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On parabolic subgroups of symplectic reflection groups

Part of: General commutative ring theory Local theory Other groups of matrices

Published online by Cambridge University Press:  10 January 2023

Gwyn Bellamy ,
Johannes Schmitt Open the ORCID record for Johannes Schmitt [Opens in a new window]  and
Ulrich Thiel
Gwyn Bellamy
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK
Johannes Schmitt*
Affiliation:
Fachbereich Mathematik, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany
Ulrich Thiel
Affiliation:
Fachbereich Mathematik, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany
*
E-mail: schmitt@mathematik.uni-kl.de
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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 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.

Keywords

Symplectic reflection groupsparabolic subgroupssymplectic resolutionsSteinberg’s fixed point theorem

MSC classification

Primary: 20H20: Other matrix groups over fields 13A50: Actions of groups on commutative rings; invariant theory
Secondary: 14B05: Singularities
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 65 , Issue 2 , May 2023 , pp. 401 - 413
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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