On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall Isomorphisms

Recently, Gross has shown that the Kakutani-Itô-Segal isomorphism theorem has an extension from the setting of Gaussian measure on a vector space to "heat kernel" measure (pt) on a simply connected Lie group (G) of compact type. The isomorphism relates L2(pt) to a certain completion of the universal enveloping algebra of g =̇ Lie(G). Gross proves this result using the Kakutani-Itô-Segal theorem and an infinite dimensional calculus associated to G-valued Brownian motion. Hijab has greatly simplified and clarified Gross′ proof. Hiiab′s proof avoids most, but not all, of the "infinite dimensional" analysis in the original proof. In this paper, we will build on Hijab′s proof to give a completely "finite dimensional" non-probabilistic proof of Gross′ isomorphism theorem. The proof given here relies heavily on Hall′s beautiful "extension" of the Segal-Bargmann transformation to the setting of compact Lie groups. This theorem relating L2(pt) to a certain L2-space of holomorphic functions (L2(GC) ∩ H) on the complexified Lie group GC will also be generalized to Lie groups of compact type. In the process, it is shown how to characterize, in terms of summability conditions on all the derivatives at the identity in GC, those holomorphic functions which are in L2(GC).