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

Abstract

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.

31 pages, comments welcome!