, G x,r is a k-anoid subgroup of G an

, The Shilov boundary of G x,r is reduced to a point that we denote ?(x, r)

, G x,r is the holomorphically convex envelope of ?(x, r)

, The k-anoid algebra of G x,r is the completion of Hopf(G) relatively to the norm | | Hopf(? e(K,k)r (G)) | Hopf(G)

, Let x be a rational point in the reduced Bruhat-Tits building of G, let r be a positive rational number and let g ? G(k), then 1. G g.x,r = gG x

, G i k /k (x),r = G x,r × M(k) M(k )

, G i k /k (x),r (k ) ? G(k) = G x,r (k)

, The map ? : BT R rat (G, k) × Q ?0 ? G an is continuous and injective

, We have to show that g.?(x, r) = ?(g.(x, r)). This is a direct consequence of 2.5.4, indeed ?(g.(x, r)) = ?(g.x, r) = g?(x, r)g ?1 = g

, Let K/k be the extension used to dene G x,r as G x,r = pr K/k ( ? e(K,k)r (G) ? ), we can assume that k ? K. We have G x,r = pr K/k ( ? e(K,k)r (G) ? ), thus G x,r = pr k /k (G x,r )

. K/k-(g-x, G i K/k (x),r by denition of G x,r and since K/k is a Galois extension

, K/k (pr ?1 k /k (G x,r )) = G i K/k (x),r . We also have pr ?1

. K/k-(g-x,

G. and K. , /k (x),r by denition of G x,r and since K/k is a Galois extension

, K/k (pr ?1 k /k (G x,r )) = pr ?1

). K/k-(g-x,r,

, The point x is a (rational) special point of BT R (G, k) and i K/k (x) ? BT R (G, K) is special for any nite extension K/k. The group G is not split over k, it is split over l. Let us make explicit the group G x,0 . We need to nd an extension K/k such that G is split over K, i K/k (x) is special, and r = 0 ? ord(K), shows that G × spec(k) spec(l) G m /l. The reduced Bruhat-Tits building BT R (G, k) is a singleton {x}

, ? inf ??l × {|?| | f ? ?(Hopf(G) ? 1) ? Hopf(G × k l)}

|. Hopf,

?. {|?|-|-f-?-?(<-l, X. ?-?-l, Y. , X. +-?-l, and Y. Hopf,

, And

?. {|?|-|-f-?-?(<-l, X. ?-?-l, Y. , X. +-?-l, and Y. Hopf,

?. , X. ?-?-l, Y. , X. +-?-l, and Y. Hopf, X 2 ? 2Y 2 ? 1 relatively to this norm, in order to simplify notation let us put || || = | | Hopf(G) | Hopf(G)

?. Since-x-?-(<-l, X. ?-?-l, Y. , X. +-?-l, and Y. Hopf,

X. ?-(<-l, ?. , X. ?-?-l, Y. , X. +-?-l et al.,

, Here we do not use a torus, the approach is algebraic. We use the explicit description of Hopf algebras of congruence groups of G. For any k ? -group scheme G, we use that Lie(G) is explicitely given by Hom k ? ?mod (I/I 2 , k ? ) where I, the situation of reductive group and it is constructed using a maximal torus, roots groups and splitting the reductive group

, Let k be a non arch. local eld, and let ? , k ? , the usual associated notations

, Let G = spec(A) be an ane smooth (thus at) k ? -group scheme. Let A be the k ? -Hopf algebra of G. Let ? : A ? k ? be the counit. Let I := ker(?) be the augmentation ideal. Let I 2 be the ideal II

, I/I 2 is a free k ? -module

, n of I/I 2 ; and choose alsog 1

, Let G = spec(A) be a at ane k ? -scheme satisfying the hypothesis of [32, Lemma 5.1]. Then 1. A contains no non-zero k ? -divisble element. 2. The ideal of augmentation I and its square power I 2 contain no nonzero k ? -divisible element

, A, I and I 2 are free k ? -modules

, Let r ? s, then 1. Let x ? A r , then there exists a positive integer N such that ? N x ? A s . 2. Let x ? I r , then there exists a positive integer N such that ? N x ? I s 3. Let x ? I r /I r 2 , then ? r?s x ? I s /I s 2

D. Jerey and . Adler, Rened anisotropic K-types and supercuspidal representations, Pacic J. Math, vol.185, issue.1, p.132, 1998.

M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck et al., of Séminaire de Géométrie Algébrique de l'Institut des Hautes Études Scientiques. Institut des Hautes Études Scientiques, vol.1, 1963.

V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean elds, volume 33 of Mathematical Surveys and Monographs, 1990.

V. G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math, issue.78, p.5161, 1993.

S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, vol.261

. Springer-verlag, A systematic approach to rigid analytic geometry, 1984.

S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol.21, issue.3

. Springer-verlag, , 1990.

