Explicit plancherel measure for PGL_2(F)

In this thesis we compute an explicit Plancherel fromula for PGL_2(F) where F is a non-archimedean local field. Let G be connected reductive group over a non-archimedean local field F. We show that we can obtain types and covers as defined by Kutzko and Bushnell for G/Z coming from types and covers of G in a very explicit way. We then compute those types and covers for GL_2(F ) which give rise to all types and covers for PGL_2(F) that are in the principal series. The Hecke algebra is a Hilbert algebra and has a measure associated to it called Plancherel measure of the Hecke algebra. We have that computing the Plancherel measure for PGL_2(F) essentially reduces to computing the Plancherel measure for the Hecke algebra for every type. We get that the Hacke algebras come in two flavors; they are either the group ring of the integers or they are a free algebra in two generators s_1, s_2 subject to the relations s_1^2=1 and s_2^2=(q^{-1/2}-q^{-1/2})s_2+1, where q is the order of the residue field. The Plancherel measure for both algebras are known, as a result we obtain the Plancherel measure for PGL_2(F).