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A generating function for Blattner's formula

Authors: Jeb F. Willenbring and Gregg J. Zuckerman
Journal: Proc. Amer. Math. Soc. 136 (2008), 2261-2270
MSC (2000): Primary 22E46; Secondary 17B10
Published electronically: January 28, 2008
MathSciNet review: 2383533
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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.

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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

Gregg J. Zuckerman
Affiliation: Department of Mathematics, Yale University, P.O. Box 208283; New Haven, Connecticut 06520-8283

Keywords: Blattner's formula, coherent continuation, discrete series
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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