Howe duality for the quantum groups Uqu(m,n), Uqu(M)

Abstract

A dual pair is defined to be a pair of mutually centralizing subgroups of the real symplectic group. Let (G,G′) be a dual pair of reductive groups in which G is also compact and consider the decomposition of the metaplectic representation of the symplectic group into GG′-irreducibles. Each such irreducible is a tensor product of an irreducible of G with one of G′, and it turns out there is a bijective correspondence between them, a particular irreducible of G only occurring with a particular one of G′ and vice versa. This so-called Howe duality is here generalized to the quantum group case (Uqu(m,n),Uqu(M)) for q not a root of unity. A metaplectic representation for this dual pair is given in terms of the q-oscillator algebra and a Uqu(m,n)×Uqu(M)-covariant Heisenberg–Weyl algebra is also constructed and realized on Fock space. The heart of the proof lies in showing that Uqu(m,n) and Uqu(M) are essentially mutual commutants on Fock space. The duality follows using the compactness of Uqu(M). The proof is independent of the classical theory. As a consequence, given any two unitary highest weight representations of a quantum pseudo-unitary group (which arise from the restriction of a metaplectic representation), the decomposition of their tensor product is the same as in the classical case.