Derived geometric Satake for $\mathrm{PGL}_2^{\times 3}/\mathrm{PGL}_2^\mathrm{diag}$

In this note, we study the local relative geometric Langlands conjecture of Ben-Zvi--Sakellaridis--Venkatesh for the spherical subgroup $\mathrm{PGL}_2^\mathrm{diag}$ of the triple product $\mathrm{PGL}_2^{\times 3}$ (and also for the spherical subgroup $\mathrm{G}_2$ of $\mathrm{SO}_8/\mu_2$), whose corresponding Langlands dual $\mathrm{SL}_2^{\times 3}$-variety can be identified with the symplectic vector space $(\mathbf{A}^2)^{\otimes 3} \cong \mathbf{A}^8$ of $2\times 2 \times 2$-cubes. Our analysis relies on a construction of Bhargava relating $2 \times 2 \times 2$-cubes to Gauss composition on quadratic forms, arising here as the moment map for the Hamiltonian $\mathrm{SL}_2^{\times 3}$-action on $(\mathbf{A}^2)^{\otimes 3}$, and the Cayley hyperdeterminant as studied by Gelfand-Kapranov-Zelevinsky.