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Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups
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  • Open Access
  • Published: 01 April 2009

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

  • Bruce K. Driver1 &
  • Maria Gordina2 

Probability Theory and Related Fields volume 147, pages 481–528 (2010)Cite this article

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.

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Acknowledgments

We are grateful to Professor Malliavin whose question during a workshop at the Hausdorff Institute (Bonn, Germany) led us to include a section on a holomorphic chaos expansion.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Authors and Affiliations

  1. Department of Mathematics, 0112, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Bruce K. Driver

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Maria Gordina

Authors
  1. Bruce K. Driver
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  2. Maria Gordina
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Corresponding author

Correspondence to Bruce K. Driver.

Additional information

B. K. Driver’s research was supported in part by NSF Grants DMS-0504608 and DMS-0804472 and the Miller Institute at the University of California, at Berkeley.

M. Gordina’s research was supported in part by NSF Grant DMS-0706784 and the Humboldt Foundation Research Fellowship.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Driver, B.K., Gordina, M. Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups. Probab. Theory Relat. Fields 147, 481–528 (2010). https://doi.org/10.1007/s00440-009-0213-y

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  • Received: 06 August 2008

  • Revised: 07 February 2009

  • Published: 01 April 2009

  • Issue Date: July 2010

  • DOI: https://doi.org/10.1007/s00440-009-0213-y

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Keywords

  • Heisenberg group
  • Holomorphic
  • Heat kernel
  • Quasi-invariance
  • Taylor map

Mathematics Subject Classification (2000)

  • Primary 35K05
  • 43A15
  • Secondary 58G32
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