  We prove an abstract Implicit Function Theorem with parameters for smooth
operators defined on sequence scales, modeled for the search of quasi-periodic
solutions of PDEs. The tame estimates required for the inverse linearised
operators at each step of the iterative scheme are deduced via a multiscale
inductive argument. The Cantor like set of parameters where the solution exists
is defined in a non inductive way. This formulation completely decouples the
iterative scheme from the measure theoretical analysis of the parameters where
the small divisors non-resonance conditions are verified. As an application, we
deduce the existence of quasi-periodic solutions for forced NLW and NLS
equations on any compact Lie group or manifold which is homogeneous with
respect to a compact Lie group, extending previous results valid only for tori.
A basic tool of harmonic analysis is the highest weight theory for the
irreducible representations of compact Lie groups.
