Smooth vector of a Lie-group representation - Imported Knowlpedia Content

Definition

Let $G$ be a finite-dimensional Lie group and let $(\pi,\mathcal H)$ be a strongly continuous unitary representation. A vector $\xi\in\mathcal H$ is a smooth vector if its orbit map

G\longrightarrow\mathcal H,\qquad g\longmapsto\pi(g)\xi,

is $C^\infty$ as a map from the smooth manifold $G$ to the Hilbert space $\mathcal H$ , with differentiability taken in the norm topology. The space of all smooth vectors is denoted $\mathcal H^\infty$ . It is a $\pi(G)$ -invariant linear subspace and is dense in $\mathcal H$ .

Differentiation

For $X$ in the Lie algebra $\mathfrak g$ and $\xi\in\mathcal H^\infty$ , the norm derivative

d\pi(X)\xi=\left.\frac{d}{dt}\right|_{t=0}\pi(\exp(tX))\xi

exists. These operators share the invariant dense domain $\mathcal H^\infty$ and assemble into the derived representation. They are generally unbounded as operators on $\mathcal H$ , so their common domain is essential data.

Density and regularization

For $f\in C_c^\infty(G)$ , the integrated vector

\pi(f)\xi=\int_G f(g)\pi(g)\xi\,dg

is smooth. Approximate identities of such functions converge strongly to the identity, proving the density of $\mathcal H^\infty$ ; this is commonly called Gårding's argument Warner, §4.4.

Relation to generators

For each $X\in\mathfrak g$ , Stone's theorem gives a self-adjoint operator $A_X$ with $\pi(\exp(tX))=e^{itA_X}$ . On smooth vectors,

d\pi(X)\xi=iA_X\xi.

Thus $d\pi(X)$ is the skew-symmetric infinitesimal action, while $A_X$ is the self-adjoint Stone generator. Neither operator should be described without its domain.

References

G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer, 1972. DOI record. Relevant: §4.4 on differentiable and smooth vectors.