# 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.

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### Logistics (To be continuously updated thorughout the semester)

• When: Fridays, 4:15-5:30 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 2.1-2.4 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.
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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 2.5-2.7 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.
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Week 3 (02/13)
Vivian (Qiyao) Yu
Representation theory of $$GL(2)$$ over a p-adic field
This week, we will start the discussion of the representation theory of GL(2) over a p-adic 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 non-archimedean 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.
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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 non-archimedean 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.
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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.
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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 p-adic 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.
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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.
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