$\newcommand{\std}{\mathrm{std}}\newcommand{\SL}{\mathrm{SL}}\newcommand{\mmod}{/\!\!/}$Fix the base field to be the complex numbers $\mathbf{C}$. Let $\std = \mathbf{A}^2$ denote the standard representation of $\SL\_2$, so that there is a natural action of $\SL\_2^{\times 3}$ on $\std^{\otimes 3}$. Let $Y$ denote the variety of pairs $(q\_1(x,y), q\_2(x,y))$ of binary quadratic forms with the same discriminant, so that it admits a natural action of $\SL\_2^{\times 2}$ (via the natural action of $\SL\_2$ on the space of binary quadratic forms). Some considerations from homotopy theory/relative Langlands suggest that there is a relationship between the stacks $\std^{\otimes 3}/\SL\_2^{\times 3}$ and $Y/\SL\_2^{\times 2}$. My basic question is whether there is some natural construction which relates these two stacks.


For instance, are they derived equivalent? Do they even have the same coarse moduli spaces? It is not too hard to show that the GIT quotient $Y\mmod\SL\_2^{\times 2} \cong \mathbf{A}^1$ via the discriminant; is it also true that the GIT quotient $\std^{\otimes 3}\mmod\SL\_2^{\times 3} \cong \mathbf{A}^1$? If so, what is the resulting map $\std^{\otimes 3} \to \mathbf{A}^1$? Or, is there some natural construction from the theory of binary quadratic forms which takes as input an $\SL\_2^{\times 2}$-orbit of pairs $(q\_1(x,y), q\_2(x,y))$ with the same discriminant and produces an $\SL\_2^{\times 3}$-orbit of $\std^{\otimes 3}$? Apologies for the somewhat vague question, and thanks in advance!



Pairs of quadratic forms and $\mathbf{A}^8/\mathrm{SL}_2^{\times 3}$
$\newcommand{\std}{\mathrm{std}}\newcommand{\SL}{\mathrm{SL}}\newcommand{\mmod}{/\!\!/}$
A lowbrow answer to your question is given by Bhargava's theory of $2 \times 2 \times 2$ cubes explained in his [Higher Composition Laws Paper](https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n1-p03.pdf).


In particular, you're correct that $\std^{\otimes 3}\mmod\SL\_2^{\times 3} \cong \mathbf{A}^1$, with the invariant being the discriminant listed at the bottom of page 220 in the paper (page 4 in the pdf).


For a higher-brow approach to this general topic you may be interested in Melanie Wood's paper [Gauss composition over an arbitrary base](https://arxiv.org/abs/1007.5285), though I don't think it talks about cubes directly.


