The Weyl character formula for a Kac-Moody Lie algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $𝔤,$

$$𝔤=𝔥\oplus \left(\underset{\alpha \in R}{⨁}{𝔤}_{\alpha }\right),\text{where}{𝔤}_{\alpha }=\{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\text{if}\hspace{0.17em}h\in 𝔥\hspace{0.17em}\text{then}\hspace{0.17em}[h,x]=\alpha \left(h\right)x\},$$

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

$$\begin{array}{cc}{R}^{+}=\{\alpha \in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}\alpha \ne 0,{𝔤}_{\alpha }\ne 0,{𝔤}_{\alpha }\subseteq {𝔫}^{+}\}\text{.}& \text{(1.1)}\end{array}$$

(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

$$\begin{array}{cc}{s}_i:{𝔥}^{*}⟶{𝔥}^{*}\text{is given by}{s}_i\lambda =\lambda -⟨\lambda ,h_i⟩{\alpha }_i\text{.}& \text{(1.2)}\end{array}$$

(see [Kac1104219, §3.7]).

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

$$\begin{array}{cc}\rho \in {𝔥}^{*}\text{such that}⟨\rho ,h_i⟩=1,\text{for}\hspace{0.17em}i=1,\dots ,n\text{.}& \text{(1.3)}\end{array}$$

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

$$\begin{array}{cc}{a}_{\rho }=e^{\rho }\prod _{\alpha \in R^{+}}{(1-e^{-\alpha })}^{\text{dim}\left({𝔤}_{\alpha }\right)},& \text{(1.4)}\end{array}$$

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

$$\text{char}\left(M\left(\lambda \right)\right)=\frac{1}{a_{\rho }}{e}^{\lambda +\rho },\text{for}\hspace{0.17em}\lambda \in {𝔥}_{ℂ}^{*}\text{.}$$

(see [Kac1104219, (9.7.2)]).

Let

$$\begin{array}{cc}{P}^{+}=\{\lambda \in {𝔥}_{ℂ}^{*}\hspace{0.17em}|\hspace{0.17em}\lambda \left(h_i\right)\in {\mathbb{Z}}_{\ge 0},\hspace{0.17em}\text{for}\hspace{0.17em}i=1,2,\dots ,n\}\text{.}& \text{(1.5)}\end{array}$$

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

$$\begin{array}{cc}\text{char}\left(L\left(\lambda \right)\right)=\frac{1}{a_{\rho }}\sum _{w\in W}\text{det}\left(w\right)e^{w(\lambda +\rho )},\text{for}\hspace{0.17em}\lambda \in P^{+},& \text{(1.6)}\end{array}$$

where

$$L\left(\lambda \right)=\frac{M\left(\lambda \right)}{N},\text{where}\hspace{0.17em}N\hspace{0.17em}\text{is the unique maximal proper submodule,}$$

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)

$$\begin{array}{cc}\sum _{w\in W}\text{det}\left(w\right)e^{w\rho -\rho }=\prod _{\alpha \in R^{+}}{(1-e^{-\alpha })}^{\text{dim}\left({𝔤}_{\alpha }\right)}\text{and}\rho -w\rho =\sum _{\alpha \in R^{+},w^{-1}\alpha \in R^{-}}\alpha \text{.}& \text{(1.7)}\end{array}$$

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

$${𝔥}_{ℝ}=\{h\in {𝔥}_{ℂ}\hspace{0.17em}|\hspace{0.17em}{\alpha }_i\left(h\right)\in ℝ,\hspace{0.17em}\text{for}\hspace{0.17em}i=1,2,\dots ,n\}\text{.}$$

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

$$C=\{h\in {𝔥}_{ℂ}\hspace{0.17em}|\hspace{0.17em}{\alpha }_i\left(h\right)\in {ℝ}_{\ge 0},\hspace{0.17em}\text{for}\hspace{0.17em}i=1,2,\dots ,n\}\text{and}X=\bigcup _{w\in W}wC$$

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

$$X+i{𝔥}_{ℝ}=\{x+iy\hspace{0.17em}|\hspace{0.17em}x\in X,y\in {𝔥}_{ℝ}\}\text{and}Y=\text{Interior}(X+i{𝔥}_{ℝ}),$$

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

$$X=\{h\in {𝔥}_{ℝ}\hspace{0.17em}|\hspace{0.17em}\delta \left(h\right)\in {ℝ}_{>0}\}\text{and}Y=\{h\in 𝔥\hspace{0.17em}|\hspace{0.17em}\text{Re}\left(\delta \left(h\right)\right)>0\},$$

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

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