Dual representations of and decomposition of Fock spaces

Abstract

In this paper we study left and right `regular representations' of , an infinite-dimensional analogue of the classical Lie group with entries indexed by the integers, on a space which we take to be the projective limit of Bargmann - Fock - Segal spaces. We decompose into an orthogonal direct sum where the hat indicates the closure of the algebraic direct sum, and where each is irreducible with respect to the joint left and right action, and the isotypic component of the left or right actions with double signatures of the type . For each , is the subspace generated by the joint action on a highest weight vector with highest weight . We start by defining the group and its Lie algebra , and discuss the `infinite wedge space', F, defined by Kac, and its transpose . We then define a kind of determinant function on , prove several identities and properties of these determinants, and use these to produce intertwining maps from tensor products of F and to , thereby obtaining irreducible subspaces and highest weight vectors. We further adapt these results to representations of on the inductive limit of Bargmann - Fock - Segal spaces .