The geometric interpretation of the Peter-Weyl theorem

Abstract: Let $K$ be a compact Lie group. I will review the construction of Mabuchi geodesic families of $K\times K-$invariant Kahler structures on $T^*K$, via Hamiltonian flows in imaginary time generated by a strictly convex invariant function on $Lie \, K$, and the corresponding geometric quantization. At infinite geodesic time, one obtains a rich mixed polarization of $T^*K$, the Kirwin-Wu polarization, which is then continuously connected to the vertical polarization of $T^*K$. The geometric quantization of $T^*K$ along this family of polarizations is described by a generalized coherent state transform that, as geodesic time goes to infinity, describes the convergence of holomorphic sections to distributional sections supported on Bohr-Sommerfeld cycles. These are in correspondence with coadjoint orbits $O_{\lambda+\rho}$. One then obtains a concrete (quantum) geometric interpretation of the Peter-Weyl theorem, where terms in the non-abelian Fourier series are directly related to geometric cycles in $T^*K$. The role of a singular torus action in this construction will also be emphasized. This is joint work with T.Baier, J. Hilgert, O. Kaya and J. Mourão.

algebraic geometrydifferential geometrysymplectic geometry

Audience: researchers in the topic

Geometria em Lisboa (IST)

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