no. 1
Principal series representations of Iwahori–Hecke algebras for Kac–Moody groups over local fields
[Représentations de la série principale des algèbres d’Iwahori–Hecke associées aux groupes de Kac–Moody sur les corps locaux]
Hébert, Auguste1
1 Institut de Mathématiques Elie Cartan Université de Lorraine B.P. 70239 54506 Vandoeuvre-lès-Nancy Cedex (France)
Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 187-259.

Des algèbres d’Iwahori–Hecke ont récemment été associées aux groupes de Kac–Moody sur les corps locaux non-archimédiens. Nous introduisons les représentations de la série principale pour ces algèbres. Nous étudions ces représentations et généralisons partiellement les critères d’irréductibilité de Kato et de Matsumoto.

Recently, Iwahori–Hecke algebras were associated with Kac– Moody groups over non-Archimedean local fields. We introduce principal series representations for these algebras. We study these representations and partially generalize irreducibility criteria of Kato and Matsumoto.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3469
Classification : 20C08, 20G25, 20G44
Keywords: Principal series representations, Kac–Moody groups, non-archimedean local fields, masures, reductive groups
Mot clés : Représentations de la série principale, groupes de Kac–Moody, corps locaux non-archimédiens, masures, groupes réductifs
Hébert, Auguste 1

1 Institut de Mathématiques Elie Cartan Université de Lorraine B.P. 70239 54506 Vandoeuvre-lès-Nancy Cedex (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Principal series representations of {Iwahori{\textendash}Hecke} algebras for {Kac{\textendash}Moody} groups over local fields},
     journal = {Annales de l'Institut Fourier},
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Hébert, Auguste. Principal series representations of Iwahori–Hecke algebras for Kac–Moody groups over local fields. Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 187-259. doi : 10.5802/aif.3469. https://aif.centre-mersenne.org/articles/10.5802/aif.3469/

[1] Abdellatif, Ramla; Hébert, Auguste Completed Iwahori–Hecke algebras and parahoric Hecke algebras for Kac–Moody groups over local fields, J. Éc. Polytech., Math., Volume 6 (2019), pp. 79-118 | DOI | Numdam | MR | Zbl

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

[3] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, 2005, xiv+363 pages | MR

[4] Bourbaki, Nicolas Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 4, 5 et 6, Masson, 1981, 290 pages | MR

[5] Braverman, Alexander; Kazhdan, David The spherical Hecke algebra for affine Kac–Moody groups I, Ann. Math. (2011), pp. 1603-1642 | DOI | MR | Zbl

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

[7] Bushnell, Colin J.; Henniart, Guy The local Langlands conjecture for GL (2), Grundlehren der Mathematischen Wissenschaften, 335, Springer, 2006, xii+347 pages | DOI | MR

[8] Cherednik, Ivan Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald’s operators, Int. Math. Res. Not. (1992) no. 9, pp. 171-180 | DOI | MR | Zbl

[9] Dyer, Matthew Reflection subgroups of Coxeter systems, J. Algebra, Volume 135 (1990) no. 1, pp. 57-73 | DOI | MR | Zbl

[10] Dyer, Matthew On the “Bruhat graph” of a Coxeter system, Compos. Math., Volume 78 (1991) no. 2, pp. 185-191 | Numdam | MR | Zbl

[11] 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

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

[13] Goodearl, Kenneth R.; Warfield, Robert B. Jr An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, 61, Cambridge University Press, 2004, xxiv+344 pages | DOI | MR

[14] Hébert, Auguste Gindikin–Karpelevich Finiteness for Kac–Moody Groups Over Local Fields, Int. Math. Res. Not., Volume 2017 (2017) no. 22, pp. 7028-7049 | DOI | MR | Zbl

[15] Hébert, Auguste Study of masures and of their applications in arithmetic. English version, Ph. D. Thesis, Université de Lyon, France (2018) (https://hal.archives-ouvertes.fr/tel-01856620/file/memoire.pdf)

[16] Hébert, Auguste A New Axiomatics for Masures, Can. J. Math., Volume 72 (2020) no. 3, pp. 732-773 | DOI | MR | Zbl

[17] Hébert, Auguste Distances on a masure, Transform. Groups, Volume 26 (2021) no. 4, pp. 1331-1363 | DOI | MR | Zbl

[18] Kac, Victor G. Infinite-dimensional Lie algebras, 44, Cambridge University Press, 1994

[19] Kapovich, Michael; Millson, John J. A path model for geodesics in Euclidean buildings and its applications to representation theory, Groups Geom. Dyn., Volume 2 (2008) no. 3, pp. 405-480 | DOI | MR | Zbl

[20] Kato, Shin-ichi Irreducibility of principal series representations for Hecke algebras of affine type, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 28 (1981) no. 3, pp. 929-943 | MR | Zbl

[21] Kazhdan, David; Lusztig, George Representations of Coxeter groups and Hecke algebras, Invent. Math., Volume 53 (1979) no. 2, pp. 165-184 | DOI | MR | Zbl

[22] Kumar, Shrawan Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204, Birkhäuser, 2002, xvi+606 pages | DOI | MR

[23] Lang, Serge Algebra, Graduate Texts in Mathematics, 211, Springer, 2002, xvi+914 pages | DOI | MR

[24] Lusztig, George Left cells in Weyl groups, Lie Group Representations I, Springer, 1983, pp. 99-111 | DOI | Zbl

[25] Matsumoto, Hideya Analyse harmonique dans les systèmes de Tits bornologiques de type affine, Lecture Notes in Mathematics, 590, Springer, 1977, i+219 pages | DOI | MR

[26] Mühlherr, Bernhard The isomorphism problem for Coxeter groups (2005) (https://arxiv.org/abs/math/0506572)

[27] Radcliffe, David G Rigidity of right-angled Coxeter groups (1999) (https://arxiv.org/abs/math/9901049)

[28] Reeder, Mark On certain Iwahori invariants in the unramified principal series, Pac. J. Math., Volume 153 (1992) no. 2, pp. 313-342 | DOI | MR | Zbl

[29] Reeder, Mark Nonstandard intertwining operators and the structure of unramified principal series representations, Forum Math., Volume 9 (1997) no. 4, pp. 457-516 | DOI | MR | Zbl

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

[31] Renard, David Représentations des groupes réductifs p-adiques., Société Mathématique de France, 2010

[32] Rogawski, Jonathan D. On modules over the Hecke algebra of a p-adic group, Invent. Math., Volume 79 (1985) no. 3, pp. 443-465 | DOI | MR | Zbl

[33] Rousseau, Guy Groupes de Kac–Moody déployés sur un corps local, immeubles microaffines, Compos. Math., Volume 142 (2006) no. 2, pp. 501-528 | DOI | MR | Zbl

[34] Rousseau, Guy Masures affines, Pure Appl. Math. Q., Volume 7 (2011) no. 3, pp. 859-921 | DOI | MR | Zbl

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

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

[37] Solleveld, Maarten Periodic cyclic homology of affine Hecke algebras (2009) (https://arxiv.org/abs/0910.1606)

[38] 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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