Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension
Tom 96 / 2011
Streszczenie
Consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain 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.