Modelling classical types: Algebraic group actions via algebras with symmetries

设G是一个单经典复李群，其李代数为g.设G中的一个标准抛物子群P，考虑它在G中幂零矩阵的幂零锥N上的共轭作用，该作用仅限于N中幂零度为x的元素n的某些簇，例如簇N（x），其中x为整数（即n^x=0）和P的李代数中的幂零矩阵簇N_P的群作用，我们想了解这些群作用。在类型A中，不失一般性地，当G=GL_n（对于G=SL_n，设置是相同的），我们可以将群作用转化为表示论：对于每个作用，我们找到轨道与具有关系的群的表示的某些同构类之间的双射。由于这些双射是由相关的纤维丛给出的，因此轨道闭包关系和余维都是保持的。我们已经发现了辛和正交G的类似平移：在这些情况下，存在到具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，到具有自对偶的代数的表示的同构类的双射。关于前面提到的作用，根据G的类型，我们想要达到三个主要目标：I有限性准则，列出了所有的情况下，其中所描述的群作用只允许有限数量的轨道。II为了详细地理解有限的情况下，例如，当参数化的轨道组合对象，通过描述退化，通过寻找奇异点的轨道闭包，并通过计算的决议的奇异性和交叉上同调。III在无限的情况下，我们打算描述通用的规范形式，我们定义和理解半不变，产生抛物半不变环。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行为，我们必须扩展对称表示理论，因此，这个项目的基本焦点就在于这个领域。一般线性群的情况提供了许多线索，说明这种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阶不是由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.