算术波前集与泛型$L$-包
Arithmetic Wavefront Sets and Generic $L$-packets
AI总结：
本文针对特征零局部域上典型群的具泛型局部$L$-参数的不可约容许表示，通过增强$L$-参数的算术结构定义算术波前集，证明其为表示不变量，并提出波前集猜想且验证了若干情形。
摘要：
设$G$是定义在特征为零的局部域$F$上的典型群。设$π$是$G(F)$的不可约容许表示，若$F$是阿基米德域，则$π$为Casselman-Wallach型表示。若$π$具有泛型局部$L$-参数，我们通过$π$的增强$L$-参数$(φ,χ)$的算术结构，定义$π$的算术波前集${\rm WF_{ari}}(π)$，它是$G(F)$的李代数$\frak{g}(F)$的$F$-有理幂零轨道的一个子集。这些算术结构是我们基于局部朗兰兹对应和局部Gan-Gross-Prasad猜想的有理性，采用增强$L$-参数的连续递降方法发现的。我们研究了${\rm WF_{ari}}(π)$的基本结构，并证明它是$π$的一个不变量（定理5.10）。此外，这些基本结构${\rm WF_{ari}}(π)^{max}$有望给出${\rm WF_{ari}}(π)^{max}$的精确$F$-有理结构，当$F$为阿基米德域时，这一点已在定理1.3和1.4（定理7.10、7.14和9.2）中得到实现。基于局部朗兰兹互反律，波前集猜想（猜想1.2和猜想5.14）断言，$L$-参数侧的波前集应与表示侧的波前集密切相关，即当$π$具有泛型局部$L$-参数时，有${\rm WF_{wm}}(π)^{max}={\rm WF_{ari}}(π)^{max}={\rm WF_{tr}}(π)^{max}$，其中代数波前集${\rm WF_{wm}}(π)$由Moeglin和Waldspurger在[MW87]中利用广义Whittaker模型定义，解析波前集${\rm WF_{tr}}(π)$由Howe在[H81, Hd85]中利用分布特征定义，[H74, HC78, BV80]也给出了相关定义。我们已对一系列有意义的情形验证了猜想1.2。
英文摘要：
Let $G$ be a classical group defined over a local field $F$ of characteristic zero. Let $π$ be an irreducible admissible representation $π$ of $G(F)$, which is of Casselman-Wallach type if $F$ is archimedean. If $π$ has a generic local $L$-parameter, we define the arithmetic wavefront set ${\rm WF_{ari}}(π)$ of $π$, which is a subset of $F$-rational nilpotent orbits of the Lie algebra $\mathfrak{g}(F)$ of $G(F)$, by means of the arithmetic structures of the enhanced $L$-parameter $(φ,χ)$ of $π$. Those arithmetic structures are discovered by using our method of consecutive descents of enhanced $L$-parameters, based on the rationality of the local Langlands correspondence and the local Gan-Gross-Prasad conjecture. We study the basic structure of ${\rm WF_{ari}(π)}$ and prove that it is an invariant of $π$ (Theorem 5.10). Furthermore, those basic structures ${\rm WF_{ari}(π)^{max}}$ are expected to yield the precise $F$-rational structure of ${\rm WF_{ari}(π)^{max}}$, which has been realized, when $F$ is archimedean, in Theorems 1.3 and 1.4 (Theorems 7.10, 7.14, and 9.2). Based on the local Langlands reciprocity, the Wavefront Set Conjecture (Conjecture 1.2 and Conjecture 5.14) asserts that the wavefront sets on the L-parameter side should be closed related to those on the representation side, namely, \[ {\rm WF_{wm}(π)^{max}}={\rm WF_{ari}(π)^{max}}={\rm WF_{tr}(π)^{max}} \] when $π$ has a generic local $L$-parameter, where the algebraic wavefront set ${\rm WF_{wm}}(π)$ is defined by Moeglin and Waldspurger in [MW87], using generalized Whittaker models and the analytic wavefront set ${\rm WF_{tr}}(π)$ is defined by Howe [H81, Hd85] using distribution characters, and also by [H74, HC78, BV80]. Conjecture 1.2 is verified for families of interesting cases.
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