STRUCTURES OF WEYL GROUPS OF REAL SEMISIMPLE LIE ALGEBRAS

<正> Let g be a real semisimple Lie algebra, h a Cartan subalgebra of g, gc and hc the respective complexifications of g and h, and a the conjugation of g with respect to g. Denote by W（hc） the Weyl group of gc acting on hc, and write Wσ（h） to be the subgoup of W（hc） with elements leaving g invariant, and then W（h） indicates the group of restrictions to h of inner automorphisms of g leaving h invariant. W（h） and Wσ（h） can be called the Weyl group and the quasi-Weyl group of g with respect to h respectively. In this paper, we give a clear expression to the strueture of groups W（h） and Wσ（h） for every Cartan subalgebra h of g （see Theorem 5）. The above two groups are calculated in detail for every class of the Cartan subalgebras of the classical simple Lie algebra g （see Table 1）.