THE SEGAL-BARGMANN COHERENT-STATE TRANSFORM FOR COMPACT LIE-GROUPS

Let K be an arbitrary compact, connected Lie group. We describe on K a n analog of the Segal-Bargmann ''coherent state'' transform, and we pr ove (Theorem 1) that this generalized coherent state transform maps L2 (K) isometrically onto the space of holomorphic functions in L2(G, mu) , where G is the complexification of K and where mu 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 an alytic continuation to the complex group G. We also define a ''K-avera ged'' 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 Sectio n 9, we obtain inversion formulas for both the coherent state transfor m and the K-averaged coherent state transform (Theorems 3, 4). As a co nsequence, we obtain formulas for certain Bergman kernels on the compl ex group G (Theorems 5, 6). In Section 10, we derive a general ''Paley -Wiener'' theorem for K, and we exhibit a family of convolution transf orms, each of which is an isometric isomorphism of L2(K) onto a certai n L2-space of holomorphic functions on G. These transforms include the K-averaged coherent state transform as a special case. We also discus s the analogous results for R1. (The results of Section 10 are Theorem s 7-10.) Finally, in Section 11, we discuss the case of quotient space s. We prove (Theorem 11) that both the coherent state transform and th e K-averaged coherent state transform pass in a natural way from K to K/H, where H is a closed, connected subgroup of K. (C) 1994 Academic P ress, Inc.