论文标题
Whittaker功能和特征上的争议性而不是$ P $
Whittaker functionals and contragredient in characteristic not $p$
论文作者
论文摘要
令$ r $为代数封闭的字段,$ \ ell $为其特征。让$ g $是一个本地涂鸦集团,在$ r $中具有紧凑的可抗可转让订单的子组。以$ n $ a $ g $的封闭亚组为$ r $的紧凑型亚组耗尽,并修复了$ n $的光滑字符$θ$。对于$π$,$ g $的不可约平滑$ r $ - 代表其矩阵系数紧凑地支持了中心的Modulo(我们称其为$ z $ -compact),我们表明dimensions $ \ mathrm {hom} $ \ mathrm {hom} _ {n}(π^\ vee,θ^{ - 1})$是相等的。我们从这个结果中得出了一些应用程序。首先,我们证明,任何$ g $ -Intertwiner从$π$到$ \ mathrm {ind} _n^g(θ)$都具有$ \ mathrm {indrm {indrm {ind} _ {zn}^g(ωπθ)$中的图像,其中$ω__π$是$π$的中心特征。其次,它适用于剩余特征$ p \ neq \ ell $的非阿基梅德本地领域的准切片组,其中$ n $是$ g $的鲍尔(Borel)亚组的一级自由基,以及一个通用字符$θ$。我们的尺寸平等证明是Rodier在Whittaker多重性方面至多对Cuspidal表示的关键使用的重要替代品。然后,通过提升参数,我们以$ r $ $价元的$(θ^{ - 1} \ otimesθ)$ - $ g $上的epivariant分布来恢复Rodier对Gelfand-kazhdan属性的概括。后一个事实与罗德尔(Rodier)的遗产属性(在我们的上下文中都是有效的),最多可导致惠特克(Whittaker)功能超过$ r $。我们还提供了其他申请,包括对Chang Yang证明的复杂表示结果的概括,最初由Dipendra Prasad猜想。
Let $R$ be an algebraically closed field and $\ell$ be its characteristic. Let $G$ be a locally profinite group having a compact open subgroup of invertible pro-order in $R$. Take $N$ a closed subgroup of $G$ exhausted by compact subgroups of invertible pro-orders in $R$ and fix a smooth character $θ$ of $N$. For $π$ an irreducible smooth $R$-representation of $G$ whose matrix coefficients are compactly supported modulo the center (we call it $Z$-compact), we show that the dimensions $\mathrm{Hom}_{N}(π,θ)$ and $\mathrm{Hom}_{N}(π^\vee,θ^{-1})$ are equal provided one of the two is finite. We derive a few applications from this result. First, we prove that any $G$-intertwiner from $π$ to $\mathrm{Ind}_N^G(θ)$ has image in $\mathrm{ind}_{ZN}^G(ω_πθ)$, where $ω_π$ is the central character of $π$, and the Whittaker space of $π$ agrees with that of its Whittaker periods. Second, it applies to quasi-split groups over non Archimedean local fields of residual characteristic $p \neq \ell$ and where $N$ is the unipotent radical of a Borel subgroup of $G$ together with a generic character $θ$. Our equality of dimensions turns out to be a good replacement for Rodier's crucial use of complex conjugation in the proof of Whittaker multiplicity at most one for cuspidal representations. Then by a lifting argument, we recover Rodier's generalization of the Gelfand-Kazhdan property for $R$-valued $(θ^{-1}\otimes θ)$-equivariant distributions on $G$. This latter fact, together with Rodier's heridity property, which is valid in our context, leads to the multiplicity at most one of Whittaker functionals over $R$. We also give other applications, including a generalization over $R$ of a result for complex representations proved by Chang Yang and initially conjectured by Dipendra Prasad.