Interpolation spaces and unitary representations

Abstract:

Let $G$ be a Lie group, $\pi$ a unitary representation of $G$ on a Hilbert space $\mathcal {H}(\pi )$, and ${\mathcal {H}^k}(\pi )$ the subspace of ${C^k}$ vectors for $\pi$. By quadratic interpolation there is a continuous scale ${\mathcal {H}^s}(\pi )$, $s > 0$, of $G$-invariant Hilbert spaces. When $G = H \cdot K$ is a semidirect product of closed subgroups, then it is proved that ${\mathcal {H}^s}(\pi ) = {\mathcal {H}^s}({\left . \pi \right |_H}) \cap {\mathcal {H}^s}({\left . \pi \right |_K})$ for $s > 0$. For solvable $G$ this gives a characterisation of ${\mathcal {H}^s}(\pi )$ in terms of smoothness along one-parameter subgroups, and an elliptic regularity result.