ON RESOLVING THE MULTIPLICITY OF TENSOR-PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF SYMPLECTIC GROUPS

The multiplicity problem of r-fold tensor products of irreducible repr esentations, of Sp(2k, C) are considered. The arbitrary r-fold tensor product is shown to be isomorphic to a subspace of the holomorphic Hil bert (Bargmann) space of n x 2k complex variables. Maps are constructe d which carry an irreducible representation of Sp(2k, C) into this sub space. An algebra of commuting operators is constructed and it is show n how eigenvalues and eigenvectors of certain of these operators can b e used to resolve the multiplicity.