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Transactions of the American Mathematical Society

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



A restriction theorem for modules having a spherical submodule

Authors: Nicolás Andruskiewitsch and Juan A. Tirao
Journal: Trans. Amer. Math. Soc. 331 (1992), 705-725
MSC: Primary 14L30; Secondary 13A30, 20G99
MathSciNet review: 1034657
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Abstract: We introduce the following notion: a finite dimensional representation $ V$ of a complex reductive algebraic group $ G$ is called spherical of rank one if the generic stabilizer $ M$ is reductive, the pair $ (G,M)$ is spherical and $ \dim \;{V^M}= 1$. Let $ U$ be another finite dimensional representation of $ G$; we denote by $ S^{\prime}(U)\;(S^{\prime}{(U)^G})$ the ring of polynomial functions on $ U$ (the ring of $ G$-invariant polynomial functions on $ U$). We characterize the image of $ S^{\prime}{(U \oplus V)^G}$ under the restriction map into $ S^{\prime}\,(U \oplus {V^M})$ as the $ W= {N_G}(M)/M$ invariants in the Rees ring associated to an ascending filtration of $ S^{\prime}{(U)^M}$. Furthermore, under some additional hypothesis, we give an isomorphism between the graded ring associated to that filtration and $ S^{\prime}{(U)^P}$, where $ P$ is the stabilizer of an unstable point whose $ G$-orbit has maximal dimension.

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Article copyright: © Copyright 1992 American Mathematical Society