# Representation Theory

### Previous Lectures

Root-reductive Lie algebras are relatively well behaved, countable-dimensional Lie algebras. In this talk, we shall discuss the definition of these Lie algebras as well as their important subalgebras---Borel subalgebras, Cartan subalgebras, and splitting maximal toral subalgebras. We shall also dwell on some representations of these Lie algebras if time permits.

I will discuss some ideas from a recent breakthrough in the field of Lie superalgebras:

https://arxiv.org/abs/1610.08470

No prior knowledge on Lie superalgebras will be assumed.

The classical Wedderburn-Artin theorem describes associative rings A for which the category of A-modules is semisimple. This turns out to be a left-right symmetric property: the category of left A-modules is semisimple if and only if the category of right A-modules is. In this talk, I will present a generalization of the Wedderburn-Artin theory to topololgical associative rings R in which open right ideals form a base of neighborhoods of zero. The talk will start with a discussion of split and semisimple abelian categories and end with a description of topological rings R for which the category of left R-contramodules is split (or equivalently, semisimple) or, equivalently, the category of discrete right R-modules is split (or equivalently, semisimple).

In my previous talk, I described the partial L-functions of cuspidal automorphic representations and said that their conjectured meromorphic continuation is important for the theory of automorphic forms. I then talked about integral representations as a method for proving this meromorphic continuation.

In this talk (which will be mostly independent of the previous one), I will describe a softened version of functoriality (which is the main theme in the Langlands program) and how L-functions help to detect and construct instances of functoriality.

In this talk, we will present an overview of the doubling construction for integral representations of L-functions. An explicit construction will be presented for quasi-split special orthogonal groups.

- Last modified: 24/10/2018