Structure of a Hecke algebra quotient

Let $W$ be a Coxeter group with Coxeter graph ${\Gamma }$. Let $\cal H$ be the associated Hecke algebra. We define a certain ideal ${\cal I}$ in $\cal H$ and study the quotient algebra ${\bar {\cal H}} = {\cal H}/{\cal I}$. We show that when ${\Gamma }$ is one of the infinite series of graphs of type $E$, the quotient is semi-simple. We examine the cell structures of these algebras and construct their irreducible representations. We discuss the case where ${\Gamma }$ is of type $B$, $F$, or $H$.