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 ๐”ค=๐”ซ+โŠ•๐”ฅโŠ•๐”ซ- be the corresponding Kac-Moody Lie algebra, where ๐”ซ+ is the Lie subalgebra generated by e1,โ€ฆ,en, and ๐”ซ- is the subalgebra generated by f1,โ€ฆ,fn. The roots label the nonzero eigenspaces of the adjoint action of ๐”ฅ on ๐”ค,

๐”ค=๐”ฅโŠ• (โจฮฑโˆˆR๐”คฮฑ), where๐”คฮฑ= { xโˆˆ๐”คโ€‰|โ€‰ifโ€‰hโˆˆ๐”ฅ โ€‰thenโ€‰[h,x]=ฮฑ (h)x } ,

for ฮฑโˆˆ๐”ฅ*. The set of positive roots of ๐”ค is

R+= { ฮฑโˆˆ๐”ฅ*โ€‰|โ€‰ ฮฑโ‰ 0,๐”คฮฑโ‰ 0 ,๐”คฮฑโŠ†๐”ซ+ } . (1.1)

(see [Kac1104219, (1.3.2)]). The Weyl group W is the subgroup of GL(๐”ฅ*) generated by s1,โ€ฆ,sn, where

si:๐”ฅ*โŸถ๐”ฅ* is given bysiฮป= ฮป-โŸจฮป,hiโŸฉ ฮฑi. (1.2)

(see [Kac1104219, ยง3.7]).

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

ฯโˆˆ๐”ฅ*such that โŸจฯ,hiโŸฉ =1,forโ€‰i=1, โ€ฆ,n. (1.3)

The Weyl denominator (see [Mac1983, III, 22] or [Kac1104219, ยง10.2]) is

aฯ=eฯ โˆฮฑโˆˆR+ (1-e-ฮฑ)dim(๐”คฮฑ), (1.4)

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

char(M(ฮป))= 1aฯeฮป+ฯ, forโ€‰ฮปโˆˆ๐”ฅโ„‚*.

(see [Kac1104219, (9.7.2)]).

Let

P+= { ฮปโˆˆ๐”ฅโ„‚*โ€‰|โ€‰ ฮป(hi)โˆˆ โ„คโ‰ฅ0,โ€‰forโ€‰i =1,2,โ€ฆ,n } . (1.5)

The set P+ indexes the irreducible integrable ๐”ค-modules L(ฮป) (see [Kac1104219, Lemma 10.1]). The Weyl character of the irreducible integrable ๐”ค-module L(ฮป) is

char(L(ฮป))= 1aฯโˆ‘wโˆˆW det(w) ew(ฮป+ฯ), forโ€‰ฮปโˆˆP+, (1.6)

where

L(ฮป)=M(ฮป)N ,whereโ€‰Nโ€‰is the unique maximal proper submodule,

so that L(ฮป) is the simple ๐”ค-module of highest weight ฮป (see [Kac1104219, ยง9.3 and Theorem 10.4]).

(a) [Mac1983, (3.20)] or [Kac1104219, (10.2.2)] If wโˆˆW then waฯ=det(w)aฯ.
(b) [Mac1983, (3.23)] or [Kac1104219, (10.4.4)] (The Weyl denominator formula)

โˆ‘wโˆˆWdet (w)ewฯ-ฯ= โˆฮฑโˆˆR+ (1-e-ฮฑ)dim(๐”คฮฑ) andฯ-wฯ= โˆ‘ฮฑโˆˆR+,w-1ฮฑโˆˆR- ฮฑ. (1.7)

In the case of an affine Lie algebra ๐”ค 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 ฯ-wฯ in (1.7) is the sum over the hyperplanes between 1 and w.

Let

๐”ฅโ„= { hโˆˆ๐”ฅโ„‚โ€‰|โ€‰ฮฑi (h)โˆˆโ„,โ€‰forโ€‰i= 1,2,โ€ฆ,n } .

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

C= { hโˆˆ๐”ฅโ„‚โ€‰|โ€‰ฮฑi (h)โˆˆโ„โ‰ฅ0,โ€‰ forโ€‰i=1,2,โ€ฆ,n } andX=โ‹ƒwโˆˆW wC

(see [Kac1104219, ยง3.12] or [KPe0750341, ยง1.1, p. 138]). The complexified Tits cone and the Weyl-Kac character convergence region are

X+i๐”ฅโ„= { x+iyโ€‰|โ€‰xโˆˆX, yโˆˆ๐”ฅโ„ } andY=Interior (X+i๐”ฅโ„),

respectively (see [Kac1104219, ยง10.6] and [KPe0750341, ยง1.1, 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 ๐”ค is affine then

X= { hโˆˆ๐”ฅโ„โ€‰|โ€‰ฮด (h)โˆˆโ„>0 } andY= { hโˆˆ๐”ฅโ€‰|โ€‰Re (ฮด(h))>0 } ,

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

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