Essays on representations of real groups The theorem of Dixmier-Malliavin

where L is the left regular representation of G. From this it can be deduced that for any v in V and f in C c (G) the vector π(f) is smooth, and more precisely that if X lies in U(g) then π(X)π(f)v = π(LXf)v. This implies that V ∞ is dense in V , since if {fn} is a Dirac sequence on G then π(fn)v → v. The subspace of V ∞ spanned by the π(f)v with f in C c (G) is called the Gårding subspace of V . It is relatively easy to show that a smooth vector may be expressed as a linear combination of π(f)v with f in C c (G) for arbitrarily highm, as explained in [Cartier:1974]. I’ll say something about this in the first section. It is considerably more difficult to see that if V is a Fréchet space then the smooth vectors and the Gårding subspace coincide. This remarkable result was proved in [Dixmier-Malliavin:1978].