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# A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type

Published:1999-08-01
Printed: Aug 1999
• Brian C. Hall
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## Abstract

I consider a two-parameter family $B_{s,t}$ of unitary transforms mapping an $L^{2}$-space over a Lie group of compact type onto a holomorphic $L^{2}$-space over the complexified group. These were studied using infinite-dimensional analysis in joint work with B.~Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases $s \rightarrow \infty$ and $s \rightarrow t/2$.
 MSC Classifications: 22E30 - Analysis on real and complex Lie groups [See also 33C80, 43-XX] 81S30 - Phase-space methods including Wigner distributions, etc. 58G11 - unknown classification 58G11