Geodesic
A New Axiomatics for Masures
Canadian journal of mathematics, Tome 72 (2020) no. 3, pp. 732-773

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/S0008414X19000051
Mots-clés : Masures, Hovels, Kac-Moody groups, Bruhat-Tits buildings, local fields 
Hébert, Auguste. A New Axiomatics for Masures. Canadian journal of mathematics, Tome 72 (2020) no. 3, pp. 732-773. doi: 10.4153/S0008414X19000051
@article{10_4153_S0008414X19000051,
     author = {H\'ebert, Auguste},
     title = {A {New} {Axiomatics} for {Masures}},
     journal = {Canadian journal of mathematics},
     pages = {732--773},
     year = {2020},
     volume = {72},
     number = {3},
     doi = {10.4153/S0008414X19000051},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000051/}
}
TY  - JOUR
AU  - Hébert, Auguste
TI  - A New Axiomatics for Masures
JO  - Canadian journal of mathematics
PY  - 2020
SP  - 732
EP  - 773
VL  - 72
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000051/
DO  - 10.4153/S0008414X19000051
ID  - 10_4153_S0008414X19000051
ER  - 
%0 Journal Article
%A Hébert, Auguste
%T A New Axiomatics for Masures
%J Canadian journal of mathematics
%D 2020
%P 732-773
%V 72
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000051/
%R 10.4153/S0008414X19000051
%F 10_4153_S0008414X19000051

[AH17] 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

[BGKP14] 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/610/12193 Google Scholar

[BK11] 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.2011.174.3.5 Google Scholar

[BKP16] 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-x Google Scholar

[BPGR16] 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.2016.285.1 Google Scholar

[BPGR17] 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.12231 Google Scholar

[Bro89] Brown, Kenneth S., Buildings. Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4612-1019-1 Google Scholar

[BT72] Bruhat, François and Tits, Jacques, Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. 41(1972), 5–251. Google Scholar

[BT84] 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

[Cha10] Charignon, Cyril, Immeubles affines et groupes de Kac-Moody. PhD thesis. Université Henri Poincaré Nancy 1, 2010. Google Scholar

[FHHK17] Freyn, Walter, Hartnick, Tobias, Horn, Max, and Köhl, Ralf, Kac-Moody symmetric spaces. To appear in Münster J. Math. . Google Scholar

[GR08] Gaussent, Stéphane and Rousseau, Guy, Kac-Moody groups, hovels and Littelmann paths. Ann. Inst. Fourier 58(2008), 2605–2657. Google Scholar

[GR14] 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.2014.180.3.5 Google Scholar

[Héb16] Hébert, Auguste, Distances on a masure (affine ordered hovel). . Google Scholar

[Héb17] 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/rnw224 Google Scholar

[HUL12] Hiriart-Urruty, Jean-Baptiste and Lemaréchal, Claude, Fundamentals of convex analysis. Springer-Verlag, Berlin, 2001. https://doi.org/10.1007/978-3-642-56468-0 Google Scholar

[Kac94] Kac, Victor G., Infinite-dimensional Lie algebras. Third edition, Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511626234 Google Scholar

[Mac71] 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

[Rém02] Rémy, Bertrand, Groupes de Kac-Moody déployés et presque déployés. Astérisque 2002, no. 277. Google Scholar

[Rou04] 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

[Rou11] Rousseau, Guy, Masures affines. Pure App. Math. Q. 7(2011), no. 3, 859–921. https://doi.org/10.4310/PAMQ.2011.v7.n3.a10 Google Scholar

[Rou16] 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.2724 Google Scholar

[Rou17] Rousseau, Guy, Almost split Kac-Moody groups over ultrametric fields. Groups Geom. Dyn. 11(2017), 891–975. https://doi.org/10.4171/GGD/418 Google Scholar

[Tit87] 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)90214-6 Google Scholar

Cité par Sources :