A new form of the Segal-Bargmann transform for Lie groups of compact type

I consider a two-parameter family B-s,B-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 --> infinity and s --> t/2.