The sum defining Ψ b converges in H − − 1 ⊂ ⊂ S ′ and depends only on the class of b in M 1 / M. On the other hand, the operators U (x) and V (y) with ' x, y in M 1 commute with all the corresponding operators for M. So M 1 leaves the subspace V 0 spanned by the Ψ b invariant. Hence the group A = M 1 acts on V 0. This action can immediately be identified with the action on V for the N -dimensional oscillator representation associated with A, since