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Totally Aspherical Parameters for Cherednik Algebras

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Abstract

We introduce the notion of a totally aspherical parameter for a rational Cherednik algebra. We get an explicit construction of the projective object defining the KZ functor for such parameters. We establish the existence of sufficiently many totally aspherical parameters for the groups G(, 1, n).

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Footnotes
1
After this paper was written, Etingof, [E], checked that this is the case for all complex reflection groups W.
 
Literature
[BEG]
go back to reference Yu. Berest, P. Etingof, V. Ginzburg, Finite-dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 2003, no. 19, 1053–1088. Yu. Berest, P. Etingof, V. Ginzburg, Finite-dimensional representations of rational Cherednik algebras. Int. Math. Res. Not. 2003, no. 19, 1053–1088.
[BE]
go back to reference R. Bezrukavnikov, P. Etingof, Parabolic induction and restriction functors for rational Cherednik algebras. Selecta Math., 14(2009), 397–425.MathSciNetCrossRef R. Bezrukavnikov, P. Etingof, Parabolic induction and restriction functors for rational Cherednik algebras. Selecta Math., 14(2009), 397–425.MathSciNetCrossRef
[BPW]
go back to reference T. Braden, N. Proudfoot, B. Webster, Quantizations of conical symplectic resolutions I: local and global structure. arXiv:1208.3863. T. Braden, N. Proudfoot, B. Webster, Quantizations of conical symplectic resolutions I: local and global structure. arXiv:1208.3863.
[E]
go back to reference P. Etingof, Proof of the Broué-Malle-Rouquier conjecture in characteristic zero (after I. Losev and I. Marin - G. Pfeiffer), arXiv:1606.08456. P. Etingof, Proof of the Broué-Malle-Rouquier conjecture in characteristic zero (after I. Losev and I. Marin - G. Pfeiffer), arXiv:1606.08456.
[CR]
go back to reference J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and\(\mathfrak {sl}_2\)-categorifications. Ann. Math. (2) 167(2008), n.1, 245–298. J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and\(\mathfrak {sl}_2\)-categorifications. Ann. Math. (2) 167(2008), n.1, 245–298.
[EG]
go back to reference P. Etingof and V. Ginzburg. Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.MathSciNetCrossRef P. Etingof and V. Ginzburg. Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.MathSciNetCrossRef
[EGL]
go back to reference P. Etingof, E. Gorsky, I. Losev, Representations of Rational Cherednik algebras with minimal support and torus knots. arXiv:1304.3412. P. Etingof, E. Gorsky, I. Losev, Representations of Rational Cherednik algebras with minimal support and torus knots. arXiv:1304.3412.
[GGOR]
go back to reference V. Ginzburg, N. Guay, E. Opdam and R. Rouquier, On the category\(\mathcal {O}\)for rational Cherednik algebras, Invent. Math., 154 (2003), 617–651.MathSciNetCrossRef V. Ginzburg, N. Guay, E. Opdam and R. Rouquier, On the category\(\mathcal {O}\)for rational Cherednik algebras, Invent. Math., 154 (2003), 617–651.MathSciNetCrossRef
[Go1]
go back to reference I. Gordon. A remark on rational Cherednik algebras and differential operators on the cyclic quiver. Glasg. Math. J. 48(2006), 145–160.MathSciNetCrossRef I. Gordon. A remark on rational Cherednik algebras and differential operators on the cyclic quiver. Glasg. Math. J. 48(2006), 145–160.MathSciNetCrossRef
[Go2]
go back to reference I. Gordon. Symplectic reflection alegebras. Trends in representation theory of algebras and related topics, 285–347, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008. I. Gordon. Symplectic reflection alegebras. Trends in representation theory of algebras and related topics, 285–347, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008.
[GL]
go back to reference I. Gordon, I. Losev, On category\(\mathcal {O}\)for cyclotomic Rational Cherednik algebras. J. Eur. Math. Soc. 16 (2014), 1017–1079.MathSciNetCrossRef I. Gordon, I. Losev, On category\(\mathcal {O}\)for cyclotomic Rational Cherednik algebras. J. Eur. Math. Soc. 16 (2014), 1017–1079.MathSciNetCrossRef
[HK]
go back to reference R. Hotta, M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra. Invent. Math. 75 (1984), 327–358.MathSciNetCrossRef R. Hotta, M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra. Invent. Math. 75 (1984), 327–358.MathSciNetCrossRef
[L1]
go back to reference I. Losev, Completions of symplectic reflection algebras. Selecta Math., 18(2012), N1, 179–251. I. Losev, Completions of symplectic reflection algebras. Selecta Math., 18(2012), N1, 179–251.
[L2]
[L3]
[L4]
go back to reference I. Losev. Wall-crossing functors for quantized symplectic resolutions: perversity and partial Ringel dualities. Pure Appl. Math. Q. 13 (2017), no. 2, 247–289.MathSciNetCrossRef I. Losev. Wall-crossing functors for quantized symplectic resolutions: perversity and partial Ringel dualities. Pure Appl. Math. Q. 13 (2017), no. 2, 247–289.MathSciNetCrossRef
[MN]
go back to reference K. McGerty and T. Nevins, Derived equivalence for quantum symplectic resolutions. Selecta Math. 20(2014), 675–717.MathSciNetCrossRef K. McGerty and T. Nevins, Derived equivalence for quantum symplectic resolutions. Selecta Math. 20(2014), 675–717.MathSciNetCrossRef
[R]
go back to reference R. Rouquier, 2-Kac-Moody algebras. arXiv:0812.5023. R. Rouquier, 2-Kac-Moody algebras. arXiv:0812.5023.
[S]
go back to reference P. Shan. Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras. Ann. Sci. Ecole Norm. Sup. 44 (2011), 147–182.MathSciNetCrossRef P. Shan. Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras. Ann. Sci. Ecole Norm. Sup. 44 (2011), 147–182.MathSciNetCrossRef
[SV]
go back to reference P. Shan and E. Vasserot, Heisenberg algebras and rational double affine Hecke algebras. J. Amer. Math. Soc. 25(2012), 959–1031.MathSciNetCrossRef P. Shan and E. Vasserot, Heisenberg algebras and rational double affine Hecke algebras. J. Amer. Math. Soc. 25(2012), 959–1031.MathSciNetCrossRef
[T]
go back to reference S. Thelin. An algebraic approach to the KZ-functor for rational Cherednik algebras associated with cyclic groups. J. Algebra, 471 (2017), 113–148.MathSciNetCrossRef S. Thelin. An algebraic approach to the KZ-functor for rational Cherednik algebras associated with cyclic groups. J. Algebra, 471 (2017), 113–148.MathSciNetCrossRef
[W]
go back to reference B. Webster. A categorical action on quantized quiver varieties. arXiv:1208.5957. B. Webster. A categorical action on quantized quiver varieties. arXiv:1208.5957.
Metadata
Title
Totally Aspherical Parameters for Cherednik Algebras
Author
Ivan Losev
Copyright Year
2022
DOI
https://doi.org/10.1007/978-3-030-82007-7_2

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