(untitled) (European Journal of Pure and Applied Mathematics, 2017)
In this work on GL(2) we have found that it is hard to compute the chamber homology groups from the quotient space Î²1GL(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 Ï€ is any irreducible admissible representation of GL(2,F) then one can find an extension E/F such that BC(Ï€) is either unramified or Steinberg.Â