# Publishing house / Banach Center Publications / All volumes

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

### Volume 96 / 2011

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

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.