In this talk we consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain D of Cⁿ and in L²(μ), the measure μ 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 D. Then we calculate these operators in the following elementary cases: the commutative groups (R, +) and (R^∗, ×), the three dimen- sional Heisenberg group and the affine group on the real line.