On resolving the multiplicity of the branching rule GL(2k+1,C) down arrow SO(2k+1,C)

Abstract

We consider the multiplicity problem of the branching rule GL(2k+1,C) down arrow SO(2k+1,C). Finite-dimensional irreducible representations of GL(2k+1,C) are realized as right translations on subspaces of the holomorphic Hilbert (Bargmann) spaces of q*(2k+1) complex variables. Maps are exhibited which carry an irreducible representation of SO(2k+1,C) into these subspaces. An algebra of commuting operators is constructed. Eigenvalues and eigenvectors of certain of these operators can then be used to resolve the multiplicity in the branching rule.