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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
##### Keywords:
prehomogeneous vector space; reductive group
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