Cohomological equations and invariant distributions on a compact Lie group

Abstract

This paper deals with two analytic questions on a connected compact Lie group G. i) Let a ∈ G and denote by γ the diffeomorphism of G given by γ (x) = ax (left translation by a). We give necessary and sufficient conditions for the existence of solutions of the cohomological equation f - f ∘ γ = g on the Fréchet space C^∞ (G) of complex C^∞ functions on G. ii) When G is the torus ${\Bbb T}^n$, we compute explicitly the distributions on ${\Bbb T}^n$ invariant by an affine automorphism γ, that is, γ (x) = A (x + a) with A ∈ GL(n, ℤ) and a ∈ ${\Bbb T}^n$. iii) We apply these results to describe the infinitesimal deformations of some Lie foliations.