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Pull-back and push-forward functors for holonomic modules over Cherednik algebras

Published 28 Feb 2024 in math.QA, math.AG, and math.RT | (2402.18210v1)

Abstract: In this article we continue the study of holonomic modules over sheaves of Cherednik algebras, initiated by the third author in [Tho18]. Working with arbitrary parameters, we first develop a theory of bb-functions to prove that push-forward along open embeddings preserves holonomicity. This implies that pull-back along closed embeddings also preserves holonomicity. We use these facts to show that both push-forward and pull-back under any melys morphism preserves holonomicity. Since duality preserves holonomicity, we deduce that extraordinary push-forward and extraordinary pull-back also exist for holonomic modules. As a consequence, we give a general classification of irreducible holonomic modules similar to the classification of irreducible holonomic D\mathscr{D}-modules as minimal extensions of integrable connections on locally closed subsets. Finally, we prove that ext-groups between holonomic modules are finite-dimensional and explore applications of our work to the classification of aspherical parameters and existence of finite-dimensional modules for sheaves of Cherednik algebras.

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References (37)
  1. A new algebraic approach to microlocalization of filtered rings. Trans. Amer. Math. Soc., 316(2):537–553, 1989.
  2. A. Beĭlinson and J. Bernstein. A proof of Jantzen conjectures. In I. M. Gel′normal-′{}^{\prime}start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPTfand Seminar, volume 16 of Adv. Soviet Math., pages 1–50. Amer. Math. Soc., Providence, RI, 1993.
  3. Y. Berest and O. Chalykh. Quasi-invariants of complex reflection groups. Compos. Math., 147(3):965–1002, 2011.
  4. R. Bezrukavnikov and P. Etingof. Parabolic induction and restriction functors for rational Cherednik algebras. Selecta Math. (N.S.), 14(3-4):397–425, 2009.
  5. J. Bertin. The punctual hilbert scheme: an introduction. In Geometric Methods in Representation Theory, I, volume 24 of Séminaires et Congrés, pages 1–100. Soc. Math. France, Paris, 2010.
  6. Algebraic D𝐷Ditalic_D-modules, volume 2 of Perspectives in Mathematics. Academic Press, Inc., Boston, MA, 1987.
  7. J-E. Björk. Rings of Differential Operators, volume 21 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1979.
  8. G. Bellamy and M. Martino. Affinity of Cherednik algebras on projective space. Algebra Number Theory, 8(5):1151–1177, 2014.
  9. N. Chriss and V. Ginzburg. Representation theory and complex geometry. Birkhäuser Boston Inc., Boston, MA, 1997.
  10. S. Coutinho and D. Levcovitz. 𝒟𝒟\mathscr{D}script_D-modules and étale morphisms. Comm. Algebra, 29(4):1487–1497, 2001.
  11. E. Dade. Compounding Clifford’s theory. Ann. of Math. (2), 91:236–290, 1970.
  12. E. Dade. Group-graded rings and modules. Math. Z., 174(3):241–262, 1980.
  13. C. Dunkl and S. Griffeth. Generalized Jack polynomials and the representation theory of rational Cherednik algebras. Selecta Math. (N.S.), 16(4):791–818, 2010.
  14. C. Dunkl and E. Opdam. Dunkl operators for complex reflection groups. Proc. London Math. Soc. (3), 86(1):70–108, 2003.
  15. P. Etingof and V. Ginzburg. Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. Invent. Math., 147(2):243–348, 2002.
  16. P. Etingof. Symplectic reflection algebras and affine Lie algebras. Mosc. Math. J., 12(3):543–565, 668–669, 2012.
  17. P. Etingof. Cherednik and Hecke algebras of varieties with a finite group action. Mosc. Math. J., 17(4):635–666, 2017.
  18. P. Gabriel. Des catégories abéliennes. Bulletin de la Société Mathématique de France, 90:323–448, 1962.
  19. On the category o𝑜oitalic_o for rational Cherednik algebras. Invent. Math., 154(3):617–651, 2003.
  20. V. Ginzburg. Lectures on D-modules. 1998.
  21. P. Grzeszczuk. On G𝐺Gitalic_G-systems and G𝐺Gitalic_G-graded rings. Proc. Amer. Math. Soc., 95(3):348–352, 1985.
  22. K. Goodearl and R. Warfield, Jr. An Introduction to Noncommutative Noetherian Rings, volume 61 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, second edition, 2004.
  23. R. Hartshorne. On the De Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math., (45):5–99, 1975.
  24. R. Hartshorne. Algebraic Geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.
  25. D𝐷Ditalic_D-modules, Perverse Sheaves, and Representation Theory, volume 236 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi.
  26. L. Illusie. Existence de résolutions globales. In Théorie des intersections et théorème de Riemann-Roch, SGA 6, volume 225 of Lecture Notes in Math., pages 160–221. Springer, Berlin, 1971.
  27. M. Kashiwara. Representation theory and D𝐷Ditalic_D-modules on flag varieties. Astérisque, (173-174):9, 55–109, 1989. Orbites unipotentes et représentations, III.
  28. I. Losev. Completions of symplectic reflection algebras. Selecta Math. (N.S.), 18(1):179–251, 2012.
  29. I. Losev. Finite-dimensional quotients of Hecke algebras. Algebra Number Theory, 9(2):493–502, 2015.
  30. I. Losev. Bernstein inequality and holonomic modules. Adv. Math., 308:941–963, 2017. With an appendix by Losev and P. Etingof.
  31. H. Matsumura. Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid.
  32. I. Marin and J. Michel. Automorphisms of complex reflection groups. Represent. Theory, 14:747–788, 2010.
  33. J. McConnell and J. Robson. Noncommutative Noetherian Rings, volume 30 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, revised edition, 2001. With the cooperation of L. W. Small.
  34. E. Opdam. Some applications of hypergeometric shift operators. Invent. Math., 98(1):1–18, 1989.
  35. The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2018.
  36. D. Thompson. Holonomic modules over Cherednik algebras, I. J. Algebra, 493:150–170, 2018.
  37. S. Wilcox. Supports of representations of the rational Cherednik algebra of type A. Adv. Math., 314:426–492, 2017.
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