Cramped subgroups and generalized Harish-Chandra modules

Author:
Ben Webster

Journal:
Proc. Amer. Math. Soc. **136** (2008), 3809-3814

MSC (2000):
Primary 17B20; Secondary 53D20

Published electronically:
June 9, 2008

MathSciNet review:
2425719

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a reductive complex Lie group with Lie algebra . We call a subgroup **cramped** if there is an integer such that each finite-dimensional representation of has a non-trivial invariant subspace of dimension less than . We show that a subgroup is cramped if and only if the moment map is surjective, where and are compact forms of and . We will use this in conjunction with sufficient conditions for crampedness given by Willenbring and Zuckerman (2004) to prove a geometric proposition on the intersections between adjoint orbits and Killing orthogonals to subgroups.

We will also discuss applications of the techniques of symplectic geometry to the generalized Harish-Chandra modules introduced by Penkov and Zuckerman (2004), of which our results on crampedness are special cases.

**[CdS01]**Ana Cannas da Silva,*Lectures on symplectic geometry*, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR**1853077****[GLS96]**Victor Guillemin, Eugene Lerman, and Shlomo Sternberg,*Symplectic fibrations and multiplicity diagrams*, Cambridge University Press, Cambridge, 1996. MR**1414677****[Kob02]**Toshiyuki Kobayashi,*Branching problems of unitary representations*, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 615–627. MR**1957069****[MFK94]**D. Mumford, J. Fogarty, and F. Kirwan,*Geometric invariant theory*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR**1304906****[PZ04]**Ivan Penkov and Gregg Zuckerman,*Generalized Harish-Chandra modules: a new direction in the structure theory of representations*, Acta Appl. Math.**81**(2004), no. 1-3, 311–326. MR**2069343**, 10.1023/B:ACAP.0000024204.22996.2c**[WZ04]**Jeb F. Willenbring and Gregg Zuckerman.

Small semisimple subalgebras of semisimple Lie algebras, 2004, arXiv:math.RT/0408302.

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Additional Information

**Ben Webster**

Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, California 94720

Address at time of publication:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 09540

Email:
bwebste@ias.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09421-5

Received by editor(s):
December 16, 2006

Received by editor(s) in revised form:
September 23, 2007

Published electronically:
June 9, 2008

Additional Notes:
This material is based upon work supported under a National Science Foundation Graduate Research Fellowship and partially supported by the RTG grant DMS-0354321.

Communicated by:
Dan M. Barbasch

Article copyright:
© Copyright 2008
American Mathematical Society