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 abstractWiener 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 skeletonmap which characterizes globally defined functions from the L2(ν)-closure of holomorphic polynomials by their values on the Cameron-Martin subgroup. © The Author(s) 2009.