Zero fibers of quaternionic quotient singularities

Abstract

We propose a generalization of Haiman's conjecture on the diagonal coinvariant rings of real reflection groups to the context of irreducible quaternionic reflection groups (also known as symplectic reflection groups). For a reflection group $W$ acting on a quaternionic vector space $V$, by regarding $V$ as a complex vector space we consider the scheme-theoretic fiber over zero of the quotient map $\pi:V \to V/W$. For $W$ an irreducible reflection group of (quaternionic) rank at least $6$, we show that the ring of functions on this fiber admits a $(g+1)^n$-dimensional quotient arising from an irreducible representation of a symplectic reflection algebra, where $g=2N/n$ with $N$ the number of reflections in $W$ and $n=\mathrm{dim}_\mathbf{H}(V)$, and we conjecture that this holds in general. We observe that in fact the degree of the zero fiber is precisely $g+1$ for the rank one groups (corresponding to the Kleinian singularities). In an appendix, we give a proof that three variants of the Coxeter number, including $g$, are integers.