Differentiable vector of order k

Let $\pi$ be a strongly continuous unitary representation of a finite-dimensional Lie group $G$ on $H$ , and let $k\geq0$ be an integer. A vector $v\in H$ is a $C^k$ vector if its orbit map

g\longmapsto\pi(g)v

is $C^k$ : in local coordinates its norm derivatives through order $k$ exist and are continuous. Write $H^k$ for this subspace. A $C^1$ vector is also called a differentiable vector.

All orders versus one order

Strong continuity gives $H^0=H$ , and

H\supseteq H^1\supseteq H^2\supseteq\cdots, \qquad H^\infty=\bigcap_{k\geq0}H^k.

The last space consists of smooth vectors. Differentiability once must not be used as a synonym for smoothness at all orders.

Generator interpretation

For a finite-dimensional Lie group, membership in $H^k$ is equivalent to existence of every ordered composition of at most $k$ infinitesimal generators on the vector. The iterated-domain theorem makes these domain requirements precise. For the modulation group, $H^k=\{f\in L^2(\mathbb R):x^k f\in L^2(\mathbb R)\}$ .

References

Karl-Hermann Neeb, On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups. Definition 3.1(b) and Theorem 9.4.