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Register a new userTowards the classification of symplectic linear quotient singularities admitting a symplectic resolution
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Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 - the symplectically primitive but complex imprimitive groups - and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving 39+9=48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.
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We study the hyperplane arrangements associated, via the minimal model programme, to symplectic quotient singularities. We show that this hyperplane arrangement equals the arrangement of CM-hyperplanes coming from the representation theory of restricted rational Cherednik algebras. We explain some of the interesting consequences of this identification for the representation theory of restricted rational Cherednik algebras. We also show that the Calogero-Moser space is smooth if and only if the Calogero-Moser families are trivial. We describe the arrangements of CM-hyperplanes associated to several exceptional complex reflection groups, some of which are free.
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Recently, Herbig--Schwarz--Seaton have shown that $3$-large representations of a reductive group $G$ give rise to a large class of symplectic singularities via Hamiltonian reduction. We show that these singularities are always terminal. We show that they are $mathbb{Q}$-factorial if and only if $G$ has finite abelianization. When $G$ is connected and semi-simple, we show they are actually locally factorial. As a consequence, the symplectic singularities do not admit symplectic resolutions when $G$ is semi-simple. We end with some open questions.
We compute the symplectic reductions for the action of Sp_2n on several copies of C^2n and for all coregular representations of Sl_2. If it exists we give at least one symplectic resolution for each example. In the case Sl_2 acting on sl_2+C^2 we obtain an explicit description of Fus and Namikawas example of two non-equivalent symplectic resolutions connected by a Mukai flop.
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