Cluster theory of the coherent Satake category

Abstract:

We study the category of $G(\mathcal {O})$-equivariant perverse coherent sheaves on the affine Grassmannian $\mathrm {Gr}_G$. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible Satake category. Instead, we use the Beilinson-Drinfeld Grassmannian to construct renormalized $r$-matrices. These are canonical nonzero maps between convolution products which satisfy axioms weaker than those of a braiding.

We also show that the coherent Satake category is rigid, and that together these results strongly constrain its convolution structure. In particular, they can be used to deduce the existence of (categorified) cluster structures. We study the case $G = GL_n$ in detail and prove that the $\mathbb {G}_m$-equivariant coherent Satake category of $GL_n$ is a monoidal categorification of an explicit quantum cluster algebra.

More generally, we construct renormalized $r$-matrices in any monoidal category whose product is compatible with an auxiliary chiral category, and explain how the appearance of cluster algebras in 4d $\mathcal {N}=2$ field theory may be understood from this point of view.