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Fix an algebraically closed field $K$, and let $G$ be a unipotent linear algebraic group over $K$ acting morphically on an affine variety $X$. According to [1, Prop. 2.5] we have the following result:

Theorem. Every $G$-orbit on $X$ is (Zariski) closed.

I am looking for a rational version of the above result. More precisely, let $k$ be an arbitrary subfield of $K$ and suppose that $G,X$ and the $G$-action on $X$ are all defined over $k$. The group of $k$-rational points $G(k)$ still acts morphically on $X$.

Is it true that $G(k)$-orbits are closed relative to the Zariski $k$-topology on $X$?

The discussion in [2, Sections 34.1 and 34.2] suggests that the proof of [1, Prop. 2.5] adapts fairly well when $k$ is either perfect or (more generally) $k$ is replaced by some purely inseparable extension in $K$. Unfortunately, I am interested in the case where $k$ is a non-archimedean local field of positive characteristic and $K$ is its algebraic closure.

References would be appreciated in case there is a positive answer to the above question. Thank you in advance.

BIBLIOGRAPHY

[1] R. Steinberg, "Conjugacy Classes in Algebraic Groups", Lecture Notes in Mathematics, Vol. 366, Springer, New York/Berlin, 1974;

[2] J. E. Humphreys, "Linear Algebraic Groups", Graduate Texts in Mathematics, Vol. 21, Springer, New York/Berlin, 1975.

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  • $\begingroup$ I guess by saying that the perfect hypothesis doesn't cover non-Archimedean local fields, you indicate that you want particularly to consider positive-characteristic such fields? Also, is the Zariski $k$-topology the subspace topology on the group of $k$-rational points, or something else? $\endgroup$
    – LSpice
    Commented Oct 24, 2019 at 1:05
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    $\begingroup$ @LSpice Precisely, I am mostly interested in the positive-characteristic case. I shall edit my question and add your observation. Also, by" Zariski $k-$topology on $X$" I mean the following: if $X$ is embedded in $K^n$ for some natural number $n$ and $X$ is defined over $k$, then take the subspace topology on $X$ induced by the topology on $K^n$ whose closed sets are the zero loci of subsets of polynomials in $K[x_1, \cdots, x_n]$ with coefficients in $k$. $\endgroup$
    – Pedro A. Matos
    Commented Oct 24, 2019 at 10:44
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    $\begingroup$ The [notes] (virtualmath1.stanford.edu/~conrad/249BW16Page/handouts/…) of Brian Conrad do this stuff (and most of the theory of reductive groups) in over a general base field $k$. $\endgroup$
    – edgarlorp
    Commented Oct 24, 2019 at 16:39

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