Statement of the Specific Result
Let $J$ denote the matrices with ones on the "second diagonal", meaning the diagonal between the (1,n) and (n,1) entry, and zeros elsewhere. So in the case $n=2$, which is all we'll be concerned with here:
$$ J=\begin{pmatrix} 0&1 \newline 1&0 \end{pmatrix} $$
Let the transpose of $g$ be denoted by ${}^tg$.
Let $F$ be a field (not necessarily alg. closed: for example, a number field), and consider the action of $G:=\mathrm{GL}_2(F)$ on the vector space $\mathcal{L}$ of matrices with entries in $F$ which are symmetric about the second diagonal, by $g\cdot X= g X J{}^tgJ$ for $g\in\mathrm{GL}_2(F)$. In explicit coordinates,
$$g=\begin{pmatrix}a&b \newline c&d\end{pmatrix} $$
$$X = \begin{pmatrix}\gamma&\beta \newline \alpha&\gamma\end{pmatrix}$$
$$J{}^tgJ=\begin{pmatrix}d&b \newline c&a\end{pmatrix} $$
so that this procedure of "taking the transpose of $g$ and conjugating by $J$" amounts to taking the transpose of $g$ along the second diagonal. (See below for Context). In several papers, I find the following stated:
- $G$ acts on $\mathcal{L}$ with an open orbit.
- Given a representative $X_0\in\mathcal{L}$ of the open orbit, the stabilizer $G_0(X_0)$ is in general reductive, and in this specific example, a one-dimensional torus.
- An point $X$ is called generic if the stabilizer $G_0(X)$ is of type $G_0(X_0)$, and an orbit is called generic if one, equiv. all, its points are generic points. A complete set of representatives of the generic orbits (with exactly one representative from each orbit) is given by the matrices $$ \begin{pmatrix} 0&\beta\\ \alpha&0 \end{pmatrix} $$ with $$\alpha,\beta\in {F^*}^2\backslash F^*$$ (so with the diagonal element $\gamma=0$ and the off-diagonal elements ranging over nonzero square classes of $F^*$ independently).
My questions about this result
What I would like to know...
- is there any tidy way of "characterizing" generic points or orbits, as referred to in items 1 and 2?
- Is there any conceptual or "coordinate-free" way of characterizing the representative set given in item 3?
I have a feeling that "standard, classical" invariant theory, especially that of the symplectic group, may give an answer to this. I am not sufficiently familiar with the invariant-theory literature to find this, so if you could point me to a specific reference that I could read and which would allow me to answer these questions, that would be great. Although I have a (partial) confirmation of these facts by brute-force matrix calculations, these are not really ideal to use for my purposes, nor is it clear that they could be carried out by anyone in higher dimensions than 2!
The context, and more on why I expect invariant theory to play a role
The context of this problem is that the symplectic group $\mathrm{Sp}_4(F)$ has a standard ("Siegel") parabolic $P$ with Levi factor $M$ isomorphic to $G$, which embeds into $\mathrm{Sp}_4$ by $\mathrm{diag}(g,J{}^t g^{-1}J)$ (according to one of the two or so common matrix models of $\mathrm{Sp}_4$). The nilpotent radical of $P$ can be identified with $\mathcal{L}$, and the conjugation action of $M$ is then identified with the action above.
If one considers the analogous situation with $\mathrm{SO}_{5}$ (say), and the parabolic $Q$ with Levi factor $\mathrm{GL_1}\times \mathrm{SO}_{3}$, the nilpotent radical of $Q$ can be identified with "row vectors of length 3", and the genericity condition can clearly be expressed as a row-vector representative having non-zero length. Then it is easy to divide "generic" vectors into different $M(F)$-conjugacy classes by the square-class of their (non-zero) lengths. So this very simple invariant-theory interpretation gives me the feeling there is something conceptual going on in the situation I have described, which I am unfortunately missing at the moment.
Thanks for reading and I will greatly appreciate any help!