Abstract:

Let K be a compact Lie group. As is well known, L²(K) can be interpreted as the "position-space'' geometric quantization of the cotangent bundle T*K. In this talk, I will describe a ``momentum-space'' quantization of T*K. I will also explain how this momentum-space quantization is linked to the position-space representation via parallel transport with respect to the Axelrod-della Pietra-Witten-Hitchin connection in a certain Hilbert bundle. In particular, it is a result of Florentino-Matias-Mourao-Nunes that parallel transport along a particular geodesic from position space to an intermediate fiber is the generalized Segal-Bargmann transform. I will explain how their result can be extended to any other interior fiber (thus obtaining generalized generalized Segal-Bargmann transforms), and moreover that when extended to momentum space, parallel transport yields the Peter-Weyl decomposition. This is joint work with S. Wu.