The double Gelfand-Cetlin system, invariance of polarization, and the Peter-Weyl theorem

Abstract

The bundle map $T^{\ast }U(n)⟶U(n)$ provides a real polarization of the cotangent bundle $T^{\ast }U(n)$, and yields the geometric quantization $Q_1(T^{\ast }U(n))=L^2(U(n))$. We use the Gelfand-Cetlin systems of Guillemin and Sternberg to show that $T^{\ast }U(n)$ has a different real polarization with geometric quantization $Q_2(T^{\ast }U(n))={⨁}_{\alpha }{V}_{\alpha }\otimes V_{\alpha }^{\ast }$, where the sum is over all dominant integral weights $\alpha$ of $U(n)$. The Peter-Weyl theorem, which states that these two quantizations are isomorphic, may therefore be interpreted as an instance of ``invariance of polarization" in geometric quantization.