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steinberg’s fixed point theorem

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On parabolic subgroups of symplectic reflection groups
Part of
Gwyn Bellamy, Johannes Schmitt, Ulrich Thiel
Journal:

Glasgow Mathematical Journal / Volume 65 / Issue 2 / May 2023

Published online by Cambridge University Press:

10 January 2023, pp. 401-413

Print publication:

May 2023
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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 analog of Steinberg’s Theorem for complex reflection groups.
Using computational results required in the proof, we show the nonexistence 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.