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

Abstract

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.1991 Mathematics subject classification (Amer. Math. Soc): primary 32P05; secondary 14M17, 20G25.

C2 building and projective space

“…So one can view 0>s 0 / T as a compactification of Y s jJ Y. See Voskuil [16]. Note that using the embedding above one has Here a 0 is the standard alcove defined by the affine roots a,, a 2 and 1 -a 0 .…”

“…The p-adic space we study here is Y s := f\eG(f) 8 ' ^U-^n t n e case when the stable points and the semistable points are the same these problems are studied by van der Put and Voskuil [12]. The case of quasi-split rank 1 group is studied by Voskuil [16]. This work started from a conversation with Voskuil in a cafe in Newtown.…”