×

On the asymptotic \(S_n\)-structure of invariant differential operators on symplectic manifolds. (English) Zbl 1268.58010

Summary: We consider the space of polydifferential operators on \(n\) functions on symplectic manifolds invariant under symplectic automorphisms, whose study was initiated by O. Mathieu in [Prog. Math. 131, 223–243 (1995; Zbl 0857.58043)]. 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 \(\leq 2d+1\)-th tensor powers of the reflection representation \({\mathfrak h}=\mathbb{C}^n\) of \(S_{n+1}\) is spanned by Poisson polynomials. We also prove a partial converse, and compute explicitly the isotypic part of \(\leq 4\)-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 fractions 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 \(\mathbb{Z}/(n+1)\) to \(S_{n+1}\) where \(n\) is viewed as a parameter.
We show that the space of invariant operators of order \(2m\) has polynomial dimension in \(n\) of degree equal to \(2m\), while the part not coming from Poisson polynomials has polynomial dimension of degree \(\leq 2m-3\). We use this to compute asymptotics of the dimension of invariant operators. We also give new bounds on the order of invariant operators for a fixed \(n\).
We apply this to the Poisson and Hochschild homology associated to the singularity \(\mathbb{C}^{2dn}/S_{n+1}\). Namely, the canonical surjection from \(\text{HP}_0({\mathcal O}_{\mathbb{C}^{2dn}/S_{n+1}},{\mathcal O}_{\mathbb{C}^{2dn}})\) to \(\text{grHH}_0(\text{Weyl}(\mathbb{C}^{2dn})^{S_{n+1}},\text{Weyl}(\mathbb{C}^{2dn}))\) (the Brylinski spectral sequence in degree zero) restricts to an isomorphism in the aforementioned isotypic part \({\mathfrak h}^{\otimes\leq 2d+1}\), and also in \({\mathfrak h}^{\otimes\leq 4}\). We prove a partial converse. Finally, the kernel of the entire surjection has dimension on the order of \(\frac{1}{n^3}\) times the dimension of the homology group.

MSC:

58D29 Moduli problems for topological structures
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D05 Symplectic manifolds (general theory)
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)

Citations:

Zbl 0857.58043

Software:

Magma
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alev, J.; Farinati, M. A.; Lambre, T.; Solotar, A. L., Homologie des invariants dʼune algèbre de Weyl sous lʼaction dʼun groupe fini, J. Algebra, 232, 564-577 (2000) · Zbl 1002.16005
[2] Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, J. Symbolic Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039
[3] Deligne, P., La catégorie des représentations du groupe symétrique \(S_t\), lorsque \(t\) nʼest pas un entier naturel, (Algebraic Groups and Homogeneous Spaces. Algebraic Groups and Homogeneous Spaces, Tata Inst. Fund. Res. Stud. Math. (2007), Tata Inst. Fund. Res.: Tata Inst. Fund. Res. Mumbai), 209-273 · Zbl 1165.20300
[4] P. Etingof, S. Gong, A. Pacchiano, Q. Ren, T. Schedler, Computational approaches to Poisson traces associated to finite subgroups of \(\operatorname{Sp}_{2 n}(\mathbb{C})\) arXiv:1101.5171; P. Etingof, S. Gong, A. Pacchiano, Q. Ren, T. Schedler, Computational approaches to Poisson traces associated to finite subgroups of \(\operatorname{Sp}_{2 n}(\mathbb{C})\) arXiv:1101.5171 · Zbl 1246.53111
[5] Etingof, P.; Schedler, T., Poisson traces for symmetric powers of symplectic varieties and reflection representations of Weyl groups (2011)
[6] Mathieu, O., The symplectic operad, (Functional Analysis on the Eve of the 21st Century, vol. 1. Functional Analysis on the Eve of the 21st Century, vol. 1, New Brunswick, NJ, 1993/Boston, MA. Functional Analysis on the Eve of the 21st Century, vol. 1. Functional Analysis on the Eve of the 21st Century, vol. 1, New Brunswick, NJ, 1993/Boston, MA, Progr. Math., vol. 131 (1995), Birkhäuser: Birkhäuser Boston), 223-243 · Zbl 0857.58043
[7] Stanley, R. P., The stable behavior of some characters of \(SL(n, C)\), Linear Multilinear Algebra, 16, 1-4, 3-27 (1984) · Zbl 0573.20042
[8] Stanley, R. P., Enumerative Combinatorics, vol. 2, Cambridge Stud. Adv. Math., vol. 62 (1999), Cambridge University Press: Cambridge University Press Cambridge, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin · Zbl 0928.05001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.