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What is Borel-de Siebenthal theory?

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    $\begingroup$ Whomever gave this a negative vote should state their objection (e.g., no context?). But I know this one. It's about studying the structure of identity components of centralizers of (possibly non-maximal, disconnected) mult.-type subgroups of a connected reductive group. It is described in terms of removing certain vertices/edges from the Dynkin diagram and inserting others. The original work of Borel and de Siebenthal was done for compact groups, but presumably nowadays the phrase refers to general (split?) connected reductive groups. Look at math.ens.fr/~gille/prenotes/bds.pdf $\endgroup$
    – BCnrd
    Commented Jun 20, 2010 at 19:59
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    $\begingroup$ @T: Presumably the poster encountered the term somewhere before asking the question; this is often sufficient context. $\endgroup$
    – François G. Dorais
    Commented Jun 20, 2010 at 20:24
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    $\begingroup$ As one click of a mouse can confirm for those who doubt it, the questioner is an expert asking an expert-level question. His question is not equivalent to asking "what is real analysis" or "please write an encyclopedia article at my command", and it is absurd to treat it as such. Most of the questions on MO, even the excellent ones, would be eliminated or shrunken by a requirement to first perform extensive self-help, and even if such legwork were always feasible, it would degrade the discussions because questioners rarely know what they SHOULD be asking about, for lack of knowledge. $\endgroup$
    – T..
    Commented Jun 20, 2010 at 20:54
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    $\begingroup$ Actually, I prefer expert questions stated concisely, rather than rambling discussion of not always relevant motivations, $\textit{precisely}$ because MO is an answer resource not an encyclopaedia. $\endgroup$
    – Victor Protsak
    Commented Jun 20, 2010 at 23:46
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    $\begingroup$ Thank you very much for all the mathematical responses, and let me apoologize to MO community for misusing the resource. I certainly did not ask for a wikipedia article (though Jim's answer is close to such an article). I did not ask for references because I just wanted to get a rough idea. I tried to google the answer before I asked. I did not think how to make the question valuable to other people though I assumed that those who are interested in Representation theory would read. $\endgroup$
    – Roman Fedorov
    Commented Jun 21, 2010 at 2:24

2 Answers 2

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I'm not sure the term "theory" is appropriate here, but the joint paper by Borel and de Siebenthal has had considerable influence in Lie theory over the years: MR0032659 (11,326d) Borel, A.; De Siebenthal, J., Les sous-groupes fermés de rang maximum des groupes de Lie clos. Comment. Math. Helv. 23, (1949). 200--221. (There was a short Comptes Rendus announcement in 1948.) This is found near the start of the Springer four-volume collected papers of Borel, with a couple of minor corrections appended.

To quote from the review by P.A. Smith: "Let $G$ be a compact Lie group, $G'$ a closed connected subgroup having the same rank as $G$. Let $Z'$ be the center of $G'$. The main object of this paper is to show that $G'$ is a connected component of the normalizator of $Z'$ in $G$." The proof involves
"a necessary and sufficient condition that a subsystem of root vectors of $G$ be the root vectors of a closed subgroup of $G$" and the subgroups of this type are found explicitly for all simple $G$. [Here $G$ is always assumed to be connected.]

The result on subsystems of root systems carries over in a natural way to the study of semisimple complex Lie (or algebraic) groups and their Lie algebras, for example the determination of subalgebras of maximal rank in the latter.

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In my head at least, part of it is this...

Let G be reductive. Consider the following algorithm:

  1. Extend one component of the Dynkin diagram to its affine diagram, by attaching the lowest root.

  2. From that component, remove one or more vertices. (If you want to stay semisimple, remove only one.)

  3. Repeat to taste. (If you want to stay semisimple, and you're at a disjoint union of $A_m$ diagrams, then you're definitely done.)

The result is the Dynkin diagram of a subgroup H of G of the same rank. Every such subgroup (up to finite factors) arises this way.

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  • $\begingroup$ I should have emphasized that the extended/affine diagrams and vertex removal is basic to the Borel-de Siebenthal classification of maximal subgroups, as well as some related developments: classification of maximal subgroups of semisimple algebraic groups or finite groups of Lie type in prime characteristic (Liebeck-Seitz, expanding on Dynkin's 1950s papers); centralizers of semisimple elements and pseudo-Levi subgroups or subalgebras (Carter-Deriziotis, Sommers); probably some of the geometry of affine Kazhdan-Lusztig cells, related in Lusztig's bijection to unipotent classes of dual groups.. $\endgroup$
    – Jim Humphreys
    Commented Jun 23, 2010 at 15:51
  • $\begingroup$ P.S. I don't follow the parenthetic remarks in Allen's numbered steps, about staying semisimple. (And change "is" to "are" in my first line. Though I ran out of characters.) $\endgroup$
    – Jim Humphreys
    Commented Jun 23, 2010 at 15:56
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    $\begingroup$ Steps 1+2 together replaces a root system by a sub root system. (Grr. I don't want to call it a sub-root system or a root subsystem. Anyway...) If you remove more than one vertex at step 2, the subsystem will be of lower rank, corresponding to a Levi subgroup of what you'd get from removing just one. E.g., if you start with G semisimple and remove all the vertices at step 2, the result is the *empty) Dynkin diagram of a maximal torus of G. As for the other remark, if you affinize and remove 1 vertex from an A_m diagram, you've done nothing. You're stuck; no more maximal rank s.s. subgroups. $\endgroup$
    – Allen Knutson
    Commented Jun 24, 2010 at 1:48
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    $\begingroup$ Expanding a little on Allen's algorithm, if you remove a single node with prime label, or two nodes with label 1, from the extended Dynkin diagram $\tilde D$, then the resulting subgroup H is maximal. Furthermore (modulo the action of $Z_{sc}$ on the extended Dynkin diagram) this classifies the maximal, reductive subgroups of the same rank as G up to conjugacy. Then you get every reductive, equal rank subgroup by iterating. However I do not know any way to keep track of conjugacy classes of the resulting subgroups. $\endgroup$
    – Jeffrey Adams
    Commented Mar 23, 2011 at 13:12

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