The tamely ramified Fundamental Local Equivalence

Abstract: Let G be a reductive group with Langlands dual G’. Gaitsgory
conjectured that affine Category O for G at a noncritical level should be
equivalent to Whittaker D-modules on the affine flag variety of G’ at the
dual level. We will provide motivation and background for this conjecture,
which is some form of geometric Satake for quantum groups. We have
proven this conjecture when the level is appropriately integral with Justin
Campbell, and for all levels satisfying a mild condition (coming from the bad
primes of the root system) with Sam Raskin. Depending on time and interest,
we will also indicate its relation to a Beilinson–Bernstein style localization
theorem for W-algebras, obtained jointly with Sam Raskin, and expected
consequences thereof.