Document Type


Date of Degree

Summer 2012

Degree Name

PhD (Doctor of Philosophy)

Degree In


First Advisor

Kutzko, Philip

First Committee Member

Kutzko, Philip

Second Committee Member

Muhly, Paul

Third Committee Member

Anderson, Daniel

Fourth Committee Member

Jorgensen, Palle

Fifth Committee Member

Zamba, Gideon K D


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).


Mathematics, Number Theory, Plancherel Measure


vii, 87 pages


Includes bibliographical references (page 87).


Copyright 2012 Carlos De la Mora

Included in

Mathematics Commons