EXPLICIT PLANCHEREL MEASURE FOR PGL2(F)

In this thesis we compute an explicit Plancherel fromula for PGL2(F) where F is a non-archimedean local field by a method developed by Busnell, Henniart and Kutzko. 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 Bushnell and Kutzko in “Smooth representations of reductive p-adic groups: structure theory via types ” Pure Appl. Math, 2009.) for G/Z coming from types and covers of G in a very explicit way. We then compute those types and covers for GL2(F) which give rise to all types and covers for PGL2(F) that are in the principal series. The Bernstein components s ̄ of PGL2(F) that correspond to the principal series are of the form [T̄, ϕ]G ̄ where T ̄ is the diagonal matrices in GL2(F) modulo the center and ϕ can be seen as a smooth character for F×. Let (J ̄ , λ̄ϕ) be a s̄-type. Then the Hecke algebra H(Ḡ, λ̄ϕ) is a Hilbert algebra and has a measure associated to it called Plancherel measure of H(Ḡ, λ̄ϕ). We show that computing the Plancherel measure for PGL2(F) essentially reduces to computing the Plancherel measure for H(Ḡ, λ̄ϕ) for every type (J ̄ , λ̄ϕ). We get that the Hilbert algebras H(Ḡ, λ̄ϕ) come in two flavors; they are either C[Z] o