On parabolic subgroups of symplectic reflection groups

Bellamy, G. , Schmitt, J. and Thiel, U. (2023) On parabolic subgroups of symplectic reflection groups. Glasgow Mathematical Journal, (doi: 10.1017/S0017089522000416) (Early Online Publication)

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.

• Item Type: Articles
• Status: Early Online Publication
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bellamy, Professor Gwyn and Schmitt, Mr Johannes
• Authors: Bellamy, G., Schmitt, J., and Thiel, U.
• College/School: College of Science and Engineering > School of Mathematics and Statistics; College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Glasgow Mathematical Journal
• Publisher: Cambridge University Press
• ISSN: 0017-0895
• ISSN (Online): 1469-509X
• Published Online: 10 January 2023

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