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I am interested in knowing various examples of non-split (added hypothesis reductive) reductive linear algebraic groups. In particular, I would like to collect the following examples in my counter-example toolbox.

  1. Given an integer $n>2$, an extension $E/F$ of degree $n$ and a connected reductive $F$-group $G$ that splits over $E$ (added following Prof. Humphreys' comment:) and $E$ is the smallest such extension minimally;

  2. Given a finite set $S$ of primes, a connected reductive group $G$ defined over $\mathbb Q$ which is quasi-split precisely over $\mathbb Q_v$ for $v \not\in S$ and not otherwise.

The motivation for this question was that I was reading Tits' article on Buildings in Corvallis and wanted to explicitly compute stuff about non-split groups. This prompted me to look for nonsplit groups, hence this question.

[I would like to make this question community-wiki since I am really asking 2 questions here and there is no unique answer, but for some reasons I can't find the CW box. Perhaps I need more reputation. ]

Abhishek Parab
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    Once source of many examples is Weil restriction of scalars. For example, take a non-trivial finite Galois extension $E/F$ and a split torus $T$ over $E$. Then $\mathrm{Res}_{E/F}(T)$ is a non-split torus over $F$ that is split by $E$ (it's non-split because its character lattice has a non-trivial Galois action). – Keenan Kidwell Feb 14 '14 at 00:35
  • Two comments: 1) It would help to specify whether you are looking for semisimple groups or other types. 2) In the first question, I suspect you want $E$ to be a minimal splitting field over $F$ (?) – Jim Humphreys Feb 14 '14 at 00:58
  • Thanks Prof. Humphreys, I edited the question - yes, in both cases. – Abhishek Parab Feb 14 '14 at 01:15
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    (2) is basically impossible: Any connected reductive group over $\mathbb{Q}$ will be quasi-split over almost all primes: The variety $\mathcal{B}$ of Borel sub-groups of any such group $G$ is smooth and projective. For almost all finite primes $p$, $G$ has a smooth reductive model over $\mathbb{Z}_p$. By Lang's theorem, combined with Hensel's lemma, $\mathcal{B}(\mathbb{Q}_p)$ is non-empty, for almost all $p$. – Keerthi Madapusi Feb 14 '14 at 01:35
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    So the correct formulation of (2) is: Given a finite set of primes $S$, find an example of a group $G$ such that $S$ consists precisely of the primes at which $G$ is not quasi-split. – Keerthi Madapusi Feb 14 '14 at 01:36
  • If $S$ has an even number of elements, then an example is the group of units in a quaternion algebra $D$ that is non-split exactly at the places in $S$. – Keerthi Madapusi Feb 14 '14 at 01:38
  • Keerthi: Sorry for the error, I really meant $v \not\in S$. Also could you give a reference for the statement "For almost all p, G has a smooth reductive model over Z_p". Thanks. – Abhishek Parab Feb 14 '14 at 01:49
  • Abhishek: Okay, I wasn't sure if it was a typo or not. As for the statement on reductiveness, one can assume that $G$ has a smooth, affine model over $\mathbb{Z}[1/n]$ for some suitable integer $n$. Now apply Corollary 2.6 from Exp. XIX of SGA 3. – Keerthi Madapusi Feb 14 '14 at 02:19
  • On the other hand, one doesn't really need reductiveness for this. Lang's theorem applies to any smooth connected algebraic group over a finite field and Hensel's lemma only needs smoothness. So all you really need is that $G$ has smooth connected fibers for almost all primes $p$. This is a much easier assertion and is true for smooth connected $\mathbb{Q}$-varieties in general. – Keerthi Madapusi Feb 14 '14 at 02:20
  • @KeerthiMadapusiPera: Without a rational point, "connected" is not "geometrically connected" for schemes over a field. If $K$ is a nontrivial finite extension of $\mathbf{Q}$ then ${\rm{Spec}}(K)$ is a smooth connected $\mathbf{Q}$-scheme with no smooth affine integral model (over $\mathbf{Z}[1/N]$ for sufficiently divisible $N$, say) having all but finitely many fibers connected. Also, without reductivity the special and generic fibers of a smooth affine group with connected fibers over a dvr may have Borels of different dimensions (Bruhat-Tits group schemes!), so Lang's theorem isn't enough. – user76758 Feb 14 '14 at 04:18
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    The first question is ill-posed: if $E, E'$ are finite (separable) extensions of $F$ and a connected semisimple $F$-group $G$ splits over $E$ and $E'$ then it generally doesn't split over $E \cap E'$. Unit groups of central simple algebras provide lots of examples over number fields (since the global splitting over a finite extension is controlled by local splitting over a finite set of places). So speaking of "the smallest such extension" doesn't quite make sense (aside from special cases like tori). – user76758 Feb 14 '14 at 04:42
  • @user76758-Thanks for the corrections! – Keerthi Madapusi Feb 14 '14 at 13:42
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    @user76758 - I was sloppy but I thought there is no confusion, since Humphreys' comment said minimally but I edited the question. Could you suggest a reference where I can find about unit groups of CSAs that you mention? (PS: $E\cap E'$ is ill-posed too :) ) – Abhishek Parab Feb 14 '14 at 14:03

