Derived representation on smooth vectors

Let $G$ be a finite-dimensional Lie group with Lie algebra $\mathfrak g$ , and let $\pi$ be a strongly continuous unitary representation on a Hilbert space $\mathcal H$ . On the common dense invariant subspace $\mathcal H^\infty$ of smooth vectors, its derived representation is

d\pi(X)v=\left.\frac{d}{dt}\right|_{t=0}\pi(\exp(tX))v, \qquad X\in\mathfrak g,\quad v\in\mathcal H^\infty.

The map $X\mapsto d\pi(X)$ is a Lie-algebra representation by generally unbounded operators sharing the domain $\mathcal H^\infty$ . Here $\mathfrak g$ is real: dependence on $X$ is real-linear, while each $d\pi(X)$ is complex-linear on $\mathcal H^\infty$ .

Domain and invariance

The smooth-vector space is preserved both by $\pi(G)$ and by every $d\pi(X)$ . Thus iterated expressions such as $d\pi(X_1)\cdots d\pi(X_k)v$ are defined on one canonical domain, rather than on an intersection chosen separately for each product. With its standard Fréchet topology, each $d\pi(X)$ is a continuous endomorphism of $\mathcal H^\infty$ . The space is a continuous module over the universal enveloping algebra $U(\mathfrak g_{\mathbb C})$ . The density and invariance of this space are basic smooth-vector results.

Relation to one-parameter generators

For fixed $X$ , the curve $t\mapsto\pi(\exp(tX))$ is a strongly continuous one-parameter unitary group. By Stone's theorem, the operator $d\pi(X)$ on $\mathcal H^\infty$ is skew-symmetric, is essentially skew-adjoint, and its closure is the infinitesimal generator of that group. Equivalently, if the self-adjoint-generator convention writes $\pi(\exp(tX))=e^{itA_X}$ , then $d\pi(X)=iA_X$ on smooth vectors. This factor of $i$ explains an important convention difference between Lie representation theory and spectral theory.

Algebraic identities and equivariance

For $X,Y\in\mathfrak g$ and $v\in\mathcal H^\infty$ ,

d\pi([X,Y])v =\bigl(d\pi(X)d\pi(Y)-d\pi(Y)d\pi(X)\bigr)v.

The group and infinitesimal actions are compatible through

\pi(g)d\pi(X)\pi(g)^{-1}v=d\pi(\operatorname{Ad}(g)X)v.

These identities hold on the smooth domain; treating the operators as everywhere-defined bounded operators is generally incorrect. If the original representation is finite-dimensional and smooth, this construction reduces to the ordinary differential of a Lie-group representation.

Concrete and geometric realizations

The Heisenberg example gives differentiation and multiplication by $ix$ on Schwartz space, with their commutator equal to $iI$ . For functions transported by inverse pullback along a smooth action, the derivative is minus the Lie derivative along the positive-exponential fundamental field.

References

Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer, 1972. Springer DOI record. Relevant: §4.4 on differentiable and smooth vectors.
V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Springer, 1984. Springer DOI record. Relevant: Chapter 4 on differentiating representations.