Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension
Volume 96 / 2011
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.
