Derived geometric Satake for $PGL_2^{× 3}/PGL_2^diag$

In this note, we study the local relative geometric Langlands conjecture of Ben-Zvi--Sakellaridis--Venkatesh for the spherical subgroup PGL₂ᵈiag of the triple product PGL₂× 3 (and also for the spherical subgroup G₂ of SO₈/μ₂), whose corresponding Langlands dual SL₂× 3-variety can be identified with the symplectic vector space (A²)⊗ 3 A⁸ of 2× 2 × 2-cubes. Our analysis relies on a construction of Bhargava relating 2 × 2 × 2-cubes to Gauss composition on quadratic forms, arising here as the moment map for the Hamiltonian SL₂× 3-action on (A²)⊗ 3, and the Cayley hyperdeterminant as studied by Gelfand-Kapranov-Zelevinsky.