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Quasi-invariants of complex reflection groups

Part of: Lie algebras and Lie superalgebras Rings and algebras arising under various constructions Algebraic geometry: Foundations

Published online by Cambridge University Press:  27 September 2010

Yuri Berest  and
Oleg Chalykh
Yuri Berest
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA (email: berest@math.cornell.edu)
Oleg Chalykh
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK (email: oleg@maths.leeds.ac.uk)
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Abstract

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We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of a (singular) affine variety Xk. We extend the main results of Berest et al. [Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337] to this setting: in particular, we show that the variety Xk and the module Qk are Cohen–Macaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(ℂ) , where n=dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced by Dunkl and Opdam [Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70–108] and is parallel to that of Berest et al. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.

Keywords

complex reflection groupCoxeter grouprational Cherednik algebraDunkl operatorHecke algebraring of differential operatorsWeyl algebra

MSC classification

Secondary: 16S38: Rings arising from non-commutative algebraic geometry 14A22: Noncommutative algebraic geometry 17B45: Lie algebras of linear algebraic groups
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 965 - 1002
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Bandlow, J. and Musiker, G., A new characterization for the m-quasi-invariants of S n and explicit basis for two row hook shapes, J. Combin. Theory Ser. A 115 (2008), 1333–1357.CrossRefGoogle Scholar
[2]Ben-Zvi, D. and Nevins, T., Cusps and 𝒟-modules, J. Amer. Math. Soc. 17 (2004), 155–179.CrossRefGoogle Scholar
[3]Berest, Yu., The problem of lacunas and analysis on root systems, Trans. Amer. Math. Soc. 352 (2000), 3743–3776.CrossRefGoogle Scholar
[4]Berest, Yu., Etingof, P. and Ginzburg, V., Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337.CrossRefGoogle Scholar
[5]Berest, Yu. and Wilson, G., Differential isomorphism and equivalence of algebraic verieties, in Topology, geometry and quantum field theory, London Mathematical Society Lecture Note Series, vol. 308 (Cambridge University Press, Cambridge, 2004), 98–126.CrossRefGoogle Scholar
[6]Bourbaki, N., Groupes et algèbres de Lie (Hermann, Paris, 1968), chs. IV, V et VI.Google Scholar
[7]Broué, M., Malle, G. and Rouquier, R., Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127–190.Google Scholar
[8]Chalykh, O., Feigin, M. and Veselov, A., Multidimensional Baker–Akhiezer functions and Huygens’ principle, Comm. Math. Phys. 206 (1999), 533–566.CrossRefGoogle Scholar
[9]Chalykh, O. A. and Veselov, A. P., Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 126 (1990), 597–611.CrossRefGoogle Scholar
[10]Cherednik, I., Lectures on Knizhnik–Zamolodchikov equations and Hecke algebras, Math. Soc. Jap. Mem. 1 (1998), 1–96.Google Scholar
[11]Chevalley, C., Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–882.CrossRefGoogle Scholar
[12]Cohen, A. M., Finite complex reflection groups, Ann. Sci. École Norm. Sup. 9 (1976), 379–436.CrossRefGoogle Scholar
[13]Dunkl, C. F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167–183.CrossRefGoogle Scholar
[14]Dunkl, C. F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213–1227.CrossRefGoogle Scholar
[15]Dunkl, C. F., De Jeu, M. F. E. and Opdam, E. M., Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237–256.CrossRefGoogle Scholar
[16]Dunkl, C. F. and Opdam, E. M., Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70–108.CrossRefGoogle Scholar
[17]Etingof, P. and Ginzburg, V., Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.CrossRefGoogle Scholar
