## Macdonald’s evaluation conjectures and difference Fourier transform.(English)Zbl 0854.22021

Invent. Math. 122, No. 1, 119-145 (1995); Erratum ibid. 125, No. 2, 391 (1996).
It is known that in the theory of Macdonald’s orthogonal symmetric polynomials depending on the (difference Fourier transform) parameters $$q$$, $$t$$ and generalizing the characters of compact simple Lie groups three main conjectures have been formulated (Macdonald, 1988). One of them, the norm conjecture, has been proved recently (Cherednik, 1995) in the framework of a new approach (Cherednik, 1992) to the Macdonald’s theory which is based on double affine Hecke algebras and related difference operators. The proof of the other two conjectures, the duality and evaluation conjectures, as well as the recurrence relations and other basic results on Macdonald’s polynomials at roots of unity are presented in the paper.
Reviewer: A.A.Bogush (Minsk)

### MSC:

 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A40 Character groups and dual objects 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
Full Text:

### References:

 [1] [BG] F.A. Berezin, I.M. Gelfand: Some remarks on the theory of spherical functions on symmetric Riemannian manifolds. Transl., II. Ser. Am. Math. Soc.21 (1962) 193-238 · Zbl 0195.42302 [2] [B] N. Bourbaki: Groupes et algèbres de Lie. Ch.4-6, Hermann, Paris (1969) · Zbl 0205.06001 [3] [C1] I. Cherednik: Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators. IMRN (Duke M.J.)9 (1992) 171-180 · Zbl 0770.17004 [4] [C2] I. Cherednik: Doubles affine Hecke algebras and Macdonald’s conjectures. Ann. Math.141, (1995), 191-216 · Zbl 0822.33008 [5] [C3] I. Cherednik: Induced representations of double affine Hecke algebras and applications. Math. Research Letters1 (1994) 319-337 · Zbl 0837.20052 [6] [C4] I. Cherednik: Difference-elliptic operators and root systems. IMRN1 (1995) 43-59 · Zbl 0824.17029 [7] [C5] I. Cherednik: Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations. Preprint RIMS-776 (1991) (Advances in Math. (1994)) [8] [C6] I. Cherednik: Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation. Commun. Math. Phys. (1995) · Zbl 0826.35100 [9] [D] C.F. Dunkl: Hankel transforms associated to finite reflections groups. Contemp. Math.138 (1992) 123-138 · Zbl 0789.33008 [10] [EK1] P.I. Etingof, A.A. Kirillov, Jr.: Macdonald’s polynomials and representations of quantum groups. Math. Res. Let.1 (1994) 279-296 · Zbl 0833.17007 [11] [EK2] O.I. Etingof, A.A. Kirillov, Jr.: Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials. Compos. Math. (1995) [12] [GH] A.M. Garsia, M. Haiman: A graded representation model for Macdonald’s polynomials. Proc. Nat. Acad. Sci. USA90, 3607-3610 · Zbl 0831.05062 [13] [J] M.F.e. de Jeu: The Dunkl transform. Invent. Math.113 (1993), 147-162 · Zbl 0789.33007 [14] [H] S. Helgason: Groups and geometric analysis. Academic Press, New York (1984) · Zbl 0543.58001 [15] [He] G.J. Heckman: An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math.103 (1991) 341-350 · Zbl 0721.33009 [16] [K] V.G. Kac: Infinite dimensional Lie algebras. Cambridge University Press, Cambridge (1990) · Zbl 0716.17022 [17] [Ki] A. Kirillov, Jr.: Inner product on conformal blocks and Macdonald’s polynomials at roots of unity. Preprint (1995) [18] [KL1] D. Kazhdan, G. Lusztig: Proof of the Deligne-Langlands conjecture for Hecke algebras. Invent. Math.87 (1987) 153-215 · Zbl 0613.22004 [19] [KL2] D. Kazhdan, G. Lusztig: Tensor structures arising from affine Lie algebras. III. J. AMS7 (1994), 335-381 · Zbl 0802.17007 [20] [KK] B. Kostant, S. Kumar: T-Equivariant K-theory of generalized flag varieties. J. Diff. Geom.32 (1990) 549-603 · Zbl 0731.55005 [21] [M1] I.G. Macdonald: A new class of symmetric functions. Publ. I. R. M. A., Strasbourg, Actes 20-e Seminaire Lotharingen, (1988) 131-171 [22] [M2] I.G. Macdonald: Orthogonal polynomials associated with root systems. Preprint. (1988) [23] [M3] I.G. Macdonald: Some conjectures for root systems. SIAM J. Math. Anal.13: 6 (1982) 988-1007 · Zbl 0498.17006 [24] [MS] G. Moore, N. Seiberg: Classical and quantum conformal field theory. Commun. Math. Phys.123 (1989) 177-254 · Zbl 0694.53074 [25] [O1] E.M. Opdam: Some applications of hypergeometric shift operators. Invent. Math.98 (1989) 1-18 · Zbl 0696.33006 [26] [O2] E.M. Opdam: Harmonic analysis for certain representations of graded Hecke algebras. Preprint. Math. Inst. Univ. Leiden W93-18 (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.