Generalized Bargmann transform and a group representation

Let FA(Cn) denote the Fock space associated with a real linear transformation A on Cn which is symmetric and positive definite relative to the real inner product Re〈z,w〉, z,w∈Cn. Let BA denote the Bargmann transform, mapping L2(Rn) unitarily onto FA(Cn). In this note, we show that one can find a group G, whose unitary irreducible representation at its base vector coincides with up to a constant multiple, where denotes the adjoint of BA and Kw denotes the reproducing kernel of FA(Cn).