dim Irr(»)wµ = dim Irr(»)µ (7.3)

where W denotes the Weyl group. These are all ¬nite dimensional subspaces:

Indeed their dimension is at most the corresponding dimension in the Verma

module Verm(»), since Irr(»)µ is a quotient space of Verm(»)µ . But Verm(»)µ

k k

has a basis consisting of those f1 1 · · · fmm v+ . The number of such elements is

the number of ways of writing

» ’ µ = k1 ±1 + · · · km ±m .

So dim Verm(»)µ is the number of m-tuplets of non-negative integers (k1 , . . . , km )

such that the above equation holds. This number is clearly ¬nite, and is known

as PK (» ’ µ), the Kostant partition function of » ’ µ, which will play a central

role in what follows.

Now every element of E is conjugate under W to an element of the closure

of the fundamental Weyl chamber, i.e. to a µ satisfying

(µ, ±i ) ≥ 0

7.3. THE VALUE OF THE CASIMIR. 117

i.e. to a µ that is dominant. We claim that there are only ¬nitely many dominant

weights µ which are », which will complete the proof of ¬nite dimensionality.

Indeed, the sum of two dominant weights is dominant, so » + µ is dominant.

On the other hand, » ’ µ = ki ±i with the ki ≥ 0. So

(», ») ’ (µ, u) = (» + µ, » ’ µ) = ki (» + µ, ±i ) ≥ 0.

So µ lies in the intersection of the ball of radius (», ») with the discrete set

of weights » which is ¬nite.

We record a consequence of (7.3) which is useful under very special circum-

stances. Suppose we are given a ¬nite dimensional representation of g with the

property that each weight space is one dimensional and all weights are conju-

gate under W. Then this representation must be irreducible. For example, take

g = sl(n + 1) and consider the representation of g on §k (Cn+1 ), 1 ¤ k ¤ n. In

terms of the standard basis e1 , . . . , en+1 of Cn+1 the elements ei1 § · · · § eik are

weight vectors with weights Li1 + · · · + Lik , Where h consists of all diagonal

traceless matrices and Li is the linear function which assigns to each diagonal

matrix its i-th entry.

These weight spaces are all one dimensional and conjugate under the Weyl

group. Hence these representations are irreducible with highest weight

ωi := L1 + · · · + Lk

in terms of the usual choice of base, h1 , . . . , hn where hj is the diagonal matrix

with 1 in the j-th position, ’1 in the j + 1-st position and zeros elsewhere.

Notice that

ωi (hj ) = δij

so that the ωi form a basis of the “weight lattice” consisting of those » ∈ h—

which take integral values on h1 , . . . , hn .

7.3 The value of the Casimir.

Recall that our basis of U (g) consists of the elements

f11 · · · fm hj1 · · · hj ek1 · · · ekm .

i im

m

1 1

The elements of U (h) are then the ones with no e or f component in their

expression. So we have a vector space direct sum decomposition

U (g) = U (h) • (U (g)n+ + n’ U (g)) ,

where n+ and n’ are the corresponding nilpotent subalgebras. Let γ denote

projection onto the ¬rst factor in this decomposition. Now suppose z ∈ Z(g),

the center of the universal enveloping algebra. In particular, z ∈ U (g)h . The

eigenvalues of the monomial above under the action of h ∈ h are

m

(ks ’ is )±s (h).

s=1

118 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

So any monomial in the expression for z can not have f factors alone. We have

proved that

z ’ γ(z) ∈ U (g)n+ , ∀ z ∈ Z(g). (7.4)

For any » ∈ h— , the element z ∈ Z(g) acts as a scalar, call it χ» (z) on the

Verma module associated to ».

In particular, if » is a dominant integral weight, it acts by this same scalar

on the irreducible ¬nite dimensional module associated to ».

On the other hand, the linear map » : h ’ C extends to a homomorphism,

which we will also denote by » of U (h) = S(h) ’ C. Explicitly, if we think of

elements of U (h) = S(h) as polynomials on h— , then »(P ) = P (») for P ∈ S(h).

Since n+ v = 0 if v is the maximal weight vector, we conclude from (7.4) that

χ» (z) = »(γ(z)) ∀ z ∈ Z(g). (7.5)

We want to apply this formula to the second order Casimir element associated

to the Killing form κ. So let k1 , . . . , k ∈ h be the dual basis to h1 , . . . , h

relative to κ, i.e.

κ(hi , kj ) = δij .

Let x± ∈ g± be a basis (i.e. non-zero) element and z± ∈ g’± be the dual basis

element to x± under the Killing form, so the second order Casimir element is

Casκ = hi ki + x± z± .

±

where the second sum on the right is over all roots. We might choose the

x± = e± for positive roots, and then the corresponding z± is some multiple of

the f± . (And, for present purposes we might even choose f± = z± for positive

±.) The problem is that the z± for positive ± in the above expression for Casκ

are written to the right, and we must move them to the left. So we write

Casκ = hi ki + [x± , z± ] + z± x± + x± z± .

±>0 ±>0 ±<0

i

This expression for Casκ has all the n+ elements moved to the right; in partic-

ular, all of the summands in the last two sums annihilate v» . Hence

γ(Casκ ) = hi ki + [x± , z± ]

±>0

i

and

χ» (Casκ ) = »(hi )»(ki ) + »([x± , z± ]).

±>0

i

For any h ∈ h we have

κ(h, [x± , z± ]) = κ([h, x± ], z± ) = ±(h)κ(x± , z± ) = ±(h)

so

[x± , z± ] = t±

7.3. THE VALUE OF THE CASIMIR. 119

where t± ∈ h is uniquely determined by

κ(t± , h) = ±(h) ∀ h ∈ h.

Let ( , )κ denote the bilinear form on h— obtained from the identi¬cation of h

with h— given by κ. Then

»([x± , z± ]) = »(t± ) = (», ±)κ = 2(», ρ)κ (7.6)

±>0 ±>0 ±>0

where

1

ρ := ±.

2 ±>0

On the other hand, let the constants ai be de¬ned by

ai κ(hi , h) ∀ h ∈ h.

»(h) =

i

ai hi under the isomorphism of h with h— so

In other words » corresponds to

(», »)κ = ai aj κ(hi , hj ).