Algebra and Geometry Seminar
The geometric Satake equivalence identifies the Satake category of a reductive group $G$ -- the category of equivariant perverse sheaves on the affine Grassmannian $Gr_G$ -- with the representation category of its Langlands dual group. In this talk we discuss recent advances in our understanding of its more complicated cousin, the category of perverse coherent sheaves on $Gr_G$. This category is not semi-simple and its monoidal product is not symmetric. However, it is rigid and admits renormalized r-matrices similar to those appearing in the theory of quantum loop or KLR algebras. From these results we further show that the coherent Satake category of $GL_n$ categorifies a specific cluster algebra appearing in several other mathematical and physical contexts. In particular, following ideas of Gaiotto-Moore-Neitzke and Kapustin-Saulina this can be interpreted as part of a correspondence between coherent IC sheaves and Wilson-‘t Hooft operators in 4d N=2 gauge theory. This is joint work with Sabin Cautis.