Asymptotic Hecke Categories

This thesis explores a conjecture of Lusztig about the correspondence of Fourier matrices combinatorially constructed for non-crystallographic Coxeter groups and \(S\)-matrices of the Drinfeld center of the asymptotic Hecke category associated to a two-sided cell \(c\) of a Coxeter group \(W\). We construct these asymptotic Hecke categories using the diagrammatic approach of Elias--Williamson inside the diagrammatic Hecke category of Soergel bimodules. Through explicit calculations in dihedral groups, and some cells in type \(H_3\), \(H_4\), and certain infinite Coxeter groups with finite cells, we verify Lusztig's conjecture by showing that the asymptotic Hecke categories are known in the literature as fusion categories with type \(A_k\) fusion rules. For smaller cells these are either Fibonacci categories or \(Z/2Z\)-graded vector spaces. Our calculations establish an algorithmic framework that, in the future, could be extended to compute the asymptotic Hecke category of the `so-called' middle cell in type \(H_4\).