The study of the representation theoretical aspect of generalized flag varieties

1标量广义Verma模间的同态我们对极大抛物子代数相关的标量广义Verma模间的同态进行了分类，并解释了如何利用极大情况下构造的算子来得到一般的算子。我们推测所有同态都是这样产生的。我们称一个抛物子代数为正规，如果每个抛物子代数都有一个共同的Levi部，并且在某些内自同构下彼此共轭。我们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字符是肯定的。第一年，首席研究员对上述猜想进行了几何证明，并去掉了“经典”假设。在第二年，我们研究了极大Gelfand-Krillov维的标量广义Verma模的子模连续Whittaker向量空间的不可约性著名的“复数1定理”告诉我们在拟分裂实线性李群的不可约容许表示上的连续Whittaker向量空间的维数不超过1。对于非拟分裂群，重性1定理失效。作为多重性1定理在非拟分裂情况下的自然推广，我提出了以下猜想。连续惠特克向量空间作为有限w代数上的模是不可约的。例如，对于A类群体，我们有一个肯定的答案。

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.