The moment map on symplectic vector space and oscillator representation

Abstract

Let G denote Sp(n,R), U(p,q), or O∗(2n). The main aim of this article is to show that the canonical quantization of the moment map on a symplectic G-vector space (W,ω) naturally gives rise to the oscillator (or Segal–Shale–Weil) representation of g:=Lie(G)⊗C. More precisely, after taking a complex Lagrangian subspace V of the complexification of W, we assign an element of the Weyl algebra for V to 〈μ,X〉 for each X∈g, which we denote by 〈μˆ,X〉. Then we show that the map X↦i〈μˆ,X〉 gives a representation of g. With a suitable choice of V in each case, the representation coincides with the oscillator representation of g.