Whenever I read "centraliser of maximal split torus", I think of...

Question

Inspired by this question I'd like to ask something more specific:

In the theory of connected reductive groups over fields, one often reads about the centraliser of a maximal split torus. Here is one example: let $k$ be a field and $D$ a skewfield containing $k$ such that $D$ is a finite-dimensional central simple $k$-algebra. Then for any $n \ge 1$, in the $k$-group ${\rm{SL}}_ n(D)$ the $k$-subgroup $T$ consisting of diagonal matrices with entries in ${\rm{GL}}_ 1$ is a maximal split $k$-torus, while the centraliser $S$ of $T$ consists of the diagonal matrices with entries in $D^{\times}$ (viewed as a $k$-group in the usual way).

Is this example typical or is it too simple-minded to capture the mysteries of these centralisers?

Answer 1

It would help to place your question in the context of the foundational 1965 paper Groupes reductifs by Borel and Tits, freely available online from NUMDAM: http://archive.numdam.org/

For example, their Section 4 studies centralizers of maximal $k$-split tori in terms of roots, parabolic subgroups, Levi subgroups. This set-up was used by Tits to codify many details of the classification of semisimple groups over fields of special interest: finite, local, algebraic number fields, etc.
Relative to a field of definition, certain Levi subgroups of parabolic subgroups are natural examples of the centralizers you want. Your proposed example needs to be placed more carefully within this Borel-Tits framework, I think.

The story about structure and classification of reductive groups over arbitrary fields is a long one, but the Tits strategy is to start with the known split groups and then adapt the Dynkin diagram to a field of definition. See his paper in the proceedings of the 1965 AMS Summer Institute at Boulder, available freely online through AMS e-math in the first part of the volume: MR0224710 (37 #309) Tits, J. Classification of algebraic semisimple groups. 1966 Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) pp. 33--62 Amer. Math. Soc., Providence, R.I., 1966. See the Web page http://www.ams.org/online_bks/pspum9/

More details were worked out by a student of Tits at Bonn: see MR0432776 (55 #5759) Selbach, Martin Klassifikationstheorie halbeinfacher algebraischer Gruppen. Diplomarbeit, Univ. Bonn, Bonn, 1973. Bonner Mathematische Schriften, Nr. 83. Mathematisches Institut der Universität Bonn, Bonn, 1976. v+140 pp.

Your group is of inner type A in the classification, using the Dieudonne determinant notation. So this really isn't so "typical", but occurs in the Tits list. The "split" data in his diagrams is somewhat independent of the ground field, but the remaining classification problem for anisotropic groups depends strongly on the field.

Answer 2

Can you give more detail about what sort of answer you're seeking? The groups arising in this way are precisely those that are anisotropic modulo centre—i.e., for which the only split tori are central. (Indeed, the construction just returns the original group if one starts with such a group.) I may have the terminology wrong, but I think that the centraliser arising in this way (which is unique up to conjugacy) is called the anisotropic kernel of the original group. Indeed, as Jim mentions, Borel and Tits have developed a huge amount of structural information for these sorts of groups; in addition to Groupes réductifs, for the case of groups over local fields I recommend Tits's Corvallis paper. There are some surprises: For example, in the $p$-adic case, the only anisotropic groups are forms of $A_n$.

A truly trivial observation (because it's essentially the definition) is that you wind up with an Abelian centraliser (equivalently, a torus) if and only if your original group was $k$-quasi-split.

UPDATE: Having just read the linked question, I see that you were requesting examples. I tend to think of one of the 2 extremes: either something like the quasi-split $\operatorname{SO}(3)$, where we get a torus, or an already-anisotropic-modulo-centre group like $\operatorname{GL}_1(D)$. Then again, all my intuition is in the $p$-adics, so, as I mentioned, there simply aren't that many interesting kinds of non-Abelian-ness that can occur.

Answer 3

The centralizer of a max. split torus is (as Loren noted) the anisotropic kernel of G. Maybe the following additional example is useful: Let k be a field and Q a non-degenerate quadratic form over k, and let G = SO(Q) (let's avoid char. 2 for simplicity...)

Then [Witt's Theorem] Q can be decomposed into an (orthogonal) sum

Q = Q_an + Q_hyp

where Q_an is an anisotropic quadratic form (has no non-trivial zeros), and where the quadratic from Q_hyp is hyperbolic ("looks like a quadratic form over an alg. closed field").

The choice of a hyperbolic basis for Q_hyp is (almost) the same as a choice of maximal split torus.

And the derived group of the centralizer of that maximal split torus is the anisotropic group SO(Q_an). [For detail on all this see e.g. [Borel, Linear Algebraic Groups 23.4] I'm sure there is an analogous reference in [Springer, LAG] but my copy of that book is elsewhere at the moment].

Of course, this is similar in spirit to your division algebra example. For a more elaborate source of examples, see the references Jim cites.