On structure constants of Iwahori–Hecke algebras for Kac–Moody groups
Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 465-490.

We consider the Iwahori–Hecke algebra I associated to an almost split Kac–Moody group G (affine or not) over a nonarchimedean local field 𝒦. It has a canonical double-coset basis (T w ) wW + indexed by a sub-semigroup W + of the affine Weyl group W. The multiplication is given by structure constants a w,v u = 0 : T w *T v = uP w,v a w,v u T u . A conjecture, by Braverman, Kazhdan, Patnaik, Gaussent and the authors, tells that a w,v u is a polynomial, with coefficients in , in the parameters q i -1,q i -1 of G over 𝒦. We prove this conjecture when w and v are spherical or, more generally, when they are said to be generic: this includes all cases of w,vW + if G is of affine or strictly hyperbolic type. In the split affine case (where q i =q i =q, i) we get a universal Iwahori–Hecke algebra with the same basis (T w ) wW + over a polynomial ring [Q]; it specializes to I when one sets Q=q.

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DOI: 10.5802/alco.163
Classification: 20G44, 20C08, 20G25, 20E42, 51E24
Keywords: Building, Hecke algebra, Kac–Moody group, masure, local field.
Bardy-Panse, Nicole 1; Rousseau, Guy 1

1 Université de Lorraine CNRS, IECL Nancy F-54000, France
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Bardy-Panse, Nicole; Rousseau, Guy. On structure constants of Iwahori–Hecke algebras for Kac–Moody groups. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 465-490. doi : 10.5802/alco.163. http://www.numdam.org/articles/10.5802/alco.163/

[1] Bardy-Panse, Nicole Systèmes de racines infinis, Mém. Soc. Math. Fr. (N.S.), 65, Soc. Math. France, 1996 | Zbl

[2] Bardy-Panse, Nicole; Charignon, Cyril; Gaussent, Stéphane; Rousseau, Guy Une preuve plus immobilière du théorème de « saturation Â» de Kapovich–Leeb–Millson, Enseign. Math. (2), Volume 59 (2013) no. 1-2, pp. 3-37 | DOI | MR | Zbl

[3] Bardy-Panse, Nicole; Gaussent, Stéphane; Rousseau, Guy Iwahori–Hecke algebras for Kac–Moody groups over local fields, Pacific J. Math., Volume 285 (2016) no. 1, pp. 1-61 | DOI | MR | Zbl

[4] Bardy-Panse, Nicole; Gaussent, Stéphane; Rousseau, Guy Macdonald’s formula for Kac–Moody groups over local fields, Proc. Lond. Math. Soc. (3), Volume 119 (2019) no. 1, pp. 135-175 | DOI | MR | Zbl

[5] Braverman, Alexander; Garland, Howard; Kazhdan, David; Patnaik, Manish An affine Gindikin–Karpelevich formula, Perspectives in representation theory (Contemp. Math.), Volume 610, Amer. Math. Soc., Providence, RI, 2014, pp. 43-64 | DOI | MR | Zbl

[6] Braverman, Alexander; Kazhdan, David The spherical Hecke algebra for affine Kac–Moody groups I, Ann. of Math. (2), Volume 174 (2011) no. 3, pp. 1603-1642 | DOI | MR | Zbl

[7] Braverman, Alexander; Kazhdan, David Representations of affine Kac–Moody groups over local and global fields: a survey of some recent results, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2013, pp. 91-117 | MR | Zbl

[8] Braverman, Alexander; Kazhdan, David; Patnaik, Manish Iwahori-Hecke algebras for p-adic loop groups, Invent. Math., Volume 204 (2016) no. 2, pp. 347-442 | DOI | MR | Zbl

[9] Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local I, Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math. (1972) no. 41, pp. 5-251 | DOI | Numdam | MR | Zbl

[10] Charignon, Cyril Immeubles affines et groupes de Kac–Moody, masures bordées, Ph. D. Thesis, Nancy I (2010) http://tel.archives-ouvertes.fr/docs/00/49/79/61/PDF/these.pdf

[11] Charignon, Cyril Structures immobilières pour un groupe de Kac–Moody sur un corps local (2010) (https://arxiv.org/abs/0912.0442, preprint Nancy)

