The Segal-Bargmann transform is an integral transform that maps isometrically onto , where is the Gaussian measure on ; can be expressed in the standard heat kernel on , , at . Here denotes the space of holomorphic functions. In the present paper a similar transform is investigated, with replaced by a compact connected Lie group , and replaced by a complexification of . Suitable heat kernels on and on are defined, and isometric isomorphisms
are obtained, for ; here is the -average of . In section 10 some Paley-Wiener type theorems are proved, in section 11 transforms on quotient spaces are considered, and in an appendix the main results are described in the language of creation operators, annihilation operators, and coherent states. The paper is well organized, and the proofs are transparent, even where things become rather technical.