»

Steinberg™s formula is a formula for n(»). To derive it, use the Weyl character

formula

A» +ρ A»+ρ

ch(Irr(» )) = , ch(Irr(»)) =

Aρ Aρ

in the above formula to obtain

ch(Irr(» ))A» = n(»)A»+ρ .

+ρ

»

128 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

Use the Kostant multiplicity formula (7.24) for » :

(’1)w PK (w » ’ µ)e(µ)

ch(Irr(» )) =

µ

w∈W

and the de¬nition

(’1)u e(u(» + ρ))

A» =

+ρ

u∈W

and the similar expression for A»+ρ to get

(’1)uw PK (w » ’ µ))e(u(» + ρ) + µ) =

µ u,w∈W

n(»)(’1)w e(w(» + ρ)).

w

»

Let us make a change of variables on the right hand side, writing

ν=w »

so the right hand side becomes

(’1)w n(w’1 ν)e(ν + ρ).

ν w

If ν is a dominant weight, then by Lemma 14 w’1 ν is not dominant if w’1 = e.

So n(w’1 ν) = 0 if w = 1 and so the coe¬cient of e(ν + ρ) is precisely n(ν)

when ν is dominant.

On the left hand side let

µ=ν’u »

to obtain

(’1)uw PK (w » ’ ν)e(ν + ρ).

» +u

ν,u,w

Comparing coe¬cients for ν dominant gives

(’1)uw PK (w » ’ ν).

n(ν) = » +u (7.25)

u,w

7.8 The Freudenthal - de Vries formula.

We return to the study of a semi-simple Lie algebra g and get a re¬nement of

the Weyl dimension formula by looking at the next order term in the expansion

we used to derive the Weyl dimension formula from the Weyl character formula.

By de¬nition, the Killing form restricted to the Cartan subalgebra h is given

by

κ(h, h ) = ±(h)±(h )

±

7.8. THE FREUDENTHAL - DE VRIES FORMULA. 129

where the sum is over all roots. If µ, » ∈ h— with tµ , t» the elements of H

corresponding to them under the Killing form, we have

(», µ)κ = κ(t» , tµ ) = ±(t» )±(tµ )

±

so

(», µ)κ = (», ±)κ (µ, ±)κ . (7.26)

±

For each » in the weight lattice L we have let e(») denote the “formal

exponential” so Zf in (L) is the space spanned by the e(») and we have de¬ned

the homomorphism

Ψρ : Zf in (Λ) ’ C[[t]], e(») ’ e(».ρ)κ t .

Let N and D be the images under Ψρ of the Weyl numerator and denominator.

So

N = Ψρ (Aρ+» ) = Ψρ+» (Aρ )

by (7.19) and

e±/2 ’ e’±/2

Aρ = q = (7.27)

±∈¦+

and therefore

e(»+ρ,±)κ t/2 ’ e’(»+ρ,±)κ t/2

N (t) =

±>0

1

(» + ρ, ±)2 t2 + · · · ]

= (» + ρ, ±)κ t[1 + κ

24

with a similar formula for D. Then N/D ’ d(») = the dimension of the

representation as t ’ 0 is the usual proof (that we reproduced above) of the

Weyl dimension formula. Sticking this in to N/D gives

N 1

[(» + ρ, ±)2 ’ (ρ, ±)2 ]t2 + · · ·

= d(») 1 + .

κ κ

D 24 ±>0

(µ, ±)2 by (7.26), where the sum is over

For any weight µ we have (µ, µ)κ = κ

all roots so

N 1

= d 1 + [(» + ρ, » + ρ)κ ’ (ρ, ρ)κ ]t2 + · · · ,

D 48

and we recognize the coe¬cient of 48 t2 in the above expression as χ» (Casκ )),

1

the scalar giving the value of the Casimir associated to the Killing form in the

representation with highest weight ».

On the other hand, the image under Ψρ of the character of the irreducible

representation with highest weight » is

1

e(µ,ρ)κ t = (1 + (µ, ρ)κ t + (µ, ρ)2 t2 + · · · )

κ

2

µ µ

130 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

where the sum is over all weights in the irreducible representation counted with

multiplicity. Comparing coe¬cients gives

1

d(»)χ» (Casκ ).

(µ, ρ)2 =

κ

24

µ

Applied to the adjoint representation the left hand side becomes (ρ, ρ)κ by

(7.26), while d(») is the dimension of the Lie algebra. On the other hand,

χ» (Casκ ) = 1 since tr ad(Casκ ) = dim(g) by the de¬nition of Casκ . So we get

1

(ρ, ρ)κ = dim g (7.28)

24

for any semisimple Lie algebra g.