Geometry and Invariant Theory in Modular Lie Theory

李论是纯数学的一个分支，它起源于挪威数学家索夫斯·李的工作。他发明了群概念的无限小类比：李代数的概念。事实上，有一类群我们可以将这样的李代数与之联系起来，这些群被称为李群。基本上，这些群具有与群结构相容的可微流形结构。用“代数变”代替“可微流形”，就得到了代数群的概念。这个概念在具有任意特征的代数闭域上是有意义的。李论是研究李群、代数群、李代数及其作用的数学领域。这些行为揭示了作用的李代数，李群或代数群的结构，但也揭示了它们作用的对象的结构。在后一种情况下，我们可以说动作是考虑到物体对称性的数学精确方法。李论已经成为数学中最核心的分支之一，与数学物理、微分方程、表示理论、环理论、代数和微分几何、组合学和数论有着密切的联系。这是一个非常活跃的研究领域，许多大数学家都做出了贡献。这一建议着重于关于约化群的作用的某些问题，主要是在素数特征域上（这就是标题中“模”一词所指的）。约化群是代数群中非常重要的一类。它们是根据根系来分类的。最基本的例子是由可逆矩阵nxn组成的一般线性群GL_n。模李理论有其自身的内在美，但也很重要，因为它与普通李理论通过约简模p的关系。通过约简模p证明了特征0的几个结果。同样值得注意的是，素数特征领域在编码理论和密码学中有重要的应用，而且它们更容易被计算机处理。模李论中最大的问题之一是吕兹提格猜想，它预测了可约群的线性作用的不可约成分是什么。还有许多其他相关的基本问题，其中一些问题在建议中得到了解决。该建议的主要思想之一是，在存在像Lusztig猜想这样的问题时，应该研究许多相关的典型特征现象，特别是最基本的特征现象。在本文的研究中，我计划研究三个问题，这些问题将在几何和不变理论以及素特征约化群的表示理论中提供新的见解。

Lie Theory is a branch of pure mathematics which has its roots in the work of the Norwegian mathematician Sophus Lie. He invented the infinitesimal analogue of the notion of a group: the notion of a Lie algebra. In fact there is a class of groups to which we can canonically associate such a Lie algebra, these are called Lie groups. Basically, these are groups endowed with the structure of a differentiable manifold which is compatible with the group structure. If one replaces "differentiable manifold" by "algebraic variety", then one obtains the notion of an algebraic group. This notion makes sense over an algebraically closed field of arbitrary characteristic. Lie theory can be described as the area of mathematics which studies Lie groups, algebraic groups, Lie algebras and their actions. These actions reveal something of the structure of the acting Lie algebra, Lie group or algebraic group, but also something of the structure of the object on which they act. In the latter case one can say that the action is a mathematically precise way to take the symmetry of the object into account. Lie theory has established itself as one of the most central branches of mathematics with strong links to mathematical physics, differential equations, representation theory, ring theory, algebraic and differential geometry, combinatorics and number theory. It is a very active area of research to which many big mathematicians have contributed. This proposal focuses on certain problems concerning actions of reductive groups, mainly over fields of prime characteristic (that is what the word "modular" in the title refers to). Reductive groups are a very important class of algebraic groups. They have been classified by means of root systems. The most basic example which is also very important in this proposal is the general linear group GL_n which consists of the invertible nxn matrices. Modular Lie theory has its own intrinsic beauty, but is also important because of its relation with ordinary Lie theory via reduction mod p. Several results in characteristic 0 are proved via reduction mod p. It is also worth noting that fields of prime characteristic have important applications in coding theory and cryptography and that they are easier for the computer to handle. One of the biggest problems in modular Lie theory is Lusztig's Conjecture which predicts what the irreducible constituents are of the linear actions of reductive groups. There are many other related fundamental problems, some of which are addressed in the proposal. One of the main ideas of the proposal is that in the presence of problems like Lusztig's Conjecture many related typical characteristic p phenomena should be studied, especially the most elementary ones. In the proposed research I plan to investigate three problems which should give new insights in the the geometry and invariant theory and also in the representation theory of reductive groups in prime characteristic.