Universal enveloping algebra action on smooth vectors

Core idea

Let $G$ be a connected finite-dimensional Lie group with Lie algebra $\mathfrak g$ , and let $\pi$ be a continuous unitary representation on $\mathcal H$ . Its derived representation on $\mathcal H^\infty$ extends complex-linearly from $\mathfrak g$ to $\mathfrak g_{\mathbb C}$ , then uniquely to a unital algebra homomorphism

d\pi:U(\mathfrak g_{\mathbb C})\longrightarrow \operatorname{End}_{\mathbb C}(\mathcal H^\infty).

This natural construction is called the canonical universal enveloping algebra action on smooth vectors. For a monomial $X_1\cdots X_r$ , it acts as the well-defined composition $d\pi(X_1)\cdots d\pi(X_r)$ on the common invariant domain $\mathcal H^\infty$ .

Why the extension exists

The differentiated operators satisfy

d\pi([X,Y])=d\pi(X)d\pi(Y)-d\pi(Y)d\pi(X)

on smooth vectors. The universal property of the universal enveloping algebra therefore produces the unique extension. Invariance of $\mathcal H^\infty$ under every $d\pi(X)$ ensures that all words have one common domain rather than a separately chosen intersection Warner, §4.4.

Equivariance and filtration

The group action and enveloping-algebra action obey

\pi(g)d\pi(u)\pi(g)^{-1}=d\pi(\operatorname{Ad}(g)u), \qquad u\in U(\mathfrak g_{\mathbb C}).

The degree filtration on $U(\mathfrak g_{\mathbb C})$ records the order of iterated differentiation. Central elements act by operators commuting with $\pi(G)$ , a fact that underlies Casimir operators and infinitesimal characters.

Example and analytic warning

For $G=\mathbb R$ , one infinitesimal generator $D$ determines the action: $U(\mathfrak g_{\mathbb C})\cong\mathbb C[X]$ , and $d\pi(p(X))=p(D)$ on the smooth vectors, which lie in the domain of every power of $D$ .

References

Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Grundlehren der mathematischen Wissenschaften 188, Springer, 1972. DOI record. Relevant: §4.4 on differentiable and smooth vectors.
Jacques Dixmier, Enveloping Algebras, Graduate Studies in Mathematics 11, American Mathematical Society, 1996. DOI record. Relevant: Chapter 2 on the universal property and filtration of enveloping algebras.