Let be a connected and compact Lie group and suppose that it is contained in its complexification . The authors introduce the Fock space associated with and give a simple proof of the unitarity of the Segal-Bargmann transform proved by B. C. Hall for the group . First, let be the -invariant Laplacian operator on . Let be a heat kernel of , i.e., a solution of the equation , , where is the delta distribution at the unit element 1 of . After recalling a result of B. C. Hall [J. Funct. Anal. 122, No. 1, 103–151 (1994; Zbl 0838.22004)], the authors assert that can be holomorphically extended to . Let be a heat kernel of the Laplacian on .
By a theorem in the book by D. W. Robinson [Elliptic operators and Lie groups, Oxford etc.: Clarendon Press (1991; Zbl 0747.47030)] we know that is a positive function for each . We define by
Now, we define the Fock space for the group by the space of all holomorphic functions such that
where denotes a Haar measure on . The authors prove that the Fock space has a reproducing kernel given by for . In fact, they prove that
holds for , .
Next, they prove that there exists a linear map which is a unitary -equivariant operator. The authors call the Segal-Bargmann transformation. To define the Weyl transform we put for and consider the space . The authors define an operator , which they call the Weyl transform, and then they prove the boundedness of this operator.