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

http://dx.doi.org/10.4153/CJM-1999-035-3
Canad. J. Math. 51(1999), 816-834

  Published:1999-08-01
 Printed: Aug 1999

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 $s \rightarrow t/2$.