Conical vectors in induced modules
Conical vectors in induced modules
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诱导模中的圆锥向量
DOI：
10.1090/s0002-9947-1975-0376786-1
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发表时间：
1975
期刊：
影响因子：
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通讯作者：
J. Lepowsky
中科院分区：
文献类型：
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作者：
J. Lepowsky
设9是实半单李代数，具有Iwasawa分解g=t(Da@n，m是e中a的中心化子，定义了8-模中的锥向量为非零men不变向量.由m的作用平凡的一维(m@a Dn)模代数诱导的8-模有锥向量的“标准生成元”。本文求出了这9-模的所有锥向量，在特殊情况下dim a=1。这些锥向量有有趣的表示为两个变量的多项式，这些多项式可分解为线性或二次因子。由于直接计算确定圆锥向量太困难，证明了元数学的“转移原理”，将有关圆锥向量的定理从一个李代数转移到另一个李代数，从而将问题简化为一个可解的特例。研究了特征为零的任意域上具有分裂Cartan子空间的半单对称李代数。对Kostant-Mostow双重传递性定理进行了阐述。
Let 9 be a real semisimple Lie algebra with Iwasawa decomposition g=t (Da@n, and let m be the centralizer of a in e. A conical vector in a 8-module is defined to be a nonzero me n-invariant vector. The 8-modules which are algebraically induced from one-dimensional (m @a Dn)modules on which the action of m is trivial have "canonical generators" which are conical vectors. In this paper, all the conical vectors in these 9-modules are found, in the special case dim a= 1. The conical vectors have interesting expressions as polynomials in two variables which factor into linear or quadratic factors. Because it is too difficult to determine the conical vectors by direct computation, metamathematical "transfer principles" are proved, to transfer theorems about conical vectors from one Lie algebra to another; this reduces the problem to a special case which can be solved. The whole study is carried out for semisimple symmetric Lie algebras with splitting Cartan subspaces, over arbitrary fields of characteristic zero. An exposition of the Kostant-Mostow double transitivity theorem is included.