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.

- compact.dvi (160596 bytes) [2000 Nov 29]
- compact.dvi.gz (60216 bytes)
- compact.ps.gz (146144 bytes)

Harm Voskuil <voskuil@math.rug.nl>