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 be a connected, semisimple Lie group with finite center and let be a maximal compact subgroup. We investigate a method to compute multiplicities of -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 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 of type split . 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

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

DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09284-8

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.