[12] Charignon, Cyril Immeubles affines et groupes de Kac–Moody, masures bordées, Éditions universitaires européennes, Sarrebruck, 2011 (Thèse Nancy, 2 juillet 2010)

[13] Cherednik, Ivan Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald’s operators, Internat. Math. Res. Notices (1992) no. 9, pp. 171-180 | DOI | MR | Zbl

[14] Cherednik, Ivan Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2), Volume 141 (1995) no. 1, pp. 191-216 | DOI | MR | Zbl

[15] Ciobotaru, Corina; Mühlherr, Bernhard; Rousseau, Guy The cone topology on masures, Adv. Geom., Volume 20 (2020) no. 1, pp. 1-28 (With an appendix by Auguste Hébert) | DOI | MR | Zbl

[16] Garland, Howard A Cartan decomposition for p-adic loop groups, Math. Ann., Volume 302 (1995) no. 1, pp. 151-175 | DOI | MR | Zbl

[17] Garland, Howard; Grojnowski, Ian Affine Hecke algebras associated to Kac–Moody groups (https://arxiv.org/abs/q-alg/9508019, preprint)

[18] Gaussent, Stéphane; Rousseau, Guy Kac–Moody groups, hovels and Littelmann paths, Ann. Inst. Fourier, Volume 58 (2008) no. 7, pp. 2605-2657 | DOI | Numdam | MR | Zbl

[19] Gaussent, Stéphane; Rousseau, Guy Spherical Hecke algebras for Kac–Moody groups over local fields, Ann. of Math. (2), Volume 180 (2014) no. 3, pp. 1051-1087 | DOI | MR | Zbl

[20] Hébert, Auguste Étude des masures et de leurs applications en arithmétique, english version, Ph. D. Thesis, Univ. Jean Monnet de Saint Etienne (Université de Lyon) (2018) (https://hal.archives-ouvertes.fr/tel-01856620)

[21] Hébert, Auguste A new axiomatics for masures, Canad. J. Math., Volume 72 (2020) no. 3, pp. 732-773 | DOI | MR | Zbl

[22] Iwahori, Nagayoshi; Matsumoto, Hideya On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. (1965) no. 25, pp. 5-48 | DOI | MR

[23] Kac, Victor G. Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990, xxii+400 pages | DOI | MR | Zbl

[24] Kapranov, Mikhail Double affine Hecke algebras and 2-dimensional local fields, J. Amer. Math. Soc., Volume 14 (2001) no. 1, pp. 239-262 | DOI | MR | Zbl

[25] Macdonald, Ian G. Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge University Press, Cambridge, 2003, x+175 pages | DOI | MR | Zbl

[26] Moody, Robert; Pianzola, Arturo On infinite root systems, Trans. Amer. Math. Soc., Volume 315 (1989) no. 2, pp. 661-696 | DOI | MR | Zbl

[27] Moody, Robert; Pianzola, Arturo Lie algebras with triangular decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1995, xxii+685 pages (A Wiley-Interscience Publication) | MR | Zbl

[28] Muthiah, Dinakar On Iwahori–Hecke algebras for p-adic loop groups: double coset basis and Bruhat order, Amer. J. Math., Volume 140 (2018) no. 1, pp. 221-244 | DOI | MR | Zbl

[29] Parkinson, James Buildings and Hecke algebras, J. Algebra, Volume 297 (2006) no. 1, pp. 1-49 | DOI | MR | Zbl

[30] Rémy, Bertrand Groupes de Kac–Moody déployés et presque déployés, Astérisque, 277, Soc. Math. France, 2002, viii+348 pages | Numdam | MR | Zbl

[31] Rousseau, Guy Masures affines, Pure Appl. Math. Q., Volume 7 (2011) no. 3, pp. 859-921 (Special Issue: In honor of Jacques Tits) | DOI | MR | Zbl

[32] Rousseau, Guy Groupes de Kac–Moody déployés sur un corps local II. Masures ordonnées, Bull. Soc. Math. France, Volume 144 (2016) no. 4, pp. 613-692 | DOI | MR | Zbl

[33] Rousseau, Guy Almost split Kac–Moody groups over ultrametric fields, Groups Geom. Dyn., Volume 11 (2017) no. 3, pp. 891-975 | DOI | MR | Zbl

[34] Tits, Jacques Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra, Volume 105 (1987) no. 2, pp. 542-573 | DOI | MR | Zbl

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