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Compactification of the Bruhat-Tits building of PGL by seminorms. (English) Zbl 1121.20024
Summary: We construct a compactification \(\overline X\) of the Bruhat-Tits building \(X\) associated to the group \(\text{PGL}(V)\) which can be identified with the space of homothety classes of seminorms on \(V\) endowed with the topology of pointwise convergence. Then we define a continuous map from the projective space to \(\overline X\) which extends the reduction map from Drinfeld’s \(p\)-adic symmetric domain to the building \(X\).

20E42 Groups with a \(BN\)-pair; buildings
20G25 Linear algebraic groups over local fields and their integers
Full Text: DOI arXiv
[1] Berkovich, V. G.: Spectral theory and analytic geometry over non-archimedean fields. Math Surveys Monographs 33. American Mathematical Society 1990
[2] Berkovich, V. G.: The automorphism group of the Drinfeld upper half-plane. C.R. Acad. Sci. Paris 321, Série I 1127-1132 (1995) · Zbl 0856.14007
[3] Borel, A.: Linear Algebraic Groups. Second edition. Springer 1991 · Zbl 0726.20030
[4] Borel, A., Serre, J.-P.: Cohomologie d?immeubles et de groupes S-arithmétiques. Topology 15, 211-232 (1976) · Zbl 0338.20055
[5] Bourbaki, N.: Groupes et algèbres de Lie. Chapitres 4, 5 et 6. Herrmann 1968 · Zbl 0186.33001
[6] Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math. IHES 41, 5-252 (1972)
[7] Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. math. France 112, 259-301 (1984) · Zbl 0565.14028
[8] Drinfeld, V. G.: Elliptic modules. Math USSR Sbornik 23, 561-592 (1974) · Zbl 0321.14014
[9] Gérardin, P.: Harmonic functions on buildings of reductive split groups. In: Operator algebras and group representations, Vol. I. Monogr. Stud. Math. 17. Pitman 1984, 208-221
[10] Goldman, O., Iwahori, N.: The space of p-adic norms. Acta math. 109, 137-177 (1963) · Zbl 0133.29402
[11] Landvogt, E.: A compactification of the Bruhat-Tits building. Lecture Notes in Mathematics 1619, Springer 1996 · Zbl 0946.20025
[12] Werner, A.: Compactification of the Bruhat-Tits building of PGL by lattices of smaller rank. Documenta Math. 6, 315-342 (2001) · Zbl 1048.20014
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