The study of the representation theoretical aspect of generalized flag varieties

1标量广义Verma模之间的同态我们对与极大抛物子代数相关的标量广义Verma模之间的同态进行了分类，并说明了如何利用在极大情形下构造的算子得到一些一般的算子。我们称一个抛物子代数为正规的，如果每个具有公共Levi部分的抛物子代数在某个内自同构下彼此共轭。我们已经证明了对于经典代数和“几乎一半”正规抛物子代数，上述猜想对于正则无穷小特征标是肯定的，在第一年，主要研究者给出了上述猜想的几何证明，并去掉了“经典”假设。第二年，我们研究了最大Gelfand-Krillov维数的标量广义Verma模的子模。2连续Whittaker向量空间的不可约性著名的“重数1定理”告诉我们，拟分裂真实的线性李群的不可约容许表示上的连续Whittaker向量空间的维数至多为1。对于非拟分裂群，重数1定理不成立。作为重数1定理到非拟分裂情形的自然推广，本文提出了如下猜想：连续Whittaker向量空间不可约为有限W-代数上的模，例如，我们对A型群有肯定的回答.

1 Homomorphisms between scalar generalized Verma modulesWe had classified the homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras and explained how to use the operators constructed in the maximal case to get some operators in general. We conjectures that all the homomorphisms arise in this way.We call a parabolic subalgebra normal, if each parabolic subalgebra which has a common Levi part is conjugate to each other under some inner automorphism. We had proved that for classical algebras and "almost half" of normal parabolic subalgebra, the above conjecture is affirmative for regular infinitesimal characters.In the first year, the principal investigator gave geometrical proof of the above conjecture and removed the assumption "classical". In the second year, we investigated submodules of scalar generalized Verma modules of maximal Gelfand-Krillov dimensions.2 Irreducibility of the space of continuous Whittaker vectorsThe famous "multiplicity one theorem" tells us that the dimension of the space of continuous Whittaker vectors on an irreducible admissible representation of a quasi-split real linear Lie group is at most one. For non quasi-split groups the multiplicity one theorem fails. As a natural extension of the multiplicity one theorem to non quasi-split case, I propose the following conjecture.The space of continuous Whittaker vectors is irreducible as a module over the finite W-algebra.For example, we have an affirmative answer for the type A groups.