First we will introduce the notion of Gelfand pair. This is an important notion in representation theory. It has applications to classical representation theory and harmonic analysis. More recently it was also applied to automorphic forms and number theory. 

Then we will discuss the connection of this notion to
invariant distributions. We will list some recent results on Gelfand
pairs and demonstrate the tools used to achieve those results on a simple example $(GL_2, GL_1)$.  

If we have time in the end we will
discuss the question of when a symmetric pair is a Gelfand pair.