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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, ln,m}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), δn,m} such that Ht,m o γn,m = 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 (Formula Presented) as well as lim (Formula Presented).

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Progress in Mathematics

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