THE INVERSE SEGAL-BARGMANN TRANSFORM FOR COMPACT LIE-GROUPS

In this paper I give a new inversion formula (Theorem 1) for the gener alized Segal-Bargmann transform introduced in B. C. Hall, J, Fuct. Ana l. 122 (1994), 103-151. The inversion formula may be viewed as a formu la for the inverse heat operator for a compact Lie group. I then use t his formula to give a new direct proof of the unitary of the K-invaria nt form of the Segal-Bargmann transform (Theorem 2). The proof of the inversion formula relies on an identity (Theorem 5) which relates the Laplacian for a compact Lie group K to the Laplacian for the non-compa ct dual symmetric space K-c/K. (C) 1997 Academic Press