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The bundle map$T^*\hspace{-2pt}\operatorname{U}(n)\longrightarrow\operatorname{U}(n)$provides a real polarization of the cotangent bundle$T^*\hspace{-2pt}\operatorname{U}(n)$, and yields the geometric quantization$Q_1(T^*\hspace{-2pt}\operatorname{U}(n)) = L^2(\operatorname{U}(n))$. We usethe Gelfand-Cetlin systems of Guillemin and Sternberg to show that$T^*\hspace{-2pt}\operatorname{U}(n)$ has a different real polarization withgeometric quantization $Q_2(T^*\hspace{-2pt}\operatorname{U}(n))=\bigoplus_\alpha V_\alpha \otimes V_\alpha^*$, where the sum is over alldominant integral weights $\alpha$ of $\operatorname{U}(n)$. The Peter-Weyltheorem, which states that these two quantizations are isomorphic, maytherefore be interpreted as an instance of ``invariance of polarization" ingeometric quantization.

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