Duke Mathematical Journal

Compatibility of t-structures for quantum symplectic resolutions

Kevin McGerty and Thomas Nevins

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Abstract

Let W be a smooth complex variety with the action of a connected reductive group G. Adapting Teleman’s stratification approach to a microlocal context, we prove a vanishing theorem for the functor of G-invariant sections—that is, of quantum Hamiltonian reduction—for G-equivariant twisted D-modules on W. As a consequence, when W is affine we establish a sufficient combinatorial condition for exactness of the global sections functors of microlocalization theory. When combined with the derived equivalence results of our recent work, this gives precise criteria for “microlocalization of representation categories” in the spirit of Kashiwara–Rouquier and many other authors.

Article information

Source
Duke Math. J., Volume 165, Number 13 (2016), 2529-2585.

Dates
Received: 21 March 2014
Revised: 16 October 2015
First available in Project Euclid: 23 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1466703957

Digital Object Identifier
doi:10.1215/00127094-3619684

Mathematical Reviews number (MathSciNet)
MR3546968

Zentralblatt MATH identifier
06650078

Subjects
Primary: 16G99: None of the above, but in this section
Secondary: 53D55: Deformation quantization, star products 53D20: Momentum maps; symplectic reduction 17B63: Poisson algebras 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22]

Keywords
quantum Hamiltonian reduction $D$-modules microlocalization $t$-exactness localization Kirwan–Ness stratification

Citation

McGerty, Kevin; Nevins, Thomas. Compatibility of $t$ -structures for quantum symplectic resolutions. Duke Math. J. 165 (2016), no. 13, 2529--2585. doi:10.1215/00127094-3619684. https://projecteuclid.org/euclid.dmj/1466703957


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References

  • [1] G. Bellamy, C. Dodd, K. McGerty, and T. Nevins, Categorical cell decomposition of quantized symplectic algebraic varieties, preprint, arXiv:1311.6804v2 [math.AG].
  • [2] G. Bellamy and T. Kuwabara, On deformation quantizations of hypertoric varieties, Pacific J. Math. 260 (2012), 89–127.
  • [3] A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497.
  • [4] T. Braden, N. Proudfoot, and B. Webster, Quantizations of conical symplectic resolutions, I: Local and global structure, preprint, arXiv:1208.3863v4 [math.RT].
  • [5] P. Etingof, W. L. Gan, V. Ginzburg, and A. Oblomkov, Harish–Chandra homomorphisms and symplectic reflection algebras for wreath-products, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 91–155.
  • [6] W. L. Gan and V. Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. IMRN 2002, no. 5, 243–255.
  • [7] V. Ginzburg, I. Gordon, and J. T. Stafford, Differential operators and Cherednik algebras, Selecta Math. (N.S.) 14 (2009), 629–666.
  • [8] I. Gordon, A remark on rational Cherednik algebras and differential operators on the cyclic quiver, Glasg. Math. J. 48 (2006), 145–160.
  • [9] I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), 222–274.
  • [10] I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves, Duke Math. J. 132 (2006), 73–135.
  • [11] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, 2nd ed., Cambridge Univ. Press, Cambridge, 1990.
  • [12] M. P. Holland, Quantization of the Marsden–Weinstein reduction for extended Dynkin quivers, Ann. Sci. Éc. Norm. Supér. (4) 32 (1999), 813–834.
  • [13] M. Kashiwara, Systems of Microdifferential Equations, Progr. Math. 34, Birkhäuser, Boston, 1983.
  • [14] M. Kashiwara, “Equivariant derived category and representation of real semisimple Lie groups” in Representation Theory and Complex Analysis, Lecture Notes in Math. 1931, Springer, Berlin, 2008, 137–234.
  • [15] M. Kashiwara and R. Rouquier, Microlocalization of rational Cherednik algebras, Duke Math. J. 144 (2008), 525–573.
  • [16] M. Kashiwara and P. Schapira, Deformation Quantization Modules, Astérisque 345, Soc. Math. France, Paris, 2012.
  • [17] G. R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), 299–316.
  • [18] A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530.
  • [19] F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Math. Notes 31, Princeton Univ. Press, Princeton, 1984.
  • [20] F. Knop, Weyl groups of Hamiltonian manifolds, I, preprint, arXiv:dg-ga/9712010v1.
  • [21] Y. Li, A geometric realization of modified quantum algebras, preprint, arXiv:1007.5384v4 [math.RT].
  • [22] Y. Li, On geometric realizations of quantum modified algebras and their canonical bases, II, preprint, arXiv:1009.0838v1 [math.RT].
  • [23] I. Losev, Symplectic slices for actions of reductive groups (in Russian), Mat. Sb. 197, no. 2 (2006), 75–86; English translation in Sb. Math. 197, no. 1–2 (2006), 213–224.
  • [24] I. Losev, Isomorphisms of quantizations via quantization of resolutions, Adv. Math. 231 (2012), 1216–1270.
  • [25] I. Losev, Abelian localization for cyclotomic Cherednik algebras, Int. Math. Res. Not. IMRN 2015, no. 18, 8860–8873.
  • [26] K. McGerty and T. Nevins, Derived equivalence for quantum symplectic resolutions, Selecta Math. (N.S.) 20 (2014), 675–717.
  • [27] K. McGerty and T. Nevins, Morse decomposition for $\mathcal{D}$-module categories on stacks, preprint, arXiv:1402.7365v1 [math.AG].
  • [28] K. McGerty and T. Nevins, Microlocal aspects of geometric Langlands, in preparation.
  • [29] I. M. Musson and M. Van den Bergh, Invariants under Tori of Rings of Differential Operators and Related Topics, Mem. Amer. Math. Soc. 136, Amer. Math. Soc., Providence, 1998.
  • [30] A. Oblomkov, Deformed Harish-Chandra homomorphism for the cyclic quiver, Math. Res. Lett. 14 (2007), 359–372.
  • [31] H. Sumihiro, Equivariant completion, II, Kyoto J. Math. 15 (1975), 573–605.
  • [32] C. Teleman, The quantization conjecture revisited, Ann. of Math. (2) 152 (2000), 1–43.
  • [33] B. Webster, A categorical action on quantized quiver varieties, preprint, arXiv:1208.5957v3 [math.AG].
  • [34] A. Yekutieli, Deformation quantization in algebraic geometry, Adv. Math. 198 (2005), 383–432.
  • [35] A. Yekutieli, Twisted deformation quantization of algebraic varieties, Adv. Math. 268 (2015), 241–305.
  • [36] H. Zheng, Categorification of integrable representations of quantum groups, Acta Math. Sin. (Engl. Ser.) 30 (2014), 899–932.