Finite-dimensional quotients of Hecke algebras

Abstract

Let $W$ be a complex reflection group. We prove that there is a maximal finite-dimensional quotient of the Hecke algebra ${\mathcal{\mathscr{H}}}_q\left(W\right)$ of $W$, and that the dimension of this quotient coincides with $\left|W\right|$. This is a weak version of a 1998 Broué–Malle–Rouquier conjecture. The proof is based on the categories $\mathcal{O}$ for rational Cherednik algebras.