An inversion formula for the Segal–Bargmann transform on a symmetric space of non-compact type

We prove analogs of the heat kernel transform inversion formulae of Golse, Leichtnam and the author [E. Leichtnam, F. Golse, M. Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. École Norm. Sup. (4) 29 (6) (1996) 669–736. MR1422988 (97h:58153), Theorems 0.3, 0.4] in the setting of a Riemannian symmetric space of Helgason's non-compact type.