Admissible representations of GL(2,Qp), the Kirillov model and application to local new vectors.

Abstract

Jacquet and Langlands (1970) showed that every infinite dimensional, irreducible, admissible representation of GL(2,Qp) has a unique Kirillov model, i.e. is isomorphic to a representation (π,K), where the space K consists of locally constant functions on Qp☓ on which π operates in some special way. The thesis will cover the proof of the above result after constructing the convenient framework. For this purpose I will start by introducing the notion of admissible representations of GL(2,Qp) and do some topology on Qp. After that I will prove the existence of the Kirillov model and prove some properties of the Bruhat-Schwartz space. Then I will prove the uniqueness part, which requires the construction of commutative operators; this part is the heart of the thesis. I shall then study a special example: the principal series representation. The last part of the thesis is dedicated to an application to the local new vectors of a representation.