Yes, your desired equality is true: regarding the left regular representation as
$$
\operatorname{Ind}_{\{0\}}^{\mathbb R}1,
$$
it becomes a special case of the characterization of smooth vectors in induced representations by N. S. Poulsen, [*On $C^\infty$-vectors and intertwining bilinear forms for representations of Lie groups*][1]. J. Functional Analysis **9** (1972), 87–120, Theorem 5.1.


  [1]: http://www.ams.org/mathscinet-getitem?mr=310137