Analytic factorization of Lie group representations

For every moderate growth representation (π,E) of a real Lie group G on a Fréchet space, we prove a factorization theorem of Dixmier–Malliavin type for the space of analytic vectors Eω. There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that Eω=Π(A(G))Eω. As a corollary we obtain that Eω coincides with the space of analytic vectors for the Laplace–Beltrami operator on G.