Title: Multiplicity one theorem for GL(n) and other Gelfand pairs.

Abstract:
First, we will introduce the notion of Gelfand pair and its connection to the invariant distributions. This is an important notion in representation theory. It has applications to classical representation theory and harmonic analysis. It was also applied more recently to automorphic forms and number theory.

Then we will discuss the multiplicity one theorem for GL(n) which is an important example of a Gelfand pair.
This theorem states that an irreducible representation of GL(n+1) "decomposes" in  a multiplicity free way to irreducible representations of GL(n).
The main step in the proof is proving that any distribution on GL(n+1) which is invariant w.r.t. conjugations by GL(n) is also invariant w.r.t. transposition.

In the end, if we have time, we will discuss the question of when a symmetric pair is a Gelfand pair.