Representation theory of p-adic groups and the local Langlands correspondence for GL(2)

Abstract

We discuss the representation theory of GL(2, F), where F is a non-archimedean local field following 'The Local Langlands Conjecture for GL(2)' by Bushnell and Henniart. Then we look at the decomposition of L^2(H\PGL(2, F)) into irreducible unitary representations, where H is a cocompact discrete subgroup. We prove a correspondence between multiplicity of spherical representations in the decomposition and eigenvalue of Hecke operator on the quotient graph of Bruhat-Tits tree. Proof uses the theory of spherical functions. Following the Bushnell and Henniart's book, we also state the local Langlands correspondence for GL(2, F), by discussing all the required machinery to understand the statement. Along with the representation theory of GL(2, F), this requires representation theory of Weil group and the theory of L-functions and local constants of these two classes of representations.