On Sobolev norms for Lie group representations

We define Sobolev norms of arbitrary real order for a Banach representation (π,E) of a Lie group, with regard to a single differential operator D=dπ(R ²+Δ). Here, Δ is a Laplace element in the universal enveloping algebra, and R>0 depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for D on the space of smooth vectors of E. The main tool is a novel factorization of the delta distribution on a Lie group.