---
date: 2022-02-23 18:45
modification date: Friday 25th March 2022 20:45:29
title: "Beuzart-Plessis, On the spectral decomposition of the Jacquet-Rallis trace formula and the Gan-Gross-Prasad conjecture for unitary groups"
aliases: ["spectral decomposition of the Jacquet-Rallis trace formula and the Gan-Gross-Prasad conjecture for unitary groups"]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #arithmetic-geometry/Langlands 
- Refs:
	- #todo/add-references
- Links:
	- [automorphic form](Unsorted/automorphic%20form.md)

---

## Raphaël Beuzart-Plessis, On the spectral decomposition of the Jacquet-Rallis trace formula and the Gan-Gross-Prasad conjecture for unitary groups.

> Reference: Raphaël Beuzart-Plessis, On the spectral decomposition of the Jacquet-Rallis trace formula and the [Gan-Gross-Prasad conjecture](Gan-Gross-Prasad%20conjecture) for unitary groups.
  BC NT/AG Seminar.

- Look at [automorphic](Unsorted/automorphic%20representation.md) [cuspidal representations](Unsorted/automorphic%20representation.md) to $\PGL_2(\FF)$ for $\FF$ a [number field](number%20field.md).

- This talk: generalizing some of these formulas:

![](attachments/image_2021-05-06-14-07-33.png)

  - See [split torus](split%20torus), [quaternion algebra](quaternion%20algebra.md).

- See [cusp forms](cusp%20forms), [period](Unsorted/period.md), [trace formulas](trace%20formulas)

- Gan-Gross-Prasad and Ichino-Ikeda conjecture for unitary groups

  - $E/\FF$ a quadratic extension of number fields, so $\Gal(E/\FF) \cong \ZZ/2 = \ts{ 1, c }$.
  - Take a nondegenerate [Hermitian form](Hermitian%20form) $h$ on $E^n$, define $U(h)$ as the [unitary group](unitary%20group) of $h$ and set $U_h \da U(h) \cross U(h \oplus h_0)$ and $h_0: E^2 \to E$ where $(x,y) \mapsto xy^c$.
  - Define an [L function](L%20function.md) :

  ![](attachments/image_2021-05-06-14-13-13.png)

- Theorem: from [the Langlands philosophy](Unsorted/Langlands.md).
  There is a [Hermitian form](Hermitian%20form) and a [cuspidal representation](Unsorted/automorphic%20representation.md) in the same [L packet](L%20packet) where $P$ is nonvanishing
.
- **Theorem/conjecture**: if $\sigma$ is [tempered](tempered.md) everywhere, there is a formula:

  ![](attachments/image_2021-05-06-14-15-00.png)

  - See [special values of L functions](special%20values%20of%20L%20functions.md).

- Proved in special cases.

- Recent result: $P_h\neq - \implies L(1/2, \Pi)\neq 0$ proved using twisted automorphic descent.
- See [regularization](regularization), [Langlands decomposition](Langlands%20decomposition), [Eisenstein series](Eisenstein%20series.md), [Levi](Unsorted/Representation%20Theory%20(Subject%20MOC).md), [GIT quotients](GIT%20quotients), [orbital integrals](orbital%20integrals).

  ![](attachments/image_2021-05-06-14-28-57.png)

- Global [distributions](distributions.md) break into an [Euler product](Euler%20product) of local distributions.
  
- Part of proof: use [zeta integrals](zeta%20integrals) of [kernel functions](kernel%20functions), fix one variable to get a function in a [Schwartz space](Schwartz%20space.md).
  - See [unipotent](unipotent) and [parabolic](parabolic) subgroups
  - Uses a "standard unfolding" process?
- See [Whittaker function](Whittaker%20function), [Petersson inner product](Petersson%20inner%20product), [Phragmen-Lindelof principle](Phragmen-Lindelof%20principle) to control one zeta integral in terms of another.

See [Serre's conjecture](Serre's%20conjecture).