Eisenstein Series Learning Seminar
This is a virtual learning seminar focused on Eisenstein series. The plan is to explain how Eisenstein series led Langlands to formulate his conjectures, explain the basic analytic properties of Eisenstein series and their role in the spectral decomposition of $L^2$, and go through the paper of BernsteinLapid which gives the meromorphic continuation and functional equational for Eisenstein series of general reductive groups using fairly soft techniques.
We will meet on Mondays from 3:00 to 4:00 PM EDT. Please email me if you’d like to be added to the seminar mailing list, or if you’d like the seminar Zoom link.
Here are the main references:

[A] Automorphic Representations and Number Theory  J. Arthur, Canadian Mathematical Society, Conf. Proc., Volume 1 (1981), available here

[GH] An Introduction to Automorphic Representations with a View Towards the Trace Formula  J. Getz, H. Hahn, (Jan. 7, 2022 draft) latest draft available here

[MW] Spectral Decomposition and Eisenstein Series  C. Moeglin, J.L. Waldspurger, Cambridge University Press

[BL] On the Meromorphic Continuation of Eisenstein Series  J. Bernstein, E. Lapid, available here

[G] Introduction à la Théorie de Langlands  R. Godement, Séminaire N. Bourbaki, 1968, exp. no 321, p. 115144, available here
Here is a rough plan outlining the topics we will cover, along with references to the relevant sections of the above material.
1. Introduction, motivation and review  

Why should you care about Eisenstein series?  [A]  Siddharth 
Some structure theory for reductive groups  [GH, §1.9], [MW, §I.1.4, §I.1.67]  Kewen 
Haar measures  [GH, §3.2, §3.6], [MW, §I.1.13]  Siddharth 
Reduction theory  [GH, §2.67], [MW, §I.2.1]  Siddharth 
Automorphic forms and automorphic representations  [GH, §6.34], [MW, §I.2.23, §I.2.1718]  Yu, Siddharth 
Automorphic forms in the $L^2$ sense, compatibility  [GH, §6.6]  Alex K. 
2. The cuspidal subspace  

The direct integral decomposition of $L^2([G])$  [GH, §3.810]  Malors 
Periods, Poincaré series, and the closure of the cuspidal spectrum  [GH, §9.12]  Matthew 
$R(f)$ restricted to the cuspidal spectrum is traceclass, reduction to the basic estimate  [GH, §9.3]  Siddharth 
Fourieranalytic proof of the basic estimate  [GH, §9.45]  Siddharth 
3. Eisenstein series  

Statement of the basic analytic theorems  [GH, §10.13], [MW, §II.1.56]  Siddharth 
Proof of convergence of Eisenstein series  [G, §3], [MW, §II.1.5]  Siddharth 
Proof of the basic properties of the intertwining operators  [MW, §II.1.6]  Sarah 
Global intertwining operators and Langlands’ insight; a sample computation in $GL_2$  [A, §4]  
Global intertwining operators and Langlands functoriality  [A, §5] 
4. Meromorphic continuation and functional equation  

Overview of the argument, statement of the main theorems, Bernstein’s principle of meromorphic continuation  [BL, §12, Appendix A] 
Application on Bernstein’s principle to Eisenstein series, the case of $SL_2$  [BL, §34] 
Cuspidal exponents and the cuspidal projection maps  [BL, §5.12], [MW, §I.3.25] 
Leading cuspidal components  [BL, §5.35] 
Uniqueness principle for Eisenstein series via leading cuspidal components  [BL, §5.69] 
Growth estimates for automorphic forms  [MW, §I.2.45, §I.2.1011] 
Local finiteness  [BL, §6] 
Conclusion of the proof in the number field case  [BL, §7] 
The function field case  [BL, §8] 