Plancherel formula for $${{\,\mathrm{\mathrm {GL}}\,}}_n(F){\backslash } {{\,\mathrm{\mathrm {GL}}\,}}_n(E)$$ and applications to the Ichino–Ikeda and formal degree conjectures for unitary groups

The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case

“…[BPLZZ21, Theorem 1.7]). Previous works had to assume extra local hypothesis on , which implied that was also simple (see [Zha14b], [Xue19], [BP21a] and [BP21c]) or only proved the direction 2. ⇒ 1. of the theorem ([GJR09], [IY19], [JZ20]).…”

“…Moreover, once again, this theorem is proved in [BPLZZ21] under the extra assumption that is cuspidal (in which case |S | = 4). Previous results in that direction includes [Zha14a], [BP21a], [BP21c] where some varying local assumptions on σ entailing the cuspidality of were imposed. In a slightly different direction, the paper [GL] establishes the above identity up to an unspecified algebraic number under some arithmetic assumptions on σ .…”

“…Motivations. -As in [Zha14b], [Zha14a], [Xue19], [BP21a], [BP21c] and [BPLZZ21], our proofs of Theorems 1.1.5.1 and 1.1.6.1 follow the strategy of Jacquet and Rallis [JR11] and are thus based on a comparison of relative trace formulas on unitary groups U h for h ∈ H n and the group G. Let's recall that these trace formulas have two different expansions: one, called the geometric side, in terms of distributions indexed by geometric classes and the other, called the spectral side, in terms of distributions indexed by cuspidal data. As usual, the point is to get enough test functions to first compare the geometric sides which gives a comparison of spectral sides.…”

A Whittaker-Plancherel Inversion Formula for SL2(ℂ)

“…Moreover, the Whittaker-Plancherel formula for a reductive p-adic group G was developed by Sakellaridis-Venkatesh [SV,§6.3] and Delorme [Del] by different methods. The proof of Sakellaridis and Venkatesh is relatively short and can be readily adapted to real groups as in the works of Beuzart-Plessis [BP1,BP2] (see [BP2,Proposition 2.142]).…”

The relative trace formula approach for spherical varieties

“…为保证上一节中所涉及的假设都成立, 本节只考虑缓和稳定 (stable) 表示 情形, 更一般情形可参见文献 [7,8]. 本节涉及的主要结果是基于 Zhang [9,14] 、Beuzart-Plessis [16,17] 、 Beuzart-Plessis 等 [7] 、Xue [18] 、恽之玮 (以及 Gordon) (参见文献 [19]) 等的工作.…”