The Inverse Segal–Bargmann Transform for Compact Lie Groups

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 theinverseheat operator for a compact Lie group. I then use this formula to give a new direct proof of the unitary of theK-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 groupKto the Laplacian for the non-compact dual symmetric space KC/K.