Representations of Finite Unipotent Linear Groups by the Method of Clusters
Representations of Finite Unipotent Linear Groups by the Method of Clusters
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发表时间：
2010-04
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通讯作者：
Ning Yan
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作者：
Ning Yan
域 K 上的一般线性群 GL(n, K) 包含一个特别突出的子群 U(n, K)，由所有上三角单能元素组成。在本文中，我们感兴趣的是 K 是有限域 F_q 的情况，我们的目标是更好地理解 U(n, F_q) 的表示理论。长期以来，对该群复杂的不可约表示的完整分类一直被认为是一项艰巨的任务。基里洛夫的轨道方法以其在 K 具有特征 0 时的成功而闻名，是直觉和猜想的天然来源，但在我们的例子中，共伴轨道和复杂表示之间的关系仍然是一个谜。在这里，我们引入轨道方法的自然变体，其中核心作用是由某些共伴轨道簇发挥的。这种“簇方法”导致在 U(n, F_q) 表示环中构造一个子环，该子环结构丰富但易于理解。簇方法还具有轨道方法原理中所期望的许多主要特征。
The general linear group GL(n, K) over a field K contains a particularly prominent subgroup U(n, K), consisting of all the upper triangular unipotent elements. In this paper we are interested in the case when K is the finite field F_q, and our goal is to better understand the representation theory of U(n, F_q). The complete classification of the complex irreducible representations of this group has long been known to be a difficult task. The orbit method of Kirillov, famous for its success when K has characteristic 0, is a natural source of intuition and conjectures, but in our case the relation between coadjoint orbits and complex representations is still a mystery. Here we introduce a natural variant of the orbit method, in which the central role is played by certain clusters of coadjoint orbits. This "method of clusters" leads to the construction of a subring in the representation ring of U(n, F_q) that is rich in structure but pleasantly comprehensible. The cluster method also has many of the major features one would expect from the philosophy of orbit method.