The center of the asymptotic Hecke category and unipotent character sheaves

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 afinite field the associated (geometric) Hecke category admits a truncation in atwo-sided Kazhdan--Lusztig cell, making it a categorification of the asymptoticalgebra (J-ring), and that the categorical center of this "asymptotic Heckecategory" is equivalent to the category of unipotent character sheavessupported in the cell. Subsequently, Lusztig noted that an asymptotic Heckecategory can be constructed for any finite Coxeter group using Soergelbimodules. Lusztig conjectured that the centers of these categories are modulartensor categories (which was then proven by Elias and Williamson) and that fornon-crystallographic finite Coxeter groups the S-matrices coincide with theFourier matrices that were constructed in the 1990s by Lusztig, Malle, andBrou\'e--Malle. If the conjecture is true, the centers may be considered ascategories of "unipotent character sheaves" for non-crystallographic finiteCoxeter groups. In this paper, we show that the conjecture is true for dihedral groups andfor some (we cannot resolve all) cells of H3 and H4. The key ingredient is themethod of H-reduction and the identification of the (reduced) asymptotic Heckecategory with known categories whose center is already known as well. Weconclude by studying the asymptotic Hecke category and its center for someinfinite Coxeter groups with a finite cell.