  We study the subregular $J$-ring $J_C$ of a Coxeter system $(W,S)$, a subring
of Lusztig's $J$-ring. We prove that $J_C$ is isomorphic to a quotient of the
path algebra of the double quiver of $(W,S)$ by a suitable ideal that we
associate to a family of Chebyshev polynomials. As applications, we use quiver
representations to study the category mod-$A_K$ of finite dimensional right
modules of the algebra $A_K=K\otimes_\Z J_C$ over an algebraically closed field
$K$ of characteristic zero. Our results include classifications of Coxeter
systems for which mod-$A_K$ is semisimple, has finitely many simple modules up
to isomorphism, or has a bound on the dimensions of simple modules.
Incidentally, we show that every group algebra of a free product of finite
cyclic groups is Morita equivalent to the algebra $A_K$ for a suitable Coxeter
system; this allows us to specialize the classifications to the module
categories of such group algebras.
