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

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