If V and W are two separable Hilbert spaces, define to be the subgroup of , the group of all invertible bounded operators of the Hilbert space which leave invariant the Hermitian form defined by the operator on
Let be the subgroup of consisting of those elements for which and are Hilbert-Schmidt operators. If , , , are in it turns out that a co-cycle, which induces a central extension of , , with
The author determines the unitary lowest weight representations of namely the irreducible components of the k-fold tensor product of the Segal-Shale-Weil representation [see e.g. G. B. Segal, Commun. Math. Phys. 80, 301-342 (1981; Zbl 0495.22017)].
Let H be a Hilbert space, let denote the Hilbert space completion of the vector space of nth powers of the symmetric algebra of H with the Hermitian form
where runs through all permutations of 1,2,...,n. Define endowed with the final topology defined by the inclusions , and the Hilbert space completion of S(H). Let endowed with the initial topology defined by the projections . and are the antilinear dual spaces of each other and with continuous and dense inclusions. is the canonical representation of the Lie algebra L(H) on S(H). By means of , a representation of is constructed. Herein L(H) is the Lie algebra of bounded operators of H and C(H) the subalgebra of all compact operators of H; and denotes the subalgebra of consisting of elements of the form , , , b and c being Hilbert-Schmidt operators. For , in , , , the co-cycle induces a central extension of
The author proves (1.6 Theorem, p. 67): The representation of can be lifted to a unitary representation of on S(V. In the case when V and W are finite-dimensional, Kashiwara and Vergne showed that this representation is completely reducible and determined all the minimal weight vectors. By methods which are purely algebraic, the author generalizes their results to the infinite-dimensional case.
In Section 2 these questions are answered for the case of the symplectic group Sp(V), V a complex Hilbert space. The decomposition of the tensor products of the Segal-Shale-Weil representation of the metaplectic group is obtained.