Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu’s by homological mirr...

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu’s by homological mirr...

Relying on recent advances in the theory of motives we develope a general formalism for derived categories with Q-coefficients perfect (ind-)schemes. As an application give motivic refinement Zhu's geometric Satake equivalence Witt vector affine Grassmannians this set-up.

Suppose that G is a connected reductive group over a p-adic field F , that K is a hyperspecial maximal compact subgroup of G(F ), and that V is an irreducible representation of K over the algebraic closure of the residue field of F . We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, Kbiequivariant functions f : G(F ) EndV . These Hecke algebras wer...

We endow the set of lattices in Qp with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake correspondence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.

We prove that the Satake-Baily-Borel compactification of certain Shimura varieties are Fano varieties, Calabi-Yau or have ample canonical divisors with mild singularities. also some variants statements, give applications and discuss various examples including new ones, for instance, moduli spaces unpolarized (log) Enriques surfaces.