Differentiable vector in a representation space

$V$ of a representation $T$ of a Lie group $G$

A vector $\xi \in V$ for which the mapping $$ g \mapsto T(g) \xi $$ is an infinitely-differentiable (of class $C^\infty$) vector function on $G$ with values in $V$. For the vector function $f : G \rightarrow V$ to be differentiable, a necessary condition (and, in the case of a locally convex quasi-complete space $V$, also a sufficient condition) is that all scalar functions of the type $F \circ f$, where $F$ is a linear continuous functional on $V$, be differentiable [1]. The Gel'fand–Gårding theorem may be stated as follows: If $T$ is a continuous representation of a Lie group $G$ in a Banach space $V$, then the set $V^\infty$ of differentiable vectors is dense in $V$. This theorem has been proved for one-parameter groups in [2], and for the general case in [3]. For a generalization of this result to a wide class of representations on locally convex spaces see [4] and [5].

The presence of differentiable vectors in the representation space of a Lie group makes it possible to construct a representation of the corresponding Lie algebra, and thus connect the theory of representations of groups with the theory of representations of Lie algebras [6].

References

[1]	A. Grothendieck, "Espaces vectoriels topologiques" , Univ. Sao Paulo (1954)
[2]	I.M. Gel'fand, "On one-parameter groups of operators in a normed space" Dokl. Akad. Nauk. SSSR , 25 (1939) pp. 713–718 (In Russian)
[3]	L. Gårding, "Note on continuous representations of Lie groups" Proc. Nat. Acad. Sci. USA , 33 (1947) pp. 331–332
[4]	R.T. Moore, "Measurable, continuous and smooth vectors for semigroups and group representations" , Amer. Math. Soc. (1968)
[5]	D.P. Zhelobenko, "On infinitely differentiable vectors in representation theory" Vestnik Moskov. Univ. Ser. Mat. , 1 (1965) pp. 3–10 (In Russian) (English summary)
[6]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)

References

[a1]	G. Warner, "Harmonic analysis on semi-simple Lie groups" , 1 , Springer (1972)