Chamber homology and K-theory for the p-adic group GL(3)

摘要

In this thesis we have got computations of chamber homology for p-adic groups GL(3). In this first chapter, some rudimentary definitions, notions, and basic facts about C*-algebra and basic K-theory are given. In the second chapter we give introductory material of p-adic analysis, local fields and reductive p-adic groups and their representations. In the third chapter we give a brief account of affine Hecke algebras. We start by Coxeter groups and show how we define Hecke algebras. We also display another way by studying td-groups (totally disconnected group) and close this chapter by representations of Hecke algebras. In the fourth chapter we study affine buildings. We start by giving account about trees which are building in GL(2) then we give a study about affine building for GL(3) and we give accounts about the chambers in GL(3), simplicial and polysimplicial complexes to construct affine building betaG of the reductive p-adic groups G = GL(n). In the fifth chapter we study chamber homology. We start by giving the general definition of homology group. We give some results and brief study of chamber homolog of G = GL(3). We also give an account about Bernstien decomposition. Finally the sixth chapter gives a work on computation of chamber homology H*(G;betaG) of GL(3). We discuss the maximal simple types (J, lambda) and semisimple types, and conclude computations of chamber homology.