This is a reading seminar about the Plancherel formula for p-adic groups. The main reference for this seminar is La Formule de Plancherel pour les Groups p-adiques by J.-L. Waldspurger. We will meet every Wednesday from 1:30 to 3:00 pm at Vincent Hall 570 starting from 4th September.
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 torus over p-adic fields.
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.
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 25th: 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 27th: Ed Karasiewicz
Time: 2:30 p.m.
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.
Section II.3: Estimations of the log norm.
Section II.4: Some further estimations of Harish-Chandra's Ξ function.
Section III.1: Square integrable representations.
Section III.2: Tempered representations.
Section III.3: The Geometric Lemma for tempered representations.
Section III.4: Subrepresentation Theorem for tempered representations.
Section III.5: The weak constant term of matrix coefficients of tempered representations.
Section III.6: Harish-Chandra's Schwartz space.
Section III.7: An elementary form for the theory of Bernstein center on tempered representations.
Section IV: Intertwining operators I: Absolute Convergence.
Section IV: Intertwining operators II: The analytic continuation.
Section V.1: Weak constant terms of matrix coefficients of induced representations.