The Weyl character formula for a Kac-Moody Lie algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 27 July 2013

The Weyl character formula for a Kac-Moody Lie algebra

In this section we review the Weyl-Kac character formula for integrable representations of Kac-Moody Lie algebras following [Kac1104219, Ch. 10].

Let C be a symmetrizable Cartan matrix and let 𝔤=𝔫+𝔥𝔫- be the corresponding Kac-Moody Lie algebra, where 𝔫+ is the Lie subalgebra generated by e1,,en, and 𝔫- is the subalgebra generated by f1,,fn. The roots label the nonzero eigenspaces of the adjoint action of 𝔥 on 𝔤,

𝔤=𝔥 (αR𝔤α), where𝔤α= { x𝔤|ifh𝔥 then[h,x]=α (h)x } ,

for α𝔥*. The set of positive roots of 𝔤 is

R+= { α𝔥*| α0,𝔤α0 ,𝔤α𝔫+ } . (1.1)

(see [Kac1104219, (1.3.2)]). The Weyl group W is the subgroup of GL(𝔥*) generated by s1,,sn, where

si:𝔥*𝔥* is given bysiλ= λ-λ,hi αi. (1.2)

(see [Kac1104219, §3.7]).

Let (see [Mac1983, III, 20] or [Kac1104219, (2.5.1)])

ρ𝔥*such that ρ,hi =1,fori=1, ,n. (1.3)

The Weyl denominator (see [Mac1983, III, 22] or [Kac1104219, §10.2]) is

aρ=eρ αR+ (1-e-α)dim(𝔤α), (1.4)

and the character of a Verma module M(λ) of highest weight λ is

char(M(λ))= 1aρeλ+ρ, forλ𝔥*.

(see [Kac1104219, (9.7.2)]).


P+= { λ𝔥*| λ(hi) 0,fori =1,2,,n } . (1.5)

The set P+ indexes the irreducible integrable 𝔤-modules L(λ) (see [Kac1104219, Lemma 10.1]). The Weyl character of the irreducible integrable 𝔤-module L(λ) is

char(L(λ))= 1aρwW det(w) ew(λ+ρ), forλP+, (1.6)


L(λ)=M(λ)N ,whereNis the unique maximal proper submodule,

so that L(λ) is the simple 𝔤-module of highest weight λ (see [Kac1104219, §9.3 and Theorem 10.4]).

(a) [Mac1983, (3.20)] or [Kac1104219, (10.2.2)] If wW then waρ=det(w)aρ.
(b) [Mac1983, (3.23)] or [Kac1104219, (10.4.4)] (The Weyl denominator formula)

wWdet (w)ewρ-ρ= αR+ (1-e-α)dim(𝔤α) andρ-wρ= αR+,w-1αR- α. (1.7)

In the case of an affine Lie algebra 𝔤 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 ρ-wρ in (1.7) is the sum over the hyperplanes between 1 and w.


𝔥= { h𝔥|αi (h),fori= 1,2,,n } .

The cone C is the the fundamental chamber and the Tits cone is the union of the W-images of C,

C= { h𝔥|αi (h)0, fori=1,2,,n } andX=wW wC

(see [Kac1104219, §3.12] or [KPe0750341, §1.1, p. 138]). The complexified Tits cone and the Weyl-Kac character convergence region are

X+i𝔥= { x+iy|xX, y𝔥 } andY=Interior (X+i𝔥),

respectively (see [Kac1104219, §10.6] and [KPe0750341, §1.1, p. 140]). The importance of the sets X and Y 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 𝔤 is affine then

X= { h𝔥|δ (h)>0 } andY= { h𝔥|Re (δ(h))>0 } ,

see [KPe0750341, Prop. 1.9, Lemma 2.3(c) and Prop. 2.5(c)].

page history