linear function » such that

hv = »(h)v ∀ h ∈ h, e± v = 0, ∀± ∈ ¦+ . (7.2)

Using irreducibility again, we conclude that

W = U (g)v.

The module is cyclic generated by v. In fact we can be more precise: Let

h1 , . . . , h be the basis of h corresponding to the choice of simple roots, let

ei ∈ g±i , fi ∈ g’±i where ±1 , . . . , ±m are all the positive roots. (We can choose

them so that each e and f generate a little sl(2).) Then

g = n’ • h • n+ ,

113

114 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

where e1 , . . . , em is a basis of n+ , where h1 , . . . , h is a basis of h, and f1 , . . . , fm

is a basis of n’ . The Poincar´-Birkho¬-Witt theorem says that monomials of

e

the form

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

i im

m

1 1

form a basis of U (g). Here we have chosen to place all the e™s to the extreme

right, with the h™s in the middle and the f ™s to the left. It now follows that the

elements

i im

f11 · · · fm v

span W . Every such element, if non-zero, is a weight vector with weight

» ’ (i1 ±1 + · · · + im ±m ).

Recall that

means that » ’ µ =

µ » ki ±i , ±i > 0,

where the ki are non-negative integers. We have shown that every weight µ of

W satis¬es

µ ».

So we make the de¬nition: A cyclic highest weight module for g is a module

(not necessarily ¬nite dimensional) which has a vector v+ such that

x+ v+ = 0, ∀ x+ ∈ n+ , hv+ = »(h)v+ ∀h ∈ h

and

V = U (g)v+ .

In any such cyclic highest weight module every submodule is a direct sum of its

weight spaces (by van der Monde). The weight spaces Vµ all satisfy

µ »

and we have

V= Vµ .

Any proper submodule can not contain the highest weight vector, and so the

sum of two proper submodules is again a proper submodule. Hence any such V

has a unique maximal submodule and hence a unique irreducible quotient. The

quotient of any highest weight module by an invariant submodule, if not zero,

is again a cyclic highest weight module with the same highest weight.

7.1 Verma modules.

There is a “biggest” cyclic highest weight module, associated with any » ∈ h—

called the Verma module. It is de¬ned as follows: Let us set

b := h • n+ .

7.2. WHEN IS DIM IRR(») < ∞? 115

Given any » ∈ h— let C» denote the one dimensional vector space C with

basis z+ and with the action of b given by

(h + xβ )z+ := »(h)z+ .

β0

So it is a left U (b) module. By the Poincar´ Birkho¬ Witt theorem, U (g) is a

e

i1 im

free right U (b) module with basis {f1 · · · f }, and so we can form the Verma

module

Verm(») := U (g) —U (b) C»

which is a cyclic module with highest weight vector v+ := 1 — z+ .

Furthermore, any two irreducible cyclic highest weight modules with the

same highest weight are isomorphic. Indeed, if V and W are two such with

highest weight vector v+ , u+ , consider V • W which has (v+ , u+ ) as a maximal

weight vector with weight », and hence Z := U (g)(v+ , u+ ) is cyclic and of

highest weight ». Projections onto the ¬rst and second factors give non-zero

homomorphisms which must be surjective. But Z has a unique irreducible

quotient. Hence these must induce isomorphisms on this quotient, V and W

are isomorphic.

Hence, up to isomorphism, there is a unique irreducible cyclic highest weight

module with highest weight ». We call it

Irr(»).

In short, we have constructed a “largest” highest weight module Verm(»)

and a “smallest” highest weight module Irr(»).

When is dim Irr(») < ∞?

7.2

If Irr(») is ¬nite dimensional, then it is ¬nite dimensional as a module over any

subalgebra, in particular over any subalgebra isomorphic to sl(2). Applied to

the subalgebra sl(2)i generated by ei , hi , fi we conclude that

»(hi ) ∈ Z.

Such a weight is is called integral. Furthermore the representation theory of

sl(2) says that the maximal weight for any ¬nite dimensional representation

must satisfy

»(hi ) = », ±i ≥ 0

so that » lies in the closure of the fundamental Weyl chamber. Such a weight is

called dominant. So a necessary condition for Irr(») to be ¬nite dimensional

is that » be dominant integral. We now show that conversely, Irr(») is ¬nite

dimensional whenever » is dominant integral.

For this we recall that in the universal enveloping algebra U (g) we have

1. [ej , fik+1 ] = 0, if i = j

116 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

2. [hj , fik+1 ] = ’(k + 1)±i (hj )fik+1

3. [ei , fik+1 ] = ’(k + 1)fik (k · 1 ’ hi )

where the ¬rst two equations are consequences of the fact that ad is a derivation

and

[ei , fj ] = 0 if i = j since ±i ’ ±j is not a root

and

[hj , fj ] = ’±j (hi )fj .

The last is a the fact about sl(2) which we have proved in Chapter II. Notice

that it follows from 1.) that ej (fik )v+ = 0 for all k and all i = j and from 3.)

that

»(h )+1

ei fi i v+ = 0

»(h )+1

so that fi i v+ is a maximal weight vector. If it were non-zero, the cyclic

module it generates would be a proper submodule of Irr(») contradicting the

irreducibility. Hence

»(h )+1

fi i v+ = 0.

»(h )

So for each i the subspace spanned by v+ , fi v+ , · · · , fi i v+ is a ¬nite dimen-

sional sl(2)i module. In particular Irr(») contains some ¬nite dimensional sl(2)i

modules. Let V denote the sum of all such. If W is a ¬nite dimensional sl(2)i

module, then e± W is again ¬nite dimensional, thus so their sum, which is a

¬nite dimensional sl(2)i module. Hence V is g- stable, hence all of Irr(»).

In particular, the ei and the fi act as locally nilpotent operators on Irr(»).

So the operators „i := (exp ei )(exp ’fi )(exp ei ) are well de¬ned and

„i (Irr(»))µ = Irr(»)si µ

so