The Segal-Bargmann "Coherent State" Transform for Compact Lie Groups

Let K be an arbitrary compact, connected Lie group. We describe on K an analog of the Segal-Bargmann "coherent state" transform, and we prove (Theorem 1) that this generalized coherent state transform maps L2(K) isometrically onto the space of holomorphic functions in L2(G, μ), where G is the complexification of K and where μ is an appropriate heat kernel measure on G. The generalized coherent state transform is defined in terms of the heat kernel on the compact group K, and its analytic continuation to the complex group G. We also define a "K-averaged" version of the coherent state transform, and we prove a similar result for it (Theorem 2). In the Appendix we describe the "coherent states" which motivate the definitions of these transforms. In Section 9, we obtain inversion formulas for both the coherent state transform and the K-averaged coherent state transform (Theorems 3, 4). As a consequence, we obtain formulas for certain Bergman kernels on the complex group G (Theorems 5, 6). In Section 10, we derive a general "Paley-Wiener" theorem for K, and we exhibit a family of convolution transforms, each of which is an isometric isomorphism of L2(K) onto a certain L2-space of holomorphic functions on G. These transforms include the K-averaged coherent state transform as a special case. We also discuss the analogous results for R1. (The results of Section 10 are Theorems 7-10.) Finally, in Section 11, we discuss the case of quotient spaces. We prove (Theorem 11) that both the coherent state transform and the K-averaged coherent state transform pass in a natural way from K to K/H, where H is a closed, connected subgroup of K.