论文标题
$ g_2 $的显式本地兰兰兹通讯
The Explicit Local Langlands Correspondence for $G_2$
论文作者
论文摘要
我们使用Hecke代数技术制定了一种通用策略，该策略通过还原到LLC来构建$ p $ adadic还原组的明确兰格兰对应关系，以构建适当的Levi子组的超舒张代表。 作为我们一般策略的一个例子，我们在非Archimedean本地领域的特殊组$ G_2 $构建了明确的本地Langlands通信，并具有显式$ L $ a $ - 包装，并在小组和Galois方面之间的明确匹配。我们还列出了我们LLC的表征属性。在\ cite {g2稳定性}中，我们使用$ l $ -packets的稳定性属性完成了唯一的特征。对于中级系列，我们基于Hecke代数的先前结果。对于主要系列，我们改善了MUIC等的先前工作，并在组和Galois方面获得更明确的描述。 此外，我们展示了$ g_2 $的非单位\ textit {singular} supercuspidal表示的存在，并在\ textit {混合} $ l $ -packets中将其展示，混合了超级柔韧性表示与非副质表示。此外，我们的LLC满足了预期属性的列表，包括与Cuspidal支持的兼容性。
We develop a general strategy for constructing the explicit Local Langlands Correspondences for $p$-adic reductive groups via reduction to LLC for supercuspidal representations of proper Levi subgroups, using Hecke algebra techniques. As an example of our general strategy, we construct the explicit Local Langlands Correspondence for the exceptional group $G_2$ over a nonarchimedean local field, with explicit $L$-packets and explicit matching between the group and Galois sides. We also give a list of characterizing properties for our LLC. In \cite{G2-stability}, we complete unique characterization using stability property of our $L$-packets. For intermediate series, we build on our previous results on Hecke algebras. For principal series, we improve previous works of Muic etc. and obtain more explicit descriptions on both group and Galois sides. Moreover, we show the existence of non-unipotent \textit{singular} supercuspidal representations of $G_2$, and exhibit them in \textit{mixed} $L$-packets mixing supercuspidal representations with non-supercuspidal ones. Furthermore, our LLC satisfies a list of expected properties, including the compatibility with cuspidal support.
