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I have a question about unipotent group actions. I was referred to Rosenlicht's papers, but I had trouble getting much out of these because I don't understand the old algebraic geometry language very well. From what I can tell Rosenlicht's results do not imply what I'm looking for. I expect the answer or a reference should be known by people more knowledgeable than myself.

Let $G$ be a connected, unipotent algebraic group over a field $k$ of characteristic zero, acting as a group of $k$-vector space automorphisms on $X = \mathbb A_k^n$.

Rosenlicht's theorem says that there exists a $k$-open set $X'$ of $X$ which is $G$-stable and for which the geometric quotient $G \backslash X'$ exists. It also says that a "cross section $k$-morphism" $G \backslash X' \rightarrow X'$ exists.

My question: let $d$ be the largest dimension of a closed orbit in $X$.

  • Does there exist a $k$-open $G$-stable set $X'$ of $X$, and a $k$-closed set $W$ of $X'$, such that $W$ is a fundamental domain for the action of $G$ on $X'$, i.e. each the orbit of each $x \in X'$ meets $W$ at exactly one point?

  • If so, can $X'$ and $W$ be chosen so that the map taking $x \in X'$ to its unique orbit representative in $W$ is the geometric quotient of Rosenlicht's theorem?

  • Does $W$ always arise from the intersection of $X'$ and a "generic" $n-d$-dimensional subspace of $\mathbb A^n$?

My question is motivated by a general collection of examples I've been looking at for the past couple of years where I have always found a positive answer.

Example:

Let

$$G = \{ \begin{pmatrix} g \\ & h \end{pmatrix} : \textrm{ $g, h \in \operatorname{GL}_n$ are upper triangular unipotent} \}$$

$$X = \begin{pmatrix} 0 & x \\ 0 & 0 \end{pmatrix} : x \in \operatorname{Mat}_n\} \cong \mathbb A_k^{n^2}$$

Then $G$ acts on $X$ by conjugation. The largest dimension of a closed orbit in $X$ is $\operatorname{Dim} G$. A fundamental domain for the action of $G$ on an open set in $X$ is the set of nonzero antidiagonal matrices

$$W = \begin{pmatrix} & & & \ast \\& & \ddots \\ & \ast \\ \ast \end{pmatrix}$$

and the conjugation map $G \times W \rightarrow X$ is an isomorphism of varieties onto an open set in $X$.

More general class of examples where I expect nice results

Let $H$ be a quasi-split group over $k$ with maximal parabolic subgroup $P = MN$ and Borel subgroup $B = TU$ with Levi factors $T \subset M$ and unipotent radicals $N \subset U$. Then $G = U \cap M$ acts as Lie algebra automorphisms of $\mathfrak n = \operatorname{Lie}(N)$ by conjugation. In many examples I have determined that a fundamental domain for the action of $G$ on an open subset $\mathfrak n$ arises from looking at subspaces spanned by certain root vectors.

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The affirmative answer to the first two question is indeed well-known. The existence of the section $W$ boils down to the vanishing of $H^1(X,\mathbf G_a)$ on an affine variety $X$.

A generic $n-d$-dimensional subspace will intersect a $G$-orbit in $D$ points where $D$ is the degree of that orbit. So, the answer to the third question is clearly negative if the generic orbits is not itself an affine subspace.

In general, Rosenlicht's theorem is very weak since it doesn't provide any control over $X'$ and $W$. For general actions of unipotent groups one cannot hope for anything better, though. Judging from your examples I presume, though, that you are interested in a much more special (and important) situation namely where $G$ is the unipotent radical of a parabolic $P$ of a reductive group $H$ and the action of $G$ is the restriction of an $H$-action. That' s what the "Local Structure Theorem" is for. It was first proved by Brion-Luna-Vust (Espaces homogènes sphériques, Invent. Math. 84 (1986) 617–632) and goes in its simplest form as follows:

Let $H$ be connected reductive acting on an affine variety $X$. Let $f\in\mathcal O(X)$ be a highest weight vector. Let $P$ (a parabolic) be the stabilizer of $\mathbb Cf$. Let $P_u$ be its unipotent radical and $M\subseteq P$ a Levi complement. Let $P_u^-\subseteq H$ be the opposite subgroup with respect to $M$. Now let $$X':=\{x\in X\mid f(x)\ne0\},$$ a $P$-stable open subset of $X$, and $$W:=\{x\in X'\mid (\xi f)(x)=0\text{ for all }\xi\in{\rm Lie P_u^ -}\},$$ a closed affine $M$-stable sub variety of $X'$. Then the canonical morphism $$P_u\times W=P\times^MW\to X':[p,x]\mapsto px$$ is an isomorphism of $P$-varieties. In particular, the action of $P_u$ on $X'$ is free and $W$ is a slice.

The big advantage of the LST over Rosenlicht's theorem is that $X'$ and $W$ are completely explicit. For example if $X$ is a vector space and $f$ is linear then $W$ is the open part of a linear subspace, as you have observed in your examples.

The LST applies a priori only to a specific unipotent radical but one can either move to a "generic" semiinvariant $f$ or iterate the construction for the action of $M$ on $W$. At the end one can manage that all unipotent elements of $M$ act trivially on $W$. But then the $U$-orbits in $X'$ are the $P_u$-orbits where $U\subseteq H$ is maximal unipotent.

This way one can show that indeed $W$ can be chosen open in a linear subspace if $X$ is a vector space. For details see my paper "Some remarks on multiplicity free spaces" (Representation theories and algebraic geometry (Montreal, PQ, 1997), 301–317, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, Kluwer Acad. Publ., Dordrecht, 1998.)

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  • $\begingroup$ Thanks very much for your answer, this is exactly the kind of thing I'm looking for. What is meant by saying that a global section $f \in \mathcal O(X)$ is a highest weight vector? $\endgroup$ – D_S Jan 31 at 19:00
  • $\begingroup$ That $f$ is $B$-semiinvariant. $\endgroup$ – Friedrich Knop Jan 31 at 19:44
  • $\begingroup$ I am trying to recover the fundamental domain of my example in $\operatorname{GL}_{2n}$ using what you have said. $X$ is acted on by the reductive group $H = \operatorname{GL}_n \times \operatorname{GL}_n$, and I am trying to determine a fundamental domain in $X$ for the action of the unipotent radical of the standard Borel of $H$. The only Borel semi-invariant function I think of on $X$ is the determinant map. The corresponding parabolic subgroup of $H$ is $H$ itself, so this unfortunately tells me nothing. How would one obtain proper parabolic subgroups for this example? $\endgroup$ – D_S Feb 3 at 22:33
  • $\begingroup$ The determinant is of no use since it is even an $H$-semiinvariant. In that case, the parabolic $P$ is all of $H$. Take as $f$ the coefficient $x_{n1}$ in the lower left corner of $x$ which is $B$- but not $H$-semiinvariant. This yields a reduction $n\to n-1$. Repeat $n-1$ times and you obtain $W$ from your example. $\endgroup$ – Friedrich Knop Feb 4 at 14:29
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The first two questions have an affirmative answer. See my paper https://arxiv.org/abs/1712.03838 for a constructive proof. The results are even true for connected solvable groups, also in positive characteristic. In the paper there are further references for special cases that should cover your setting. Your third questions is not covered, though, since the setting in the papers is an action on an affine or quasi-affine variety. I'd have to think about the third question, but I have doubts. Even for an additive group action, it would mean that a local slice can be found "generically" in degree 1.

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