Let denote , , or . The main aim of this article is to show that the canonical quantization of the moment map on a symplectic -vector space naturally gives rise to the oscillator (or Segal–Shale–Weil) representation of . More precisely, after taking a complex Lagrangian subspace of the complexification of , we assign an element of the Weyl algebra for to for each , which we denote by . Then we show that the map gives a representation of . With a suitable choice of in each case, the representation coincides with the oscillator representation of .
"The moment map on symplectic vector space and oscillator representation." Kyoto J. Math. 57 (3) 553 - 583, September 2017. https://doi.org/10.1215/21562261-2017-0006