Consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain 𝓓 of ℂ 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. 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 (ℝ,+) and (ℝ* = ℝ-0,×).
B) The multiplicative group M of 2×2 complex invertible matrices and some subgroups of M.
C) The three-dimensional Heisenberg group.