AI 中文总结
本文研究非阿基米德局部域上的扭曲雅克比模，证明其与斯佩表示诱导的相关性质，确定例外参数集，并给出辛二重覆盖及大特征有限域上的对应结果。
AI 中文摘要
由斯佩表示诱导的艾森斯坦级数的傅里叶系数是蔡、弗里德伯格、金兹堡和卡普兰提出的广义加倍方法的核。金兹堡和苏德里将这些系数与尖点形式积分，得到由更短的斯佩表示和任意尖点数据诱导的艾森斯坦级数。我们确定特征为零的非阿基米德局部域F上的对应局部模，设q_F为其剩余域的基数，J_s是由Δ(τ,m+i)|det|^s诱导的表示的扭曲雅克比模，它带有G×H的作用，其中G和H是分裂辛群或特殊正交群。设σ是G的有限长度可允许表示，我们证明，在依赖于τ和σ的有限组q_F^{-s}值之外，(J_s⊗σ)_G由Δ(τ,i)|det|^s和σ的固定共轭诱导，无需对σ做任何一般性假设。对于不可约σ，这确定了J_s的所有形如σ^∨⊠π的不可约商。该证明计算了每个轨道贡献，包括那些因尖点性而全局消失的贡献，还给出了包含例外参数的显式有限集，除非G是分裂群SO₂，否则对于不可约超尖点σ，该集为空。我们对辛二重覆盖证明了类似结果，在大特征有限域上，当τ在GL_n上是尖点且n大于G的秩时，给出了扭曲雅克比模的完全分解。
英文摘要
Fourier coefficients of Eisenstein series induced from Speh representations are the kernels of the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Ginzburg and Soudry integrated these coefficients against cusp forms to obtain Eisenstein series induced from shorter Speh representations and arbitrary cuspidal data. We determine the corresponding local modules over a non-archimedean local field $F$ of characteristic zero. Let $q_F$ be the cardinality of its residue field. Let $J_s$ be the twisted Jacquet module of the representation induced from $Δ(τ,m+i)|\det|^s$. It carries an action of $G\times H$, where $G$ and $H$ are split symplectic or special orthogonal groups. Let $σ$ be an admissible representation of $G$ of finite length. We prove that $(J_s\otimesσ)_G$ is induced from $Δ(τ,i)|\det|^s$ and a fixed conjugate of $σ$, outside a finite set of values of $q_F^{-s}$ depending on $τ$ and $σ$. No genericity assumption on $σ$ is needed. For irreducible $σ$, this determines all irreducible quotients of $J_s$ of the form $σ^\vee\boxtimesπ$. The proof computes every orbit contribution, including those that vanish globally by cuspidality. It also gives an explicit finite set containing the exceptional parameters. This set is empty for irreducible supercuspidal $σ$ unless $G$ is the split group $\mathrm{SO}_2$. We prove the analogous result for the symplectic double cover. Over finite fields of large characteristic, we give the complete decomposition of the twisted Jacquet module when $τ$ is cuspidal on $\mathrm{GL}_n$ and $n$ is larger than the rank of $G$.
Comments51 pages
