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Proceedings of the American Mathematical Society

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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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Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak g$. We call a subgroup $H\subset G$ cramped if there is an integer $b(G,H)$ such that each finite-dimensional representation of $G$ has a non-trivial invariant subspace of dimension less than $b(G,H)$. We show that a subgroup is cramped if and only if the moment map $T^*(K/L)\to \mathfrak k^*$ is surjective, where $K$ and $L$ are compact forms of $G$ and $H$. 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.

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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
MR Author ID: 794563

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