\(C^{\infty}\)-vectors of irreducible representations of exponential solvable Lie groups (Q2468454)

To study unitary representations and harmonic analysis for solvable Lie groups, the orbit method could provide us with a powerful tool. In particular, when we construct an irreducible monomial representation \(\pi= \text{ind}^G_B\chi\) for a connected and simply connected nilpotent Lie group \(G\), the space \({\mathcal K}^\infty_\pi\) of \(C^\infty\)-vectors of \(\pi\) coincides as Fréchet space with the space \({\mathcal S}(\mathbb{R}^m)\), \(m= \dim(G/B)\), of rapidly decreasing smooth Schwartz functions. This well known fact is very useful and has many applications. Now, when we leave the nilpotent case, the determination of the space \({\mathcal K}^\infty_\pi\) is very difficult and it turns out to be one of the obstacles in the concrete analysis of unitary representations for exponential solvable Lie groups. This paper tries to attack this difficult problem. It defines some spaces of rapidly decreasing smooth functions on an exponential solvable Lie group \(G\), and proves that this space becomes the space of \(C^\infty\)-vectors of an irreducible representation of a certain exponential solvable Lie group.