  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 use
the Gelfand-Cetlin systems of Guillemin and Sternberg to show that
$T^*\hspace{-2pt}\operatorname{U}(n)$ has a different real polarization with
geometric quantization $Q_2(T^*\hspace{-2pt}\operatorname{U}(n))=
\bigoplus_\alpha V_\alpha \otimes V_\alpha^*$, where the sum is over all
dominant integral weights $\alpha$ of $\operatorname{U}(n)$. 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.
