Abstract
In this note we draw consequences of theorems of Kashiwara–Schmid, Casselman, and Schneider–Stuhler. Canonical globalizations of Harish–Chandra modules can be considered as coefficient modules for cohomology groups with respect to cocompact discrete subgroups or nilpotent Lie algebras. We obtain finiteness and comparison theorems for these cohomology groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borel, A. and Serre, J. P.: Cohomologie d'immeubles et de groupes S-arithmétiques, Topology 15 (1976), 211–232.
Borel, A. and Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Princeton University Press, Princeton, NJ, 1980.
Bratten, T.: A comparison theorem for Lie algebra homology groups, Preprint, 1995.
Bunke, U. and Olbrich, M.: Gamma-cohomology and the Selberg zeta function, J. reine angew. Math. 467 (1995), 199–219.
Bunke, U. and Olbrich, M.: Resolutions of distribution globalizations of Harish-Chandra modules and cohomology, J. reine angew. Math. (to appear).
Deitmar, A.: On some zeta functions attached to compact Shimura manifolds, Preprint, 1995.
Franke, J.: Harmonic analysis in weighted L 2-spaces, Preprint, 1995.
Hecht, H. and Schmid, W.: Characters, asymptotics, and n-homology of Harish-Chandra modules, Acta Math. 151 (1983), 49–151.
Hecht, H. and Taylor, J. L.: A comparison theorem for n-homology, Compositio Math. 86 (1993), 189–207.
Kashiwara, M. and Schmid, W.: Quasi-equivariant \(\mathcal{D}\)-modules, equivariant derived category, and representations of reductive Lie groups, in Brylinski, J. L., Brylinski, R., Guillemin, V. and Kac, V. (eds.), Lie Theory and Geometry. In Honor of Bertram Kostant, Progr. in Math., Vol. 123, Birkhäuser, Boston, 1994, pp. 457–488.
Knapp, A. W.: Lie Groups, Lie Algebras, and Cohomology, Princeton University Press, Princeton, NJ, 1988.
Schmid, W.: Boundary value problems for group invariant differential equations, Soc. Math. de France, Asterisque, hors série, 1985, 311–321.
Schneider, P. and Stuhler, U.: Representation theory and sheaves on the Bruhat-Tits building, Preprint, 1995.
Wallach, N. R.: Real Reductive Groups, Academic Press, San Diego, CA, 1988.
Wallach, N. R.: Real Reductive Groups II, Academic Press, San Diego, CA, 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bunke, U., Olbrich, M. Cohomological Properties of the Canonical Globalizations of Harish–Chandra Modules. Annals of Global Analysis and Geometry 15, 401–418 (1997). https://doi.org/10.1023/A:1006538211648
Issue Date:
DOI: https://doi.org/10.1023/A:1006538211648