Let $(W,S)$ be an arbitrary Coxeter system and let $G$ be its Coxeter diagram. We recall Lusztig's construction of the asymptotic Hecke algebra $J$ of $(W,S)$, an associative algebra closely related to the Iwahori--Hecke algebra of $(W,S)$, and present some results on a subalgebra $J_C$ of $J$ that we call the subregular $J$-ring. We show that while products in $J$ are defined in terms of Kazhdan--Lusztig polynomials, they can be computed in $J_C$ by a simple combinatorial algorithm centered around a certain truncated Clebsch--Gordan rule. As applications, we relate $J_C$ to the path algebra of a quiver whose underlying graph is $G$ and deduce some results on the structure and representations of $J_C$.

This is joint work with Ivan Dimitrov, Charles Paquette, and David Wehlau.