×

zbMATH — the first resource for mathematics

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 \(X\).

MSC:
20E42 Groups with a \(BN\)-pair; buildings
20G25 Linear algebraic groups over local fields and their integers
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.