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
.