Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

TL;DR

This paper studies square integrable holomorphic functions on a new class of infinite-dimensional Heisenberg-like groups, establishing a unitary equivalence with algebraic structures and characterizing functions via a skeleton map.

Contribution

It introduces a novel class of infinite-dimensional Heisenberg-like groups and connects holomorphic functions to algebraic completions, expanding analysis in infinite-dimensional settings.

Findings

Established a unitary equivalence with a universal enveloping algebra

Constructed a skeleton map using heat kernel measure quasi-invariance

Characterized globally defined functions via Cameron-Martin subgroup

Abstract

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure $\mu$ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the "Lie algebra" of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the $L^{2}(\nu)$-closure of holomorphic polynomials by their values on the Cameron-Martin subgroup.