Geometry and Invariant Theory in Modular Lie Theory

李理论是纯数学的一个分支，它起源于挪威数学家索菲斯·李的工作。他发明了无穷小类似的概念，一组：概念的李代数。事实上，有一类群，我们可以规范地关联这样的李代数，这些被称为李群。基本上，这些是具有与群结构相容的可微流形结构的群。如果用“代数簇”代替“可微流形”，那么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。李理论可以被描述为研究李群，代数群，李代数及其作用的数学领域。这些作用揭示了作用李代数、李群或代数群的某些结构，也揭示了它们作用于其上的对象的某些结构。在后一种情况下，我们可以说，这种作用是一种数学上精确的方式，以考虑物体的对称性。李理论已经确立了自己作为一个最中心的分支数学与强大的联系，数学物理，微分方程，表示论，环理论，代数和微分几何，组合数学和数论。这是一个非常活跃的研究领域，许多大数学家作出了贡献。这个建议集中在某些问题上的行动，约化集团，主要是在领域的主要特点（这是什么词“模块”在标题中所指的）。约化群是一类非常重要的代数群。它们是通过根系来分类的。最基本的例子是由可逆n × n矩阵组成的一般线性群GL_n，这在这个建议中也是非常重要的。模李理论有其自身的内在美，但也是重要的，因为它与普通李理论通过减少模p.几个结果在特征0证明通过减少模p.这也是值得注意的是，领域的主要特征有重要的应用在编码理论和密码学，他们更容易为计算机处理。模李理论中最大的问题之一是Lusztig猜想，它预测了约化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，其中一些在提案中得到了解决。 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p现象，特别是最基本的p现象。在拟议的研究中，我计划调查三个问题，这应该给新的见解的几何和不变理论，也在表示理论的约化群的主要特征。

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.