The center of the asymptotic Hecke category and unipotent character sheaves
Liam Rogel, Ulrich Thiel
TL;DR
The paper advances a program to categorify Lusztig’s asymptotic Hecke algebras for Coxeter groups by constructing and analyzing the asymptotic Hecke category from Soergel bimodules, and then identifying its Drinfeld center with modular tensor categories that generalize unipotent character sheaves to non-crystallographic cases. A key methodological tool is H-reduction, which reduces center computations from a J-cell to an H-cell, enabling explicit identifications with categories like Coh$_G(X\times X)$ and their centers, and linking the S-matrix to Lusztig’s Fourier matrices. The authors prove Lusztig’s conjecture for dihedral groups and obtain substantial progress for cells in $H_3$ and $H_4$, with a detailed treatment of the dihedral case via the adjoint Verlinde category and Temperley–Lieb data. They also extend considerations to certain infinite Coxeter groups with finite cells, showing that centers often reduce to known fusion/categorical data and thus yield concrete S-matrices consistent with Fourier-transform-type matrices. Overall, the work provides a cohesive framework connecting asymptotic Hecke categories, their centers, and (unipotent) character-like sheaf theories across crystallographic and non-crystallographic contexts, enriching the understanding of spetses and modular tensor structures in this setting.
Abstract
In 2015, Lusztig [Bull. Inst. Math. Acad. Sin. (N.S.)10(2015), no.1, 1-72] showed that for a connected reductive group over an algebraic closure of a finite field the associated (geometric) Hecke category admits a truncation in a two-sided Kazhdan--Lusztig cell, making it a categorification of the asymptotic algebra (J-ring), and that the categorical center of this "asymptotic Hecke category" is equivalent to the category of unipotent character sheaves supported in the cell. Subsequently, Lusztig noted that an asymptotic Hecke category can be constructed for any finite Coxeter group using Soergel bimodules. Lusztig conjectured that the centers of these categories are modular tensor categories (which was then proven by Elias and Williamson) and that for non-crystallographic finite Coxeter groups the S-matrices coincide with the Fourier matrices that were constructed in the 1990s by Lusztig, Malle, and Broué--Malle. If the conjecture is true, the centers may be considered as categories of "unipotent character sheaves" for non-crystallographic finite Coxeter groups. In this paper, we show that the conjecture is true for dihedral groups and for some (we cannot resolve all) cells of H3 and H4. The key ingredient is the method of H-reduction and the identification of the (reduced) asymptotic Hecke category with known categories whose center is already known as well. We conclude by studying the asymptotic Hecke category and its center for some infinite Coxeter groups with a finite cell.