Cartan subgroups of p-adic groups.

Question

Does anyone know about any reference describing the structure of Cartan subgroups in the case of connected p-adic reductive (or let's say semi-simple) groups? I would like to know how different is this from the real case.

Thanks in advance, Jim Riel

Answer 1

Hi Jim,

The situation is quite different in the p-adic case, although there are similarities. There are many things one can say about Cartan subgroups (what p-adic people usually call maximal tori) in connected reductive p-adic groups. However, the picture is not completely understood, which is one reason why you might be having difficulty finding good references.

For semisimple groups, U.R. Walker has a nice exposition of rational conjugacy classes of maximal tori in semisimple groups : https://www.math.lsu.edu/gradfiles/walker.pdf

Lawrence Morris explicitly describes elliptic tori (maximal tori that are anisotropic mod center) in classical groups and their groups of similitudes in this paper : "Some tamely ramified supercuspidal representations of symplectic groups", the proposition in Section 1.3.

Along a different line, Stephen DeBacker has parameterized all unramified maximal tori (maximal tori that split over an unramified extension) in this paper : "Parameterizing conjugacy classes of maximal unramified tori via Bruhat-Tits theory".

I believe that Stephen and Jeffrey Adler are currently working on extending their results to try to parameterize all tamely ramified tori (tori that split over a tamely ramified extension), though I'm not sure what the current status of that is.

The situation over the reals is much simpler, as you probably know. Any real Cartan subgroup is a product of copies of $S^1$, $\mathbb{C}^{\times}$, and $\mathbb{R}^{\times}$. One apparent reason for the higher level of complication for maximal tori in p-adic groups is the existence of more algebraic extensions in the p-adic setting. One similarity with p-adic groups is that the tori (that appear in real groups) also appear in p-adic groups.

To get one started, here are some basic examples :

1) Let $E/F$ be a quadratic extension of p-adic fields. Then $E^{\times}$ embeds in $GL(2,F)$ in the same way that $\mathbb{C}^{\times}$ embeds in $GL(2,\mathbb{R})$. Moreover, if $E/F$ is a separable degree $n$ extension of p-adic fields, then $E^{\times}$ embeds in $GL(n,F)$ as a maximal torus, in the analogous way.

2) Let $E/F$ be a quadratic extension of p-adic fields. Then $E^1$, the norm 1 elements from $E$ to $F$, embeds in $SL(2,F)$ in the same way that $S^1$ embeds in $SL(2,\mathbb{R})$.

3) For something slightly different : Let $E/F$ be the quadratic unramified extension of a p-adic field $F$. Let $K/F$ be the quartic unramified extension. Then $K_E^1$, the norm 1 elements from $K$ to $E$, embeds in $Sp(4,F)$. A general class of examples like this is discussed in the paper of Morris above.

Best,

Moshe

Answer 2

just stumbled upon this old thread and decided to contribute my 2 cents (though I completely agree with Moshe's answer): J.-L. Waldspurger has a very explicit description of conjugacy classes of semisimple elements in the classical groups, in his Asterisque volume "Integrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifies", I believe in Section I.7. This is, essentially, Moshe's examples 1 --3 but done in a systematic way for every classical group. One can get a parametrization of tori from this description of semisimple conjugacy classes by looking at centralizers.