Symplectic-orthogonal theta lifts of generic discrete series
Symplectic-orthogonal theta lifts of generic discrete series
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通用离散级数的辛正交 theta 提升
DOI：
10.1215/s0012-7094-00-10128-7
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发表时间：
2000
影响因子：
2.5
通讯作者：
Gordan Savin
中科院分区：
文献类型：
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作者：
G. Muić;Gordan Savin
0.导论.设F是特征为零的非阿基米德局部域。本文研究了辛群Sp（n，F）与特殊偶正交分裂群SO（2 r，F）的表示之间的对应关系，其中r ≥ 2.设ωn，r是Sp（2nr，F）的Weil表示，它附加在F的一个非平凡可加特征标F上.我们证明了将Weil表示ωn，r限制为Sp（n，F）×SO（2 r，F）所产生的对应对于一般平方可积表示是函子的.更精确地说，设T是Sp（n，F）的光滑不可约表示。设2（T，r）是ωn，r的最大T-同构商。满足2（T，r）6= 0的最小r称为T的首现指数。现在假设T是一个类属离散级数。(See（1.1）为定义。设L（s，T）为标准L函数，如[Sh 1]中所示。然后我们得到以下结果。如果L（0，T）=∞，则第一出现指数为n。设τ ′是2（T，n）的不可约商.则τ ′是SO（2n，F）的一个广义离散级数表示，对GL（m，F）的任意离散级数表示δ（m任意），有L（s，δ×T）= L（s，δ）L（s，δ×τ ′）.如果L（0，T）6= ∞，则第一个出现指数是n+1。则2（T，n+1）有唯一的不可约′-通有商τ ′.此外，τ ′是SO（2n+2，F）的离散级数表示，对GL（m，F）的任意离散级数表示δ（m任意），有L（s，δ×τ ′）= L（s，δ）L（s，δ×T）.对于SO（2n，F）的′-一般离散级数，我们也有类似的结果.我们请读者参阅第2节的精确陈述。我们的结果有一个理论解释如下。考虑对偶群SO（2n，C）<$SO（2n+1，C）<$SO（2n+2，C）的包含。设W ′（F）是F的Weil-Deligne群. T的代数朗兰兹参数是一个容许同态（参见[Bo]）φ：W ′（F）−→ SO（2n+1，C）。
0. Introduction. Let F be a non-Archimedean local field of characteristic zero. In this paper we study a correspondence between representations of symplectic groups Sp(n,F ) and special even-orthogonal split groups SO(2r,F ), where r ≥ 2. Letωn,r be the Weil representation of Sp(2nr,F ) attached to a nontrivial additive characterψF of F . We show that the correspondence arising by restricting the Weil representationωn,r to Sp(n,F )×SO(2r,F ) is functorial for generic square integrable representations. More precisely, let T be a smooth, irreducible representation of Sp(n,F ). Let 2(T, r) be the maximal T-isotypic quotient of ωn,r . The smallest r such that2(T, r) 6= 0 is called the first occurrence index of T. Now assume that T is a ψ-generic discrete series. (See (1.1) for the definition of ψ .) Let L(s,T) be the standard Lfunction attached to T as in [Sh1]. Then we have the following results. IfL(0,T)=∞, then the first occurrence index is n. Let τ ′ be an irreducible quotient of 2(T,n). Then τ ′ is a ψ ′-generic discrete series representation of SO(2n,F ), and for any discrete series representation δ of GL(m,F ) (m arbitrary), we have L(s,δ×T)= L(s,δ)L(s,δ×τ ′). If L(0,T) 6= ∞, then the first occurrence index is n+1. Then 2(T,n+1) has the unique irreducible ψ ′-generic quotient τ ′. Furthermore, τ ′ is a discrete series representation of SO(2n+2,F ), and for any discrete series representation δ of GL(m,F ) (m arbitrary), we have L(s,δ×τ ′)= L(s,δ)L(s,δ×T). We also have analogous results for ψ ′-generic discrete series of SO(2n,F ). We refer the reader to Section 2 for precise statements. Our results have a conjectural interpretation as follows. Consider inclusions of dual groups SO(2n,C)⊂ SO(2n+1,C)⊂ SO(2n+2,C). Let W ′(F ) be the Weil-Deligne group of F . The conjectural Langlands parameter of T is an admissible homomorphism (see [Bo]) φ :W ′(F )−→ SO(2n+1,C).
DOI：
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发表时间：
2007
期刊：
影响因子：
--
作者：
Senda;A.;Koyama;K.;Murakami;H.;Suzaku GC-Team;Ji-Yeon Seok;Hiroshi Yamashita
通讯作者：
Hiroshi Yamashita