q=

and this is precisely the denominator in the Weyl character formula:

w

w∈W (’1) e(w(» + ρ))

WCF chIrr(») = (7.18)

w

w∈W (’1) e(wρ)

7.5 The Weyl dimension formula.

For any weight, µ we de¬ne

(’1)w e(wµ).

Aµ :=

w∈W

Then we can write the Weyl character formula as

A»+ρ

chIrr(») = .

Aρ

For any weight µ de¬ne the homomorphism Ψµ from the ring Z¬n (L) into

the ring of formal power series in one variable t by the formula

Ψµ (e(ν)) = e(ν,µ)κ t

(and extend linearly). The left hand side of the Weyl character formula belongs

to Z¬n (L), and hence so does the right hand side which is a quotient of two

elements of Z¬n (L). Therefore for any µ we have

Ψµ (Aρ+» )

Ψµ (chIrr(») ) = .

Ψµ (Aρ )

Ψµ (Aν ) = Ψν (Aµ ) (7.19)

for any pair of weights. Indeed,

(’1)w e(µ,wν)κ t

Ψµ (Aν ) =

w

’1

(’1)w e(w µ,ν)κ t

=

w

(’1)w e(wµ,ν)κ t

=

= Ψν (Aµ ).

126 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

In particular,

Ψρ (A» ) = Ψ» (Aρ )

= Ψ» (q)

(e(±/2) ’ e(’±/2))

= Ψ»

e(»,±)κ t/2 ’ e’(»,±)κ t/2)

=

±∈¦+

+

(», ±)κ t#¦ + terms of higher degree in t.

=

Hence

Ψρ (A»+ρ ) (» + ρ, ±)κ

Ψρ (chIrr(») ) = = + terms of positive degree in t.

Ψρ (Aρ ) (ρ, ±)κ

Now consider the composite homomorphism: ¬rst apply Ψρ and then set t =

0. This has the e¬ect of replacing every e(µ) by the constant 1. Hence applied to

the left hand side of the Weyl character formula this gives the dimension of the

representation Irr(»). The previous equation shows that when this composite

homomorphism is applied to the right hand side of the Weyl character formula,

we get the right hand side of the Weyl dimension formula:

±∈¦+ (» + ρ, ±)κ

dim Irr(») = . (7.20)

±∈¦+ (ρ, ±)κ

7.6 The Kostant multiplicity formula.

Let us multiply the fundamental equation (7.17) by pe(’ρ) and use the fact

(7.16) that qpe(’ρ) = 1 to obtain

(’1)w pe(’ρ)e(w(» + ρ)).

chIrr(») =

w∈W

But

pe(’ρ)e(w(» + ρ)) = p(· ’ w(» + ρ) + ρ)

or, in more pedestrian terms, the left hand side of this equation has, as its

coe¬cient of e(µ) the value

p(µ + ρ ’ w(» + ρ)).

On the other hand, by de¬nition,

chIrr(») = dim(Irr(»)µ e(µ).

We thus obtain Kostant™s formula for the multiplicity of a weight µ in the

irreducible module with highest weight »:

(’1)w p(µ + ρ ’ w(» + ρ)).

KMF dim (Irr(»))µ = (7.21)

w∈W

7.7. STEINBERG™S FORMULA. 127

It will be convenient to introduce some notation which simpli¬es the appearance

of the Kostant multiplicity formula: For w ∈ W and µ ∈ L (or in E for that

matter) de¬ne

w µ := w(µ + ρ) ’ ρ. (7.22)

This de¬nes another action of W on E where the “origin of the orthogonal

transformations w has been shifted from 0 to ’ρ”. Then we can rewrite the

Kostant multiplicity formula as

(’1)w PK (w » ’ µ)

dim(Irr(»))µ = (7.23)

w∈W

or as

(’1)w PK (w » ’ µ)e(µ),

ch(Irr(»)) = (7.24)

µ

w∈W

where PK is the original Kostant partition function.

For the purposes of the next section it will be useful to record the following

lemma:

Lemma 14 If ν is a dominant weight and e = w ∈ W then w ν is not

dominant.

Proof. If ν is dominant, so lies in the closure of the positive Weyl chamber,

then ν + ρ lies in the interior of the positive Weyl chamber. Hence if w = e,

then w(ν + ρ)(hi ) < 0 for some i, and so w ν = w(ν + ρ) ’ ρ is not dominant.

QED

7.7 Steinberg™s formula.

Suppose that » and » are dominant integral weights. Decompose Irr(» ) —

Irr(» ) into irreducibles, and let n(») = n(», » — » ) denote the multiplicity of

Irr(») in this decomposition into irreducibles (with n(») = 0 if Irr(») does not

appear as a summand in the decomposition). In particular, n(ν) = 0 if ν is not

a dominant weight since Irr(ν) is in¬nite dimensional in this case, so can not

appear as a summand in the decomposition. In terms of characters, we have