Semisimple elements of a parabolic subgroup are contained in some Levi
$begingroup$
Let $P$ be a parabolic $k$-subgroup of a connected, reductive group $G$ over a perfect field $k$. Let $N$ be the unipotent radical of $P$. It is defined over $k$. Let $M$ be a Levi $k$-subgroup of $P$.
Let $g in P(k)$ be a semisimple element. Do we always have $pgp^{-1} in M(k)$ for some $p in P(k)$?
I believe this should be true. It is a general result that if $M'$ is another Levi $k$-subgroup of $P$, then there exists an $n in N(k)$ such that $nM'n^{-1} = M$.
The question then becomes whether every semisimple element of $P(k)$ is contained in a Levi subgroup of $P$. Equivalently, every maximal torus of $P(k)$ which is defined over $k$ is contained in a Levi $k$-subgroup of $P(k)$.
algebraic-groups reductive-groups
asked Jan 8 at 2:10
D_SD_S
14.2k61755
$endgroup$
add a comment |
$begingroup$
Let $P$ be a parabolic $k$-subgroup of a connected, reductive group $G$ over a perfect field $k$. Let $N$ be the unipotent radical of $P$. It is defined over $k$. Let $M$ be a Levi $k$-subgroup of $P$.
Let $g in P(k)$ be a semisimple element. Do we always have $pgp^{-1} in M(k)$ for some $p in P(k)$?
I believe this should be true. It is a general result that if $M'$ is another Levi $k$-subgroup of $P$, then there exists an $n in N(k)$ such that $nM'n^{-1} = M$.
The question then becomes whether every semisimple element of $P(k)$ is contained in a Levi subgroup of $P$. Equivalently, every maximal torus of $P(k)$ which is defined over $k$ is contained in a Levi $k$-subgroup of $P(k)$.
algebraic-groups reductive-groups
asked Jan 8 at 2:10
D_SD_S
14.2k61755
$endgroup$
$begingroup$
The connected centralizer of some regular semisimple element $gamma$ in $P(k)$ is a maximal torus $T_{gamma}$ and $T_{gamma}$ is the almost-direct prod. of its anisotropic and split parts $T_{an}T_{sp}=T_{gamma}$. Taking the connected centralizer of $T_{sp}$ gives a Levi of $P$ that contains $T_{gamma}$
$endgroup$
– wsokursk
Jan 8 at 16:00
add a comment |
$begingroup$
Let $P$ be a parabolic $k$-subgroup of a connected, reductive group $G$ over a perfect field $k$. Let $N$ be the unipotent radical of $P$. It is defined over $k$. Let $M$ be a Levi $k$-subgroup of $P$.
Let $g in P(k)$ be a semisimple element. Do we always have $pgp^{-1} in M(k)$ for some $p in P(k)$?
I believe this should be true. It is a general result that if $M'$ is another Levi $k$-subgroup of $P$, then there exists an $n in N(k)$ such that $nM'n^{-1} = M$.
The question then becomes whether every semisimple element of $P(k)$ is contained in a Levi subgroup of $P$. Equivalently, every maximal torus of $P(k)$ which is defined over $k$ is contained in a Levi $k$-subgroup of $P(k)$.
algebraic-groups reductive-groups
asked Jan 8 at 2:10
D_SD_S
14.2k61755
$endgroup$
Let $P$ be a parabolic $k$-subgroup of a connected, reductive group $G$ over a perfect field $k$. Let $N$ be the unipotent radical of $P$. It is defined over $k$. Let $M$ be a Levi $k$-subgroup of $P$.
Let $g in P(k)$ be a semisimple element. Do we always have $pgp^{-1} in M(k)$ for some $p in P(k)$?
I believe this should be true. It is a general result that if $M'$ is another Levi $k$-subgroup of $P$, then there exists an $n in N(k)$ such that $nM'n^{-1} = M$.
The question then becomes whether every semisimple element of $P(k)$ is contained in a Levi subgroup of $P$. Equivalently, every maximal torus of $P(k)$ which is defined over $k$ is contained in a Levi $k$-subgroup of $P(k)$.
algebraic-groups reductive-groups
algebraic-groups reductive-groups
asked Jan 8 at 2:10
D_SD_S
14.2k61755
asked Jan 8 at 2:10
D_SD_S
14.2k61755
asked Jan 8 at 2:10
D_SD_S
14.2k61755
asked Jan 8 at 2:10
D_SD_S
14.2k61755
asked Jan 8 at 2:10
D_SD_S
14.2k61755
14.2k61755
$begingroup$
The connected centralizer of some regular semisimple element $gamma$ in $P(k)$ is a maximal torus $T_{gamma}$ and $T_{gamma}$ is the almost-direct prod. of its anisotropic and split parts $T_{an}T_{sp}=T_{gamma}$. Taking the connected centralizer of $T_{sp}$ gives a Levi of $P$ that contains $T_{gamma}$
$endgroup$
– wsokursk
Jan 8 at 16:00
add a comment |
$begingroup$
The connected centralizer of some regular semisimple element $gamma$ in $P(k)$ is a maximal torus $T_{gamma}$ and $T_{gamma}$ is the almost-direct prod. of its anisotropic and split parts $T_{an}T_{sp}=T_{gamma}$. Taking the connected centralizer of $T_{sp}$ gives a Levi of $P$ that contains $T_{gamma}$
$endgroup$
– wsokursk
Jan 8 at 16:00
$begingroup$
The connected centralizer of some regular semisimple element $gamma$ in $P(k)$ is a maximal torus $T_{gamma}$ and $T_{gamma}$ is the almost-direct prod. of its anisotropic and split parts $T_{an}T_{sp}=T_{gamma}$. Taking the connected centralizer of $T_{sp}$ gives a Levi of $P$ that contains $T_{gamma}$
$endgroup$
– wsokursk
Jan 8 at 16:00
$begingroup$
The connected centralizer of some regular semisimple element $gamma$ in $P(k)$ is a maximal torus $T_{gamma}$ and $T_{gamma}$ is the almost-direct prod. of its anisotropic and split parts $T_{an}T_{sp}=T_{gamma}$. Taking the connected centralizer of $T_{sp}$ gives a Levi of $P$ that contains $T_{gamma}$
$endgroup$
– wsokursk
Jan 8 at 16:00
add a comment |
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$begingroup$
The connected centralizer of some regular semisimple element $gamma$ in $P(k)$ is a maximal torus $T_{gamma}$ and $T_{gamma}$ is the almost-direct prod. of its anisotropic and split parts $T_{an}T_{sp}=T_{gamma}$. Taking the connected centralizer of $T_{sp}$ gives a Levi of $P$ that contains $T_{gamma}$
$endgroup$
– wsokursk
Jan 8 at 16:00