Cohomological equations and invariant distributions on a compact Lie group

This paper deals with two analytic questions on a connected compact Lie group G. i) Let a ε 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 - fo γ = g on the Fréchet space C∞(G) of complex C∞ functions on G. ii) When G is the torus Tn, we compute explicitly the distributions on Tn invariant by an affine automorphism γ, that is, γ(x) = A(x + a) with A ε GL(n; ℤ) and a ε Tn. iii) We apply these results to describe the infinitesimal deformations of some Lie foliations.