Discrete series representations and K multiplicities for U(p,q). User's guide

TL;DR

This paper provides a detailed guide and algorithm for computing K-multiplicities in discrete series representations of U(p,q), utilizing Blattner's formula and Kostant partition functions.

Contribution

It introduces a practical algorithm for calculating multiplicities in U(p,q) discrete series, supported by mathematical background and implementation details.

Findings

Algorithm effectively computes K-multiplicities

Utilizes Blattner's formula and Kostant partition functions

Highlights properties of piecewise polynomial multiplicity functions

Abstract

This document is a companion for the Maple program : Discrete series and K-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explain an algorithm to compute the multiplicities of an irreducible representation of U(p)x U(q) in a discrete series of U(p,q). It is based on Blattner's formula. We recall the general mathematical background to compute Kostant partition functions via multidimensional residues, and we outline our algorithm. We also point out some properties of the piecewise polynomial functions describing multiplicities based on Paradan's results.