Bruhat–Tits theory from Berkovich’s point of view. Analytic filtrations
Annales Henri Lebesgue, Volume 5 (2022), pp. 813-839.


Keywords Berkovich spaces, Bruhat–Tits buildings, Moy–Prasad filtrations


We define filtrations by affinoid groups, in the Berkovich analytification of a connected reductive group, related to Moy–Prasad filtrations. They are parametrized by a cone, whose basis is the Bruhat–Tits building and whose vertex is the neutral element, via the notions of Shilov boundary and holomorphically convex envelope.


[Ber90] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990 | MR | Zbl

[Ber93] Berkovich, Vladimir G. Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 78 (1993), pp. 5-161 | DOI | Numdam | MR | Zbl

[BT84] Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Publ. Math., Inst. Hautes Étud. Sci. (1984) no. 60, pp. 197-376 | MR | Zbl

[Cor20] Cornut, Christophe Filtrations and buildings, Memoirs of the American Mathematical Society, 1296, American Mathematical Society, 2020 | DOI | MR | Zbl

[DFN15] Berkovich spaces and applications (Ducros, Antoine; Favre, Charles; Nicaise, Johannes, eds.), Lecture Notes in Mathematics, 2119, Springer, 2015 (based on a workshop, Santiago de Chile, Chile, January 2008 and a summer school, Paris, France, June 2010) | DOI | MR | Zbl

[Kim07] Kim, Ju-Lee Supercuspidal representations: an exhaustion theorem, J. Am. Math. Soc., Volume 20 (2007) no. 2, pp. 273-320 | DOI | MR | Zbl

[KP21] Kaletha, Tasho; Prasad, Gopal Bruhat–Tits theory: a new approach (2021) (in preparation)

[May19] Mayeux, Arnaud On the constructions of supercuspidal representations, Ph. D. Thesis, Université Sorbonne Paris Cité (2019) (, HAL_ID=tel-02866443, HAL_VERSION=v1)

[MP94] Moy, Allen; Prasad, Gopal Unrefined minimal K-types for p-adic groups, Invent. Math., Volume 116 (1994) no. 1-3, pp. 393-408 | DOI | MR | Zbl

[MP96] Moy, Allen; Prasad, Gopal Jacquet functors and unrefined minimal K-types, Comment. Math. Helv., Volume 71 (1996) no. 1, pp. 98-121 | DOI | MR | Zbl

[MP21] Maculan, Marco; Poineau, Jérôme Notions of Stein spaces in non-Archimedean geometry, J. Algebr. Geom., Volume 30 (2021) no. 2, pp. 287-330 | DOI | MR | Zbl

[MRR20] Mayeux, Arnaud; Richarz, Timo; Romagny, Matthieu Néron blowups and low-degree cohomological applications (2020) (

[PP15] Poineau, Jérôme; Pulita, Andrea The convergence Newton polygon of a p-adic differential equation II: Continuity and finiteness on Berkovich curves, Acta Math., Volume 214 (2015) no. 2, pp. 357-393 | DOI | MR | Zbl

[RTW10] Rémy, Bertrand; Thuillier, Amaury; Werner, Annette Bruhat–Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings, Ann. Sci. Éc. Norm. Supér., Volume 43 (2010) no. 3, pp. 461-554 | DOI | Numdam | MR | Zbl

[Thu05] Thuillier, Amaury Théorie du potentiel sur les courbes en géométrie analytique non archimédienne : applications à la théorie d’Arakelov, Ph. D. Thesis, Université Rennes 1, Rennes, France (2005) (

[Yu01] Yu, Jiu-Kang Construction of tame supercuspidal representations, J. Am. Math. Soc., Volume 14 (2001) no. 3, pp. 579-622 | DOI | MR | Zbl

[Yu15] Yu, Jiu-Kang Smooth models associated to concave functions in Bruhat–Tits theory, Autour des schémas en groupes. École d’Été “Schémas en groupes”. Vol. III (Edixhoven, Bas et al., eds.) (Panoramas et Synthèses), Volume 47, Société Mathématique de France, 2015, pp. 227-258 | MR | Zbl