arXiv 0707.01442007-07-21math.RTmath.NT

可约群中的维数数据、局部和整体共轭

Dimension Data, Local and Global Conjugacy in Reductive Groups

Song Wang

备注：: (July 21st): Slight revision

摘要：

设G是（$\mathbb{C}$上）连通可约群，H是连通半单子群. H的维数数据（相对于它在G中的给定嵌入）是数$\{{\rm dim} V^{H}\}$的集合，其中V遍历G的所有有限维表示。根据Larsen-Pink（[L-P90]）的一个定理，维数数据确定H直到同构，如果G = GL（n）甚至直到共轭. Langlands教授提出了一个问题，即强（共轭）结果是否对任意G成立。本文给出了如下（否定的）答案：如果H是A_{4 n}，$B_{2 n} (n \geq 2)$，$C_{2 n} (n \geq 2)$，E_{6}，E_{8}，F_{4}和G_{2}型单，则（对于适当的$N$）存在H到$G = SO (2 N)$的嵌入i和i'（H）对，使得图像i（H）和i'（H）具有相同的维数但不共轭.事实上，我们已经证明i（H）和i'（H）是\emph{locally conjugate}，即，对所有半单$h \in H$，i（h）和i'（h）在G中共轭。如果假定函子性，这个结果将提供这样的G在整体域上的自守形式的重数1的失败。在不连通的情况下，特别是当H是有限的时候，这些事情是已知的，如Blasius [Blasius 94]对$SL (n) (n \geq 3)$和Gan-Gurevich-Jiang 2002（[GAN]）对G_{2}的工作。

英文摘要：

Let G be a connected reductive group (over $\mathbb{C}$) and H a connected semisimple subgroup. The dimension data of H (realative to its given embedding in G) is the collection of the numbers $\{{\rm dim} V^{H}\}$, where V runs over all the finite dimensional representations of G. By a Theorem of Larsen-Pink ([L-P90]), the dimension data determines H up to isomorphism, and if G = GL (n) even up to conjugacy. Professor Langlands raised the question as to whether the strong (conjugacy) result holds for arbitrary G. In this paper We provided the following (negative) answer: If H is simple of type A_{4 n}, $B_{2 n} (n \geq 2)$, $C_{2 n} (n \geq 2)$, E_{6}, E_{8}, F_{4} and G_{2}, then there exist (for suitable $N$) pairs of embeddings i and i' of H into $G = SO (2 N)$ such that there image i (H) and i' (H) have the same dimension data but are not conjugate. In fact we have shown that i (H) and i' (H) are \emph{locally conjugate}, i.e., that i (h) and i' (h) are conjugate in G for all semisimple $h \in H$. If one assumes functoriality, this result will furnish the failure of multiplicity one for automorphic forms on such G over global fields. Such things are known in the disconnected cases, especially when H is finite, as in the works of Blasius [Blasius94] for $SL (n) (n \geq 3)$ and Gan-Gurevich-Jiang2002 ([Gan]) for G_{2}.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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