Conical Vectors in Induced Modules

Abstract

Let $\mathfrak {g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n}$, and let $\mathfrak {m}$ be the centralizer of $\mathfrak {a}$ in $\mathfrak {k}$. A conical vector in a $\mathfrak {g}$-module is defined to be a nonzero $\mathfrak {m} \oplus \mathfrak {n}$-invariant vector. The $\mathfrak {g}$-modules which are algebraically induced from one-dimensional $(\mathfrak {m} \oplus \mathfrak {a} \oplus \mathfrak {n})$-modules on which the action of $\mathfrak {m}$ is trivial have “canonical generators” which are conical vectors. In this paper, all the conical vectors in these $\mathfrak {g}$-modules are found, in the special case $\dim \mathfrak {a} = 1$. 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.