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On the action of the unitary group on the projective plane over a local field

Part of: Linear algebraic groups and related topics Special varieties Non-Archimedean analysis

Published online by Cambridge University Press:  09 April 2009

Harm Voskuil
Harm Voskuil
Affiliation:
School of Mathematics and Statistics University of SydneyNSW 2006Australia e-mail: [email protected]
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Abstract

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Let G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

MSC classification

Secondary: 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem) 14M17: Homogeneous spaces and generalizations 20G25: Linear algebraic groups over local fields and their integers
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 3 , June 1997 , pp. 371 - 397
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Bosch, S., Güntzer, U. and Remmert, R., Non-archimedean analysis (Springer, Berlin, 1984).CrossRefGoogle Scholar
[2]Fresnel, J. and van der Put, M., Geométrie analytique rigide et applications, Prog. Math. 18 (Birkhäuser, Boston, 1981).Google Scholar
[3]Bruhat, F. and Tits, J., ‘Groupes Réductifs sur un corps local I: Données radicielles valuées’, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251.CrossRefGoogle Scholar
[4]van der Put, M. and Voskuil, H., ‘Symmetric spaces associated to split algebraic groups over a local field’, J. ReineAngew. Math. 433 (1992), 69–100.Google Scholar
[5]Mustafin, G. A., ‘Nonarchimedean uniformization’, Math. USSR-Sb. 34 (1978), 187–214.CrossRefGoogle Scholar
[6]Oda, T., Convex bodies and algebraic geometry (Springer, Berlin, 1988).Google Scholar
[7]Tits, J., ‘Reductive groups over local fields’, Proc. Amer. Math. Soc. Symp. Pure Math. 33 (1979), 29–69.CrossRefGoogle Scholar
[8]Voskuil, H., Non-archimedean Hopf Surfaces, Séminaire de Théorie des Nombres de Bordeaux 3(1991), 405–466.Google Scholar
[9]Voskuil, H., ‘P-adic symmetric spaces: The unitary group acting on the projective plane’, in: Algebraic geometry symposium at Kinosaki, 1993 (ed. profMaruyama, ) (Kyoto University Press, Kyoto, 1994), pp. 58–76.Google Scholar