Gelfand Pairs and Automorphic L-functions

本文的目的是研究约化群G上自同构型的某些积分。 积分是在G的子群H上的，H在G中是大的，在这个意义上，在基场的代数闭包上，群H在G的旗簇中有一个开轨道。 积分可以被认为是周期附加到自同构和原来是，在所有情况下，有趣的数字。特别地，在许多情况下，周期的平方被证明是自守L-函数的中心值。这种现象的例子是Waldspurger和其他人的结果以及Gross-Prasad的成果。在其他情况下，积分不为零的形式本身就很有趣。研究者利用迹公式的一个变体，称为相对迹公式，研究了这些周期及其性质，该建议侧重于所谓朗兰兹纲领的一个方面。该程序的目标是研究自守对象（称为自守表示）;它们可以用某些算子的谱来描述（广义傅立叶分析）。附加到这些对象的数据被编码在一个复杂变量的函数族中，即自守L函数。这些函数的约束属性包含了自同构表示的所有可能的信息。 此外，来自几何或丢番图问题的其他L-函数也应该是自守L-函数。自守L-函数在一个特定点（中心值）的值是一个特别有趣的数字，很难计算。在许多情况下，它被证明，这个数字是非负的，但这是不能直接建立。 研究人员正在使用一种新的方法（相对迹公式），通过将它们表示为积分的平方来获得有关某些数据的信息。这是一种将自守L-函数与自守表示联系起来的新方法。

The goal of the proposal is to study certain integrals of automorphicforms on a reductive group G. The integrals are over a subgroup H of Gwhich is large in G, in the sense that over the algebraic closure of theground-field, the group H has an open orbit in the flag variety of G. The integrals may be thought of being periods attached to automorphicforms and turn out to be, in all cases, interesting numbers. Inparticular, in many cases, the square of the periods are conjectured to bethe central values of automorphic L-functions. Examples of such aphenomenon are results of Waldspurger and others and conjectures ofGross-Prasad. In other cases, the forms for which the integrals arenon-zero are interesting in their own right. The investigator studiestheses periods and their properties by using a variant of the traceformula, called the relative trace formula.The proposal focuses on one aspect of the so called Langlands program. Thegoal of the program is to study the automorphic objects (calledautomorphic representations); they can be described in terms of thespectrum of certain operators (generalized Fourier analysis). The dataattached to those objects is encoded in a family of functions of onecomplex variable, the automorphic L-functions. The conjectured propertiesof these functions contain all possible information on the automorphicrepresentations. Moreover, other L-functions coming from geometry orDiophantine problems should be automorphic L-functions. The value of theautomorphic L-functions at a particular point (central value) is aespecially interesting number which is difficult to compute. In manycases, it is conjectured that this number is non-negative and yet this iscannot be established directly. The investigator is using a new method(the relative trace formula) to obtain information about some of thesenumbers by representing them as square of integrals. This a novel way toconnect the automorphic L-functions and the automorphic representations.