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On representations of rational Cherednik algebras of complex rank

Author: Inna Entova Aizenbud
Journal: Represent. Theory 18 (2014), 361-407
MSC (2010): Primary 16S99, 18D10
Published electronically: November 24, 2014
MathSciNet review: 3280664
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Abstract: We study a family of abelian categories $ \underline {\mathcal {O}}_{\text { } c,\nu }$ depending on complex parameters $ c, \nu $ which are interpolations of the category $ \mathcal {O}$ for the rational Cherednik algebra $ H_c(\nu )$ of type $ A$, where $ \nu $ is a positive integer. We define the notion of a Verma object in such a category (a natural analogue of the notion of Verma module).

We give some necessary conditions and some sufficient conditions for the existence of a non-trivial morphism between two such Verma objects. We also compute the character of the irreducible quotient of a Verma object for sufficiently generic values of parameters $ c, \nu $, and prove that a Verma object of infinite length exists in $ \mathcal {O}_{\text { } c,\nu }$ only if $ c \in \mathbb{Q}_{<0}$. We also show that for every $ c \in \mathbb{Q}_{<0}$ there exists $ \nu \in \mathbb{Q}_{<0}$ such that there exists a Verma object of infinite length in $ \mathcal {O}_{\text { } c,\nu }$.

The latter result is an example of a degeneration phenomenon which can occur in rational values of $ \nu $, as was conjectured by P. Etingof.

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  • [1] Roman Bezrukavnikov and Pavel Etingof, Parabolic induction and restriction functors for rational Cherednik algebras, Selecta Math. (N.S.) 14 (2009), no. 3-4, 397-425. MR 2511190 (2010e:20007),
  • [2] Yuri Berest, Pavel Etingof, and Victor Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), no. 2, 279-337. MR 1980996 (2004f:16039),
  • [3] Yuri Berest, Pavel Etingof, and Victor Ginzburg, Finite-dimensional representations of rational Cherednik algebras, Int. Math. Res. Not. 19 (2003), 1053-1088. MR 1961261 (2004h:16027),
  • [4] Emmanuel Briand, Rosa Orellana, and Mercedes Rosas, The stability of the Kronecker product of Schur functions, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), Discrete Math. Theor. Comput. Sci. Proc., AN, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010, pp. 557-567 (English, with English, French and Spanish summaries). MR 2673866 (2012m:05404)
  • [5] Jonathan Comes and Victor Ostrik, On blocks of Deligne's category $ \underline {\rm Re}{\rm p}(S_t)$, Adv. Math. 226 (2011), no. 2, 1331-1377. MR 2737787 (2012b:20020),
  • [6] P. Deligne, La catégorie des représentations du groupe symétrique $ S_t$, lorsque $ t$ n'est pas un entier naturel, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007, pp. 209-273 (French, with English and French summaries). MR 2348906 (2009b:20021)
  • [7] Richard Dipper and Gordon James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 54 (1987), no. 1, 57-82. MR 872250 (88m:20084),
  • [8] Karin Erdmann and Daniel K. Nakano, Representation type of Hecke algebras of type $ A$, Trans. Amer. Math. Soc. 354 (2002), no. 1, 275-285 (electronic). MR 1859276 (2002j:20011),
  • [9] Pavel Etingof, Representation theory in complex rank, I, Transform. Groups 19 (2014), no. 2, 359-381. MR 3200430,
  • [10] P. Etingof, X. Ma, Lecture notes on Cherednik algebras; arXiv:1001.0432v4 [math.RT].
  • [11] Pavel Etingof and Emanuel Stoica, Unitary representations of rational Cherednik algebras, Represent. Theory 13 (2009), 349-370. With an appendix by Stephen Griffeth. MR 2534594 (2011c:16081),
  • [12] William Fulton and Joe Harris, Representation theory: A first course; Readings in Mathematics, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. MR 1153249 (93a:20069)
  • [13] Victor Ginzburg, Nicolas Guay, Eric Opdam, and Raphaël Rouquier, On the category $ \mathcal {O}$ for rational Cherednik algebras, Invent. Math. 154 (2003), no. 3, 617-651. MR 2018786 (2005f:20010),
  • [14] Eugene Gorsky, Alexei Oblomkov, Jacob Rasmussen, and Vivek Shende, Torus knots and the rational DAHA, Duke Math. J. 163 (2014), no. 14, 2709-2794. MR 3273582,
  • [15] I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), no. 1, 222-274. MR 2183255 (2008i:14006),
  • [16] Friedrich Knop, A construction of semisimple tensor categories, C. R. Math. Acad. Sci. Paris 343 (2006), no. 1, 15-18 (English, with English and French summaries). MR 2241951 (2007b:18008),
  • [17] I. Losev, Towards multiplicities for categories O of cyclotomic rational Cherednik algebras, arXiv:1207.1299v2 [math.RT].
  • [18] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144 (96h:05207)
  • [19] Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316 (2001g:20006)
  • [20] Akhil Mathew, Categories parametrized by schemes and representation theory in complex rank, J. Algebra 381 (2013), 140-163. MR 3030515,
  • [21] Raphaël Rouquier, $ q$-Schur algebras and complex reflection groups, Mosc. Math. J. 8 (2008), no. 1, 119-158, 184 (English, with English and Russian summaries). MR 2422270 (2010b:20081)
  • [22] Raphaël Rouquier, Representations of rational Cherednik algebras, Infinite-dimensional aspects of representation theory and applications, Contemp. Math., vol. 392, Amer. Math. Soc., Providence, RI, 2005, pp. 103-131. MR 2189874 (2007d:20006),

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Additional Information

Inna Entova Aizenbud
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Keywords: Deligne categories, rational Cherednik algebra
Received by editor(s): March 17, 2014
Received by editor(s) in revised form: June 14, 2014, and September 17, 2014
Published electronically: November 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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