Purity of stable pieces in compactifications of semisimple groups

理解几何对象最有成效的技术之一是给它们附加线性不变量。对这些不变量的研究常常导致表征理论中的一个问题。这些不变量的一个重要问题是它们是否具有纯变量的性质。一个正的答案允许控制几何对象的退化。一个重要的线性不变量是De Rham上同调。对于某些在数论中起中心作用的几何对象（例如，阿贝尔变异、曲线或正特征的p可分群），它们的De Rham上同调带有所谓的截断晶体的结构。它们可以以两种方式被视为表征理论领域内的对象：作为Crawley-Boevey意义上的特定氏族的表征或作为射影线性群紧化中的稳定块。该项目的目的是研究De Rham上同是否是纯粹的，重点是第二种解释。

One of the most fruitful techniques to understand geometric objects is to attach to them linear invariants. The study of these invariants often leads to a problem within representation theory. One important question for these invariants is whether they have the property to be pure. A positive answer allows to control the degeneration of geometric objects. An important linear invariant is the De Rham cohomology. For certain geometric objects playing a central role in number theory (e.g., abelian varieties, curves, or p-divisible groups in positive characteristic) their De Rham cohomology carries the structure of a so-called truncated crystal. These can be considered in two ways as objects within the area of representation theory: as representation of certain clans in the sense of Crawley-Boevey or as stable pieces in a compactification of the projective linear group. The goal of this project is to study whether the De Rham cohomology is pure focussing on the second interpretation.