The subregular part of Lusztig's asymptotic Hecke algebra

Given an arbitrary Coxeter system (W,S), Lusztig defined its asymptotic Hecke algebra J, an associative algebra closely related to the usual Hecke algebra and the category of Soergel bimodules for (W,S). The algebra J decomposes as a direct sum of subalgebras indexed by the 2-sided Kazhdan-Lusztig cells of W, and we present some results on the subalgebra $J_c$ corresponding to a particular cell c known as the subregular cell. We show that products in $J_c$ can be computed by repeated use of (variations of) the Clebsch-Gordan formula arising from the representation theory of $sl_2$, and we use this multiplication rule to obtain alternative descriptions of $J_c$ for Coxeter systems of certain types (such as all simply-laced ones).