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A theorem of Chevalley type for prehomogeneous vector spaces. (English) Zbl 0846.17006
Let \(G\) be a complex reductive group, acting linearly on \(V= \mathbb{C}^n\). Assume that \(V\) has a dense \(G\)-orbit. Let \(\mathbb{C} [V]_{G, \varphi}= \{f\in \mathbb{C}[ V]\mid f(gv)= \varphi (g) f(v)\}\) and \(f\) its non-zero element. Then it is known that there exists a unique \(G\)-orbit \(O_1\) which is closed in \(V\setminus f^{-1} (0)\). Let \(T\) be a maximal torus of the isotropy subgroup \(G_{v_1}\) of \(G\) at \(v_1\in O_1\), \(N_G (T)\) the normalizer of \(T\) in \(G\), \(G'= N_G (T)/T\), and \(V'= \{v\in V\mid tv= v\) for any \(t\in T\}\). We can show that \(\varphi\) induces a character of \(G'\), which we denote by the same letter \(\varphi\). Define \(\mathbb{C}[ V' ]_{G', \varphi}\) in the same way as above. Among other things, the following results are proved:
(1) The \(G'\)-orbit of \(v_1\) is open in \(V'\). (2) The isotropy subgroup \(G'_{v_1}\) of \(G'\) at \(v_1\) is finite. (3) The restriction induces an isomorphism \(\mathbb{C}[ V]_{G, \varphi} \widetilde {\to} \mathbb{C}[ V' ]_{G', \varphi}\).
Reviewer: A.Gyoja (Kyoto)

MSC:
17B20 Simple, semisimple, reductive (super)algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
20G05 Representation theory for linear algebraic groups
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