Abstract

In this talk we introduce a hybrid quantum group at root of unity. We consider its category O and discuss some basic properties including linkage principle and BGG reciprocity. Then we show that there is an isomorphism between the center of a block (of arbitrary singular type) of the category O with the cohomology ring of the partial affine flag variety (of the corresponding parahoric type). A key ingredient is an abelian equivalence between the Steinberg block of O and the category of equivariant coherent sheaves on the Springer resolution. If time permitted, we will discuss a deformed version of these results.