论文标题
还原组和功能性的封面
Covers of reductive groups and functoriality
论文作者
论文摘要
对于当地字段$ f $上的准切片连接的还原$ g $，我们定义了一个紧凑的Abelian Group $ \tildeπ_1（g）$和一个扩展名$ 1 \ to \tildeπ_1（g）\至G（f）至G（f）_ \ f）_ \ f）\ fo（f）\ fo（f）\ fo（f）\ fo（f）\ 1 $ $ g_ \ textrm {sc}（f）$。任何字符$ x：\tildeπ_1（g）\toμ_n（\ mathbb {c}）$都会导致$ n $ -fold cover $ g（f）_x _x $ g（f）$ g（f）$ a pussout。我们为此封面定义了$ l $ -group $^lg_x $，通常是$ \ textrm {gal}（f^s/f）$ by $ \ hat g $的非切片扩展。我们证明了$ g（f）_x $的精致本地兰兰兹信件，假设它以与$ g $相同的伴随组的连接还原组而闻名。 这种构建的动机来自兰兰兹的功能性猜想的考虑，其中$ \ MATHCAL {h} \ subset {^lg} $的$ l $ g $ g $的$ g $ groups不需要$ l $ l $ - 其他还原组的组。如果这样的子组已满并在连接的最大排名的还连接的还原亚组中与$ \ hat g $相交，则我们构建一个自然三重$（H，x，x，ξ）$由quasi-split连接的还原型组$ h $，$ h（f）$ h（f）_x $和$ l $ l $ - $ embed $} $}这是$ \ Mathcal {H} $的同构。我们希望$ h（f）_x $转移的真实表示为$ g（f）$。 在内窥镜检查的特殊情况下，我们表明，当使用内窥镜组的自然双重覆盖$ h（f）_x $时，转移因子的构建简化了。转移因子成为两个不取决于辅助选择的两个天然不变的乘积。其中之一与科特威茨（Kottwitz）关于谎言代数转移因子的工作密切相关。另一个不是内窥镜检查情况的特异性，并且可能在一般功能问题中发挥作用。我们的工作是由亚当斯（Adams）和沃根（Vogan）的实际数字进行的。
For a quasi-split connected reductive group $G$ over a local field $F$ we define a compact abelian group $\tildeπ_1(G)$ and an extension $1 \to \tildeπ_1(G) \to G(F)_\infty \to G(F) \to 1$ of topological groups equipped with a splitting over $G_\textrm{sc}(F)$. Any character $x : \tildeπ_1(G) \to μ_n(\mathbb{C})$ leads to an $n$-fold cover $G(F)_x$ of $G(F)$ via pushout. We define an $L$-group $^LG_x$ for this cover that is generally a non-split extension of $\textrm{Gal}(F^s/F)$ by $\hat G$. We prove a refined local Langlands correspondence for $G(F)_x$, assuming it is known for connected reductive groups with the same adjoint group as $G$. Motivation for this construction comes from considerations of Langlands' functoriality conjecture, where subgroups $\mathcal{H} \subset {^LG}$ of the $L$-group of $G$ arise that need not be $L$-groups of other reductive groups. If such a subgroup is full and intersects $\hat G$ in a connected reductive subgroup of maximal rank, we construct a natural triple $(H,x,ξ)$ consisting of a quasi-split connected reductive group $H$, a double cover $H(F)_x$, and an $L$-embedding $ξ: {^LH}_x \to {^LG}$ that is an isomorphism onto $\mathcal{H}$. We expect that genuine representations of $H(F)_x$ transfer functorially to representations of $G(F)$. In the special case of endoscopy, we show that the construction of transfer factors simplifies when the natural double cover $H(F)_x$ of the endoscopic group is used. The transfer factor becomes the product of two natural invariants that do not depend on auxiliary choices. One of them is closely related to Kottwitz's work on transfer factors for Lie algebras. The other one is not specific to the case of endoscopy, and will likely play a role in general functoriality questions. Our work is motivated by work of Adams and Vogan over the real numbers.
