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ON THE SMOOTHNESS OF CENTRES OF RATIONAL CHEREDNIK ALGEBRAS IN POSITIVE CHARACTERISTIC

Published online by Cambridge University Press:  01 October 2013

GWYN BELLAMY  and
MAURIZIO MARTINO
GWYN BELLAMY
Affiliation:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, United Kingdom e-mail: [email protected]
MAURIZIO MARTINO
Affiliation:
Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Germany e-mail: [email protected]
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Abstract

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In this paper we study rational Cherednik algebras at t = 1 in positive characteristic. We study a finite-dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite series G(m, d, n), we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.

Keywords

16G9916W70
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue A: New developments in Noncommutative Algebra and its applications. A conference in honour of Kenny Brown's and Toby Stafford's 60th birthdays. , October 2013 , pp. 27 - 54
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Balagovic, M. and Chen, H., Category $\mathcal{O}$ for rational Cherednik algebras H t,c(GL 2(F p),h) in characteristic p, J. Pure Appl. Algebra 217 (9) (2013), 1683–1699.Google Scholar
2.Balagovic, M. and Chen, H., Representations of rational Cherednik algebras in positive characteristic, J. Pure Appl. Algebra 217 (4) (2013), 716–740.Google Scholar
3.Bellamy, G., On singular Calogero-Moser spaces, Bull. London Math. Soc. 41 (2) (2009), 315–326.Google Scholar
4.Bellamy, G., The Calogero–Moser partition for G(m,d,n), Nagoya Math. J. 207 (2012), 47–77.CrossRefGoogle Scholar
5.Bezrukavnikov, R. and Etingof, P., Parabolic induction and restriction functors for rational Cherednik algebras, Selecta Math. (NS) 14 (3–4) (2009), 397–425.Google Scholar
6.Bezrukavnikov, R., Finkelberg, M. and Ginzburg, V., Cherednik algebras and Hilbert schemes in characteristic p, Represent. Theory 10 (2006), 254–298,. With an appendix by Pavel Etingof.Google Scholar
7.Brown, K. A. and Changtong, K., Symplectic reflection algebras in positive characteristic, Proc. Edinb. Math. Soc., 53 (1) (2010), 61–81.Google Scholar
8.Brown, K. A. and Gordon, I. G., The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z. 238 (4) (2001), 733–779.CrossRefGoogle Scholar
9.Brown, K. A., Gordon, I. G. and Stroppel, C. H., Cherednik, Hecke and quantum algebras as free Frobenius and Calabi-Yau extensions, J. Algebra 319 (3) (2008), 1007–1034.Google Scholar
10.Curtis, C. W. and Reiner, I., Methods of representation theory, Vol. I; with applications to finite groups and orders, Pure and Applied Mathematics series (John Wiley, New York, NY, 1981).Google Scholar
11.Dunkl, C. and Griffeth, S., Generalized Jack polynomials and the representation theory of rational Cherednik algebras, Selecta Math. (NS) 16 (4) (2010), 791–818.Google Scholar
12.Dunkl, C. F. and Opdam, E. M., Dunkl operators for complex reflection groups, Proc. London Math. Soc., 86 (1) (2003), 70–108.Google Scholar
13.Eisenbud, D., Commutative algebra, Graduate Texts in Mathematics, vol. 150. (Springer-Verlag, New York, 1995). With a view toward algebraic geometry.Google Scholar
14.Etingof, P. and Ginzburg, V., Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2) (2002), 243–348.Google Scholar
15.Geck, M. and Pfeiffer, G., Characters of finite coxeter groups and Iwahori–Hecke algebras, London Mathematical Society Monographs New Series, vol. 21. (Clarendon Press/Oxford University Press, New York, NY, 2000).Google Scholar
16.Ginzburg, V. and Kaledin, D., Poisson deformations of symplectic quotient singularities, Adv. Math. 186 (1) (2004), 1–57.Google Scholar
17.Gordon, I. G., Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc. 35 (3) (2003), 321–336.Google Scholar
18.Gordon, I. G., Quiver varieties, category $\mc{O}$ for rational Cherednik algebras and Hecke algebras, International Mathematics Research Papers (IMRP), vol. 2008(3) (2008), Art. ID: rpn006, 69; doi:10.1093/imrp/rpn006.Google Scholar
19.Griffeth, S., Orthogonal functions generalizing Jack polynomials, Trans AMS 362 (2010), 6131–6157.Google Scholar
20.Griffeth, S., Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n), Proc. Edinb. Math. Soc., 53 (2) (2010), 419–445.CrossRefGoogle Scholar
21.Holmes, R. R. and Nakano, D. K., Brauer-type reciprocity for a class of graded associative algebras, J. Algebra 144 (1) (1991), 117–126.Google Scholar
22.James, G. and Kerber, A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16 (Addison-Wesley, Reading, MA, 1981) (with a foreword by Cohn, P. M., and an introduction by Gilbert de, B. Robinson).Google Scholar
23.Katz, N. M., Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. (39) (1970), 175–232.Google Scholar
24.Kemper, G. and Malle, G., The finite irreducible linear groups with polynomial ring of invariants, Transform. Groups, 2 (1) (1997), 57–89.Google Scholar
25.Lehrer, G. I. and Taylor, D. E., Unitary reflection groups, Australian Mathematical Society Lecture Series, vol. 20 (Cambridge University Press, Cambridge, UK, 2009).Google Scholar
26.Martino, M., Stratifications of Marsden–Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities, Math. Z. 258 (1) (2008), 1–28.Google Scholar
27.Martino, M., Blocks of restricted rational Cherednik algebras for G(m,d,n) (to appear) J. Algebra.Google Scholar
28.Martino, M., The Calogero–Moser partition and Rouquier families for complex reflection groups, J. Algebra 323 (2010), 193–205.CrossRefGoogle Scholar
29.Mc Connell, J. C. and Robson, J. C., Noncommutative noetherian rings, Graduate Studies in Mathematics, vol. 30 (American Mathematical Society, Providence, RI, 2001, revised edition) (with the cooperation of L. W. Small).Google Scholar
30.Premet, A., Irreducible representations of Lie algebras of reductive groups and the Kac–Weisfeiler conjecture, Invent. Math. 121 (1) (1995), 79–117.Google Scholar
31.Premet, A. and Skryabin, S., Representations of restricted Lie algebras and families of associative $\mathcal{L}$-algebras, J. Reine Angew. Math. 507 (1999), 189–218.Google Scholar
32.Pushkarev, I. A., On the theory of representations of the wreath products of finite groups and symmetric groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 240 (1997), 229–244; Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2 (1997), 294–295.Google Scholar
33.Smith, L., Polynomial invariants of finite groups, Research Notes in Mathematics, vol. 6 (A K Peters, Wellesley, MA, 1995).CrossRefGoogle Scholar
34.Tikaradze, A., An analogue of the Kac-Weisfeiler conjecture, J. Algebra 383 (2013), 168–177.CrossRefGoogle Scholar
35.Ju. Veĭsfeĭler, B. and Kac, V. G., The irreducible representations of Lie p-algebras, Funkcional. Anal. i Priložen. 5 (2) (1971), 28–36.Google Scholar