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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A restriction theorem for semisimple Lie groups of rank one

Author: Juan A. Tirao
Journal: Trans. Amer. Math. Soc. 279 (1983), 651-660
MSC: Primary 22E46
MathSciNet review: 709574
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Abstract: Let $ {\mathfrak{g}_{\mathbf{R}}} = {\mathfrak{f}_{\mathbf{R}}} + {\mathfrak{p}_{\mathbf{R}}}$ be a Cartan decomposition of a real semisimple Lie algebra $ {\mathfrak{g}_{\mathbf{R}}}$ and let $ \mathfrak{g} = \mathfrak{f} + \mathfrak{p}$ be the corresponding complexification. Also let $ {\mathfrak{a}_{\mathbf{R}}}$ be a maximal abelian subspace of $ {\mathfrak{p}_{\mathbf{R}}}$ and let $ \mathfrak{a}$ be the complex subspace of $ \mathfrak{p}$ generated by $ {\mathfrak{a}_{\mathbf{R}}}$. We assume $ \dim {\mathfrak{a}_{\mathbf{R}}} = 1$. Now let $ G$ be the adjoint group of $ \mathfrak{g}$ and let $ K$ be the analytic subgroup of $ G$ with Lie algebra $ {\text{ad}}_\mathfrak{g}(\mathfrak{f})$. If $ S^\prime(\mathfrak{g})$ denotes the ring of all polynomial functions on $ \mathfrak{g}$ then clearly $ S^\prime(\mathfrak{g})$ is a $ G$-module and a fortiori a $ K$-module. In this paper, we determine the image of the subring $ S^\prime{(\mathfrak{g})^K}$ of $ K$-invariants in $ S^\prime(\mathfrak{g})$ under the restriction map $ f \mapsto f{\vert _{\mathfrak{f} + \mathfrak{a}}}(f \in S^\prime{(\mathfrak{g})^K})$.

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Article copyright: © Copyright 1983 American Mathematical Society