Instructor: Prof. Clifton Cunningham
Email: clifton@automorphic.ca
Registration Deadline: January 17th, 2022
Lecture Times: Tuesdays | 12:00 - 2:00 pm and Thursdays | 1:00 - 2:00 pm (EST)
Course Dates: January 11th - April 7th, 2022
Mid-Semester Break: February 22nd - 25th, 2022
Registration Fee: PSU Students + PIMS Students - FREE | Other Students - $100 
Prerequisites: We assume a first course in algebraic geometry, familiarity with p-adic fields and representation theory, exposure to homological algebra and harmonic analysis. We will introduce perverse sheaves and their microlocal vanishing cycles through examples; no previous exposure to these topics is assumed.  
Evaluation: During the course we will calculate examples of perverse sheaves, L-packets and A-packets; write up one of these calculations for a problem set, worth 40% of your final grade. The remaining grade will be based on a written project on an approved a topic of your choosing.  

Format: Hybrid.
Lectures will be held in-person or online, according to the schedule below. In-person lectures will also be broadcast allowing for remote participation. 

• Toronto: January 11 & 13, February 8 & 10, March 8 & 10, April 5 & 7 
• Calgary: January 18 & 20, February 15 & 17, March 15 & 17, March 29 & 31 
• Online only: January 25 & 27, February 1 & 3, March 1 & 3, March 22 & 24 

Course Overview

This course presents Vogan's geometric perspective on L-packets and A-packets for p-adic groups. We will see how the geometry of the moduli space of Langlands parameters (with fixed infinitesimal parameter) determines ABV-packets, which are conjectured to generalize A-packets. We will also see how Arthur parameters appear as regular conormal vectors to the moduli space of Langlands parameters. The course is firmly rooted in examples, and we will develop the tools and techniques needed to calculate perverse sheaves and ABV-packets, both for classical and exceptional p-adic groups, focusing mainly on unipotent representations. We will study certain categories of unipotent representations of p-adic groups and compare them with certain categories of equivariant perverse sheaves on the moduli space of parameters, providing a categorical perspective on the Langlands correspondence. 

References

• Jeffrey Adams, Dan Barbasch, and David A. Vogan Jr., The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhaüser Boston, Inc., Boston, MA, 1992
• James Arthur, The endoscopic classification of representations, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups
• Alexander Beilinson, Joseph Bernstein, and Pierre Deligne, Faisceaux pervers, Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171
• Clifton Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek, and Bin Xu, Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples, Memoirs of the American Mathematical Society 276 (2022), No. 1353. DOI: https://doi.org/10.1090/memo/1353
• David A. Vogan Jr., The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 305–379