Recall that a prehomogeneous vector space, is a representation $V$ of a linear algebraic group $G$ having an open $G$-orbit. Let $Z$ be the neutral connected component of the stabilizer of a point of the open orbit. Assume that $G$ and $Z$ are both reductive. Is it always true that the canonical homomorphism $Z/[Z,Z]\to G/[G,G]$ has finite kernel? (This homomorphism is interesting because its cokernel is responsible for irreducible components of the complement of the open orbit)
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1$\begingroup$ The statement is clear if $G$ is a torus. On the other hand, how common are prehomogeneous vector spaces with $G$ semisimple and $Z$ reductive? Is there a classification? Among the standard examples of prehomogeneous spaces for $G={\rm GL}_n$, the spaces $M_{n,k}$ of $n\times k$ matrices, $k<n$, with left multiplication action and $\bigwedge^2 k^n$, $n$ odd, (and their duals) have non-reductive $Z$, whereas $\bigwedge^2 k^n$, $n$ even, and $S^2 k^n$ (and their duals) have reductive $Z$ and the requested property holds. $\endgroup$– Victor ProtsakCommented Apr 4, 2018 at 3:00
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$\begingroup$ Victor Protsak, I do not think there are any prehomogeneous vector bundles with G semisimple. However, there are a lot of examples with G being reductive. Take any sl_2 triple {n_-,h,n} and let G be the centralizer of h, V be the 2-eigenspace of h. $\endgroup$– Roman FedorovCommented Apr 5, 2018 at 18:31
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