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I start with a hopelessly broad question: what is known about the structure of the automorphism group of a (smooth, connected) unipotent group (over a field), and particularly about the structure of diagonalisable subgroup schemes of the automorphism group? It would be nice, but is not essential, if I could assume a non-algebraically closed field of definition.

I have two, I think sufficiently specifically focussed, questions. Suppose that $\Gamma$ is a diagonalisable group scheme acting on a unipotent group $U$.

  1. Can we extend the action of $\Gamma$ to some torus? (That is, are there a torus $T$ acting on $U$ and a map $\Gamma \to T$ such that the obvious diagram commutes?)

  2. Can we extend the action of $\Gamma$ to some diagonalisable group scheme $\tilde\Gamma$ with the property that there is some cocharacter $\tilde\lambda$ of $\tilde\Gamma$ such that $\langle\tilde\chi, \tilde\lambda\rangle > 0$ for all non-0 weights $\tilde\chi$ of $\smash{\tilde\Gamma}^\circ$ on $\mathrm{Lie}(U)$? (If necessary, I can assume here that $\Gamma$ is a torus.)

EDIT: nfdc23 points out that this is nearly a question about split unipotent groups, since wound groups admit no non-trivial torus actions. In fact, this is not far off my original motivation. In §15.13(b) of his famous book, Borel mentions a 1968 article (MR) with Springer in which he investigates rationality questions for algebraic groups. Particularly, §9 of that paper says quite a bit about the structure of unipotent groups equipped (most of the time) with a fixed-point-free torus action. (The result quoted by Borel says, in particular, that they must be split.) In most cases, it seems that the proofs work just as well if we have instead just an action by a diagonalisable group scheme. I say this on the basis just of working line-by-line through the proofs, but a positive answer to (1) would save such (as someone, I think Bushnell and Kutzko, memorably put it in another context) "lame and automatically suspect" reasoning.

In the course of these investigations, I realised that I'd been behaving as if arbitrary unipotent groups behaved like unipotent radicals of parabolic subgroups, in the sense at least that they were of the form $U(\lambda)$ (cf. Springer, §3.2.15). It's easy to see that this isn't true, but I was wondering how badly it could fail. That's the motivation for (2).

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    $\begingroup$ Unipotent groups don't have diagonalizable subgroups, so the first paragraph is a bit confusing. In #1 do you mean to write $\Gamma \rightarrow T$? Is your smooth connected unipotent group $U$ perhaps also split (e.g., over a perfect field, or the unipotent radical of a parabolic $k$-subgroup of a connected reductive $k$-group)? In general $U$ contains a normal $k$-split smooth connected $k$-subgroup $U_s$ such that $U/U_s$ is "$k$-wound" (opposite of split, roughly speaking), and torus actions on $k$-wound unipotent groups are always trivial. It may help to hear more about the motivation. $\endgroup$ – nfdc23 Jul 26 '16 at 2:02
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    $\begingroup$ In positive characteristic, Aut-functors of smooth connected unipotent $U\ne 1$ are never represented by a scheme or algebraic space (they satisfy the criterion in EGA IV$_3$ 8.14.2 to be locally of finite presentation but have infinite-dimensional tangent space at the identity via an argument I omit). Thus, speaking of a "diagonalizable subgroup" of the Aut-group is delicate. Curiously, the Aut-functor of any affine finite type $k$-group scheme is a directed union of subfunctors represented by affine group schemes of finite type (adapt the proof of Lemma A.8.13 in "Pseudo-reductive groups"). $\endgroup$ – nfdc23 Jul 26 '16 at 3:35
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    $\begingroup$ In 1967 Tits taught a course at Yale in which he did a lot with torus actions on smooth connected unipotent groups, getting better results than in Borel & Springer (fewer commutativity hypotheses, no condition on $p$-power multiples for Cor. 9.11, etc.). This is discussed in Appendix B of "Pseudo-reductive groups". The methods of Borel & Springer generally do not handle actions by $\mu_p$ in characteristic $p$ (e.g., the 2nd sentence of the proof of Lemma 9.7 breaks down for non-smooth diagonalizable groups), and the proof of Thm. 9.8 breaks down beyond tori. Does that kill your incentive? $\endgroup$ – nfdc23 Jul 26 '16 at 3:53
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    $\begingroup$ The method of proof of 9.8 involves passage to a 1-dimensional split torus (among other things); hard to see how that can adapt when the initial diagonalizable group is finite. As for your question about smoothness of fixed-point schemes, the answer is affirmative for a diagonalizable group acting on any smooth scheme over any ring, by using the infinitesimal criterion. This goes back to SGA3 in some cases (as in Lemma 2.2.4 of the article on reductive group schemes in the proceedings of the Luminy summer school on SGA3), and in general see A.8.10 in [CGP] (also see A.8.12 there too). $\endgroup$ – nfdc23 Jul 26 '16 at 16:14
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    $\begingroup$ I am only saying that the method of proof looks like it doesn't go through more generally (without some new idea). I'm not passing judgement on the questions you have posed. $\endgroup$ – nfdc23 Jul 27 '16 at 2:11

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