Toward quasi-classical Fundamental Local Equivalence.
Toward quasi-classical Fundamental Local Equivalence.
Fundamental Local Equivalence of Gaitsgory and Lurie is an instance of quantum local geometric Langlands correspondence. It asserts an equivalence between the affine category O and the category of twisted Whittaker D-modules on the affine flag variety for the dual group. A version of this conjecture was proved by Campbell, Dhillon, Raskin. In this talk I will explain our work in progress with Dhillon where we study the quasi-classical version of this equivalence: instead of modules over quantizations the categories we care about are those of equivariant modules over suitable commutative algebras. Our main result is a construction of a functor from the category O side to the Whittaker side that has several properties expected of the equivalence -- but we do not know how to prove that our functor is an equivalence. I will try to keep the exposition reasonably elementary. The talk is based on work in progress with Gurbir Dhillon.