 
Summary: Cycles in the chamber homology for GL(3)
AnneMarie Aubert, Samir Hasan and Roger Plymen
Abstract
Let F be a nonarchimedean local field and let GL(N) = GL(N, F).
We prove the existence of parahoric types for GL(N). We construct
representative cycles in all the homology classes of the chamber ho
mology of GL(3).
1 Introduction
Let F be a nonarchimedean local field and let G = GL(N) = GL(N, F).
The enlarged building 1
G of G is a polysimplicial complex on which G acts
properly. We select a chamber C 1
G. This chamber is a polysimplex, the
product of an nsimplex by a 1simplex:
C = n × 1.
To this datum we will attach a homological coefficient system, see [13,
p.11]. To each simplex in n we attach the representation ring R(G())
of the stabilizer G(), and to each inclusion we attach the induction
map:
Ind
