**Title:**Multiplicity one theorems in representation theory

**Speaker**: Shiv Prakash Patel (IIT, Delhi)

**Date**: Tuesday, February 25, 2020

**Time**: 4:00 pm

**Venue**: Seminar Room, School of Physical Sciences (SPS)

ABSTRACT

A representation $\pi$ of $G$ is called multiplicity free if the dimension of the vector space $Hom_{G}(\pi, \sigma)$ is at most 1 for all irreducible representation $\sigma$ of $G$. Let $H$ be a subgroup of $G$ and $\psi$ an irreducible representation of $H$. The triple $(G,H, \psi)$ is called Gelfand triple if the induced representation $Ind_{H}^{G} (\psi)$ of $G$ is multiplicity free. There is a geometric way to prove if some triple is Gelfand triple, which is called Gelfand's trick. Multiplicity free representations play an important role in representations theory and number theory, e.g. the use of Whittaker models. We will discuss Gelfand's trick and its use in a simple cases for Whittaker models of the representations of the group $GL_{n}(R)$ where $R$ is finite local ring.