Holomorphic discrete series and harmonic series unitary irreducible representations of non-compact Lie groups: Sp(2n, R), U(p, q) and SO*(2n)

Publisher Website
Google Scholar
Abstract
Generalised characters of the infinite dimensional, holomorphic, discrete series, unitary, irreducible representations of the non-compact groups U(p, q), Sp(2n, R) and SO*(2n) are explicitly expressed in terms of characters of finite dimensional unitary group representations. These formulae are remarkably succinct despite involving certain infinite series of Schur functions. Similar formulae are derived for harmonic series unitary representation of both U(p, q) and Sp(2n, R). Consideration of the branching rules from U(p, q) to U(q)*(p) and from Sp(2n, R) to U(n) enables holomorphic representations to be identified as a subset of the harmonic representations. The branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations. In the case of the former a known result is recast in terms of closed formulae involving Schur functions and for the latter various generalisations of these formulae are given. A conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of Sp(2n, R) and U(p, q). Illustrative examples are presented.

This publication has 26 references indexed in Scilit: