Irreducible representations of rational Cherednik algebras for exceptional Coxeter groups, Part I

This paper and its sequel describe the irreducible representations of the rational Cherednik algebra $H_c(W)$ for a finite Coxeter group $W$ of type $H_4$, $F_4$ with equal parameters, $E_6$, $E_7$, and $E_8$, when $c$ is not a half-integer. Herein appear the decomposition matrices of Category ${\mathcal{O}}_c(W)$ for $W=H_4$, $E_6$, and $E_7$, as well as a classification of finite-dimensional representations of $H_c(W)$ for these groups and for $H_c(F_4)$ with equal parameters.