Integral representation of Langlands L functions

PI计划继续他的工作(与David Ginzburg合作)，通过研究出现在例外群中的几个新例子来研究朗兰兹L函数的积分表示。其中包括GL(5)的伴随L函数的积分表示，以及所谓的倍增方法在例外群G2上的推广。朗兰兹L函数是在数论和表示论的交叉点上，出现在自同构形学科中的单复变函数。最简单的例子是Riemann Zeta函数，它在估计小于给定数的素数时起着至关重要的作用。第二个最简单的例子是狄里克莱特L函数，它们在估计算术级数中素数的分布方面扮演着类似的角色。其他例子是使用模形式和椭圆曲线定义的，导致函数的性质与这些经典例子几乎相同。上个世纪出现了从表示论的角度研究模形式和椭圆曲线的可能性，表示论本质上是研究作用于向量空间的线性算子的集合。与此同时，在表示论中自然而然地产生了一些与上面讨论的经典数论函数具有几乎相同性质的复函数。这些都是朗兰德L的函数。研究这些函数的主要策略是通过积分表示。这意味着，简单地说，一个人试图取一个两个变量的函数，其中一个是复数，然后将它积分到另一个上。结果是一个复变量的函数，人们希望通过把一切都安排好，可以把它安排成朗兰兹L函数。到目前为止，没有人确切地知道需要什么才能解决这个问题，尽管已经提出了一些有用的启发式方法。PI希望继续研究这一耐人寻味的现象。

The PI plans to continue his work (joint with David Ginzburg) on integral representation of Langlands L functions with the study of several new examples which appear in exceptional groups. These include an integral representation of the adjoint L function of GL(5), and an extension of the so-called doubling method to the exceptional group G2. Langlands L functions are certain functions of a single complex variable which appear in the subject of automorphic forms, at the intersection of number theory and representation theory. The simplest example is the Riemann zeta function, which plays a crucial role in estimating the number of prime integers less than a given number. The next simplest examples are the Dirichlet L functions, which play a similar role in estimating the distribution of primes in an arithmetic progression. Other examples were defined using modular forms and elliptic curves, resulting in functions with properties nearly identical to these classical examples. In the last century it emerged that it was possible to study modular forms and elliptic curves from the point of view of representation theory, which is essentially the study of collections of linear operators acting on a vector space. At the same time, certain complex functions with nearly the same properties as the classical number theoretic functions discussed above arose naturally in representation theory. These are Langlands L functions. The main strategy for studying these functions is via integral representations. This means, simply, that one tries to take a function of two variables, one of which is complex, and integrate it over the other one. The result is a function of one complex variable, and one hopes that by setting everything up right one can arrange for it to be a Langlands L function. To date, no one really knows exactly what it takes for this to work out, although a number of useful heuristics have been suggested. The PI wishes to continue his study of this intriguing phenomenon.