2 Answers2

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Standard examples are given by central simple algebras, as mentioned in the comments. Namely, let $A$ be a central simple algebra over a field $F$ of dimension $n^2$. Then $G = A^\times$ is an inner form of $GL(n)$, and it will be ramified (non-split) at a finite set of places. Further $A$ splits over an extension $E/F$ of degree $n$, and over global or local fields, one can take $E/F$ to be cyclic. If $n=2$, you get the case of quaternion algebras, which you are probably familiar with.

Just like the case of quaternion algebras over global fields, which must be ramified at an even number of places, one cannot make the ramification of CSAs arbitrary (for quaternion algebras, local there are only 2 choices, but for higher degree CSAs, there are more.) In particular one cannot let $S$ have cardinality 1--you need to be ramified at at least 2 places (and you can get by with 2 in higher degree as well).

There are many references one could give. From a number theory perspective, there's Weil's Basic Number Theory, Platonov-Rapinchuk, and I believe Milne's notes on Class Field Theory have some stuff. Standard references, from a less number theoretic point of view, are

  • Reiner, Maximal Orders
  • Pierce, Associative Algebras
  • Gille-Szamuely, Central Simple Algebras and Galois Cohomology

Personally, I like Reiner and Pierce, and I think Platonov-Rapinchuk gives a nice (relatively short) overview in the beginning. The latter 3 references should all give explanations of how to explicitly construct such CSAs as cyclic algebras.

Similarly, one can come up with other examples by looking at non-split forms of classical groups over global fields.

Kimball
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    Crap! I temporarly thought this year was 2014, and this question was recent. Better late than never, maybe? – Kimball Feb 26 '15 at 07:57
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If you allow yourself to work over fields of positive characteristic then finite reductive groups give you a raft of examples for (1). Indeed, let $G$ be a connected reductive algebraic group over $\mathbb{K} = \overline{\mathbb{F}_p}$ an algebraic closure of prime order $p>0$. Fix $T \leqslant B$ a maximal torus and Borel subgroup of $G$ and let $\{x_{\alpha} \mid \alpha \in \Delta\}$ be a pinning of $G$ with $\Delta$ the simple roots determined by $T \leqslant B$. For any $q$ a power of $p$ we can define a split Frobenius endomorphism $F_q : G \to G$ by $F_q(t) = t^q$ for all $t \in T$ and $F_q(x_{\alpha}(c)) = x_{\alpha}(c^q)$. Now let $\sigma : G \to G$ be any graph automorphism of $G$, i.e., $\sigma(x_{\alpha}(c)) = x_{\rho(\alpha)}(c)$ for some permutation $\rho$ of the roots. Then $F = \sigma \circ F_q = F_q\circ \sigma$ is also a Frobenius endomorphism of $G$ which makes $G$ into an $\mathbb{F}_q$-group. Indeed, we have the affine algebra $\mathbb{A}[G]$ is of the form $\mathbb{K}\otimes_{\mathbb{F}_q} A_0$ where $A_0 = \{a \in \mathbb{A}[G] \mid F^*(a) = a^q\}$ and $F^*$ is the comorphism. The minimal field over which $G$ splits will then be $\mathbb{F}_{q^n}$ where $n$ is the order of the automorphism $\sigma$.

One can get an example for any $n$ by taking $G = G_1 \times \cdots \times G_n$ a product of isomorphic factors and $\sigma$ acting by cyclic permutation. An example with $n>2$ and $G$ simple comes from the case of $\mathrm{D}_4$ and $\sigma$ the triality automorphism. But maybe these examples are not what you had in mind.

Jay Taylor
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