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Proceedings of the American Mathematical Society

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Interpolation spaces and unitary representations

Author: Roe Goodman
Journal: Proc. Amer. Math. Soc. 83 (1981), 153-158
MSC: Primary 22E25; Secondary 22E15, 46L99, 46M35
MathSciNet review: 620003
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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\vert _H}) \cap {\mathcal{H}^s}({\left. \pi \right\vert _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.

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Article copyright: © Copyright 1981 American Mathematical Society