## Cluster theory of the coherent Satake category

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Sabin Cautis and Harold Williams
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## 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.

## References

## Additional Information

**Sabin Cautis**- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
- MR Author ID: 712430
- Email: cautis@math.ubc.ca
**Harold Williams**- Affiliation: Department of Mathematics, University of California, Davis, California 95616
- MR Author ID: 1010471
- Email: hwilliams@math.ucdavis.edu
- Received by editor(s): February 17, 2018
- Received by editor(s) in revised form: January 11, 2019
- Published electronically: April 10, 2019
- Additional Notes: The first author was supported by an NSERC Discovery/accelerator grant.

The second author was supported by NSF Postdoctoral Fellowship DMS-1502845 and NSF grant DMS-1702489. - © Copyright 2019 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**32**(2019), 709-778 - MSC (2010): Primary 13F60, 22E67; Secondary 14F05
- DOI: https://doi.org/10.1090/jams/918
- MathSciNet review: 3981987