  Let G be an absolutely almost simple algebraic group defined over a
non-archimedean local field K. Let X be a projective homogeneous variety for G
and let L be an ample line bundle on X. Then there exists a unique
G-linearisation of L. We regard X as a rigid analytic variety. We consider the
open rigid analytic subspace Y (resp. Y') of X that consists of the points x in
X that are stable (resp. semistable) for all maximal K-split tori in G. Here we
take for each maximal K-split torus S in G the S-linearisation of L obtained by
restricting the G-linearisation of L to S. Then G(K) acts on both Y and Y'.
Furthermore, Y is a subspace of Y'. We define a G(K)-equivariant map I that
maps each point x in Y' to a convex subset I(x) of the affine building B of the
group G(K).
  The map I is used to construct a compactification Z of Y in the following
sense. The rigid analytic space Z is the generic fibre of a formal scheme over
Spf(R) such that the closed fibre of the formal scheme consists of proper
components that correspond 1-1 to the vertices of the building. Here R denotes
the ring of integers of the field K. The space Z contains Y and is itself
contained in Y'. Furthermore, some parabolic subgroup P of G(K) acts on Z (but
G(K) itself does not act on Z, unless Z=Y=Y'). In fact, one can describe Z as
the set of points x in X that are stable for all maximal K-split tori S of G
that are contained in P, using the restriction to S of a suitable
P-linearisation of the line bundle L.
