Title

The segal–bargmann transform on compact symmetric spaces and their direct limits

Document Type

Article

Publication Date

1-1-2013

Abstract

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).

Publication Source (Journal or Book title)

Progress in Mathematics

First Page

225

Last Page

253

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