P. Broussous and B. Lemaire, Building of GL(m, D) and centralizers, Transform. Groups, vol.7, issue.1, p.1550, 2002.

F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math, issue.41, p.5251, 1972.

F. Bruhat and J. Tits, Groupes réductifs sur un corps local

, Schémas en groupes. Existence d'une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math, issue.60, p.197376, 1984.

F. Bruhat and J. Tits, Schémas en groupes et immeubles des groupes classiques sur un corps local, vol.112, p.259301, 1984.

C. J. Bushnell, Eective local Langlands correspondence, Automorphic forms and Galois representations, vol.1, p.102134, 2014.

C. J. Bushnell and G. Henniart, The essentially tame Jacquet-Langlands correspondence for inner forms of GL(n), Special Issue: In honor of Jacques Tits, vol.7, p.469538, 2011.

C. J. Bushnell and P. C. Kutzko, The admissible dual of GL(N ) via compact open subgroups, Annals of Mathematics Studies, vol.129, 1993.

H. Carayol, Représentations cuspidales du groupe linéaire, Ann. Sci. École Norm. Sup, vol.17, issue.4, p.191225, 1984.

Z. Chang, The lie algebras of ane group schemes

, Cristophe Cornut. Filtrations and buildings, 2017.

M. Demazure and A. Grothendieck, Schémas en groupes

, Séminaire de Géométrie Algébrique de l'Institut des Hautes Études Scientiques, Institut des Hautes Études Scientiques, 1962.

A. Ducros, Espaces analytiques p-adiques au sens de Berkovich
URL : https://hal.archives-ouvertes.fr/hal-00368963

, Exp. No. 958, viii, 137176, 2005.

, Berkovich spaces and applications, vol.2119, 2015.

J. Fintzen, Types for tame p-adic groups, 2018.

I. Gelfand, D. Raikov, and G. Shilov, Commutative normed rings. Translated from the Russian, with a supplementary chapter, 1964.

A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math, issue.32, pp.1965-1967

A. Grothendieck, M. Artin, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, vol.269, 1972.

J. Hakim and F. Murnaghan, Distinguished tame supercuspidal representations, Art. ID rpn005, vol.166, 2008.

R. E. Howe, Tamely ramied supercuspidal representations of GL n, Pacic J. Math, vol.73, issue.2, p.437460, 1977.

I. Kaplansky, Projective modules, Ann. of Math, vol.68, issue.2, p.372377, 1958.

J. Kim, Supercuspidal representations: an exhaustion theorem, J. Amer. Math. Soc, vol.20, issue.2, p.273320, 2007.

J. Kim and J. Yu, Construction of tame types, Representation theory, number theory, and invariant theory, vol.323, p.337357, 2017.

A. Moy and G. Prasad, Unrened minimal K-types for p-adic groups, Invent. Math, vol.116, issue.1-3, p.393408, 1994.

A. Moy and G. Prasad, Jacquet functors and unrened minimal K-types, Comment. Math. Helv, vol.71, issue.1, p.98121, 1996.

J. Neukirch, Algebraic number theory, vol.322

. Springer-verlag, Translated from the 1992 German original and with a note by Norbert Schappacher, 1999.

G. Prasad and J. Yu, On quasi-reductive group schemes

, J. Algebraic Geom, vol.15, issue.3, 2006.

B. Rémy, A. Thuillier, and A. Werner, Bruhat-Tits theory from Berkovich's point of view. I. Realizations and compactications of buildings, Ann. Sci. Éc. Norm. Supér, vol.43, issue.4, p.461554, 2010.

D. Renard, Représentations des groupes réductifs p-adiques

. Société-mathématique-de-france, , 2010.

G. Rousseau, Immeubles des groupes réductifs sur les corps locaux

U. E. Mathématique, Publications Mathématiques d'Orsay, 1977.

G. Rousseau, Euclidean buildings, Géométries à courbure négative ou nulle, groupes discrets et rigidités, vol.18, p.77116, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00094363

R. Steinberg, Torsion in reductive groups, Advances in Math, vol.15, p.6392, 1975.

A. Thuillier, Théorie du potentiel sur les courbes en géométrie analytique non archimédienne : applications à la théorie, 2005.

J. Waldspurger, Algèbres de Hecke et induites de représentations cuspidales, pour GL(N ), J. Reine Angew. Math, vol.370, p.127191, 1986.

C. William and . Waterhouse, Introduction to ane group schemes, Graduate Texts in Mathematics, vol.66, 1979.

J. Yu, Construction of tame supercuspidal representations, J. Amer. Math. Soc, vol.14, issue.3, p.579622, 2001.

J. Yu, Bruhat-Tits theory and buildings, Ottawa lectures on admissible representations of reductive p-adic groups, vol.26, p.5377, 2009.

J. Yu, Smooth models associated to concave functions in Bruhat-Tits theory, Autour des schémas en groupes, vol.III, p.227258, 2015.