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

Abstract

The bundle map T⁎U(n)⟶U(n) provides a real polarization of the cotangent bundle T⁎U(n), and yields the geometric quantization Q1(T⁎U(n))=L2(U(n)). We use the Gelfand–Cetlin systems of Guillemin and Sternberg to show that T⁎U(n) has a different real polarization with geometric quantization Q2(T⁎U(n))=⨁αVα⊗Vα⁎, where the sum is over all dominant integral weights α 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.