Macdonald-Koornwinder polynomials and affine Hecke algebras

[AW85] R. Askey, J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize the

Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), no. 319.

[B68b]

N. Bourbaki, Groupes et alg`ebres de Lie : Chapitres 4, 5 et 6, Hermann (1968).

[C92a]

I. Cherednik, Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators, Int. Math. Res. Not., 9 (1992), 171–179.

[C92b]

I. Cherednik, Quantum Knizhnik-Zamolodchikov equations and affine root systems, Commun.

Math. Phys., 150 (1992), 109–136.

[C92c]

I. Cherednik, Double Affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators, Int. Math. Res. Not., 9 (1992), 171–179.

[C94]

I. Cherednik, Induced representations of double affine Hecke algebras and applications, Math.

Res. Lett., 1 (1994), 319–337.

[C95a]

I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. Math., 141

(1995), 191–216.

[C95b]

I. Cherednik, Non-symmetric Macdonald polynomials, Int. Math. Res. Not., 10 (1995), 483–515.

[C95c]

I. Cherednik, Macdonald’s evaluation conjectures and difference Fourier transform, Inv. Math.,

122 (1995), 119–145

[C97a]

I. Cherednik, Intertwining operators of double affine Hecke algebras, Sel. Math. new series, 3

(1997), 459–495.

[C97b]

I. Cherednik, Difference Macdonald-Mehta conjecture, Int. Math. Res. Not., 1997 (1997), no.

10, 449–467.

[C05]

I. Cherednik, Double affine Hecke algebras, London Math. Soc. Lecture Note Series, 319,

Cambridge Univ. Press, Cambridge, 2005.

[C09]

I. Cherednik, Whittaker limits of difference spherical functions, Int. Math. Res. Not., 2009

(2009), no. 20, 3793–3842.

[Chi21]

M. Chihara, Demazure slices of type A2l , Algebr. Represent. Theory, 25 (2022), 491–519.

(2)

[FH+ 15] B. Feigin, A. Hoshino, M. Noumi, J. Shibahara, J. Shiraishi, Tableau Formulas for One-Row

Macdonald Polynomials of Types Cn and Dn , SIGMA Symmetry Integrability Geom. Methods

Appl., 11 (2015), Paper 100, 21 pp.

[GR04]

G. Gasper, M. Rahman, Basic hypergeometric series, 2nd. ed., Encyclopedia of Mathematics

and its Applications 96, Cambridge University Press (2004).

[H06]

M. Haiman, Cherednik algebras, Macdonald polynomials and combinatorics, International

Congress of Mathematicians. Vol. III, 843–872, Eur. Math. Soc., Z¨

urich, 2006.

119

[HS18]

A. Hoshino, J. Shiraishi, Macdonald polynomials of type Cn with one-column diagrams and

deformed Catalan numbers, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018),

Paper No. 101, 33 pp.

[HS20]

A. Hoshino, J. Shiraishi, Branching rules for Koornwinder polynomials with one column diagrams and matrix inversions, SIGMA Symmetry Integrability Geom. Methods Appl., 16 (2020),

Paper No. 084, 28 pp.

[I03]

B. Ion, Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J., 116

(2003), no. 2, 299–318.

[Ka90]

V. G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge,

1990.

[KLS10] Koekoek, Lesky, Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues,

Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg, 2010

[Ko92]

T. H. Koornwinder, Askey-Wilson polynomials for root systems of type BC, Contemp. Math.,

138 (1992), 189–204.

[Ko]

T. H. Koornwinder, Charting the Askey and q-Askey schemes, Lecture in AMS Special Session

on The Legacy of Dick Askey, 6-9 January, 2021. The slide is available from https://staff.

fnwi.uva.nl/t.h.koornwinder/art/sheets/

[L76]

E. Looijenga, Root systems and Elliptic curves, Inv. Math., 38 (1976), 17–32.

[L89]

G. Lusztig, Affine Hecke algebras and their graded version, J, Amer. Math. Soc., 2 (1989),

599–635.

[M71]

I. G. Macdonald, Affine root systems and Dedekind’s η-function, Inv. Math., 15 (1972), 161–

174.

[M87]

I. G. Macdonald, Orthogonal polynomials associated with root systems, preprint, 1987; typed

and published in the Sminaire

Lotharingien de Combinatoire, 45 (2000), B45a; available from

arXiv:math/0011046.

[Ma95]

I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Univ., Press,

1995.

[M98]

I. G. Macdonald, Symmetric functions and orthogonal polynomials, University Lecture Series

vol. 12, Amer. Math. Soc., 1998.

[M03]

I. G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in

Mathematics 157, Cambridge Univ. Press, 2003.

[N95]

M. Noumi, Macdonald-Koornwinder polynomials and affine Hecke rings (in Japanese), in Various Aspects of Hypergeometric Functions (Kyoto, 1994), RIMS Kˆokyˆ

uroku 919, Research

Institute for Mathematical Sciences, Kyoto Univ., Kyoto, 1995, 44–55.

