M.Sc Thesis
M.Sc Student Beit-Aharon Or A Kernel Formula for the Action of the Weyl Element,in the Kirillov Model of Representation in the Unitary Principle Series of SL(2,C) Department of Mathematics Professor Ehud Moshe Baruch

Abstract

We consider representations in the Unitary Principal Series of SL(2,C). We present several models of these representations among them the Kirillov model. The main goal of this work is to describe the action of the Weyl element in the Kirillov model as a kernel formula with an explicit kernel function. We begin by proving the existence of a kernel formula for the action of the Weyl element on functions  in the representation space of the Kirrilov model that are smooth with compact support. The kernel function that we get in the proof of the existence is not explicit and it is given as an integral over some K-Bessel function. In the next part of the work we give an explicit expression for the kernel function as a combination of Bessel functions. The explicit formula that we get for the integral that defines the kernel function is a new formula for an integral of Bessel functions. In order to obtain this formula we  calculate a different function that allows us to recover the kernel function. The new function is obtained by modifying the integral that defines the kernel function and it has the property of being an entire function, a fact that helps us calculating it explicitly. We conclude by using the result we got in order to obtain a similar kernel formula for the action of the Weyl element in representations of GL(2,C).