arXiv subjects

Colton Sandvik

Publications and source records attributed to Colton Sandvik.

5 recordsLinked to original sources

arXiv2026

Soergel calculus for monodromic Hecke categories

We introduce two 2-categories which categorify the monodromic Hecke algebra. The first is algebraic in nature and generalizes Abe's theory of Soergel bimodules. The second is a diagrammatic category defined via generators and relations which generalizes the Elias-Williamson diagrammatic calculus. As our first main result, we prove that these algebraic and diagrammatic categorifications are equivalent, extending an earlier theorem of Abe. Furthermore, we relate these new categorifications to a third categorification via parity sheaves which was previously studied by the author. More precisely, we provide a monodromic analogue of a theorem of Riche and Williamson to show that the diagrammatic category is equivalent to the monodromic Hecke category of parity sheaves associated to a reductive group. Finally, we show that these monodromic Hecke categories can be described by unipotent Hecke categories associated to endoscopic Coxeter groups.

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arXiv2026

Relative Serre duality for Coxeter groups

It was conjectured by Gorsky, Hogancamp, Mellit, and Nakagane that the left and right adjoints of the parabolic induction functor between homotopy categories of Soergel bimodules associated to a finite Coxeter group are related by the relative full twist. Several cases of this conjecture are known including for symmetric groups, crystallographic Coxeter groups, and dihedral groups. We prove this conjecture in complete generality using the theory of Abe-Bott-Samelson bimodules and the Achar-Riche-Vay mixed derived category.

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arXiv2026

A Betti geometric Casselman-Shalika equivalence

Whittaker sheaves are ubiquitous in geometric representation theory; however, their definition requires one to restrict the sheaf-theoretic setting to either \'etale sheaves or $D$-modules. Gaitsgory and Lysenko proposed a solution to this problem called the Kirillov model, which is well-defined for many sheaf theories. In this paper, we advance the study of the Kirillov model in the setting of Betti sheaves with a particular emphasis on developing a theory of Iwahori-Whittaker sheaves on the affine Grassmannian. Using this framework, we prove a Betti geometric Casselman-Shalika equivalence, which relates Iwahori-Whittaker perverse sheaves on the affine Grassmannian with the Satake category.

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arXiv2025

Endoscopy for Modular Hecke Categories

Generalizing the theory of parity sheaves on complex algebraic stacks due to Juteau-Mautner-Williamson, we develop a theory of twisted equivariant parity sheaves. We use this formalism to construct a modular incarnation of Lusztig and Yun's monodromic Hecke category. We then give two applications: (1) a modular categorification of the monodromic Hecke algebra, and (2) a monoidal equivalence between the monodromic Hecke category of parity sheaves and the ordinary Hecke category of parity sheaves on the endoscopic group.

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arXiv2024

Modular Character Sheaves on Reductive Lie Algebras

Mirkovi\'c introduced the notion of character sheaves on a Lie algebra. Due to their simple geometric characterization, character sheaves on Lie algebras can be thought of as a simplified model for Lusztig's theory of character sheaves on algebraic groups. We extend the theory to the case of modular coefficients. Along the way, we will reprove some of Mirkovi\'c's results and provide connections with the modular generalized Springer correspondence of Achar, Juteau, Riche, and Williamson.

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