The construction of the Kac Moody generalizations of the Steinberg groups has not included the infinite dimensional versions of the great Lie algebras. This is because the construction has assumed the existence of representations with dominant integral highest weights, an assumption which does not hold for those Lie algebras. An extension of the definition of the Weyl group to the adjoint of the Cartan subalgebra permits the construction of the Steinberg groups with Tits systems for that case.
1997 Academic Press
The theory of Kac Moody Lie algebras (now sometimes called Kac Moody algebras) originated with the work of Wonenburger and her students, Berman, Marcuson, and Moody [4,5,12,15], and independently with that of Kac [6]. The importance of the subject grew after Macdonald [11] pointed out connections with Dedekind's '-function. In the ensuing decades, hundreds of papers and several books have appeared on Kac Moody theory. This interest stems from both the depth of the results [7,8,22] and the breadth of the applications [3,10,14].
From the start, the groups generalizing Chevalley and Steinberg groups received their share of attention. Moody and his student Teo were the first to study the Kac Moody versions of the Chevalley (adjoint) groups [16], and Marcuson initiated the theory of Kac Moody Steinberg (nonadjoint) groups [13]. The thrust of this research was to show the existence of Tits systems, [21], or B-N pairs, in the groups. Peterson and Kac [18] and Morita [17] have also considered the Kac Moody Steinberg groups. Interest in these groups has continued up until the present day [1,2,8,9,14,19,22].
In the construction of the Kac Moody versions of Steinberg's groups, it was assumed that the highest weight of the module on which the group acts was the dominant integral. We show that the Kac Moody Lie algebras which generalize the great Lie algebras A n , B n , C n , and D n admit no representation with dominant integral highest weight. This necessitates an extension of the action of the Weyl group to the Cartan subalgebra. The article no. AI971674 458