The inverse Segal-Bargmann transform for compact Lie groups
In a preceding paper [ibid. 122, 103-151 (1994; Zbl 0838.22004)] the author introduced a generalized Segal-Bargmann transform on an arbitrary compact connected Lie group \(K\), together with an inversion formula. Here he describes a new inversion formula which is obtained by taking a holomorphic function \(F(g)\) on the complexification \(K_\mathbb{C}\) of \(K\), fixing the real part of \(g\), and integrating over the imaginary axis with respect to a certain heat kernel measure. In a sense, a formula for the inverse heat operator on a compact Lie group is obtained in this way. This formula allows to give a new proof of the unitary character of the \(K\)-invariant form of the Segal-Bargmann transform.
scientific article; zbMATH DE number 997635