On Parabolic Subgroups of Symplectic Reflection Groups

TL;DR

This paper proves that parabolic subgroups of finite symplectic reflection groups are themselves symplectic reflection groups and explores implications for symplectic resolutions and singular locus properties.

Contribution

It establishes the symplectic analogue of Steinberg's Theorem for symplectic reflection groups and analyzes the existence of symplectic resolutions for certain quotient singularities.

Findings

Parabolic subgroups are symplectic reflection groups.

Non-existence of symplectic resolutions for three exceptional groups.

Singular locus of the quotient is pure of codimension two.

Abstract

Using Cohen's classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group are themselves symplectic reflection groups. This is the symplectic analogue of Steinberg's Theorem for complex reflection groups. Using computational results required in the proof, we show the non-existence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.