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This document is a companion for the Maple program : Discrete series andK-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explainan algorithm to compute the multiplicities of an irreducible representation ofU(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 partitionfunctions via multidimensional residues, and we outline our algorithm. We alsopoint out some properties of the piecewise polynomial functions describingmultiplicities based on Paradan's results.

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