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chluo
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relative rank two group: structure of parabolic subgroup-- high-level Jacob MorozovJacobson--Morozov sl_2 triple

Given a parabolic subgroup $P=MN$ of a connected reductive group $G$ defined over a local field $F$, let $W_M$ be the relative Weyl group of $M$ in $G$, assume that the reduced roots relative to $M$ in $G$ form a root system of rank two, i.e. $$W_M=\left< S_\alpha, S_\beta \right>,$$ where $\alpha$ and $\beta$ are the simple roots of M in G, and $S_\alpha$ and $S_\beta$ are the corresponding simple reflections which satisfy $$S_\alpha.M=M \mbox{ and } S_\beta.M=M.$$

For simple root $\alpha$ (resp. $\beta$), we denote by $M_\alpha$ (resp. $M_\beta$) the Levi subgroup of its associated maximal parabolic subgroup. Denote by $M'_\alpha$ the derived group of $M_\alpha$, similarly $M'_\beta$.

The question about high-level Jacob MorozovJacobson--Morozov SL_2 triple is as follows:

at least one of $M'_\alpha$ and $M'_\beta$ is isogenous to $SL(n,D)$?

where $D$ is a division algebra over $F$.

If $G$ is quasi-split and $P$ is a minimal parabolic subgroup, then it is isogenous to $Res_{E/F}SL_2$ or $SU(2,1)_{F(\sqrt{d})/F}$.

Thank you so much for your help.

relative rank two group: structure of parabolic subgroup-- high-level Jacob Morozov sl_2 triple

Given a parabolic subgroup $P=MN$ of a connected reductive group $G$ defined over a local field $F$, let $W_M$ be the relative Weyl group of $M$ in $G$, assume that the reduced roots relative to $M$ in $G$ form a root system of rank two, i.e. $$W_M=\left< S_\alpha, S_\beta \right>,$$ where $\alpha$ and $\beta$ are the simple roots of M in G, and $S_\alpha$ and $S_\beta$ are the corresponding simple reflections which satisfy $$S_\alpha.M=M \mbox{ and } S_\beta.M=M.$$

For simple root $\alpha$ (resp. $\beta$), we denote by $M_\alpha$ (resp. $M_\beta$) the Levi subgroup of its associated maximal parabolic subgroup. Denote by $M'_\alpha$ the derived group of $M_\alpha$, similarly $M'_\beta$.

The question about high-level Jacob Morozov SL_2 triple is as follows:

at least one of $M'_\alpha$ and $M'_\beta$ is isogenous to $SL(n,D)$?

where $D$ is a division algebra over $F$.

If $G$ is quasi-split and $P$ is a minimal parabolic subgroup, then it is isogenous to $Res_{E/F}SL_2$ or $SU(2,1)_{F(\sqrt{d})/F}$.

Thank you so much for your help.

relative rank two group: structure of parabolic subgroup-- high-level Jacobson--Morozov sl_2 triple

Given a parabolic subgroup $P=MN$ of a connected reductive group $G$ defined over a local field $F$, let $W_M$ be the relative Weyl group of $M$ in $G$, assume that the reduced roots relative to $M$ in $G$ form a root system of rank two, i.e. $$W_M=\left< S_\alpha, S_\beta \right>,$$ where $\alpha$ and $\beta$ are the simple roots of M in G, and $S_\alpha$ and $S_\beta$ are the corresponding simple reflections which satisfy $$S_\alpha.M=M \mbox{ and } S_\beta.M=M.$$

For simple root $\alpha$ (resp. $\beta$), we denote by $M_\alpha$ (resp. $M_\beta$) the Levi subgroup of its associated maximal parabolic subgroup. Denote by $M'_\alpha$ the derived group of $M_\alpha$, similarly $M'_\beta$.

The question about high-level Jacobson--Morozov SL_2 triple is as follows:

at least one of $M'_\alpha$ and $M'_\beta$ is isogenous to $SL(n,D)$?

where $D$ is a division algebra over $F$.

If $G$ is quasi-split and $P$ is a minimal parabolic subgroup, then it is isogenous to $Res_{E/F}SL_2$ or $SU(2,1)_{F(\sqrt{d})/F}$.

Thank you so much for your help.

Source Link
chluo
chluo
  • 301
  • 1
  • 8

relative rank two group: structure of parabolic subgroup-- high-level Jacob Morozov sl_2 triple

Given a parabolic subgroup $P=MN$ of a connected reductive group $G$ defined over a local field $F$, let $W_M$ be the relative Weyl group of $M$ in $G$, assume that the reduced roots relative to $M$ in $G$ form a root system of rank two, i.e. $$W_M=\left< S_\alpha, S_\beta \right>,$$ where $\alpha$ and $\beta$ are the simple roots of M in G, and $S_\alpha$ and $S_\beta$ are the corresponding simple reflections which satisfy $$S_\alpha.M=M \mbox{ and } S_\beta.M=M.$$

For simple root $\alpha$ (resp. $\beta$), we denote by $M_\alpha$ (resp. $M_\beta$) the Levi subgroup of its associated maximal parabolic subgroup. Denote by $M'_\alpha$ the derived group of $M_\alpha$, similarly $M'_\beta$.

The question about high-level Jacob Morozov SL_2 triple is as follows:

at least one of $M'_\alpha$ and $M'_\beta$ is isogenous to $SL(n,D)$?

where $D$ is a division algebra over $F$.

If $G$ is quasi-split and $P$ is a minimal parabolic subgroup, then it is isogenous to $Res_{E/F}SL_2$ or $SU(2,1)_{F(\sqrt{d})/F}$.

Thank you so much for your help.