.
Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan, 31 (1979), 331–357.
Ibid., Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), 307–320.
Miličić, D.: Localization and Representation Theory of Reductive Lie Groups, text in preparation, Mathematics Department, University of Utah.
Pontryagin, L.: Topological Groups, Gordon and Breach Science Publishers, 3rd., 1986.
Sally, P. and Vogan, D.: Ph.D thesis of W. Schmid in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, No. 31 in Mathematical Surveys and Monographs, Amer. Math. Soc., 1989.
Schmid, W.: Boundary value problems for group invariant differential equations, Proc. Cartan Symposium, Astérique, 1985.
Schmid, W. and Wolf, J.: Geometric quantization and derived functor modules for semisimple Lie groups, J. Func. Anal. 90 (1990), 48–112.
Serre, J.: Un théoréme de dualité, Comment. Math. Helv. 29 (1955), 9–26.
Ibid., Géométrie algébraique et géométrie analytique, Ann. Inst. Fourier 6 (1956), 1–42.
Taylor, J.: Czech and Dolbeault, types notes, Mathematics Department, University of Utah.
Varadarajan, V.: Lie Groups, Lie Algebras and their Representations, Springer-Verlag, 1984.
Vogan, D.: Representations of Real Reductive Lie Groups, No. 15 in Progress inMath., Birkhauser, 1981.
Ibid., Unitary Representations of Reductive Lie Groups, No. 118 in Annals of Math. Studies, Princeton Univ. Press, 1987.
Warner, G.: Harmonic Analysis on Semi-Simple Lie Groups, Vol. 1, Springer-Verlag, 1972.
Wong, H.: Dolbeault Cohomologies Associated with Finite Rank Representations, Ph.D. thesis, Harvard University, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
BRATTEN, T. Realizing representations on generalized flag manifolds. Compositio Mathematica 106, 283–319 (1997). https://doi.org/10.1023/A:1000126010326
Issue Date:
DOI: https://doi.org/10.1023/A:1000126010326