and therefore

ρ, ±i = 1, i = 1, . . . , := dim(h).

In other words

1

φ = »1 + · · · + » .

ρ= (7.13)

2

φ∈¦+

The Kostant partition function, PK (µ) is de¬ned as the number of sets

of non-negative integers, kβ such that

µ= kβ β.

β∈¦+

(The value is zero if µ can not be expressed as a sum of positive roots.)

For any module N and any µ ∈ h— , Nµ denotes the weight space of weight µ.

For example, in the Verma module, Verm(»), the only non-zero weight spaces

are the ones where µ = » ’ β∈¦+ kβ β and the multiplicity of this weight

space, i.e. the dimension of Verm(»)µ is the number of ways of expressing in

this fashion, i.e.

dim Verm(»)µ = PK (» ’ µ). (7.14)

In terms of the character notation introduced in the preceding section we

can write this as

PK (» ’ µ)e(µ).

chVerm(») =

To be consistent with Humphreys™ notation, de¬ne the Kostant function p by

p(ν) = PK (’ν)

and then in succinct language

chVerm(») = p(· ’ »). (7.15)

7.4. THE WEYL CHARACTER FORMULA. 123

Observe that if

f= f (µ)e(µ)

then

f · e(») = f (ν ’ »)e(ν).

f (µ)e(» + µ) =

We can express this by saying that

f · e(») = f (· ’ »).

Thus, for example,

chVerm(») = p(· ’ ») = p · e(»).

Also observe that if

1

:= 1 + e(’±) + e(’2±) + · · ·

f± =

1 ’ e(’±)

then

(1 ’ e(’±))f± = 1

and

f± = p

±∈¦+

by the de¬nition of the Kostant function.

De¬ne the function q by

(e(±/2) ’ e(’±/2)) = e(ρ) (1 ’ e(’±))

q :=

±∈¦+

since e(ρ) = e(±/2). Notice that

±∈¦+

wq = (’1)w q.

It is enough to check this on fundamental re¬‚ections, but they have the property

that they make exactly one positive root negative, hence change the sign of q.

We have

qp = e(ρ). (7.16)

Indeed,

(1 ’ e(’±)) e(ρ)pe(’ρ)

qpe(’ρ) =

(1 ’ e(’±)) p

=

(1 ’ e(’±))

= f±

= 1.

Therefore,

qchVerm(») = qpe(») = e(ρ)e(») = e(» + ρ).

124 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

Let us now multiply both sides of (7.12) by q and use the preceding equation.

We obtain

qchIrr(») = b(µ)e(µ + ρ)

where the sum is over all µ » satisfying (7.10), and the b(µ) are coe¬cients

we must determine.

Now ch Irr(») is invariant under the Weyl group W , and q transforms by

(’1)w . Hence if we apply w ∈ W to the preceding equation we obtain

(’1)w qchIrr(») = b(µ)e(w(µ + ρ)).

This shows that the set of µ + ρ with non-zero coe¬cients is stable under W

and the coe¬cients transform by the sign representation for each W orbit. In

particular, each element of the form µ = w(»+ρ)’ρ has (’1)w as its coe¬cient.

We can thus write

(’1)w e(w(» + ρ)) + R

qchV (») =

w∈W

where R is a sum of terms corresponding to µ + ρ which are not of the form

w(» + ρ). We claim that there are no such terms and hence R = 0. Indeed, if

there were such a term, the transformation properties under W would demand

that there be such a term with µ + ρ in the closure of the Weyl chamber, i.e.

µ + ρ ∈ Λ := L © D

where

D = Dg = {» ∈ E|(», φ) ≥ 0 ∀φ ∈ ∆+ }

and E = h— denotes the space of real linear combinations of the roots. But we

R

claim that

», (µ + ρ, µ + ρ) = (» + ρ, » + ρ), & µ + ρ ∈ Λ =’ µ = ».

µ

Indeed, write µ = » ’ π, π = k± ±, k± ≥ 0 so

(» + ρ, » + ρ) ’ (µ + ρ, µ + ρ)

0 =

(» + ρ, » + ρ) ’ (» + ρ ’ π, » + ρ ’ π)

=

= (» + ρ, π) + (π, µ + ρ)

≥ (» + ρ, π) since µ + ρ ∈ Λ

≥ 0

since » + ρ ∈ Λ and in fact lies in the interior of D. But the last inequality is

strict unless π = 0. Hence π = 0. We will have occasion to use this type of

argument several times again in the future. In any event we have derived the

fundamental formula

(’1)w e(w(» + ρ)).

qchIrr(») = (7.17)

w∈W

7.5. THE WEYL DIMENSION FORMULA. 125

Notice that if we take » = 0 and so the trivial representation with character

1 for V (»), (7.17) becomes