Since κ(hi , kj ) = δij we have

»(ki ) = ai .

Combined with »(hi ) = aj κ(hj , hi ) this gives

j

(», »)κ = »(hi )»(ki ). (7.7)

i

Combined with (7.6) this yields

χ» (Casκ ) = (» + ρ, » + ρ)κ ’ (ρ, ρ)κ . (7.8)

We now use this innocuous looking formula to prove the following: We let

L = Lg ‚ h— denote the lattice of integral linear forms on h, i.e.

R

(µ, φ)

L = {µ ∈ h— |2 ∈ Z ∀φ ∈ ∆}. (7.9)

(φ, φ)

L is called the weight lattice of g.

For µ, » ∈ L recall that

µ »

if » ’ µ is a sum of positive roots. Then

120 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

Proposition 26 Any cyclic highest weight module Z(»), » ∈ L has a composi-

tion series whose quotients are irreducible modules, Irr(µ) where µ » satis¬es

(µ + ρ, µ + ρ)κ = (» + ρ, » + ρ)κ . (7.10)

In fact, if

d= dim Z(»)µ

where the sum is over all µ satisfying (7.10) then there are at most d steps in

the composition series.

Remark. There are only ¬nitely many µ ∈ L satisfying (7.10) since the set

of all µ satisfying (7.10) is compact and L is discrete. Each weight is of ¬nite

multiplicity. Therefore d is ¬nite.

Proof by induction on d. We ¬rst show that if d = 1 then Z(») is irreducible.

Indeed, if not, any proper submodule W , being the sum of its weight spaces,

must have a highest weight vector with highest weight µ, say. But then

χ» (Casκ ) = χµ (Casκ )

since W is a submodule of Z(») and Casκ takes on the constant value χ» (Casκ )

on Z(»). Thus µ and » both satisfy (7.10) contradicting the assumption d = 1.

In general, suppose that Z(») is not irreducible, so has a submodule, W and

quotient module Z(»)/W . Each of these is a cyclic highest weight module,

and we have a corresponding composition series on each factor. In particular,

d = dW + dZ(»)/W so that the d™s are strictly smaller for the submodule and the

quotient module. Hence we can apply induction. QED

For each » ∈ L we introduce a formal symbol, e(») which we want to think

of as an “exponential” and so the symbols are multiplied according to the rule

e(µ) · e(ν) = e(µ + ν). (7.11)

The character of a module N is de¬ned as

dim Nµ · e(µ).

chN =

In all cases we will consider (cyclic highest weight modules and the like) all

these dimensions will be ¬nite, so the coe¬cients are well de¬ned, but (in the

case of Verma modules for example) there may be in¬nitely many terms in the

(formal) sum. Logically, such a formal sum is nothing other than a function on

L giving the “coe¬cient” of each e(µ).

In the case that N is ¬nite dimensional, the above sum is ¬nite. If

f= fµ e(µ) and g = gν e(ν)

are two ¬nite sums, then their product (using the rule (7.11)) corresponds to

convolution:

fµ e(µ) · gν e(ν) = (f g)» e(»)

7.4. THE WEYL CHARACTER FORMULA. 121

where

(f g)» := fµ gν .

µ+ν=»

So we let Z¬n (L) denote the set of Z valued functions on L which vanish outside

a ¬nite set. It is a commutative ring under convolution, and we will oscillate

in notation between writing an element of Z¬n (L) as an “exponential sum”

thinking of it as a function of ¬nite support.

Since we also want to consider in¬nite sums such as the characters of Verma

modules, we enlarge the space Z¬n (L) by de¬ning Zgen (L) to consist of Z val-

ued functions whose supports are contained in ¬nite unions of sets of the form

» ’ ± 0 k± ±. The convolution of two functions belonging to Zgen (L) is well

de¬ned, and belongs to Zgen (L). So Zgen (L) is again a ring.

It now follows from Prop.26 that

chZ(») = chIrr(µ)

where the sum is over the ¬nitely many terms in the composition series. In

particular, we can apply this to Z(») = Verm(»), the Verma module. Let us

µj ’ i ¤ j. Then

order the µi » satisfying (7.10) in such a way that µi

for each of the ¬nitely many µi occurring we get a corresponding formula for

chVerm(µi ) and so we get collection of equations

chVerm(µj ) = aij ch Irr(µi )

where aii = 1 and i ¤ j in the sum. We can invert this upper triangular matrix

and therefore conclude that there is a formula of the form

chIrr(») = b(µ)chVerm(µ) (7.12)

where the sum is over µ » satisfying (7.10) with coe¬cients b(µ) that we shall

soon determine. But we do know that b(») = 1.

7.4 The Weyl character formula.

We will now prove

Proposition 27 The non-zero coe¬cients in (7.12) occur only when

µ = w(» + ρ) ’ ρ

where w ∈ W , the Weyl group of g, and then

b(µ) = (’1)w .

Here

(’1)w := det w.

122 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

We will prove this by proving some combinatorial facts about multiplication of

sums of exponentials.

We recall our notation: For » ∈ h— , Irr(») denotes the unique irreducible

module of highest weight, »,and Verm(») denotes the Verma module of highest

weight », and more generally, Z(») denotes an arbitrary cyclic module of highest

weight ». Also

1

ρ := φ

2 + φ∈¦

is one half the sum of the positive roots. Let »i , i = 1, . . . , dim h be the basis of

the weight lattice, L dual to the base ∆. So

»i (h±j ) = »i , ±j = δij .

Since si (±i ) = ’±i while keeping all the other positive roots positive, we saw

that this implied that