Trilinear forms of gl2
Trilinear forms of gl2
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gl2 的三线性形式
DOI：
10.2140/pjm.2001.197.119
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发表时间：
2001
影响因子：
0.6
通讯作者：
Hung Yean Loke
中科院分区：
文献类型：
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作者：
Hung Yean Loke
1.1.本文研究了GL ~ 2（F）的三个主级数表示张量积的不变线性形式，其中F是阿基米德域。当F是p-adic域时，通过Prasad [Pa 1]的工作，已知存在不变三线性形式。他表明，空间的不变形式是最多一维，它存在的，当且仅当某个因子是1。他的工作部分是由[Re]驱动的。他还考虑了F = R的情况，我们将在下面更详细地描述他的结果。设F = R或C，G = GL ~ 2（F），K是极大紧子群.设g为其李代数。设πi（i = 1，2，3）是G的不可约无限维Harish-Chandra模.假设πi的三个中心特征标的乘积是平凡的。若F = R，则πi是主级数或离散级数表示。设H是R上的四元数除代数，我们用U2来标识它的非零元素子集H。如果πi是一个离散级数，我们记π′ i为H ∞的不可约有限维表示，它具有与πi相同的无穷小特征标和中心特征标。当F = C时，πi总是在主级数中。我们记得πi对应于F的Weil群WF的一个表示σi。对于F的一个非平凡特征标σ和WF的一个表示σ，我们将一个σ因子[JL]
1.1. In this paper we study the invariant linear forms on the tensor products of three principal series representations of GL2(F ) where F is an archimedean field. When F is a p-adic field, the existence of invariant trilinear forms is known through the work of Prasad [Pa1]. He shows that the space of invariant forms is at most one dimensional and that it exists if and only if a certain epsilon factor is 1. His work was partly motivated by [Re]. He also considers the case when F = R and we will describe his result in more detail below. Let F = R or C and let G = GL2(F ) with maximal compact subgroup K. Let g be its Lie algebra. Let πi (i = 1, 2, 3) be an irreducible infinite dimensional Harish-Chandra module of G. Assume that the product of three central characters of πi is trivial. If F = R then πi is either a principal series or discrete series representation. Let H be the quaternion division algebra over R and we identify its subset of non-zero elements H∗ with U2. If πi is a discrete series, we denote π′ i to be the irreducible finite dimensional representation of H ∗ with the same infinitesimal character and central character as πi. When F = C, πi is always in the principal series. We recall that πi corresponds to a representation σi of the Weil group WF of F . For a non-trivial character ψ of F and a representation σ of WF , we associate an epsilon factor [JL]