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## The coherent Satake category Williams, Harold | CIRM H

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Research talks;Algebraic and Complex Geometry;Mathematical Physics

The geometric Satake equivalence identifies the Satake category of a reductive group $G$ - that is, the category of equivariant perverse sheaves on the affine Grassmannian $G_{rG}$ - with the representation category of its Langlands dual group $G^∨$. While the Satake category is topological in nature, it has a poorly understood algebro-geometric cousin: the category of perverse coherent sheaves on $G_{rG}$. This category is not semi-simple and its monoidal product is not symmetric. We show however that it is rigid and admits renormalized r-matrices similar to those appearing in the theory of quantum loop or KLR algebras. Applying the framework developed by Kang-Kashiwara-Kim-Oh in their proof of the dual canonical basis conjecture, we use these results to show that the coherent Satake category of $GL_n$ is a monoidal cluster categorification in the sense of Hernandez-Leclerc. This clarifies the physical meaning of the coherent Satake category: simple perverse coherent sheaves correspond to Wilson-’t Hooft operators in $\mathcal{N} = 2$ gauge theory, just as simple perverse sheaves correspond to ’t Hooft operators in $\mathcal{N} = 4$ gauge theory following the work of Kapustin-Witten. Our results also explain the appearance of identical quivers in the work of Kedem-Di Francesco on $Q$-systems and in the context of BPS quivers. More generally, our construction of renormalized r-matrices works in any chiral $E_1$-category, providing a new way of understanding the ubiquity of cluster algebras in $\mathcal{N} = 2$ field theory: the existence of renormalized r-matrices, hence of iterated cluster mutation, is a formal feature of such theories after passing to their holomorphic-topological twists. This is joint work with Sabin Cautis (arXiv:1801.08111). The geometric Satake equivalence identifies the Satake category of a reductive group $G$ - that is, the category of equivariant perverse sheaves on the affine Grassmannian $G_{rG}$ - with the representation category of its Langlands dual group $G^∨$. While the Satake category is topological in nature, it has a poorly understood algebro-geometric cousin: the category of perverse coherent sheaves on $G_{rG}$. This category is not semi-simple and ...

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