The Segal-Bargmann transform on compact symmetric spaces and their direct limits

We study the Segal-Bargmann transform, or the heat transform, Ht for a compact symmetric space M = U/K. We give a new proof, using representation theory and the restriction principle, of the fact that the map Ht is a unitary isomorphism Ht: L2(M) → Ht(Mℂ). We show that the Segal- Bargmann transform behaves nicely under propagation of symmetric spaces. If {Mn = Un/Kn}n is a direct family of compact symmetric spaces such that Mm propagates Mn, m ≥ n, then this gives rise to direct families of Hilbert spaces {L2(Mn), γn,m} and {Ht(Mnℂ), δm,n} such that Ht,m o δn,m o Ht,n. We also consider similar commutative diagrams for the Kn-invariant case. These lead to isometric isomorphisms between the Hilbert spaces lim → L2(Mn) ≃ lim → H(Mnℂ) as well as lim → L2(Mn)Kn ≃ lim → H(Mnℂ)Knℂ