There is both theoretical and numerical evidence that the set of irreducible representations of a reductive group over local or finite fields is naturally partitioned into families according to analytic properties of representations. Examples of such properties are the rate of decay at infinity of “matrix coefficients” in the local field setting, and the order of magnitude of “character ratios” in the finite field situation.In these notes we describe known results, new results, and conjectures in the theory of “size” of representations of classical groups over finite fields (when correctly stated, most of them hold also in the local field setting), whose ultimate goal is to classify the above mentioned families of representations and accordingly to estimate the relevant analytic properties of each family.Specifically, we treat two main issues: the first is the introduction of a rigorous definition of a notion of size for representations of classical groups, and the second issue is a method to construct and obtain information on each family of representation of a given size.In particular, we propose several compatible notions of size that we call U-rank, tensor rank and asymptotic rank, and we develop a method called eta correspondence to construct the families of representation of each given rank.Rank suggests a new way to organize the representations of classical groups over finite and local fields—a way in which the building blocks are the “smallest” representations. This is in contrast to Harish-Chandra’s philosophy of cusp forms that is the main organizational principle since the 60s, and in it the building blocks are the cuspidal representations which are, in some sense, the “largest”. The philosophy of cusp forms is well adapted to establishing the Plancherel formula for reductive groups over local fields, and led to Lusztig’s classification of the irreducible representations of such groups over finite fields. However, the understanding of certain analytic properties, such as those mentioned above, seems to require a different approach.

中文翻译：

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表征理论中的等级和对偶

从理论和数值上都有证据表明，根据表示的解析性质，在局部或有限域上的还原性组的不可约表示的集合自然分为几类。这样的属性的示例是局部场设置中“矩阵系数”的无穷大处的衰减率，以及有限场情况下“字符比”的量级。在这些注释中，我们描述了已知结果，新结果以及古典领域在有限域上的表示的“大小”理论中的猜想（正确陈述时，它们中的大多数也都在局部场设置中），其最终目标是对上述表示族进行分类，并据此估计每个家庭的相关分析属性。具体来说，我们处理两个主要问题：U秩，张量秩和渐近秩，我们开发了一种称为eta对应的方法Rank提出了一种在有限域和局部域上组织经典群体表示的新方法，即构建基块是“最小”表示的一种方法。这与哈里什·钱德拉（Harish-Chandra）的尖尖形式哲学形成鲜明对比，尖尖形式哲学是60年代以来的主要组织原则，在其中，构建块是尖锐的代表，从某种意义上说，它们是“最大”的代表。尖峰形式的哲学非常适合于建立Plancherel公式用于局部场上的归约组，并导致Lusztig对此类域在有限域上的不可约表示进行分类。但是，对某些分析属性（例如上面提到的那些属性）的理解似乎需要不同的方法。