Relations among various versions of the Segal-Bargmann transform

TL;DR

This paper explores the relationships among different versions of the Segal-Bargmann transform, establishing new identities for Coxeter groups and Lie groups, and highlighting a key determinant role of Version C by Version A.

Contribution

It provides new relations among Versions A, B, and C of the Segal-Bargmann transform for Coxeter groups and Lie groups, including a counterexample and a key determinative result.

Findings

Relations among Versions A, B, and C are established for Coxeter groups.

Analogous relations are shown for Lie groups, except for one identity.

Version C of the transform is determined by Version A in both contexts.

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.