Gan-Gross-Prasad conjecture: References
Here I give a comprehensive list of existing works on Gan-Gross-Prasad conjectures, for
- local GGP,
- global GGP,
- refined global GGP (i.e. Ichino-Ikeda type formulas).
I’ll try to include as many as works, not just restricted to classical groups (e.g. some works on metaplectic groups).
The project is inspired from this table for relative Langlands duality by Jonathan Wang.
Here are some notes on how-to-read the table:
- references without years are the papers on arXiv or author’s website that are not published yet.
- some of published references are given arXiv link instead, if the former link is not freely available.
- references with
° are results that formulated the conjecture for the first time.
- references with
* are results with certain assumptions: local assumptions, theta lifts, only covers one direction, etc.
- references with
** remove certain technical assumptions of previous works.
- if one proved refined global result that implies unrefined version, then I only include it in the last column.
- for local results, I put ${p}$ (resp. ${\infty}$) for non-archimedean (resp. archimedean) results.
- for local results, higher codimension cases follows from “basic” cases (i.e. codimension 0 or 1), by Gan–Gross–Prasad, Theorem 19.1.
| Type |
Case |
Local |
Global |
Refined |
| \(\mathrm{SO}_{n} \times \mathrm{SO}_{n+1}\) |
all \(n\) |
Gross–Prasad (1992)\({}^{\circ}\), Waldspurger (2012)\({}^{p}\text{*}\), Mœglin–Waldspurger (2012)\({}^{p}\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\), Ginzburg–Rallis–Soudry (2011)\(\text{*}\) |
Ichino–Ikeda (2010)\({}^{\circ}\text{*}\) |
| |
\(n=2\) |
|
Jacquet (1986) |
Waldspurger (1985), Krishna (2016) |
| |
\(n=3\) |
|
|
Ichino (2008) |
| |
\(n=4\) |
|
|
Gan–Ichino (2010)\(\text{*}\) |
| |
\(n=5\) |
|
|
Ichino–Ikeda (2010)\({}^{\circ}\text{*}\) |
| \(\mathrm{SO}_{n} \times \mathrm{SO}_{n+2r+1}\) |
all \((n,r)\) |
Gross–Prasad (1994)\({}^{\circ}\), Mœglin–Waldspurger (2012)\({}^{p}\) + Gan–Gross–Prasad (2012), Luo\({}^{p\infty}\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\), Jiang–Zhang (2020)\(\text{*}\) |
Liu (2014)\({}^{\circ}\text{*}\) |
| |
\(n=2,r=1\) |
|
Furusawa–Martin (2014)\(\text{*}\) |
Liu (2014)\({}^{\circ}\text{*}\), Corbett (2016)\(\text{*}\), Qiu (2013)\(\text{*}\), Murase–Narita (2016)\(\text{*}\) |
| |
\(n=3,r=1\) |
|
|
Liu (2014)\({}^{\circ}\text{*}\) |
| |
\(n=2\), all \(r\) |
|
Furusawa–Morimoto (2017)\(\text{*}\) |
Furusawa–Morimoto (2018)\(\text{*}\) |
| \(\mathrm{U}_{n} \times \mathrm{U}_{n+1}\) |
all \(n\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\), Beauzart–Plessis (2016)\({}^{p}\), Beauzart–Plessis (2020)\({}^{\infty}\) |
Ginzburg–Rallis–Soudry (2011)\(\text{*}\), Zhang (2014)\(\text{*}\) |
Harris (2014)\({}^{\circ}\text{*}\), Zhang (2014)\(\text{*}\), Xue (2017)\(\text{**}\), Beauzart-Plessis–Liu–Zhang–Zhu (2021)\(\text{**}\) Beauzart-Plessis–Chaudouard–Zydor (2022)\(\text{**}\) |
| |
\(n=1, 2\) |
|
|
Harris (2014)\({}^{\circ}\text{*}\) |
| \(\mathrm{U}_{n} \times \mathrm{U}_{n+2r+1}\) |
all \((n,r)\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\), Beauzart–Plessis (2016)\({}^{p}\) + Gan–Gross–Prasad (2012), Xue (2023)\({}^{\infty}\text{*}\), Xue\({}^{\infty}\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\), Jiang–Zhang (2020)\(\text{*}\) |
Beuzart-Plessis–Chaudouard\({}^{\circ}\) |
| |
\(n=0\) |
|
Ginzburg–Rallis–Soudry (2011)\(\text{*}\) |
|
| |
\(n=1\) |
|
|
Furusawa–Morimoto |
| \(\mathrm{U}_{n} \times \mathrm{U}_{n}\) |
all \(n\) |
Gan–Ichino (2016)\({}^{p}\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\), Xue (2014) |
Xue (2016)\(\text{*}\) |
| \(\mathrm{U}_{n} \times \mathrm{U}_{n + 2r}\) |
all \((n,r)\) |
Gan–Ichino (2016)\({}^{p}\) + Gan–Gross–Prasad (2012), Xue\({}^{\infty}\) |
|
Boisseau–Lu–Xue |
| \(\mathrm{Mp}_{2n} \times \mathrm{Sp}_{2n}\) |
all \(n\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\), Atobe (2018)\({}^{p}\text{*}\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\) |
Xue (2017)\({}^{\circ}\text{*}\) |
| \(\mathrm{Mp}_{2n} \times \mathrm{Sp}_{2n + 2r}\) |
all \((n, r)\) |
Atobe (2018)\({}^{p}\text{*}\) + Gan–Gross–Prasad (2012)\({}^{\circ}\) |
Gan–Gross–Prasad (2012)\({}^{\circ}\) |
Xue (2017)\({}^{\circ}\text{*}\) |
| |
\((n,r)=(1,1)\) |
|
|
Xue (2017)\({}^{\circ}\text{*}\) |
| \(\mathrm{GSpin}_{n} \times \mathrm{GSpin}_{n+1}\) |
all \(n\) |
Emory–Takeda (2023)\({}^{p}\text{*}\) |
|
Emory (2020)\({}^{\circ}\text{*}\) |
| |
\(n \leq 4\) |
|
|
Emory (2020)\({}^{\circ}\text{*}\) |
I may finish with introducing one more great article by Gross that covers the history of the conjecture.
Please let me know if there are any errors or possible updates - this table is supposed to be continuously updated.
Tags:
math