Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension

Volume 96 / 2011

Hélène Airault, Habib Ouerdiane Banach Center Publications 96 (2011), 9-34 MSC: Primary 58J65; Secondary 60J60, 60H07. DOI: 10.4064/bc96-0-1

Abstract

Consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain $ {\cal D}$ of $\mathbb{C}$ and in $L^2(\mu)$, the measure $\mu$ being the unitarizing measure of the representation. On finite-dimensional examples, we show that this unitarizing measure is also the invariant measure for some differential operators on ${\cal D}$. We calculate these operators and we develop the concepts of unitarizing measure and invariant measure for an OU operator $($differential operator associated to the representation$)$ in the following elementary cases:

A) The commutative groups $(\mathbb{R}, +)$ and $(\mathbb{R}^\ast=\mathbb{R}-{0}, \times)$.

B) The multiplicative group $M$ of $2\times 2$ complex invertible matrices and some subgroups of $M$.

C) The three-dimensional Heisenberg group.

Authors

  • Hélène AiraultUniversité de Picardie Jules Verne, INSSET
    48, rue Raspail, 02100 Saint-Quentin (Aisne)
    UMR6140-CNRS, 33, rue Saint-Leu, 80039 Amiens, France
    e-mail
  • Habib OuerdianeDepartment of Mathematics, Faculty of Science of Tunis
    University of Tunis El Manar, Campus Universitaire, 1060, Tunis, Tunisia
    e-mail

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