Modelling classical types: Algebraic group actions via algebras with symmetries

设G是具有李代数g的简单经典复李群。我们在G中固定一个标准抛物子群P，并考虑它在相同大小的幂零矩阵的幂零锥N上的共轭作用。这个作用仅限于N中的某些变种，例如，x是整数的变种N(X)，幂零度为x的元素n的变种(即，n^x=0)，以及P的李代数中的幂零矩阵的变种N_P。我们想了解这些群作用。在类型A中，当不损失通用性=G_n(对于G=SL_n，设置相同)时，我们可以将群动作转化为表示理论：对于每个动作，我们找到了具有关系的箭图的轨道和某些同构表示类之间的双射。由于这些双射是由相关的纤维丛给出的，轨道闭合关系和余维保持不变。我们发现了辛和正交G的一个类似的平移：在这些情况下，有关系的对称箭图的对称表示的某些对称同构类的双射，即具有自对偶的代数的同构表示类的双射。考虑到前面提到的作用，取决于G的类型，我们想要达到三个主要目标：一是列出所描述的群作用只允许有限数目的轨道的所有情况的有限准则；二是详细地理解有限的情况，例如，通过描述退化的组合对象来参数化轨道时，通过寻找轨道闭包中的奇点，通过计算奇点和交上同调的分解。III在无限情况下，我们意图描述一般范式，我们用它来定义和理解产生抛物线半不变环的半不变量。我们将计算显式商和等变上同调。对于A型，有许多已知的表示论结果可以用来检验上述目标。为了理解其他经典类型的作用，我们必须扩展对称表示理论，因此，本项目的基本焦点在于这一领域。一般线性群的情况提供了许多线索，这样的展开可能是有用的和可能的；但我们也知道，新的现象是可以预期的。例如，我们证明了偶正交情形下的退化序不是由A型诱导的。

Let G be a simple classical complex Lie group with Lie algebra g. We fix a standard parabolic subgroup P in G and consider its conjugation action on the nilpotent cone N of nilpotent matrices in g of the same size.This action restricts to certain varieties in N, for example to the variety N(x), where x is an integer, of elements n of nilpotency degree x (that is, n^x=0) and to the variety N_P of nilpotent matrices in the Lie algebra of P. We want to understand these group actions.In type A, when without loss of generality G=GL_n (for G=SL_n, the setup is the same), we can translate the group actions to Representation Theory: For each action, we find a bijection between the orbits and certain isomorphism classes of representations of a quiver with relations. Since these bijections are given by associated fibre bundles, orbit closure relations and codimensions are preserved.We have found a similar translation for symplectic and orthogonal G: in these cases there are bijections to certain symmetric isomorphism classes of symmetric representations of symmetric quivers with relations, that is, to isomorphism classes of representations of algebras with self-duality.Concerning the actions mentioned before, depending on the type of G, we want to reach three main goals:I A finiteness criterion which lists all cases in which the described group action only admits a finite number of orbits.II To understand the finite cases in detail, for example while parametrizing the orbits by combinatorial objects, by describing degenerations, by finding singularities in the orbit closures, and by calculation of resolutions of singularities and intersection cohomology.III In the infinite cases, we intent to describe generic normal forms with which we define and understand semi-invariants which generate the parabolic semi-invariant rings. Explicit quotients and equivariant cohomology will be calculated.For type A, there are many known representation-theoretic results which we can use for the examination of the mentioned goals. To understand the actions for the other classical types, we have to extend the symmetric representation theory and the basic focus of this project, thus, lies in this area. The case of the general linear group provides many clues in which way such expansion might be useful and possible; but we also know that new phenomena are to be expected. For example, we proved that the degeneration order in the even orthogonal case is not induced by type A.