Non-archimedean flag domains and semistability I, by Harm Voskuil

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 <>