The Weyl character formula for a Kac-Moody Lie algebra

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 $𝔤={𝔫}^{+}\oplus 𝔥\oplus {𝔫}^{-}$ be the corresponding Kac-Moody Lie algebra, where ${𝔫}^{+}$ is the Lie subalgebra generated by ${e}_{1},\dots ,{e}_{n},$ and ${𝔫}^{-}$ is the subalgebra generated by ${f}_{1},\dots ,{f}_{n}\text{.}$ The roots label the nonzero eigenspaces of the adjoint action of $𝔥$ on $𝔤,$

$𝔤=𝔥⊕ (⨁α∈R𝔤α), where𝔤α= { x∈𝔤 | if h∈𝔥 then [h,x]=α (h)x } ,$

for $\alpha \in {𝔥}^{*}\text{.}$ The set of positive roots of $𝔤$ is

$R+= { α∈𝔥* | α≠0,𝔤α≠0 ,𝔤α⊆𝔫+ } . (1.1)$

(see [Kac1104219, (1.3.2)]). The Weyl group $W$ is the subgroup of $GL\left({𝔥}^{*}\right)$ generated by ${s}_{1},\dots ,{s}_{n},$ 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)])

$ρ∈𝔥*such that ⟨ρ,hi⟩ =1,for i=1, …,n. (1.3)$

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

$aρ=eρ ∏α∈R+ (1-e-α)dim(𝔤α), (1.4)$

and the character of a Verma module $M\left(\lambda \right)$ of highest weight $\lambda$ is

$char(M(λ))= 1aρeλ+ρ, for λ∈𝔥ℂ*.$

(see [Kac1104219, (9.7.2)]).

Let

$P+= { λ∈𝔥ℂ* | λ(hi)∈ ℤ≥0, for i =1,2,…,n } . (1.5)$

The set ${P}^{+}$ indexes the irreducible integrable $𝔤\text{-modules}$ $L\left(\lambda \right)$ (see [Kac1104219, Lemma 10.1]). The Weyl character of the irreducible integrable $𝔤\text{-module}$ $L\left(\lambda \right)$ is

$char(L(λ))= 1aρ∑w∈W det(w) ew(λ+ρ), for λ∈P+, (1.6)$

where

$L(λ)=M(λ)N ,where N is the unique maximal proper submodule,$

so that $L\left(\lambda \right)$ is the simple $𝔤\text{-module}$ of highest weight $\lambda$ (see [Kac1104219, §9.3 and Theorem 10.4]).

 (a) [Mac1983, (3.20)] or [Kac1104219, (10.2.2)] If $w\in W$ then $w{a}_{\rho }=\text{det}\left(w\right){a}_{\rho }\text{.}$ (b) [Mac1983, (3.23)] or [Kac1104219, (10.4.4)] (The Weyl denominator formula)

$∑w∈Wdet (w)ewρ-ρ= ∏α∈R+ (1-e-α)dim(𝔤α) andρ-wρ= ∑α∈R+,w-1α∈R- α. (1.7)$

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 $\rho -w\rho$ in (1.7) is the sum over the hyperplanes between $1$ and $w\text{.}$

Let

$𝔥ℝ= { h∈𝔥ℂ | αi (h)∈ℝ, for i= 1,2,…,n } .$

The cone $C$ is the the fundamental chamber and the Tits cone is the union of the $W\text{-images}$ of $C,$

$C= { h∈𝔥ℂ | αi (h)∈ℝ≥0, for i=1,2,…,n } andX=⋃w∈W wC$

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

$X+i𝔥ℝ= { x+iy | x∈X, y∈𝔥ℝ } andY=Interior (X+i𝔥ℝ),$

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

$X= { h∈𝔥ℝ | δ (h)∈ℝ>0 } andY= { h∈𝔥 | Re (δ(h))>0 } ,$

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