Orbits on flag varieties and moment maps

在数学中，对称性通常用群作用来描述。 如果对称群很大，那么对称性就被认为是大而“美丽”的。 在这个意义上，最美的对象之一是同质的变体。 如果对称群是一个约化代数群，且齐性空间是紧的，则称它为旗簇。 在本研究中，我们利用矩图来研究旗型品种的各种轨道，主要结果如下。 (i)我们构造了许多有限型的双旗簇，在很好的情况下，我们成功地对这类簇进行了分类。 (ii)对于一个真实的约化群的不可约酉表示，可以考虑一个相关簇。 定义了幂零轨道之间的余维1连通性，并构造了相应簇在余维1中最大连通的不可约表示。

In mathematics, symmetry is often described by group actions. If the symmetry group is large, the symmetry is considered to be big and "beautiful". One of the most beautiful object in that sense is a homogeneous variety. If the symmetry group is a reductive algebraic group and the homogeneous space is compact, it is called a flag variety. In this research, we studied various orbits on flag varieties using moment maps.Main results are the followings. (i) We constructed many double flag varieties of finite type, and in good cases, we succeeded in classification of such varieties. (ii) For an irreducible unitary representation of a real reductive group, one can consider an associated variety. We defined codimension one connectedness among nilpotent orbits, and constructed irreducible representations whose assocaited varieties are maximally connected in cosimension one.