Seminar: Automorphic Forms & Representations (Spring 2023)
Topic: Automorphic Representations
In this seminar, we will be talking about basics of automorphic forms and representation. Using Daniel Bump's "Automorphic Forms and Representations" as our main reference, we will start with automorphic forms and representations of \(GL(2, \mathbb R)\), followed by representation of \(GL(2)\) over \(p\)adic field. Finally, we talk about representation of automorphic forms in general. This is an introductory seminary that aims to cover the basis of automorphic forms, so everyone are welcome to participate.
Tentative Syllabus:
See here
Logistics (To be continuously updated thorughout the semester)

When: Fridays, 4:155:30 PM ET

Where: Math 520

Organizer: Jiahe Shen, Qiyao (Vivian) Yu, Rafah Hajjar, Xiaorun Wu

References:

Main References

Daniel Bump, Book "Automorphic Forms and Representations"
Schedule

Week 1 (02/03)

Jiahe Shen

Introduction to Maass forms and Spectral Problems

This week, I will briefly go through chapter 2.12.4 of Bump's "Automorphic Forms and Representations". We will be first talking about Maass forms and the Spectral Problem. After that, we will be reviewing about basic Lie theory, then I wil briefly talk about discreteness of the spectrum. Finally, I will talk about some basic representation theory.

notes here

Week 2 (02/06)

Jiahe Shen

Irreducible \( (\mathfrak g, K)\)modules, unitaricity and interwining integrals, and spectral problem

This week, I will briefly go thourgh chapter 2.52.7 of Bump's "Automorphic Forms and Representations". First we will be talk about irreducible \( (\mathfrak g, K)\)modules for \(GL(2, \mathbb R)\), then we will mention about unitaricity and interwining integrals. Next, we will be discussing about representations and spectral problem.

notes here

Week 3 (02/13)

Vivian (Qiyao) Yu

Representation theory of \(GL(2)\) over a padic field

This week, we will start the discussion of the representation theory of GL(2) over a padic field. We will begin by introducing smooth and admissible representations and their basic properties. If time permits, we will introduce some tools like sheaves and distributions that lay the foundation for upcoming talks.

Week 4 (02/20)

Vivian (Qiyao) Yu

Whittaker models and Jacquet functor

This week, we will introduce Whittaker models for smooth representations of GL(n,F), where F is a nonarchimedean local field. We will prove the uniqueness of Whittaker models (also known as local multiplicity one theorem) when n=2. Furthermore, we will introduce the Jacquet functor and prove some of its important properties.

notes here

Week 5 (02/27)

Vivian (Qiyao) Yu

Principal Series Representation for \(GL(2,F)\)

This week, we will introduce the principal series representation for \(GL(2,F)\), where \(F\) is a nonarchimedean local field. We will start by defining representations of \(GL(2,F)\) induced from representations of the Borel \(B(F)\) and move on to discuss conditions for irreducibility of the induce representations. We will also show directly that the induce representations admit at most one Whittaker model. We will transition into next week’s talk by defining spherical representations.

notes here

Week 6 (03/06)

Vivian (Qiyao) Yu

Principal Series & Special Representations

This week, we will talk about a special subset of irreducible admissible representations, the spherical representations. Spherical representations are extremely important for us to understand the decomposition of an automorphic representation into local representations. We will classify the irreducible admissible spherical representations using the principal series representations we discussed last week. After studying the spherical Whittaker functional and the spherical function, we will move on to discuss the local functional equation, which is the analogue of the local functional equation appeared in Tate’s thesis.

notes here

Week 7 (04/03)

Rafah Hajjar Muñoz

Automorphic representations associated to classical objects

After studying in detail the representations of GL(2) over both ℝ and a padic field and their connection to automorphic forms, we will now introduce automorphic representations of the adele group, which will unify the local representations into a global theory.
In this lecture, we will review Tate's thesis, which is essentially the theory of automorphic representations of GL(1) over the adeles, and we will show how some automorphic representations of GL(2) arise from the classical theory of modular forms and Maass forms.

notes here

Week 8 (04/17)

Rafah Hajjar Muñoz

Automorphic representations of GL(n)

After finishing the review of the relation between automorphic forms of GL(2,R) and the classical theory of modular forms and Maass forms, we will introduce the general notion of automorphic representations of GL(n,A), for A the adele ring of a number field.

notes here