Probability Theory and Related Fields

, Volume 147, Issue 3–4, pp 481–528

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

Open Access

DOI: 10.1007/s00440-009-0213-y

Cite this article as:
Driver, B.K. & Gordina, M. Probab. Theory Relat. Fields (2010) 147: 481. doi:10.1007/s00440-009-0213-y


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 L2(ν)-closure of holomorphic polynomials by their values on the Cameron–Martin subgroup.


Heisenberg group Holomorphic Heat kernel Quasi-invariance Taylor map 

Mathematics Subject Classification (2000)

Primary 35K05 43A15 Secondary 58G32 

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of Mathematics, 0112University of CaliforniaSan Diego, La JollaUSA
  2. 2.Department of MathematicsUniversity of ConnecticutStorrsUSA

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