Poisson homology in degree 0 for some rings of symplectic invariants

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Schedler²

2012

Abstract. Let X ⊂ C 3 be a surface with an isolated singularity at the origin, given by the equation Q(x, y, z) = 0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = C 2 /G for G < SL 2 (C) finite. Let Y := S n X be the n-th symmetric power of X. We compute the zeroth Poisson homology HP 0 (Y ), as a graded vector space with respect to the weight grading. In the Kleinian case, this confirms a conjecture of Alev, that), where Weyl 2n is the Weyl algebra on 2n generators. That is, the Brylinski spectral sequence degenerates in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, A γ , for all but countably many parameters γ in the elliptic curve.

Section: Introductionsupporting

confidence: 82%

“…The case (Z/2) n ⋊ S n then identifies with the Weyl groups of type B n (and also with type C n ). We do not know whether this conjecture holds for general Weyl groups of types other than B (or C), although it was verified in types D 2 = A 1 × A 1 and D 3 = A 3 in [But08].…”

Section: Introductionmentioning

confidence: 94%

Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Schedler²

2012

“…In some special cases, the lower bound is attained, i.e., the surjection is an isomorphism. For example, HP 0 (O G V ) ∼ = gr HH 0 (D G X ) is known to hold when V = C 2 , and in [ES09b], the first and last authors generalized this to the case V = C 2n = (C 2 ) ⊕n and G = S n ⋉ K n for K < SL 2 (C) (certain cases were shown previously in [But09], and this result was conjectured by Alev [But09,Remark 40]). In [ES], the same authors will show that HP 0 (O G V ) ∼ = gr HH 0 (D G X ) when G = S n+1 is a Weyl group of type A n acting on its reflection representation V = C 2n (but not for the D n case).…”

mentioning

confidence: 99%

Computational Approaches to Poisson Traces Associated to Finite Subgroups of

Experimental Mathematics

Gong

Pacchiano

et al. 2012

Abstract. We reduce the computation of Poisson traces on quotients of symplectic vector spaces by finite subgroups of symplectic automorphisms to a finite one, by proving several results which bound the degrees of such traces as well as the dimension in each degree. This applies more generally to traces on all polynomial functions which are invariant under invariant Hamiltonian flow. We implement these approaches by computer together with direct computation for infinite families of groups, focusing on complex reflection and abelian subgroups of GL2(C) < Sp 4 (C), Coxeter groups of rank ≤ 3 and A4, B4 = C4, and D4, and subgroups of SL2(C).

Poisson Traces for Symmetric Powers of Symplectic Varieties

International Mathematics Research Notices

Schedler

2013

We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities T * C n−1 /Sn associated to the type A Weyl group Sn and its reflection representation C n−1 . We also compute the full structure of the natural D-module, previously defined by the authors, whose solution space over algebraic distributions identifies with the space of Poisson traces. As a consequence, we deduce bounds on the numbers of finite-dimensional irreducible representations and prime ideals of quantizations of these varieties. Finally, motivated by these results, we pose conjectures on symplectic resolutions, and give related examples of the natural D-module. In an appendix, the second author computes the Poisson traces and associated D-module for the quotients T * C n /Dn associated to type D Weyl groups. In a second appendix, the same author provides a direct proof of one of the main theorems.