Explicit and l-modular theta correspondence

假设有一场盛大的婚礼：假设法国总统萨科齐和卡拉•布吕尼要结婚了。在这一生中最重要的一天到来之前，这对订婚夫妇正在准备两场派对：单身派对和单身派对。我们为《Hello》杂志工作，我们的两位记者豪和沃尔德斯伯格发现，在这两场派对上，参加的人数完全相同，而且，参加男性派对的每个男人都会和参加女性派对的一个女人约会（反之亦然）。《Hello》杂志的编辑们（亨尼亚特、哈里斯……）要求了解更多关于这场婚礼的信息。如果可能的话，我们想知道每个派对都有哪些人，谁和谁出去了。我们为什么对这种事感兴趣？答案很简单：不可能进入萨科齐的派对（由于安全问题），但我们的一名摄影师可能更容易进入母鸡的派对。如果我们看到凯特参加了单身派对，而我之前已经证明凯特和威廉出去了，那么我们就会推断出威廉参加了萨科齐的派对！但是，正如你所看到的，这是一个非常困难的问题。在不知道每个派对都有谁（凯特和威廉）的情况下，怎么能说出谁在和谁约会（凯特和威廉）呢？策略如下：首先，我们简化问题，比如金发男人和黑发女人在一起，或者学数学的男人和学哲学的女人在一起……然后我们要给每个类别的人命名，最后我们要证明谁和谁在一起。在这个类比中，每一方都是一个p进群体，其中的人是这些群体的“表征”——这些抽象的数学对象与数论（本质上是对最基本的数学对象的研究：自然数1,2,3，…）有着深刻的联系。双方之间的匹配被称为θ对应，问题是要使其明确：哪些表征出现，它们与谁配对？第一步是给每个p进群的表示命名——这是一个分类问题。对某些群体来说这比其他群体容易，这就是对应的重要性理解了一个群体的表征，我们就可以用对应来推断出另一个群体的表征信息。在过去的150年里，theta对应及其前身在数学上有着重要的应用；对它的明确理解将带来更多。

Suppose there is a big wedding: let us say that President Sarkozy of France and Carla Bruni are getting married. Before this most important day in their lives, the betrothed are preparing two parties: the hen party and the stag party. We are working for Hello magazine and two of our reporters, Howe and Waldspurger, have discovered that, at these two parties, there will be exactly the same number of people and, moreover, each man in the stag party is going out with one woman in the hen party (and vice versa).The editors of Hello magazine (Henniart, Harris,...) have asked for more information about this wedding. If possible we would like to know which people are in each party and who is going out with whom. Why are we interested in such a thing? The answer is very simple: it will be impossible to enter Sarkozy's party (due to the security) but it may be easier for one of our photographers to get into the hen party. If we see that Kate is at the hen party and I have proved before that Kate is going out with William then we will deduce that William is at Sarkozy's party!But, as you might see, this is a very difficult problem. How can one say who is going out with whom (Kate with William) without knowing who is in each party (Kate and William)? The strategy is the following: first we simplify the problem with arguments like blond-haired men are with black-haired women or men who study mathematics are with women whostudy philosophy ... Then we have to put names to people in each category and finally we have to prove who is with whom.In the analogy, each party is a p-adic group and the people in the party are ``representations'' of these groups -- these are abstract mathematical objects which have deep connections with Number Theory (which is essentially the study of the most basic mathematical objects: the natural numbers 1,2,3,...). The matching between the two parties is called the theta correspondence and the problem is to make it explicit: which representations appear and with whom are they paired? The first step is to put names to the representations of each p-adic group -- this is a classification problem. For some groups this is easier than for others, and this is the importance of the theta correspondence: having understood about the representations one group, we can use the theta correspondence to deduce information about therepresentations of the other.The theta correspondence and its predecessors have had major (mathematical) applications through the last 150 years; an explicit understanding of it will lead to many more.