Gaitsgory's central sheaves
A theorem of Bernstein identifies the center of the affine Hecke algebra of a reductive group GG with the Grothendieck ring of the tensor category of representations of the dual group G∨G∨. Gaitsgory constructed a functor which categrorifies this result. This functor sends Satake sheaves on the affine Grassmannian of GG to Iwahori-equivariant perverse sheaves on the affine flag variety, and the sheaves in the image lie in the center of this category. The functor is given as nearby cycles for a family over a curve whose general fiber is the affine Grassmannian times the finite flag variety and whose special fiber is the affine flag variety. As a result, the functor carries an important additional structure, an endomorphism coming from monodromy of nearby cycles.