Cramped subgroups and generalized Harish-Chandra modules
HTML articles powered by AMS MathViewer
- by Ben Webster PDF
- Proc. Amer. Math. Soc. 136 (2008), 3809-3814 Request permission
Abstract:
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.
References
- Ana Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR 1853077, DOI 10.1007/978-3-540-45330-7
- Victor Guillemin, Eugene Lerman, and Shlomo Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge, 1996. MR 1414677, DOI 10.1017/CBO9780511574788
- 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
- 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, DOI 10.1007/978-3-642-57916-5
- 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, DOI 10.1023/B:ACAP.0000024204.22996.2c
- Jeb F. Willenbring and Gregg Zuckerman. Small semisimple subalgebras of semisimple Lie algebras, 2004, arXiv:math.RT/0408302.
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
- Email: bwebste@ias.edu
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 3809-3814
- MSC (2000): Primary 17B20; Secondary 53D20
- DOI: https://doi.org/10.1090/S0002-9939-08-09421-5
- MathSciNet review: 2425719