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Suppose $K$ is a (not necessarily algebraically closed) field, and $G_1$ and $G_2$ are split semisimple algebraic groups over $K$ which become isomorphic over $\bar{K}$, the algebraic closure of $K$. Are $G_1$ and $G_2$ isomorphic over $K$? What about if the $G$s are reductive?

It seems like this should follow (at least in the semisimple case) from Tits' general structure theorem for semisimple groups over a not-necessarily algebraically closed field; as explained in section 35.5 of Humphreys' Linear Algebraic Groups, a semisimple algebraic group is determined by its $\bar{K}$ isomorphism class, its anisotropic kernel (which looks like it is trivial for a split group) and its `index' (for which there again only seems to be one choice for a split group). But I am not expert enough to completely trust this argument...

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  • $\begingroup$ Does the existence of isotropic quadratic forms over Q which are not similar give a counterexample (via the orthogonal groups)? $\endgroup$
    – Pete L. Clark
    Commented Feb 15, 2010 at 22:07
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    $\begingroup$ Pete: I think these groups aren't going to be split in general. $\endgroup$
    – Kevin Buzzard
    Commented Feb 15, 2010 at 23:16
  • $\begingroup$ @Kevin: Yes, I see it now. I was thinking of the three-dimensional case for part of my argument and of the > three-dimensional case for another part. $\endgroup$
    – Pete L. Clark
    Commented Feb 15, 2010 at 23:33
  • $\begingroup$ If I am underanding things correctly now, the orthogonal group of a quadratic form will be split iff the quadratic form has maximal Witt index $\lfloor \operatorname{dim}(q)/2 \rfloor$? $\endgroup$
    – Pete L. Clark
    Commented Feb 16, 2010 at 7:27
  • $\begingroup$ @Pete: yes, that is correct. $\endgroup$
    – George McNinch
    Commented Apr 22, 2010 at 16:40

2 Answers 2

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The answer is yes, for arbitrary split connected reductive groups over any field. The main point is that the Existence, Isomorphism, and Isogeny Theorems (relating split connected reductive groups and root data) are valid over any field. One reference is SGA3 near the end (which works over any base scheme), but in Appendix A.4 of the book "Pseudo-reductive groups" there is given a direct proof over fields via faithfully flat descent, taking as input the results over algebraically closed fields (since for some reason the non-SGA3 references always seem to make this restriction).

[Caveat: that A.4 gives a complete treatment for the Isomorphism and Isogeny Theorems over general ground fields, and that is what the question is really about anyway; for the Existence Theorem in the case of exceptional types I don't know a way to "pull it down" from an algebraic closure, instead of having to revisit the constructions to make them work over prime fields or $\mathbf{Z}$.]

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  • $\begingroup$ In short: over any field you like there's a canonical bijection between iso classes of based root data (this is just a bunch of linear algebra) and iso classes of split reductive groups. If you just want to see statements then I've always found the articles by Springer and Borel in Corvallis to be quite precise (although for proofs you'll have to dig deeper e.g. following up Brian's references). $\endgroup$
    – Kevin Buzzard
    Commented Feb 15, 2010 at 23:19
  • $\begingroup$ Kevin, where in Corvallis does anyone state the result for split connected reductive groups over a general field? I just flipped through Springer's article; did I miss it? Apart from that, it is useful that the classification is a bit better than isomorphism classes: one can capture isogenies as well (and distinguish central from Frobenius from the weird types that only exist in small nonzero characteristics). For years it drove me up the wall that I couldn't find a non-SGA3 reference allowing a general ground field; now that problem is solved. :) $\endgroup$
    – BCnrd
    Commented Feb 15, 2010 at 23:46
  • $\begingroup$ I can't find a clean reference. How about this: Remark 2.10 gives you the existence in the semi-simple case (over any field). Some homework now gives you existence in the reductive case. G split implies that the map he calls "mu_G" is trivial. Now say G and H are both split forms of the same conn red group over k-bar. The remark at the end of 3.2 implies G,H are inner forms of one another. Now some more homework implies G and H are isomorphic. Most definitely not as good as I had hoped in terms of the "reference" stakes... $\endgroup$
    – Kevin Buzzard
    Commented Feb 16, 2010 at 11:21
  • $\begingroup$ I'll remark that when Toby Gee and I at some point wanted such statements (at some point in our pre-preprint we wanted the L-group of a conn red group to be a group over Q) we invoked SGA3... $\endgroup$
    – Kevin Buzzard
    Commented Feb 16, 2010 at 11:46
  • $\begingroup$ Section II.1 of Jantzen's book Representations of Algebraic Groups contains an account of the existence and isomorphism theorems (but not the isogeny theorem) for split reductive groups G over a PID k (or rather, over Spec(k)), following arguments of Takeuchi. $\endgroup$
    – George McNinch
    Commented Apr 22, 2010 at 17:08
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Here is another reference which seems very readable: A. Borel, J. Tits, Groupes réductifs Publ. Math. IHES , 27 (1965) pp. 55–150, Theorem 2.13: Two reductive $K$-split groups $G$ and $G'$ which are isomorphic over $\bar{K}$ are already isomorphic over $K$.

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