Last update: 27 July 2013
In this section we review the Weyl-Kac character formula for integrable representations of Kac-Moody Lie algebras following [Kac1104219, Ch. 10].
Let be a symmetrizable Cartan matrix and let be the corresponding Kac-Moody Lie algebra, where is the Lie subalgebra generated by and is the subalgebra generated by The roots label the nonzero eigenspaces of the adjoint action of on
for The set of positive roots of is
(see [Kac1104219, (1.3.2)]). The Weyl group is the subgroup of generated by where
(see [Kac1104219, §3.7]).
Let (see [Mac1983, III, 20] or [Kac1104219, (2.5.1)])
The Weyl denominator (see [Mac1983, III, 22] or [Kac1104219, §10.2]) is
and the character of a Verma module of highest weight is
(see [Kac1104219, (9.7.2)]).
The set indexes the irreducible integrable (see [Kac1104219, Lemma 10.1]). The Weyl character of the irreducible integrable is
so that is the simple of highest weight (see [Kac1104219, §9.3 and Theorem 10.4]).
|(a)||[Mac1983, (3.20)] or [Kac1104219, (10.2.2)] If then|
|(b)||[Mac1983, (3.23)] or [Kac1104219, (10.4.4)] (The Weyl denominator formula)|
In the case of an affine Lie algebra where the elements of the affine Weyl group are identified with alcoves, identification of the affine Weyl group with alcoves, the right hand side of the expansion of in (1.7) is the sum over the hyperplanes between and
The cone is the the fundamental chamber and the Tits cone is the union of the of
(see [Kac1104219, §3.12] or [KPe0750341, §1.1, p. 138]). The complexified Tits cone and the Weyl-Kac character convergence region are
respectively (see [Kac1104219, §10.6] and [KPe0750341, §1.1, p. 140]). The importance of the sets and is that they are the sets on which the Weyl numerator, the Weyl denominator, and the Weyl character converge (see [Kac1104219, Prop. 11.10, Prop. 10.6(d)]).
If is affine then
see [KPe0750341, Prop. 1.9, Lemma 2.3(c) and Prop. 2.5(c)].