The Segal–Bargmann transform for noncompact symmetric spaces of the complex type

We consider the generalized Segal–Bargmann transform, defined in terms of the heat operator, for a noncompact symmetric space of the complex type. For radial functions, we show that the Segal–Bargmann transform is a unitary map onto a certain L2 space of meromorphic functions. For general functions, we give an inversion formula for the Segal–Bargmann transform, involving integration against an “unwrapped” version of the heat kernel for the dual compact symmetric space. Both results involve delicate cancellations of singularities. © 2005 Elsevier Inc. All rights reserved.