A generating function for Blattner’s formula

Willenbring, Jeb; Zuckerman, Gregg

doi:10.1090/S0002-9939-08-09284-8

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.

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