K-theory, Chamber Homology and Base Change for GL(2)

Abstract

In this work on GL(2) we have found that it is hard to compute the chamber homology groups from the quotient space β1GL(2)/GL(2) (Mobius band), so we introduced a new way to compute the chamber homology groups by restricting to the original quotient space (edge) before taking the real line R. We have not yet given a full description of what happening under base change when we work on the cuspidal representation but, we somehow, gave a way to compute the base change effect of some type of cuspidal representations which are the admissible pairs. The base change of a principal series representations is always a principal series. Similarly, the base change of a twist of Steinberg representation is again a twist of Steinberg. However, an irreducible Galois representation can certainly restrict to a reducible one. Thus it is possible for the base change of a cuspidal to be principal series. In fact, if π is any irreducible admissible representation of GL(2,F) then one can find an extension E/F such that BC(π) is either unramified or Steinberg.