Theta dichotomy for unitary groups
Theta dichotomy for unitary groups
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DOI：
10.1090/s0894-0347-96-00198-1
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发表时间：
1996
影响因子：
3.9
通讯作者：
M. Harris;S. Kudla;W. Sweet
中科院分区：
文献类型：
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作者：
M. Harris;S. Kudla;W. Sweet
Gross和Prasad[14]最近的一些工作表明，局部域F的Weil-Deligne群的某些辛表示的根数控制着F上正交群表示的某些分支规则。在全局水平上，这一局部现象应该对某些L感兴趣的算术函数对称中心值的结构产生影响[12]。在某种程度上，这个猜想是基于Tunnell[47]和Waldspurger[49]的经典工作以及三重乘积L函数[13，34，17]的情况。在所有这些例子中，局部根数检测特定类型的不变线性泛函的存在。对于厄米特形式或四元数-厄米特形式的等距群，可以建立类似的猜想。事实证明，根数在局部和全局的theta对应中也起到了作用。粗略地说，某些局部根数应该控制在相同大小的组之间的局部theta对应中的表示的出现，例如，对于形式为(Sp(N)，O(2n+1)和(U(N)，U(N))的对偶)。如将看到的，这种对的theta对应与(S，χ)中在么正轴上的点S=0处的某个诱导表示相关联。相反，Prasad[35]讨论的对(Sp(N)，O(2n))和(Sp(N)，O(2n+2))的对应关系与点±12处类似的诱导表示的行为有关，在这种情况下不会产生epsilon因子。本文考虑非阿基米德情形下酉群的局部theta对应。设F是特征不等于2的非阿基米德局部域，E是F的二次扩张。有关更多注释，请参阅本简介末尾的注释部分。设V和W是分别具有厄米特形式(，)：V×V−→E和斜厄米特形式<，>：W×W−→E的m和n维E向量空间，则V和W的等距群G(V)和G(W)在辛群Sp(W)中形成对偶还原对，其中W=V⊗E，被视为2Mn维赋有辛形式的F向量空间
Some recent work of Gross and Prasad [14] suggests that the root numbers attached to certain symplectic representations of the Weil-Deligne group of a local field F control certain branching rules for representations of orthogonal groups over F . On a global level, this local phenomenon should have implications for the structure of the value at the center of symmetry for certain L-functions of arithmetic interest [12]. This conjectural picture is based, to some extent, on the now classic work of Tunnell [47] and Waldspurger [49] as well as on the case of the triple product L-function [13, 34, 17]. In all of these examples, the local root number detects the existence of a certain type of invariant linear functional. It is possible to set up analogous conjectures for the isometry groups of Hermitian or quaternion-hermitian forms. It turns out that root numbers also play a role in the local and global theta correspondence. Roughly speaking, certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the ‘same size’, e.g., for dual pairs of the form (Sp(n), O(2n+1)) and (U(n), U(n)). As will be seen, the theta correspondence for such pairs is connected with a certain induced representation In(s, χ) at the point s = 0 on the unitary axis. By contrast, the correspondence for the pairs (Sp(n), O(2n)) and (Sp(n), O(2n+ 2)), discussed by Prasad [35], is connected to the behavior of a similar induced representation at the points ± 12 , in which case no epsilon factor arises. In this paper we consider the local theta correspondence for unitary groups in the non-archimedean case. Let F be a non-archimedean local field of characteristic not equal to 2, and let E be a quadratic extension of F . For further notation, see the notation section at the end of this introduction. Let V and W be E vector spaces of dimensions m and n equipped, respectively, with a Hermitian form ( , ) : V × V −→ E and a skew-Hermitian form 〈 , 〉 : W ×W −→ E. Then the isometry groups G(V ) and G(W ) of the spaces V and W form a dual reductive pair in the symplectic group Sp(W), where W = V ⊗E W , viewed as an F vector space of dimension 2mn and equipped with the symplectic form