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# how to readoutrecognize subgroups fromthrough Dynkin diagram?

Fix $\mathbb{C}$ as the base field, and reductive groups are assumed to be connected.

Consider the example $SO_N\subset SL_N$. $SO_N$ is its own normalizer in $SL_N$, and the rank is much smaller , namely $[\dfrac{N-1}{2}]$, than the rank of $SL_N$. How can this be reflected in the Dynkin diagram of $SL_N$?

In http://mathoverflow.net/questions/18764 one sees that Levi subgroups $L$ of a given semi-siimple group $G$ can be recovered from Dynkin subdiagrams of the diagram of $G$, and subgroups of maximal rank can be recovered similarly from extended Dynkin diagrams. As is commented below, having maximal rank means being of the same rank as $G$ is, namely containing a maximal torus of $G$.

Contrary to the discussions in loc.cit, one considers a reductive subgroup $H\subset G$, such that the normalizer $N(H,G)$ of $H$ in $G$ is of lower rank than $G$. Its connected component, denoted as $N^\circ$, is not covered in loc.cit. Say $SO_{2n}\subset SL_{2n}$, the extended Dynkin diagram of $SL_{2n}$ looks like a loop with dots, while the one for $SO_{2n}$ contains branching vertices. It is not clear that the latter is produced from the former by removing vertices.

And furthermore when one passes to a general base field, say perfect or of characteristic zero for simplicity, with separable closure $\bar{k}$, it becomes more complicated if one restricts to the notion of $k$-rank. Consider a reductive $k$-subgroup $H\subset G$ that is self-normalizing, in the sense that it equals the neutral connected component of $N(H,G)$. By comparing the $k$-ranks and $\bar{k}$-ranks of $H$ and $G$, one is led to several different cases. Does one still have arguments similar to the operations on Dynkin diagrams as is in http://mathoverflow.net/questions/18764 ?

Sorry for any misunderstanding about the cited mathoverflow discussions above, and references on extended Dynkin diagrams are also welcome.

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In http://mathoverflow.net/questions/18764 one sees that Levi subgroups $L$ of a given semi-siimple group $G$ can be recovered from Dynkin subdiagrams of the diagram of $G$, and subgroups of maximal rank can be recovered similarly from extended Dynkin diagrams. Does the notion of As is commented below, having maximal rank here means that the subgroups in question have being of the same rank as $G$ does?

If is, namely containing a maximal torus of $G$.

Contrary to the discussions in loc.cit, one works over considers a field reductive subgroup $k$ not algebraically closedH\subset G$, such that the normalizer$N(H,G)$of$H$in$G$is there a relative version of lower rank than$G$. Its connected component, denoted as$N^\circ$, is not covered in loc.cit. Say$SO_{2n}\subset SL_{2n}$, the extended Dynkin diagram available for this kind of inclusion of subgroups$SL_{2n}$looks like a loop with dots, especially while the one for those subgroup$H$such SO_{2n}$ contains branching vertices. It is not clear that the normalizer latter is produced from the former by removing vertices.

And furthermore when one passes to a general base field, say perfect or of characteristic zero for simplicity, with separable closure $N(H,G)$ are \bar{k}$, it becomes more complicated if one restricts to the notion of lower rank than$k$-rank. Consider a reductive$k$-subgroup$H\subset G$itselfthat is self-normalizing, in the sense that it equals the neutral connected component of$N(H,G)$. By comparing the$k$-ranks and$\bar{k}$-ranks of$H$and$G$, one is led to several different cases. Does one still have arguments similar to the operations on Dynkin diagrams as is in http://mathoverflow.net/questions/18764 ? 1 # how to read out subgroups from Dynkin diagram? Fix$\mathbb{C}$as the base field, and reductive groups are assumed to be connected. Consider the example$SO_N\subset SL_N$.$SO_N$is its own normalizer in$SL_N$, and the rank is much smaller, namely$[\dfrac{N-1}{2}]$, than the rank of$SL_N$. How can this be reflected in the Dynkin diagram of$SL_N$? In http://mathoverflow.net/questions/18764 one sees that Levi subgroups$L$of a given semi-siimple group$G$can be recovered from Dynkin subdiagrams of the diagram of$G$, and subgroups of maximal rank can be recovered similarly from extended Dynkin diagrams. Does the notion of maximal rank here means that the subgroups in question have the same rank as$G$does? If one works over a field$k$not algebraically closed, is there a relative version of extended Dynkin diagram available for this kind of inclusion of subgroups, especially for those subgroup$H$such that the normalizer$N(H,G)$are of lower rank than$G\$ itself?

Sorry for any misunderstanding about the cited mathoverflow discussions above, and references on extended Dynkin diagrams are also welcome.