## 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!