Dimension Data, Local and Global Conjugacy in Reductive Groups

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dc.creatorWang, Song
dc.date2007-07-02
dc.date2007-07-21
dc.date.accessioned2026-07-07T08:19:19Z
dc.date.available2026-07-07T08:19:19Z
dc.descriptionLet 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}.
dc.description(July 21st): Slight revision
dc.identifierhttps://arxiv.org/abs/0707.0144
dc.identifierhttp://arxiv.org/abs/0707.0144
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/134724
dc.subjectRepresentation Theory
dc.subjectNumber Theory
dc.subject11F99
dc.titleDimension Data, Local and Global Conjugacy in Reductive Groups
dc.typetext

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