ON RESOLVING THE MULTIPLICITY OF THE BRANCHING RULE GL(2K+1,C)DOWN-ARROW-SO(2K+1,C)

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 x (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.