Reading Seminar on the Plancherel formula for p-adic groups

Fall 2024

The main reference for this seminar is La Formule de Plancherel pour les Groupes p-adiques by J.-L. Waldspurger.

We meet every Wednesday from 1:30 to 3:00 pm at Vincent Hall 570, starting September 4, 2024.

Talk Schedule

• September 4: Zhaolin Li
  • Introduction to the Plancherel formula.
  • Section I.1: Basic definitions of reductive groups and Haar measures with examples.
  • Section I.2: Finite functions on split tori over p-adic fields.
• September 11: Zhaolin Li
  • Section I.3: Representations of p-adic groups: Jacquet modules, induced representations, Frobenius reciprocity, and the Geometric Lemma.
  • Section I.4: The asymptotic behavior of matrix coefficients.
• September 18: Zhaolin Li
  • Section I.5: A family of admissible representations and a generalization of the contragredient property of Jacquet modules.
  • Section I.6: A p-adic analog of K-finite vectors in regular representations.
• September 25: Sagnik Mukherjee
  • Section II.1: Basic properties of Harish-Chandra's Ξ function.
  • Section II.2: The representation on the equivariant line bundle on the flag variety induced from the modular character.
• September 27: Ed Karasiewicz (National University of Singapore)
  • Time: 2:30 pm
  • Place: Vincent Hall 1
  • Title: Stable Wavefront Sets for Theta Representations
  • Abstract: The Fourier coefficients of theta functions have featured prominently in numerous number theory applications and constructions in the Langlands program. For example, they play an important role in the recent work of Friedberg-Ginzburg generalizing the theta correspondence to higher covering groups. For their construction one wants to know the wavefront set of the theta representations, i.e. the largest nilpotent orbit with nonvanishing Fourier coefficient.
    
    To investigate these Fourier coefficients it can be valuable to study the analogous local question. In this talk we consider local depth 0 theta representations and describe how to compute their stable wavefront set. This is joint work with Emile Okada and Runze Wang.
• October 2: Sagnik Mukherjee
  • Section II.3: Estimations of the log norm.
• October 9: Sagnik Mukherjee
  • Section II.4: Some further estimations of Harish-Chandra's Ξ function.
• October 16: Junyi Zhai
  • Section III.1: Square integrable representations.
  • Section III.2: Tempered representations.
• October 23: Junyi Zhai
  • Section III.3: The Geometric Lemma for tempered representations.
  • Section III.4: Subrepresentation Theorem for tempered representations.
• October 30: Junyi Zhai
  • Section III.5: The weak constant term of matrix coefficients of tempered representations.
  • Section III.6: Harish-Chandra's Schwartz space.
• November 6: Junyi Zhai
  • Section III.7: An elementary form for the theory of Bernstein center on tempered representations.
• November 13: Guodong Xi
  • Section IV: Intertwining operators I: Absolute Convergence.
• November 20: Guodong Xi
  • Section IV: Intertwining operators II: The analytic continuation.
• December 4: Yuxuan Ge
  • Section V.1: Weak constant terms of matrix coefficients of induced representations.