A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups

TL;DR

This paper establishes a subelliptic Taylor isomorphism for holomorphic functions on infinite-dimensional Heisenberg-like groups, linking these functions to algebraic structures via a composition of restriction and Taylor maps.

Contribution

It introduces a novel subelliptic Taylor isomorphism for infinite-dimensional Heisenberg groups, connecting square integrable holomorphic functions to the universal enveloping algebra.

Findings

Proves a unitary equivalence between holomorphic functions and algebraic completions.

Defines the isomorphism as a composition of restriction and Taylor maps.

Extends the classical finite-dimensional theory to infinite-dimensional settings.

Abstract

Let $G$ denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on $G$ that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite dimensional projections are smooth measures. We prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the "Cameron-Martin" Lie subalgebra. The isomorphism defining the equivalence is given as a composition of restriction and Taylor maps.