Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

Bruce Driver, Maria Gordina

Published 2008-09-29Version 1

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.