Some Orbital Integrals and a Technique for Counting Representations of GL 2(F)

Abstract

Let F be a local field of characteristic zero, with q elements in its residue field, ring of integers uniformizer ωF and maximal ideal . Let GF = GL2(F). We fix Haar measures dg and dz on GF and ZF, the centre of GF, so that meas(K) = meas where K = GL2() is a maximal compact subgroup of GF. If T is a torus in GF we take dt to be the Haar measure on T such that means(TM)=1 where TM denotes the maximal compact subgroup of T.