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A New Axiomatics for Masures

Part of: Structure and classification of infinite or finite groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  29 January 2019

Auguste Hébert
Auguste Hébert*
Affiliation:
Université de Lyon, UJM-Saint-Etienne CNRS, UMR CNRS 5208, F-42023, Saint-Etienne, France Email: [email protected]
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Abstract

Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau to study Kac–Moody groups over ultrametric fields that generalize reductive groups. Rousseau gave an axiomatic definition of these spaces. We propose an equivalent axiomatic definition, which is shorter, more practical, and closer to the axiom of Bruhat–Tits buildings. Our main tool to prove the equivalence of the axioms is the study of the convexity properties in masures.

Keywords

MasuresHovelsKac-Moody groupsBruhat-Tits buildingslocal fields

MSC classification

Primary: 20E42: Groups with a $BN$-pair; buildings
Secondary: 20G44: Kac-Moody groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 3 , June 2020 , pp. 732 - 773
Copyright
© Canadian Mathematical Society 2019

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Footnotes

The author was partially supported by ANR grant ANR-15-CE40-0012.

References

Abdellatif, Ramla and Hébert, Auguste, Completed Iwahori-Hecke algebras and parahorical Hecke algebras for Kac-Moody groups over local fields. J. Éc. Polytech. Math. 6(2019), 79–118.Google Scholar
Braverman, Alexander, Garland, Howard, Kazhdan, David, and Patnaik, Manish, An affine Gindikin-Karpelevich formula. Perspectives in representation theory Contemp. Math. 610(2014), 43–64. https://doi.org/10.1090/conm/610/12193Google Scholar
Braverman, Alexander and Kazhdan, David, The spherical Hecke algebra for affine Kac-Moody groups I. Ann. of Math. 174(2011), no. 3, 1603–1642. https://doi.org/10.4007/annals.2011.174.3.5Google Scholar
Braverman, Alexander, Kazhdan, David, and Patnaik, Manish M., Iwahori–Hecke algebras for p-adic loop groups. Invent. Math. 204(2016), no. 2, 347–442. https://doi.org/10.1007/s00222-015-0612-xGoogle Scholar
Bardy-Panse, Nicole, Gaussent, Stéphane, and Rousseau, Guy, Iwahori-Hecke algebras for Kac-Moody groups over local fields. Pacific J. Math. 285(2016), 1–61. https://doi.org/10.2140/pjm.2016.285.1Google Scholar
Bardy-Panse, Nicole, Gaussent, Stéphane, and Rousseau, Guy, Macdonald’s formula for Kac-Moody groups over local fields. Proc. Lond. Math. Soc. (3) 119(2019), 135–175. https://doi.org/10.1112/plms.12231Google Scholar
Brown, Kenneth S., Buildings. Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4612-1019-1Google Scholar
Bruhat, François and Tits, Jacques, Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. 41(1972), 5–251.Google Scholar
Bruhat, François and Tits, Jacques, Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. 60(1984), no. 1, 5–184.Google Scholar
Charignon, Cyril, Immeubles affines et groupes de Kac-Moody. PhD thesis. Université Henri Poincaré Nancy 1, 2010.Google Scholar
Freyn, Walter, Hartnick, Tobias, Horn, Max, and Köhl, Ralf, Kac-Moody symmetric spaces. To appear in Münster J. Math. arxiv:1702.08426.Google Scholar
Gaussent, Stéphane and Rousseau, Guy, Kac-Moody groups, hovels and Littelmann paths. Ann. Inst. Fourier 58(2008), 2605–2657.Google Scholar
Gaussent, Stéphane and Rousseau, Guy, Spherical Hecke algebras for Kac-Moody groups over local fields. Ann. of Math. 180(2014), 1051–1087. https://doi.org/10.4007/annals.2014.180.3.5Google Scholar
Hébert, Auguste, Distances on a masure (affine ordered hovel). arxiv:1611.06105.Google Scholar
Hébert, Auguste, Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields. Int. Math. Res. Not. IMRN 2017, no. 22, 7028–7049. https://doi.org/10.1093/imrn/rnw224Google Scholar
Hiriart-Urruty, Jean-Baptiste and Lemaréchal, Claude, Fundamentals of convex analysis. Springer-Verlag, Berlin, 2001. https://doi.org/10.1007/978-3-642-56468-0Google Scholar
Kac, Victor G., Infinite-dimensional Lie algebras. Third edition, Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511626234Google Scholar
Macdonald, Ian Grant, Spherical functions on a group of p-adic type. Publ. Ramanujan Inst., 2, Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras, 1971.Google Scholar
Rémy, Bertrand, Groupes de Kac-Moody déployés et presque déployés. Astérisque 2002, no. 277.Google Scholar
Rousseau, Guy, Euclidean buildings. In: Géométries à courbure négative ou nulle, groupes discrets et rigidité. Sémin. Congr., 18, Société Mathématique de France, Paris, 2004, pp. 77–116.Google Scholar
Rousseau, Guy, Masures affines. Pure App. Math. Q. 7(2011), no. 3, 859–921. https://doi.org/10.4310/PAMQ.2011.v7.n3.a10Google Scholar
Rousseau, Guy, Groupes de Kac-Moody déployés sur un corps local II. Masures ordonnées. Bull. Soc. Math. France 144(2016), no. 4, 613–692. https://doi.org/10.24033/bsmf.2724Google Scholar
Rousseau, Guy, Almost split Kac-Moody groups over ultrametric fields. Groups Geom. Dyn. 11(2017), 891–975. https://doi.org/10.4171/GGD/418Google Scholar
Jacques, Tits, Uniqueness and presentation of Kac-Moody groups over fields. J. Algebra 105(1987), no. 2, 542–573. https://doi.org/10.1016/0021-8693(87)90214-6Google Scholar