The Segal–Bargmann Transform on a Symmetric Space of Compact Type

We study the Segal–Bargmann transform on a symmetric space X of compact type, mapping L2(X) into holomorphic functions on the complexification XC. We invert this transform by integrating against a “dual” heat kernel measure in the fibers of a natural fibration of XC over X. We prove that the Segal–Bargmann transform is an isometry from L2(X) onto the space of holomorphic functions on XC which are square integrable with respect to a natural measure. These results extend those of B. Hall in the compact group case.