## Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory

Last update: 28 April 2014

## Notes and References

This is an excerpt of the paper Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory by Ivan Cherednik, in collaboration with Etsuro Date, Kenji Iohara, Michio Jimbo, Masaki Kashiwara, Tetsuji Miwa, Masatoshi Noumi, and Yoshihisa Saito.

## Bibliography

[Aom1988] K. Aomoto, A note on holonomic $q\text{-difference}$ systems, Algebraic Analysis, 1, Eds. M. Kashiwara, T. Kawai, Academic Press. San diego (1988), 22-28.

[AIs1983] R. Askey and M.E.H. Ismail, A generalization of ultraspherical polynomials, in Studies in Pure Mathematics (ed. P. Erdös), Birkhäuser (1983), 55-78.

[Bir1969] J. Birman, On Braid groups, Commun. Pure Appl. Math., 22 (1969), 41-72.

[Che1989] I.V. Cherednik, Generalized braid groups and local $r\text{-matrix}$ systems, Doklady Akad. Nauk SSSR, 307, No. 1, 27-34 (1989).

[Che1991] I. Cherednik, Monodromy representations for generalized Knizhnik-Zamolodchikov equations and Hecke algebras, Publ. RIMS, Kyoto Univ. 27 (1991), 711–726.

[Che1991-3] I.V. Cherednik, Affine extensions of Knizhnik-Zamolodchikov equations and Lusztig's isomorphisms, in: 'Special Functions'. ICM-90 Satellite Conference Proceedings, Eds. M. Kashiwara and T. Miwa, Springer (1991), 63-77.

[Che1991-4] I.V. Cherednik, Integral solutions of trigonometric Knizhnik-Zamolodchikov equations and Kac-Moody algebras, Publ. RIMS 27 (1991), 727-744.

[Che1991-2] I. Cherednik, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), 411-431. MR 93b:17040

[Che1992-2] I.V. Cherednik, Quantum Knizhnik-Zamolodchikov equations and affine root systems, Commun. Math. Phys. 150 (1992), 109-136.

[Che1992] I.V. Cherednik, Double affine Hecke algebras and Khniznik-Zamolodchikov equations and Macdonalds operators, Int. Math. Res. Notices 9 (1992) 171-180.

[Che1993] I.V. Cherednik, The Macdonald constant-term conjecture, Int. Math. Res. Notices 6 (1993), 165-177.

[Che1994-2] I.V. Cherednik, Induced representations of double affine Hecke algebras and applications, Math. Res. Lett. 1 (1994), 319-337.

[Che1994-3] I.V. Cherednik, Integration of quantum many-body problems by an affine Knizhnik-Zamolodchikov equations, Preprint RIMS-776 (1991), Adv. Math. 106 (1994), 65-95.

[Che1995-2] I.V. Cherednik, Macdonald's evalutation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119-145.

[Che1995-3] I.V. Cherednik, Nonsymmetric Macdonald polynomials, Int. Math. Res. Notices 10 (1995), 483-515.

[Che1995-4] I.V. Cherednik, Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation, Commun. Math. Phys. 169 No. 2 (1995), 441-461.

[Che1995-5] I.V. Cherednik, Difference-elliptic operators and root systems, IMRN 1 (1995), 43-59.

[Che1996] I.V. Cherednik, Intertwining operators for double affine Hecke algebras, Preprint RIMS-1079 (1996), Selecta Math.

[Dri1986-2] V.G. Drinfel'd, Degenerate affine Hecke algebras and Yangians, Funktsion. Anal. Prilozhen., 20, No. 1, 69-70 (1986).

[Dun1989] C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. AMS. 311 (1989) 167-183.

[EKi1993] P. Etingof and A. Kirillov Jr., Representations of affine Lie algebras, parabolic equations, and Lamé functions, Duke Math. J., (1993).

[EKi1995] P. Etingof and A. Kirillov Jr., Representation-theoretic proof of the inner product and symmetry identities for Macdonald's polynomials, Compositio Mathematica (1995).

[FRe1991] I.B. Frenkel and N.Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Commun. Math. Phys. (1991).

[HOp1987] G.J. Heckman and E.M. Opdam, Root systems and hypergeometric functions I, Comp. Math. 64 (1987), 329-352.

[Hec1987] G.J. Heckman, Root systems and hypergeometric functions II, Comp. Math. 64 (1987), 353-373.

[Hec1991] G.J. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, Invent. math. 103 (1991), 341-350.

[Kat1981] S. Kato, Irreducibility of principal series representations for Hecke algebras of affine type, J. Fac. Sci. Univ. Tokyo Sec. IA 28 (1981) 929–943.

[Kir1995] A. Kirillov, Jr., Inner product on conformal blocks and Macdonald's polynomials at roots of unity, Preprint (1995).

[KLu0862716] D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153–215, MR0862716.

[KZa1984] V.G. Knizhnik and A.B. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions, Nucl. Physics B247 (1984) 83-103.

[Koh1987] T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier Grenouble 37 (1987) 139-160.

[Lek1981] H. van der Lek, Extended Artin groups, Singularities, Part 2 (Arcata, Calif., 1981), 117–121, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983.

[Lus1989] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599–635. MR 90e:16049

[Mac0011046] I.G. Macdonald, Orthogonal polynomials associated with root systems, Sém. Lotharingien Combinatoire 45 (2000/01), Art. B45a (electronic) text of 1978 preprint. MR1817334 and arXiv:math.QA/0011046

[Mac1423624] I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki 1994/95, Astérisque 237 (1996) Exp. No. 797, 4, 189-207. MR1423624

[Mac1995] I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, New York (second edition, 1995).

[Mat1992] A. Matsuo, Integrable connections related to zonal spherical functions, Invent. Math. 110 (1992) 96-121.

[Nou1992] M. Noumi, Macdonald's symmetric functions on some quantum homogeneous spaces, preprint (1992).

[OPe1983] M.A. Olshanetsky and A.M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.

[Opd1995] E. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75–121. MR 98f:33025

[Rog1985] J.D. Rogawski, On modules over the Hecke algebra of a $p\text{-adic}$ group, Invent. Math. 79 (1985) 443–465. MR 86j:22028

[Sai1985] K. Saito, Extended affine root systems I, Publ. RIMS. Kyoto Univ. 21 No. 1 (1985), 75-179.

[Smi1986] F. Smirnov, General formula for solution formfactors in Sine-Gordon, Model. J. Phys. A Math. Gen. 19 (1986). L575-L580.

[Sut1971] B. Sutherland, Exact results for a quantum many-body problem in one-dimension, Phys. Rev. A4 (1971), 2019-2021. Phys. Rev. A5 (1971), 1372-1376.

[TKa1988] A. Tsuchiya and Y. Kanie, Vertex operators in conformal field theory on ${P}^{1}$ and monodromy representations of braid groups, Adv. Stud. Pure. Math. 16 (1988), pp. 297-372.