Abstract
Let HGL(V ) be a connected complex reductive group where V is a finite- dimensional complex vector space. Let v 2 V and let G = fg 2 GL(V ) jgHv = Hvg. Following Ra¨os (Ra¨o07) we say that the orbit Hv is characteristic for H if the identity component of G is H. If H is semisimple, we say that Hv is semi-characteristic for H if the identity component of G is an extension of H by a torus. We classify the H-orbits which are not (semi)-characteristic in many cases.
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