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YCor
YCor
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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)

Recall that a prehomogeneous vector space, is a representation $V$ of a linear 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)

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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Roman Fedorov
Roman Fedorov
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Prehomogeneous vector spaces for reductive groups

Recall that a prehomogeneous vector space, is a representation $V$ of a linear 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)