Algèbre de Hecke affine

En mathématiques, une algèbre de Hecke affine est une algèbre associée à un groupe de Weyl affine et peut être utilisée pour prouver la conjecture de Macdonald portant sur les polynômes éponymes.

Définition

Soit $V$ un espace euclidien de dimension finie et $\Sigma$ un système de racines affine sur $V$ . Une algèbre de Hecke affine est une certaine algèbre associative qui est une déformation de l'algèbre de groupe $\mathbb {C} [W]$ du groupe de Weyl $W$ de $\Sigma$ (le groupe de Weyl affine). Elle est généralement notée $H(\Sigma ,q)$ , où $q:\Sigma \rightarrow \mathbb {C}$ est la fonction dite « de multiplicité », qui joue le rôle de paramètre de déformation. En effet, pour $q\equiv 1$ , l'algèbre de Hecke affine $H(\Sigma ,q)$ n'est autre que l'algèbre de groupe $\mathbb {C} [W]$ .

Généralisations

Ivan Cherednik a introduit des généralisations des algèbres affines de Hecke appelées algèbres de Hecke doublement affines (double affine Hecke algebras, d'où l'acronyme DAHA sous lequel elles sont connues). Grâce à cet outil, il a pu donner une preuve de la conjecture de Macdonald pour les polynômes éponymes (en s'appuyant sur les travaux d'Eric Opdam (en)). Une autre source importante d'inspiration de Cherednik pour introduire les algèbres de Hecke doublement affines était les équations de Knizhnik–Zamolodchikov quantiques (en).

Références

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Affine Hecke algebra » (voir la liste des auteurs).
(en) Ivan Cherednik, Double affine Hecke algebras, vol. 319, Cambridge University Press, coll. « London Mathematical Society Lecture Note Series », 2005 (ISBN 978-0-521-60918-0, DOI 10.1017/CBO9780511546501, MR 2133033)
(en) Nagayoshi et Hideya, « On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups », Publications mathématiques de l'IHÉS, vol. 25,‎ 1965, p. 5–48 (DOI 10.1007/bf02684396, MR 185016, zbMATH 0228.20015, S2CID 4591855, lire en ligne)
(en) Kazhdan et Lusztig, « Proof of the Deligne-Langlands conjecture for Hecke algebras », Inventiones Mathematicae, vol. 87, n^o 1,‎ 1987, p. 153–21 (DOI 10.1007/BF01389157, Bibcode 1987InMat..87..153K, MR 862716, S2CID 122648418)
(en) Kirillov, « Lectures on affine Hecke algebras and Macdonald's conjectures », Bulletin of the American Mathematical Society, vol. 34, n^o 3,‎ 1997, p. 251-292 (DOI 10.1090/S0273-0979-97-00727-1, MR 1441642, lire en ligne)
(en) George Lusztig, « Notes on affine Hecke algebras : Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, June 28 - July 6, 1999 », dans Ivan Yavor (dir.), Iwahori-Hecke Algebras and their Representation Theory, vol. 1804, coll. « Lecture Notes in Mathematics », 110 p. (DOI 10.1007/978-3-540-36205-0_3, MR 1979925), p. 71-103
(en) George Lusztig (notes d'un cours donné au MIT en 1999), « Lectures on affine Hecke algebras with unequal parameters », (non publié),‎ août 2001 (arXiv 0108172)
(en) I. G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, vol. 157, Cambridge University Press, coll. « Cambridge Tracts in Mathematics », 2003 (ISBN 0-521-82472-9, MR 1976581)

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