Speaker:  Ilya Dumanski (MIT)

Abstract: Cautis and Williams considered the category of perverse coherent sheaves on the affine Grassmanian and proved for GL and conjectured for other types that this category has a cluster structure. I will speak about partial progress towards the proof of this conjecture for simply-laced types. Our approach is based on relating the coherent Satake category with the category of finite-dimensional modules over the affine quantum group. The bridge between these two categories is provided by the notion of Feigin-Loktev fusion product for modules over the current algebra. In particular, it helps to construct cluster short exact sequences of perverse coherent sheaves using the existence of exact sequences of modules over the quantum affine group, called the Q-systems.