关于单李代数的Coxeter-Killing变换

<正> 决定复单李群(代数)的Betti数是个经典的问题.大家知道,它们就是该李群(代数)的Poincare多项式

ON COXETER ELEMENTS OF SIMPLE LIE ALGEBRAS

It is known that a Coxeter element w of the Weyl group W of a simple Lie algebrag has the following elegant property: If w is a Coxeter element of W, then itseigenvalues are just e~mj(2πi/h) (j = 1,2,…,l), where m_j(j= 1, 2,…, l) are the Poin care exponents of g,and h = 1 + o, where o is the order of the highest root of g.This note aims at proving the followingTheorem: w ∈W is a Coxeter element of W if and only if it satisfies: (1) 1 is not the eigenvalue of w, and, (2) the number of the cycles of w is at most l.Using similar method, we also give a simple proof of the two important results due to B. Kostant.