PRODUCTS AND SYMMETRIZED POWERS OF IRREDUCIBLE REPRESENTATIONS OF SO-ASTERISK(2N)

The calculation of branching rules, tensor products and plethysms of t he infinite-dimensional harmonic series unitary irreducible representa tions of the non-compact group SO(2n) is considered and the duality b etween SO(2n) and Sp(2k) exploited. The branching rule for the restri ction of an arbitrary harmonic series irreducible representation of SO (2n) to U(n) is derived, and the decomposition is given explicitly fo r each of the infinite number of fundamental harmonic series irreducib le representations, H-m, of SO(2n) whose direct sum constitutes the m etaplectic representation, H, of SO(2n). A concise expression for the decomposition of tensor products is derived and a complete analysis o f the terms in both H-m x H-m' and H x H is given. A general formula f or plethysms of arbitrary irreducible representations of SO(2n) is de rived and its implementation illustrated both by means of a detailed g eneric example and by a complete determination of the symmetric and an tisymmetric terms of H x H. Finally, relationships that arise from the embedding of the product groups SO(2n) x Sp(2k) and Sp(2n, R) x O(2k ) in the metaplectic group Mp(4nk) are discussed.