Derived geometric Satake for PGL₂^ 3/PGL₂ᵈiag

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 SO₈/₂), 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.