The moment map on symplectic vector space and oscillator representation (Q2403965)

The main result of the paper is the following theorem. Let \(G=\mathrm{Sp}(n,\mathbb{R})\) or \(U(p,q)\) or \(O^*(2n)\). Let \((W,\omega)\) be the real symplectic \(G\)-vector space \(W=\mathbb{R}^{2n}\) or \((\mathbb{C}^{p+q})_{\mathbb{R}}\) or \((\mathbb{C}^{2n})_{\mathbb{R}}\) equipped with \(\omega(u,v)= ^t\nu)J_nw\) for \(W=\mathbb{R}^{2n}\) or \(\mathrm{Im}(v^*I_{p,q} w)\) for \(W=(\mathbb{C}^{p+q})_{\mathbb{R}}\) or \(\mathrm{Im}(v^*I_{n,n} w)\) for \(W=\mathbb{C}^{2n})_{\mathbb{R}}\) where \(J_n\) is the standard symplectic form on \(\mathbb{R}^{2n}\) and \(I_{p,q}\) is the standard quadratic form on \((\mathbb{C}^{p+q})\). Then with a certain choice of the complex Lagrangian subspace of the complexification \(W_{\mathbb{C}}\) of \(W\) the canonical quantization of the moment map \(\mu: W\rightarrow g_0^*\) yields the oscillator representations of \(g=sp_n, gl_{p+q},o_{2n}\), respectively.