Hyperplane arrangements associated to symplectic quotient singularities

Gwyn Bellamy, Ulrich Thiel

Published 2017-02-16Version 1

We study the hyperplane arrangements associated, via the minimal model programme, to symplectic quotient singularities. It is expected that this hyperplane arrangement equals the arrangement of "essential hyperplanes" coming from the representation theory of restricted rational Cherednik algebras. We show that this statement would follow from a rather innocuous conjecture about the number of torus fixed points on the universal Poisson deformation of a $\mathbb{Q}$-factorial terminalization. We explain some of the interesting consequences of this conjecture for the representation theory of restricted rational Cherednik algebras. We also show that the Calogero-Moser space is singular if and only if the Calogero-Moser families are trivial. By computing the arrangement of essentially hyperplanes associated to many exceptional complex reflection groups, we produce interesting new arrangements, some of which are free.