# Nonarchimedean Flag Domains and Semistability

## Author

**Harm Voskuil**

## Status

Research Report 98-35

Date: 23 December 1998

## Abstract

Let G be a simply connected absolutely almost simple linear algebraic
group defined over a nonarchimedean local field K.
Let X be a projective homogeneous variety for G.

We consider the analytic subset Y of X that consists of the
points that are semistable for all maximal K-split tori of G with
respect to a fixed ample line bundle. The linearization of the torus
action is obtained by restricting the unique G-linearization of this
linebundle. We define a map that associates to each point of Y a
convex subset of the building. This map is defined by using semistability
over the ring of integers of K.

For split groups we give a partial description of the image of
this map. We prove that in certain cases this is actually a complete
description of the image of the map.

## Key phrases

nonarchimedean local field. semisimple groups. affine buildings.
projective homogeneous varieties. semistability.

## AMS Subject Classification (1991)

Primary: 22E35

Secondary: 32P05, 14G20

## Content

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