A generating function for Blattner’s formula
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- by Jeb F. Willenbring and Gregg J. Zuckerman PDF
- Proc. Amer. Math. Soc. 136 (2008), 2261-2270 Request permission
Abstract:
Let $G$ be a connected, semisimple Lie group with finite center and let $K$ be a maximal compact subgroup. We investigate a method to compute multiplicities of $K$-types in the discrete series using a rational expression for a generating function obtained from Blattner’s formula. This expression involves a product with a character of an irreducible finite-dimensional representation of $K$ and is valid for any discrete series system. Other results include a new proof of a symmetry of Blattner’s formula, and a positivity result for certain low rank examples. We consider in detail the situation for $G$ of type split $\rm G_2$. The motivation for this work came from an attempt to understand pictures coming from Blattner’s formula, some of which we include in the paper.References
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Additional Information
- Jeb F. Willenbring
- Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 0413, Milwaukee, Wisconsin 53201-0413
- MR Author ID: 662347
- Email: jw@uwm.edu
- Gregg J. Zuckerman
- Affiliation: Department of Mathematics, Yale University, P.O. Box 208283; New Haven, Connecticut 06520-8283
- Email: gregg.zuckerman@yale.edu
- Received by editor(s): April 26, 2007
- Published electronically: January 28, 2008
- Additional Notes: The first author was supported in part by NSA Grant # H98230-05-1-0078.
- Communicated by: Gail R. Letzter
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2261-2270
- MSC (2000): Primary 22E46; Secondary 17B10
- DOI: https://doi.org/10.1090/S0002-9939-08-09284-8
- MathSciNet review: 2383533