Counting irreducible representations of rational Cherednik algebras of given support

Given a finite complex reflection group $W$ with reflection representation $V$, one can consider the associated rational Cherednik algebra $H_c(W, V)$ and its representation category $O_c(W, V)$, depending on a parameter $c$. The irreducible representations in $O_c(W, V)$ are in natural bijection with the irreducible representations of W, and each representation $M$ in $O_c(W, V)$ has an associated support, a closed subvariety of $V$. Via the $KZ$ functor, the irreducible representations in $O_c(W, V)$ of full support are in bijection with the irreducible representations of the Hecke algebra $H_q(W)$. In the case that $W$ is a finite Coxeter group, I will explain how to count irreducible representations in $O_c(W, V)$ of arbitrary given support by introducing a functor $KZ_L$, generalizing the $KZ$ functor and depending on a finite-dimensional representation $L$ of a rational Cherednik algebra attached to a parabolic subgroup of $W$.
This is joint work with Ivan Losev.