Bernhard Muehlherr, Holger P. Petersson, Richard M. Weiss: 'Descent in Buildings' Annals of Mathematics Studies, vol. 190, Princeton University Press, Princeton, 2015,352 pp.

摘要

The Context Roughly speaking, buildings are (products of) simplicial complexes admitting suitable group actions. These actions are used to understand better some groups from the structural, sometimes representation-theoretic, viewpoint. This idea works for specific classes: by and large, the starting point (at least in classical situations) is given by matrix groups with arbitrary fields of coefficients. Before going into details, let us mention that, already in this quite general case, it leads to deep results on the structure of the latter groups [6]. For instance such an action, with its strong transitivity properties, can be used to prove in a uniform way the simplicity of most classical matrix groups. In this case the involved buildings are called spherical since they are union of tilings of spheres, glued along subtilings of codimension≥ 1.