A new linear quotient of C⁴ admitting a symplectic resolution

We show that the quotient C^4/G admits a symplectic resolution for G = (Q_8 x D_8)/(Z/2) < Sp(4,C). Here Q_8 is the quaternionic group of order eight and D_8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements -1 of each. It is equipped with the tensor product of the defining two-dimensional representations of Q_8 and D_8. This group is also naturally a subgroup of the wreath product group of Q_8 by S_2. We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C^4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V / G admitting symplectic resolutions

Computational approaches to Poisson traces associated to finite subgroups of Sp(2n,C)

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 GL(2,C) < Sp(4,C), Coxeter groups of rank <= 3 and A_4, B_4=C_4, and D_4, and subgroups of SL(2,C).Comment: 37 pages, 6 figure

A new linear quotient of C^4 admitting a symplectic resolution

We show that the quotient C^4/G admits a symplectic resolution for G = (Q_8 x D_8)/(Z/2) < Sp(4,C). Here Q_8 is the quaternionic group of order eight and D_8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements -1 of each. It is equipped with the tensor product of the defining two-dimensional representations of Q_8 and D_8. This group is also naturally a subgroup of the wreath product group of Q_8 by S_2. We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C^4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V / G admitting symplectic resolutions.Comment: 15 page

A new linear quotient of C 4 admitting a symplectic resolution

C2 C2 ∼= C4.We show that the quotient C[superscript 4]/G admits a symplectic resolution for G = Q[subscript 8] x [subscript Z/2]D[subscript 8] < Sp[subscript 4](C). Here Q[subscript 8] is the quaternionic group of order eight and D[subscript 8] is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements −Id of each. It is equipped with the tensor product representation C[superscript 2] ⊠ C[superscript 2] ≅ C[superscript 4]. This group is also naturally a subgroup of the wreath product group Q[superscript 8][subscript 2] ⋊ S[subscript 2] < Sp[subscript 4](C). We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C[superscript 4]/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions

Poisson-de Rham homology of hypertoric varieties and nilpotent cones

We prove a conjecture of Etingof and the second author for hypertoric varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson-de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson-de Rham-Poincare polynomial, and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham. We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.Comment: 25 page

Poisson traces for symmetric powers of symplectic varieties

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 C^{2n-2}/S_n associated to the type A Weyl group S_n 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 C^{2n}/D_n associated to type D Weyl groups. In a second appendix, the same author provides a direct proof of one of the main theorems.Comment: 28 page

On the asymptotic S_n-structure of invariant differential operators on symplectic manifolds

We consider the space of polydifferential operators on n functions on symplectic manifolds invariant under symplectic automorphisms, whose study was initiated by Mathieu in 1995. Permutations of inputs yield an action of S_n, which extends to an action of S_{n+1}. We study this structure viewing n as a parameter, in the sense of Deligne's category. For manifolds of dimension 2d, we show that the isotypic part of this space of <= 2d+1-th tensor powers of the reflection representation h=C^n of S_{n+1} is spanned by Poisson polynomials. We also prove a partial converse, and compute explicitly the isotypic part of <= 4-th tensor powers of the reflection representation. We give generating functions for the isotypic parts corresponding to Young diagrams which only differ in the length of the top row, and prove that they are rational functions whose denominators are related to hook lengths of the diagrams obtained by removing the top row. This also gives such a formula for the same isotypic parts of induced representations from Z/(n+1) to S_{n+1} where n is viewed as a parameter. We apply this to the Poisson and Hochschild homology associated to the singularity C^{2dn}/S_{n+1}. Namely, the Brylinski spectral sequence from the zeroth Poisson homology of the S_{n+1}-invariants of the n-th Weyl algebra of C^{2d} with coefficients in the whole Weyl algebra degenerates in the 2d+1-th tensor power of h, as well as its fourth tensor power. Furthermore, the kernel of this spectral sequence has dimension on the order of 1/n^3 times the dimension of the homology group.Comment: v2: 47 pages; removed what was part (ii) of Theorem 1.3.45 since its proof was invalid. Nothing else was affected. v3: Several corrections; final version to be published in J. Algebr

A Singular Theta Lift and the Shimura Correspondence

Modular forms play a central and critical role in the study of modern number theory. These remarkable and beautiful functions have led to many spectacular results including, most famously, the proof of Fermat's Last Theorem. In this thesis we find connections between these enigmatic objects. In particular, we describe the construction and properties of a singular theta lift, closely related to the well known Shimura correspondence. We first define a (twisted) lift of harmonic weak Maass forms of weight 3/2-k, by integrating against a well chosen kernel Siegel theta function. Using this, we obtain a new class of automorphic objects in the upper-half plane of weight 2-2k for the group Gamma_{0}(N). We reveal these objects have intriguing singularities along a collection of geodesics. These singularities divide the upper-half plane into Weyl chambers with associated wall crossing formulas. We show our lift is harmonic away from the singularities and so is an example of a locally harmonic Maass form. We also find an explicit Fourier expansion. The Shimura/Shintani lifts provided very important correspondences between half-integral and even weight modular forms. Using a natural differential operator we link our lift to these. This connection then allows us to derive the properties of the Shimura lift. The nature of the singularities suggests we formulate all of these ideas as distributions and finally we consider the current equation encompassing them. This work provides extensions of the theta lifts considered by Borcherds (1998), Bruinier (2002), Bruinier and Funke (2004), Hövel (2012) and Bringmann, Kane and Viazovska (2013)