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Deleting proxy re-direction from MR link that I blindly C&P'd (sorry!)
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edit approved Jun 10, 2016 at 0:24
LSpice
edit approved Jun 10, 2016 at 0:24
LSpice
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Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see, e.g., D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341MR1650341), which motivates my question.

Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see, e.g., D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341), which motivates my question.

Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see, e.g., D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341), which motivates my question.

Added links to the paper by Poguntke and its MR
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edit approved Jun 9, 2016 at 23:14
LSpice
edit approved Jun 9, 2016 at 23:14
LSpice
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Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see, e.g. D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341MR1650341), which motivates my question.

Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see e.g. D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341), which motivates my question.

Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see, e.g., D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341), which motivates my question.

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Guntram
Guntram
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The normalizer of a reductive subgroup

Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see e.g. D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341), which motivates my question.