Hokkaido Mathematical Journal

OHTA Takuya,

An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants.

Hokkaido Mathematical Journal, 37 (2008) pp.437-454

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Abstract

Let V be a finite dimensional vector space over the complex number field C. Suppose that, by the adjoint action, a reductive subgroup ˜G of GL(V ) acts on a subspace \˜L of End(V ) and a closed subgroup G of \˜G acts on a subspace L of \˜L. In this paper, we give a sufficient condition on the inclusion (G, L) \hookrightarrow ( \˜G, \˜L) for which the orbits correspondence L/G \to \˜L/˜G (O \to ˜O := Ad(\˜G) \dot O) is injective. Moreover we show that the ring C[L]^G of G-invariants on L is the integral closure of C[\˜L]^{\˜G}|_L in its quotient field. Then, if the ring C[\˜L]^{\˜G}|L is normal, the restriction map rest : C[\˜L]^{\˜G} \to C[L]^G (f \to f|_L) is surjective. By using this, we give some examples for which L/G \to \˜L/\˜G is injective and rest : C[\˜L]^{\˜G} \to C[L]^G is surjective.

MSC(Primary)13A50
MSC(Secondary)14R20, 14L35
Uncontrolled Keywordsinclusion theorem between sets of orbits, the restriction map of rings of invariants.