Gelfand Pairs and Automorphic Representations

[9619766 . Jacquet]该项目的目标是研究某些周期积分。设G是数域F上的约化群；设H是G的一个子群；我们假设在F的代数闭包上，群H在G的标志簇上有一个开轨道。如果F是G上的自同构形式，而t是H的一个合适的特征，则相应的周期积分是F乘以t对H的幂商的积分。在许多情况下，我们推测在F的每一处，三重体（G，H,t）满足Gelfand多重性1条件。进一步，我们推测周期积分与自同构l函数在对称中心的值有关。例如，l函数的值本质上可能是周期积分的平方。在其他情况下，从朗兰兹程序的角度来看，允许一个积分为非零的向量f的自同构表示是特别有趣的。在本文中，研究周期积分的工具是轨迹公式的一种变体，即相对轨迹公式。l函数在对称中心的值会有一些应用。在某些情况下，有可能证明该值是非负的。其他结果将更直接地与Langlands程序相关，并将提供有关l包的功能原理或结构的信息。一般来说，这些信息不能从标准迹公式或对偶对理论中得到。局部谐波分析中出现的问题可能会引起独立的兴趣。一个显著的例子是某些Kloosterman积分（或和）之间的推测同一性。这个建议涉及朗兰兹纲领，而朗兰兹纲领是数论的一部分。数论是对整数性质的研究，是数学中最古老的分支。从一开始，数论中的问题就为在该学科的其他分支中创造新的数学提供了动力。朗兰程序是一种将数论与微积分联系起来的普遍哲学；它体现了研究整数的现代方法。现代数论是非常技术性和深奥的，但它在理论计算机科学和编码理论等领域有着惊人的应用。

9619766 Jacquet The goal of the project is to study certain period integrals. Let G be a reductive group over a number field F; let H be a subgroup of G; we assume that over the algebraic closure of F the group H has an open orbit in the flag variety of G. If f is an automorphic form on G and t a suitable character of H the corresponding period integral is the integral of f times t taken over the adelic quotient of H. In many cases, it is conjectured that at every place of F the triple (G,H,t) satisfies Gelfand multiplicity one condition. Furthermore, it is conjectured that the period integral is related to the value of an automorphic L-function at its center of symmetry. For instance, the value of the L-function may be essentially the square of a period integral. In other cases, automorphic representations that admit a vector f for which the integral is non-zero are especially interesting from the point of view of the Langlands program. In this proposal, the tool for studying the period integrals is a variant of the trace formula, the relative trace formula. There will be applications to the value of the L-functions at their center of symmetry. In some cases, it might be possible to prove that this value is non-negative. Other results will be more directly related to the Langlands program, and will provide information on the principle of functoriality or the structure of the L-packets. In general, this information cannot be obtained from the standard trace formula or the theory of dual pairs. The questions in local harmonic analysis which arise may be of independent interest. A notable example is a conjectural identity between certain Kloosterman integrals (or sums). This proposal deals with the Langlands program, and the Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diver se parts of the discipline. The Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.