On refined filtration by supports for rational Cherednik categories $\mathcal {O}$

Abstract

For a complex reflection group W with reflection representation $\mathfrak {h}$, we define and study a natural filtration by Serre subcategories of the category $\mathcal {O}_c(W, \mathfrak {h})$ of representations of the rational Cherednik algebra $H_c(W, \mathfrak {h})$. This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov–Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When W is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori–Hecke algebras. This yields a method for counting the number of irreducible objects in $\mathcal {O}_c(W, \mathfrak {h})$ of given support. We apply these techniques to count the number of irreducible representations in $\mathcal {O}_c(W, \mathfrak {h})$ of given support for all exceptional Coxeter groups W and all parameters c, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of $\mathcal {O}_c(W, \mathfrak {h})$ for exceptional Coxeter groups W in many new cases.