Organizer: Shaoyun Yi

Speaker: Jiawei Yang (杨嘉维)

Participant: Chuhui Gao (高楚辉)

Reference:
An Introduction to Automorphic Representations - With a view toward trace formulae (GTM 300), Jayce R. Getz and Heekyoung Hahn

Tentative Schedule:

No.	Date	Sections	Topics
1	3/11	Automorphic Representation Lecture 1	Abstract
2	3/20 (F)	Automorphic Representation Lecture 2	Abstract
3	4/1	Automorphic Representation Lecture 3	Abstract
4	4/8	Automorphic Representation Lecture 4	Abstract
5	4/15	Automorphic Representation Lecture 5	Abstract
6	4/22	Automorphic Representation Lecture 6	Abstract
7	5/13	Automorphic Representation Lecture 7	Abstract
8	5/20	Automorphic Representation Lecture 8	Abstract
9	5/27	Automorphic Representation Lecture 9	Abstract

Abstracts

Automorphic Representation Lecture 1

For the first session of our second seminar on automorphic representation, I will give a rough overview of our seminar, which consists of Langlands' decomposition of the $L^2$-spectrum of the adelic quotient group for the first half, and the (relative) trace formula for the second half. We will start from the cuspidal spectrum of the $L^2$-spectrum.

Automorphic Representation Lecture 2

This time, we will develop the theory of Poincaré series and use this technique to prove the closedness of the cuspidal $L^2$-spectrum in the whole $L^2$-spectrum.

Automorphic Representation Lecture 3

This time, we will take up the discussion on the deduction of the discreteness of the spectrum, and give a sufficient condition for an operator to be of trace class, which is crucial in the (number field case) proof of the second part of Theorem 1.2 discussed in the next section.

Automorphic Representation Lecture 4

This time, we will first finish the proof of the sufficient condition for an operator to be of trace class. After that, we will begin to prove our main theorem for the number field case via this condition using a basic estimate.

Automorphic Representation Lecture 5

We will still work on the basic estimate.

Automorphic Representation Lecture 6

We will finish the estimate in the number field case. After that, we will turn to the function field case, which is much easier than before obviously. Hopefully, we will take about rapidly decreasing functions.

Automorphic Representation Lecture 7

We start to study Eisenstein series. Under Harish-Chandra philosophy, we should first study the induced representations and the intertwining operators, which characterize the relation between induced representations. After that, we will define what is an Eisenstein series and give some properties roughly.

Automorphic Representation Lecture 8

We will first study the Constant Terms of an automorphic form along parabolic subgroups, which provide global analogues of Jacquet modules in representation theory of $p$-adic groups. The philosophy is that the growth rate of an automorphic form is controlled by its constant terms (since parabolics form the boundaries of the locally symmetric domain). We will analyze and give an expansion of the constant term of an Eisenstein series along a different parabolic subgroup with the help of intertwining operators and certain Weyl group elements. For the second part, we will introduce the decomposition of the $L^2$-spectrum of the automorphic quotient throughout the entire semester using the association classes of parabolic subgroups.

Automorphic Representation Lecture 9

Motivated by Langlands' spectral decomposition of the automorphic $L^2$-spectrum, which expresses the spectrum in terms of the discrete spectra of Levi subgroups and Eisenstein series, we introduce isobaric representations as representations built from subgroups, both locally and globally. We then describe how these representations enter the Langlands classification and explain the resulting description of the discrete spectrum of $\mathrm{GL}_n$.