Algebra and Geometry Seminar

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.