A generating function for Blattner's formula
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 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.
Cite
@article{arxiv.math/0612266,
title = {A generating function for Blattner's formula},
author = {Jeb F. Willenbring and Gregg J. Zuckerman},
journal= {arXiv preprint arXiv:math/0612266},
year = {2007}
}
Comments
12 pages, 4 figures, added section on a positivity result