The inverse Segal-Bargmann transform for compact Lie groups

Abstract

In this paper I give a new inversion formula (Theorem 1) for the generalized Segal-Bargmann transform introduced in B. C. Hall, J. Funct. Anal. 122 (1994), 103-151. The inversion formula may be viewed as a formula for the inverse heat operator for a compact Lie group. I then use this formula to give a new direct proof of the unitary of the K-invariant 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-compact dual symmetric space Kℂ/K. © 1997 Academic Press.