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A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras. (English) Zbl 0725.20012
Some generalizations of the Lusztig-Lascoux-Schützenberger operators for affine Hecke algebras are considered. As corollaries we obtain Lusztig’s isomorphisms from affine Hecke algebras to their degenerate versions, a “natural” interpretation of the Dunkl operators and a new class of differential-difference operators generalizing Dunkl’s ones and the Knizhnik-Zamolodchikov operators from the two dimensional conformal field theory.

20C35 Applications of group representations to physics and other areas of science
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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[1] [BGG] Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Schubret cells and the cohomology of G/P. Russ. Math. Surv.28, 1-26 (1973) · Zbl 0289.57024 · doi:10.1070/RM1973v028n03ABEH001557
[2] [Bo] Bourbaki, N.: Groupes et algébres de Lie, Chapter 6. Paris: Hermann 1969
[3] [Ch1] Cherednik, I.V.: Generalized braid groups and localr-matrix systems. Dokl. Akad. Nauk SSSR,307:1, 27-34 (1989)
[4] [Ch2] Cherednik, I.V.: Monodromy representations for generalized Knizhnik-Zamolodchikov equations and Hecke algebras. ITP-89-74E, Kiev. (Preprint 1989); Publ. Res. Inst. Math. Sci. (to appear)
[5] [Ch3] Cherednik, I.V.: A new interpretation of Gelfand-Tzetlin bases. Duke Math. J.54:2, 563-577 (1987) · Zbl 0645.17006 · doi:10.1215/S0012-7094-87-05423-8
[6] [Ch4] Cherednik, I.V.: On special bases of irreducible representations of the degenerate affine Hecke algebra. Funct. Anal. Appl.20:1, 87-89 (1986) · Zbl 0599.20050 · doi:10.1007/BF01077327
[7] [Ch5] Cherednik, I.V.: Factorized particles on the half-line and root systems. Theor. Math. Phys.61:1, 35-43 (1984) · Zbl 0575.22021 · doi:10.1007/BF01038545
[8] [De] Demazure, M.: Désingularisation des variétès de Schubert généralisés. Ann. Éc. Norm. Supér.7, 53-88 (1974)
[9] [Dr] Drinfeld, V.G.: Degenerate affine Hecke algebras and Yangians. Funct. Anal. Appl.20:1, 69-70 (1986)
[10] [Du] Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc.311, 167-183 (1989) · Zbl 0652.33004 · doi:10.1090/S0002-9947-1989-0951883-8
[11] [He1] Heckman, G.J.: An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math. (Submitted) · Zbl 0721.33009
[12] [He2] Heckman, G.J.: A remark on the Dunkl differential-difference operators. Proceedings of Bowdoin Conference on harmonic analysis in reductive groups, 1989. (To appear)
[13] [HO] Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functionsI. Compos. Math.64, 329-352 (1987) · Zbl 0656.17006
[14] [Ka] Kato, S.: Irreducibility of principal series representations for Hecke algebras of affine type. J. Fac. Sci., Univ. Tokyo, IA,28:3, 929-943 (1983) · Zbl 0499.22018
[15] [KL] Kazhdan, D., Lusztig, G.: Proof of Deligne-Langlands conjecture for Hecke algebras. Invent. Math.87, 153-215 (1987) · Zbl 0613.22004 · doi:10.1007/BF01389157
[16] [LS1] Lascoux A., Schützenberger M.-P.: Non-commutative Schubert polynomials. Funct. Anal. Appl.23:3, 63-64 (1989)
[17] [LS2] Lascoux A., Schützenberger M.-P.: Symmetrization operators in polynomial rings. Funct. Anal. Appl.21:4, 77-78 (1987) · Zbl 0625.20056 · doi:10.1007/BF01077996
[18] [Lu1] Lusztig, G.: Affine Hecke algebras and their graded version. J. Am. Math. Soc.2:3, 599-685 (1989) · Zbl 0715.22020 · doi:10.1090/S0894-0347-1989-0991016-9
[19] [Lu2] Lusztig, G.: Cuspidal local systems and graded Hecke algebras I. Publ. Math., Inst. Hautes Étud. Sci.67, 145-202 (1988) · Zbl 0699.22026 · doi:10.1007/BF02699129
[20] [Lu3] Lusztig, G.: EquivariantK-theory and representations of Hecke algebras. Proc. Am. Math. Soc.94:2, 337-342 (1985) · Zbl 0571.22014
[21] [Ma] Matsumoto, H.: Analyse harmonique dans les systems de Tits bornologiques de type affine. (Lect. Notes Math. vol. 590) Berlin Heidelberg New York: Springer 1979
[22] [Mu] Murphy, G.E.: A new construction of Young’s seminormal representation of the symmetric group. J. Algebra69, 287-297 (1981) · Zbl 0455.20007 · doi:10.1016/0021-8693(81)90205-2
[23] [Ro] Rogawski, J.D.: On modules over the Hecke algebra of ap-adic group. Invent. Math.79, 443-465 (1985) · Zbl 0579.20037 · doi:10.1007/BF01388516
[24] [Ze] Zelevinsky, A.V.: Induced representations of reductivep-adic groupsII. Ann. Sci. Éc. Norm. Supér. Sér. IV. Sér.13, 165-210 (1980) · Zbl 0441.22014
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