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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. J. Functional Analysis 9 (1972), 87–120, Theorem 5.1.