AI 中文总结
本文针对幺正对偶问题中极小主系列表示搜索空间过大的挑战，以E₈(8)为例，通过引入互换算子极限理论证明FPP不等式，确定了特定极小主系列的幺正集结构。
AI 中文摘要
幺正对偶问题的挑战之一是需要考虑的可能表示数量极为庞大。本文描述了一种用于缩小极小主系列表示搜索空间的方法，该方法基于“基本平行六面体”（简称FPP），即基本权的线性组合构成的集合，其系数属于区间[0,1]。作者此前提出的一个猜想（该猜想源于Barbasch的工作，后被Vogan大幅推广，最近由Davis和Mason-Brown证明）断言，FPP包含所有支配无穷小特征，对于这些特征，极小主系列具有幺正可约商。我们提出一种技术，用于在特定例子中证明该“FPP不等式”，这里以分裂实形式E₈(8)为例，通过引入当ν沿基本权方向趋于无穷时的互换算子极限理论。该极限理论受Wilfried Schmid关于霍奇结构变分的工作启发。作为应用，我们证明某一极小主系列的幺正集由单个开房的闭包组成。
英文摘要
One of the challenges of the unitary dual problem is the daunting number of possible representations to consider. This article describes an approach to narrowing the search space for minimal principal series representations, in terms of the ``fundamental parallelepiped'' (or ``FPP''): the set of linear combinations of fundamental weights with coefficients in the interval $[0,1]$. An earlier conjecture of the author, which was rooted in work of Barbasch and then subsequently vastly generalized by Vogan (and recently proven by Davis and Mason-Brown), asserts that the FPP houses all dominant infinitesimal characters for which minimal principal series have a unitarizable quotient. We present a technique to prove this ``FPP inequality'' in specific examples, demonstrated here for the split real form $E_{8(8)}$, by introducing a limiting theory of intertwining operators as $ν$ approaches $\infty$ in directions of fundamental weights. This limiting theory is inspired by Wilfried Schmid's work on variation of Hodge structure. As an application, we show that the unitary set for a particular minimal principal series consists of the closure of a single open alcove.
Comments23 pages
