Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

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 μ 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 (ν)-closure of holomorphic polynomials by their values on the Cameron-Martin subgroup.

Heat kernel analysis on semi-infinite Lie groups

“…Moreover, this theorem gives good bounds on the L p -norms of the Radon-Nikodym derivatives. These results are important for future applications to spaces of holomorphic functions on G, as in [13]. We also show in Theorem 5.7 that a logarithmic Sobolev inequality holds for polynomial cylinder functions on G.…”

“…We collect here some properties of the heat kernel measure on G. The following two propositions are completely analogous to Corollary 4.9 of [11] and Proposition 4.6 in [13]. The proofs are included here for the convenience of the reader.…”

An Inversion Formula for the Gaussian Radon Transform for Banach Spaces

“…The map S B : L 2 (B, µ) → HL 2 (H C ) is unitary. For more details and proofs of these results, see [10], [4], [6] or [3].…”

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

“…However, some Lie groups carry so-called heat kernel measures which behave in many respects like Haar measure with a heat kernel density (cf. [DG08]).…”