[NS04]

M. Noumi, J. V. Stokman, Askey-Wilson polynomials: an affine Hecke algebraic approach,

in Laredo Lectures on Orthogonal Polynomials and Special Functions (R. Alvarez-Nodarse, F.

Marcellan and W. Van Assche, Eds), pp. 111-144, Nova Science Publishers, 2004.

[NSh]

M. Noumi, J. Shiraishi, A direct approach to the bispectral problem for the RuijsenaarsMacdonald q-difference operators, preprint (2012), arXiv:1206.5364.

[OS18]

D. Orr, M. Shimozono, Specializations of nonsymmetric Macdonald-Koornwinder polynomials,

J. Algebraic Combin., 47 (2018), no. 1, 91–127.

[Ra06]

A. Ram, Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux,

Pure Appl. Math. Q., 2 (4) (2006), 963–1013.

120

[RY11]

A. Ram, M. Yip, A combinatorial formula for Macdonald polynomials, Adv. Math., 226 (2011),

309–331.

[Ro21]

H. Rosengren, Proofs of some partition identities conjectured by Kanade and Russell, Ramanujan J., (2021), published online; arXiv:1912.03689.

[R87]

S. N. M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and elliptic

function identities, Commun. Math. Phys., 110 (1987), 191–213.

[Sa99]

S. Sahi, Nonsymmetric Koornwinder polynomials and duality, Ann. Math., 150 (1999), 267–

282.

[Sa00]

S. Sahi, Some properties of Koornwinder polynomials, Contemp. Math., 254 (2000), 395–411.

[San00]

Y. B. Sanderson, On the Connection Between Macdonald Polynomials and Demazure Characters, J. Alg. Comb., 11 (2000), 269–275.

[St00]

J. V. Stokman, Koornwinder Polynomials and Affine Hecke Algebras, Int. Math. Res. Not., 19

(2000), 1005–1042.

[St02]

J. V. Stokman, An expansion formula for the Askey-Wilson function, J. Approx. Theory, 114

(2002), 308–342.

[St03]

J. V. Stokman, Difference Fourier transforms for nonreduced root systems, Selecta Math., 9

(2003), 409–494.

[St04]

J. V. Stokman, Lecture Notes on Koornwinder polynomials, in Laredo Lectures on Orthogonal

Polynomials and Special Functions, 145–207, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ. Hauppauge, NY, 2004.

[St14]

J. V. Stokman, The c-function expansion of a basic hypergeometric function associated to root

systems, Ann. Math., 179 (2014), 253–299.

[St20]

J. V. Stokman, Macdonald-Koornwinder polynomials, in Encyclopedia of Special Functions:

The Askey-Bateman Project: Volume 2, Multivariable Special Functions, Cambridge University

Press, 2020; provisional version available from arXiv:1111.6112.

[T10]

Y. Takeyama, Differential equations compatible with boundary rational qKZ equation, in New

trends in quantum integrable systems, pp. 421–450 (2010); arXiv:0908.2288.

[vD96]

J. van Diejen, Self-dual Koornwinder-Macdonald polynomials, Invent. Math., 126 (1996), no.

2, 319–339.

[vMS09] M. van Meer, J. V. Stokman, Double Affine Hecke Algebras and Bispectral Quantum KnizhnikZamolodchikov Equations, Int. Math. Res. Not., 6 (2010), 969–1040.

[vM11]

M. van Meer, Bispectral quantum Knizhnik-Zamolodchikov equations for arbitrary root systems,

Sel. Math. New Ser., 17 (2011), 183–221.

[Ya22]

K. Yamaguchi, A Littlewood–Richardson rule for Koornwinder polynomials, J. Algebraic Combin., 56 (2022), 335–381, https://doi.org/10.1007/s10801-022-01114-5;

arXiv:2009.13963.

[YY22]

K. Yamaguchi, S. Yanagida, Specializing Koornwinder polynomials to Macdonald polynomials

of type B, C, D and BC, J. Algebraic Combin., (2022), online published: https://doi.org/

10.1007/s10801-022-01165-8

[YY]

K. Yamaguchi, S. Yanagida, A review of rank one bispectral correspondence of quantum affine

KZ equations and Macdonald-type eigenvalue problems, proceeding draft of “Recent developments in Combinatorial Representation Theory” at RIMS, Kyoto University, November 2022,

arXiv:2211.13671.

121

[Yi12]

M. Yip, A Littlewood–Richardson rule for Macdonald polynomials, Math. Z., 272 (2012), 1259–

1290.

[青 13]

青本和彦, 直交多項式入門, 数学書房, 2013.

[三 04]

三町勝久, ダイソンからマクドナルドまで–マクドナルド多項式入門–, 代数学百科 I 群論の進化

第 4 章, 335–437, 朝倉書店, 2004.

122

...