Simplicial complexes lying equivariantly over the affine building of GL(<Emphasis Type="Italic">N</Emphasis>)

Let G=GL(N,K), K a non-archimedean local field and X be the semisimple affine building of G over K. We construct a projective tower of G-sets with X(0)=X. They are obtained by using a minor modification in Bruhat and Tits original construction (an idea due to P. Schneider). Using the structure of X as an abstract building, we construct a projective tower of simplicial G-complexes such that, for each r, X_(r) is canonically a geometrical realization of X_r. In the case N=2, X_r has a natural two-sheeted covering _r and we show that the supercuspidal part of the cohomology space is characterized by a nice property.Mathematics Subject Classification (2000): 14R25, 20E42, 20G25, 55U10, 57S25