Relations Among Various Versions of the Segal-Bargmann Transform

Abstract

We present various relations among Versions A, B and C of the Segal-Bargmann transform. We get results for the Segal-Bargmann transform associated to a Coxeter group acting on a finite dimensional Euclidean space. Then analogous results are shown for the Segal-Bargmann transform of a connected, compact Lie group for all except one of the identities established in the Coxeter case. A counterexample is given to show that the remaining identity from the Coxeter case does not have an analogous identity for the Lie group case. A major result is that in both contexts the Segal-Bargmann transform for Version C is determined by that for Version A. (2000): Primary 45H05, 44A15; Secondary 46E15 Keywords: Segal-Bargmann transfrom, Coxeter group, Dunkl heat kernel 1 Research partially supported by CONACYT (Mexico) project 49187. Mathematics Subject ClassificationWe will use the holomorphic Dunkl kernel function E µ :We will also be using the analytic continuation of the Dunkl heat kernel, which is given for z, w ∈ C N by( 1.1) This kernel arises in the solution of the initial value problem of the heat equation associated with the Dunkl Laplacian operator. (See [13].) We next define the kernel functions of the versions of the Segal-Bargmann transform associated to a Coxeter group for z ∈ C N and q ∈ R N by). (1.4) See [1], [4] and [15] for the origins of this theory in the case µ ≡ 0. The versions of the Segal-Bargmann transform are given as follows. (See [2], [5], [16], [17] and [20].) Versions A and C are defined byand ω µ,t is the density of a measure on R N . Version B is defined bywhere z ∈ C N , f ∈ L 2 (R N , m µ,t ) and m µ,t is the density of a measure on R N . Associated to these versions there are reproducing kernel Hilbert spaces of holomorphic functions f : C N → C, denoted A µ,t , B µ,t and C µ,t respectively, such that A µ,t : L 2 (R N , ω µ,t ) → A µ,t B µ,t : L 2 (R N , m µ,t ) → B µ,t C µ,t : L 2 (R N , ω µ,t ) → C µ,t are unitary isomorphisms. It turns out that A µ,t = B µ,t as Hilbert spaces.