论文标题
谐波分析和伽马在符号组上起作用
Harmonic Analysis and Gamma Functions on Symplectic Groups
论文作者
论文摘要
在特征零的$ p $ -adic本地字段$ f $上，我们在扩展的符号组$ g = {\ mathbb g} _m \ times {\ mathrm sp} _ {2n} $上开发了一种新型的谐波分析。它与langlands $γ$符合函数相关，该功能附加在$ g（f）$的任何不可约束的可允许的表示$χ\otimesπ$和二组$ g^\ vee的标准表示$ρ$（{\ Mathbb c}）$中，并确认了本地理论中的一系列建议 - 在本地理论中进行了预知。同时，我们在$ {\ rm gl} _1（f）$上开发了一种新型的谐波分析，该$ {\ rm gl} _1（f）$与$γ$ -function $β_ψ（χ_s）$（$ n+1 $ n+1 $某些ABELIAN $γ$ - 功能）相关。我们在$ {\ rm gl} _1（f）$上的工作在我们在$ g（f）$上开发的开发中起着必不可少的作用。这两种类型的谐波分析都专门研究泰特论文中著名的本地理论，当时$ n = 0 $。该方法是使用$ {\ rm sp} _ {2n} $的压实在格拉曼尼亚种类中，包括$ {\ rm sp} _ {4n} $，我们能够利用piatetski-shapiro和许多其他$ lallis的本地理论来利用良好开发的本地local lasta $ lalliant $ lalliant $ lalliant $ lalliant $ lalliand $ $ {\ rm sp} _ {2n} $。 该方法可以看作是$ {\ rm gl} _n $的标准$ l $功能的Godement-jacquet工作的扩展，并有望在所有古典组中使用。我们将在未来的工作中考虑阿基梅德本地理论和全球理论。
Over a $p$-adic local field $F$ of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group $G={\mathbb G}_m\times{\mathrm Sp}_{2n}$. It is associated to the Langlands $γ$-functions attached to any irreducible admissible representations $χ\otimesπ$ of $G(F)$ and the standard representation $ρ$ of the dual group $G^\vee({\mathbb C})$, and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on ${\rm GL}_1(F)$, which is associated to a $γ$-function $β_ψ(χ_s)$ (a product of $n+1$ certain abelian $γ$-functions). Our work on ${\rm GL}_1(F)$ plays an indispensable role in the development of our work on $G(F)$. These two types of harmonic analyses both specialize to the well-known local theory developed in Tate's thesis when $n=0$. The approach is to use the compactification of ${\rm Sp}_{2n}$ in the Grassmannian variety of ${\rm Sp}_{4n}$, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis and many other works) on the doubling local zeta integrals for the standard $L$-functions of ${\rm Sp}_{2n}$. The method can be viewed as an extension of the work of Godement-Jacquet for the standard $L$-function of ${\rm GL}_n$ and is expected to work for all classical groups. We will consider the archimedean local theory and the global theory in our future work.