[18]Etingof, P. and Ginzburg, V., On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 2 (2002), 555–566.CrossRefGoogle Scholar
[19]Feigin, M. and Veselov, A. P., Quasi-invariants of Coxeter groups and m-harmonic polynomials, Int. Math. Res. Not. 10 (2002), 521–545.CrossRefGoogle Scholar
[20]Felder, G. and Veselov, A. P., Action of Coxeter groups on m-harmonic polynomials and KZ equations, Mosc. Math. J. 3 (2003), 1269–1291.CrossRefGoogle Scholar
[21]Garsia, A. M. and Wallach, N., Some new applications of orbit harmonics, Sém. Lothar. Combin. 50 (2003/04), 47 pp. (electronic).Google Scholar
[22]Garsia, A. M. and Wallach, N., The non-degeneracy of the bilinear form of m-quasi-invariants, Adv. Appl. Math. 37 (2006), 309–359.CrossRefGoogle Scholar
[23]Ginzburg, V., On primitive ideals, Selecta Math. (New series) 9 (2003), 379–407.CrossRefGoogle Scholar
[24]Ginzburg, V., Guay, N., Opdam, E. and Rouquier, R., On the category 𝒪 for rational Cherednik algebras, Invent. Math. 154 (2003), 617–651.CrossRefGoogle Scholar
[25]Gordon, I., On the quotient ring by diagonal invariants, Invent. Math. 153 (2003), 503–518.CrossRefGoogle Scholar
[26]Gordon, I. and Stafford, J. T., Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), 222–274.CrossRefGoogle Scholar
[27]Grothendieck, A., Eléments de geometrie algébrique IV, Publ. Math. Inst. Hautes. Études Sci. 32 (1967), Paris.Google Scholar
[28]Heckman, G. J., A remark on the Dunkl differential-difference operators, in Harmonic analysis on reductive groups, Progress in Mathematics, vol. 101 (Birkhäuser, Boston, 1991), 181–191.CrossRefGoogle Scholar
[29]Joseph, A., The Enright functor on the Bernstein–Gelfand–Gelfand category 𝒪, Invent. Math. 67 (1982), 423–445.CrossRefGoogle Scholar
[30]Knop, F., Graded cofinite rings of differential operators, Michigan Math. J. 54 (2006), 3–23.CrossRefGoogle Scholar
[31]Kohno, T., Integrable connections related to Manin and Schechtman’s higher braid groups, Illinois J. Math. 34 (1990), 476–484.CrossRefGoogle Scholar
[32]McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings (John Wiley & Sons, New York, 1987).Google Scholar
[33]Montgomery, S., Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818 (Springer, Berlin, 1980).CrossRefGoogle Scholar
[34]Opdam, E. M., Complex reflection groups and fake degrees, Preprint, arXiv:math.RT/9808026.Google Scholar
[35]Opdam, E. M., Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), 333–373.Google Scholar
[36]Opdam, E. M., A remark on the irreducible characters and fake degrees of finite real reflection groups, Invent. Math. 120 (1995), 447–454.CrossRefGoogle Scholar
[37]Opdam, E. M., Lecture notes on dunkl operators for real and complex reflection groups. With a preface by Toshio Oshima, MSJ Memoirs, vol. 8 (Mathematical Society of Japan, Tokyo, 2000).Google Scholar
[38]Rouquier, R., q-Schur algebras and complex reflection groups, Mosc. Math. J. 8 (2008), 119–158.CrossRefGoogle Scholar
[39]Shephard, G. C. and Todd, J. A., Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.CrossRefGoogle Scholar
[40]Smith, S. P. and Stafford, J. T., Differential operators on an affine curve, Proc. London Math. Soc. 56 (1988), 229–259.CrossRefGoogle Scholar
[41]Stafford, J. T. and Van den Bergh, M., On quasi-invariants of complex reflection groups, Private notes (2004).Google Scholar
[42]Steinberg, R., Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392–400.CrossRefGoogle Scholar
[43]Vale, R., On category 𝒪 for the rational Cherednik algebra of the complex reflection group (ℤ/ℓℤ)≀S n, PhD thesis, University of Glasgow (2006).Google Scholar
[44]Van den Bergh, M., Invariant differential operators on semi-invariants for Tori and weighted projective space, Lecture Notes in Mathematics, vol. 1478 (Springer, New-York, 1991), 255–272.Google Scholar
[45]Veselov, A. P., Styrkas, K. L. and Chalykh, O. A., Algebraic integrability for the Schrödinger equation and finite reflection groups, Theoret. Math. Phys. 94 (1993), 253–275.CrossRefGoogle Scholar