Explicit plancherel measure for PGL_2(F)

Abstract

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 [T, φ]Ḡ 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 , λφ) be a s-type. Then the Hecke algebra H(Ḡ, λφ) is a Hilbert algebra and has a measure associated to it called Plancherel measure of H(Ḡ, λφ). We show that computing the Plancherel measure for PGL2(F) essentially reduces to computing the Plancherel measure for H(Ḡ, λφ) for every type (J , λφ). We get that the Hilbert algebras H(Ḡ, λφ) come in two flavors; they are either C[Z] or they are a free algebra in two generators s1, s2 subject to the relations s 2 1 = 1 and s2 = (q −1/2 − q)s2 + 1 and we denote this algebra by H(q, 1). The Plancherel measure for C[Z] as well as the Plancherel measure for H(q, 1) are known.