Summary: Let \(G\) be a noncompact semi-simple Lie group with finite center and \(\mu\) a probability measure on \(G\). We consider (i) the semigroup \(S_{\mu}\) generated by the support of \(\mu\) (with the assumption that \(\mathrm{int} S_{\mu}\neq \emptyset\)); (ii) The spectral radii \(r_{\lambda}\) of the operators \(U_{\lambda}\left( \mu \right)\) where \(U_{\lambda}\) is a (nonunitary) representation of \(G\) induced by a real character and (iii) the moment Lyapunov exponents \(\gamma \left( \lambda,x\right)\) of the i.i.d. random product on \(G\) defined by \(\mu\). The equality \(r_{\lambda}=\gamma \left( \lambda,x\right)\) holds in many cases. We give a necessary and sufficient condition to have \(S_{\mu}=G\) in terms of the analyticity of the map \(\lambda \mapsto r_{\lambda}\). The condition is applied to measures obtained by solutions of invariant stochastic differential equations on \(G\) yielding a necessary and sufficient condition for the controllability of invariant control systems on \(G\) in terms of the largest eigenvalues of second order differential operators.