关于标准模和Vogan L-包的注记
A note on Standard Modules and Vogan L-packets
摘要：
设$F$是特征$0$的非阿基米德局部域，$G$是定义在$F$上的连通约化群的$F$ -有理点群，$G'$是其拟分裂内部形式的$F$ -有理点群.给定分别用于$G$和$G'$的标准模$I(τ,ν)$和$I(τ',ν')$，其中$τ'$具有一般的调和表示，使得$τ$的超尖点支撑中的表示的Harish-Chandra $μ$ -函数和$τ'$的某些Jacquet模中的一般的本质平方积分表示的Harish-Chandra $μ$ -函数一致（经过适当的识别$ν=ν'$下的基本空间），我们表明，$I(τ,ν)$是不可约的，只要$I(τ',ν')$。如果分别为$I(τ,ν)$和$I(τ',ν')$的Langlands等价物$J(τ,ν)$和$J(τ',ν')$位于相同的Vogan $L$ -包中，则满足条件（每当定义这个Vogan $L$ -包时），证明对于任何Vogan $L$ -包，其朗兰兹商等于Vogan $L$ -包的成员的所有标准模是不可约的，当且仅当这个Vogan $L$包包含一个通用表示。Moeglin-Waldspurger证明了拟分裂正交群和辛群的一般Vogan $L$ -包的结果，并将其用于证明这些群的局部Gan-Gross-Prasad拓扑的一般情况。
英文摘要：
Let $F$ be a non-Archimedean local field of characteristic $0$, let $G$ be the group of $F$-rational points of a connected reductive group defined over $F$ and let $G'$ be the group of $F$-rational points of its quasi-split inner form. Given standard modules $I(τ,ν)$ and $I(τ',ν')$ for $G$ and $G'$ respectively with $τ'$ a generic tempered representation, such that the Harish-Chandra's $μ$-functions of a representation in the supercuspidal support of $τ$ and of a generic essentially square-integral representation in some Jacquet module of $τ'$ agree (after a suitable identification of the underlying spaces under which $ν=ν'$), we show that $I(τ,ν)$ is irreducible whenever $I(τ',ν')$ is. The conditions are satisfied if the Langlands quotients $J(τ,ν)$ and $J(τ',ν')$ of respectively $I(τ,ν)$ and $I(τ',ν')$ lie in the same Vogan $L$-packet (whenever this Vogan $L$-packet is defined), proving that, for any Vogan $L$-packet, all the standard modules whose Langlands quotient is equal to a member of the Vogan $L$-packet are irreducible, if and only if this Vogan $L$-packet contains a generic representation. The result for generic Vogan $L$-packets of quasi-split orthogonal and symplectic groups was proven by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these Groups.
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