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). We show that the convex subset I(x) of B is bounded if and only
if the point x is contained in Y. Furthermore, the subset I(x) of B consists
of a single point for all x in X if and only if the line bundle L is such
that the notions of stability and semistability coincide (i.e. iff. Y=Y').
If G(K)=SL(n,K) and X is the projective space of dimension n-1, then Y=Y' is
Drinfeld's symmetric space. In this case the map I is the reduction map
defined by Drinfeld.
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.
Harm Voskuil <email@example.com>