In multiplying all of these terms together there is a huge cancellation and what

is left for the dimension of this fundamental representation is

(2n + 1)(2n ’ 2)

.

2

Notice that this equals

2n

’ 1 = dim §2 V ’ 1.

2

More generally this dimension argument will show that the fundamental repre-

sentations are the kernels of the contraction maps i(„¦) : §k ’ (V ) §k’2 (V )

where „¦ is the symplectic form.

For Bn it is easy to check that ωi := L1 + · · · + Li (i ¤ n ’ 1), and ωn =

1

2 (L1 + · · · + Ln ) are the basic weights and the Weyl dimension formula gives

2n + 1

for j ¤ n ’ 1 as the dimensions of the irreducibles with

the value

j

these weight, so that they are §j (V ), j = 1, . . . n ’ 1 while the dimension of the

irreducible corresponding to ωn is 2n . This is the spin representation which we

will study later.

Finally, for Dn = o(2n) the basic weights are

ωj = L1 + · · · + Lj , j ¤ n ’ 2,

and

1 1

(L1 + · · · + Ln’1 + Ln ) and ωn := (L1 + · · · + Ln’1 ’ Ln ).

ωn’1 :=

2 2

The Weyl dimension formula shows that the the ¬rst n ’ 2 fundamental repre-

sentations are in fact the representation on §j (V ), j = 1, . . . , n ’ 2 while the

last two have dimension 2n’1 . These are the half spin representations which we

will also study later.

7.10 Equal rank subgroups.

In this section we present a generalization of the Weyl character formula due

to Ramond-Gross-Kostant-Sternberg. It depends on an interpretation of the

Weyl denominator in terms of the spin representation of the orthogonal group

O(g/h), and so on some results which we will prove in Chapter IX. But its

logical place is in this chapter. So we will quote the results that we will need.

You might prefer to read this section after Chapter IX.

134 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

Let p be an even dimensional space with a symmetric bilinear such that

p = p+ • p’

is a direct sum decomposition of p into two isotropic subspaces. In other words

p+ and p’ are each half the dimension of p, and the scalar product of any two

vectors in p+ vanishes, as does the scalar product of any two elements of p’ .

For example, we might take p = n+ • n’ and the symmetric bilinear form to

be the Killing form. Then p± = n± is such a desired decomposition.

The symmetric bilinear form then puts p± into duality, i.e. we may identify

p’ with p— and vice versa. Suppose that we have a commutative Lie algebra

+

h acting on p as in¬nitesimal isometries, so as to preserve each p± , that the e+i

are weight vectors corresponding to weights βi and that the e’ form the dual

i

basis, corresponding to the negative of these weights ’βi . In particular, we have

a Lie algebra homomorphism ν from h to o(p), and the two spin representations

of o(p) give two representations of h. By abuse of language, let us denote these

two representations by Spin+ν and Spin’ν . We can also consider the characters

of these representations of h. According to equation (9.22) (to be proved in

Chapter IX) we have

1 1

chSpin+ ν ’ chSpin’ ν = e( βj ) ’ e(’ βj ) .

2 2

j

In the case that h is the Cartan subalgebra of a semi-simple Lie algebra and

and p± = n± we recognize this expression as the Weyl denominator.

Now let g be a semi-simple Lie algebra and r ‚ g a reductive subalgebra of

the same rank. This means that we can choose a Cartan subalgebra of g which

is also a Cartan subalgebra of r. The roots of r form a subset of the roots of g.

The Weyl group Wg acts simply transitively on the Weyl chambers of g each of

which is contained in a Weyl chamber for r. We choose a positive root system

for g, which then determines a positive root system for r, and the positive Weyl

chamber for g is contained in the positive Weyl chamber for r.

Let

C ‚ Wg

denote the set of those elements of the Weyl group of g which map the positive

Weyl chamber of g into the positive Weyl chamber for r. By the simple transi-

tivity of the Weyl group actions on chambers, we know that elements of C form

coset representatives for the subgroup Wr ‚ Wg . In particular, the number of

elements of C is the same as the index of Wr in Wg .

Let

ρg and ρr

denote half the sum of the positive roots of g and r respectively. For any

dominant weight » of g the weight » + ρg lies in the interior of the positive

Weyl chamber for g. Hence for each c ∈ C, the element c(» + ρg ) lies in the

interior for r and hence

c • » := c(» + ρg ) ’ ρr

7.10. EQUAL RANK SUBGROUPS. 135

is a dominant weight for r, and each of these is distinct.

Let V» denote the irreducible representation of g with highest weight ». We

can consider it as a representation of the subalgebra r. Also the Killing form

(or more generally any ad invariant symmetric bilinear form) on g induces an

invariant form on r. Let p denote the orthogonal complement of r in g. We

thus get a homomorphism of r into the orthogonal algebra o(g/r), which is an

even dimensional orthogonal algebra, and hence has two spin representations.

To specify which of these two spin representations we shall denote by S+ and

which by S’ , we note that there is a one dimensional weight space with weight

ρg ’ ρr , and we let S+ denote the spin representation which contains that one

dimensional space. The spaces S± are o(g/r) modules, and via the homomor-

phism r ’ o(g/r) we can consider them as r modules.

Finally, for any dominant integral weight µ of r we let Uµ denote the irre-

ducible module of r with highest weight µ.

With all this notation we can now state

Theorem 16 [G-K-R-S] In the representation ring R(r) we have

(’1)c Uc•» .

V » — S+ ’ V » — S’ = (7.31)

c∈C

Proof. To say that the above equation holds in the representation ring of r

means that when we take the signed sums of the characters of the representations

occurring on both sides we get equality. In the special case that r = h, we have

observed that (7.31) is just the Weyl character formula:

(’1)w e(w(» + ρg )).

χ(Irr(»)(χ(S+g/h ) ’ χ(S’g/h )) =

w∈Wg

The general case follows from this special case by dividing both sides of this

equation by χ(S+r/h ) ’ χ(S’r/h ). The left hand side becomes the character of

the left hand side of (7.31) because the weights that go into this quotient via

(9.22) are exactly those roots of g which are not roots of r. The right hand side

becomes the character of the right hand side of (9.22) by reorganizing the sum

and using the Weyl character formula for r. QED

136 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

Chapter 8

Serre™s theorem.

We have classi¬ed all the possibilities for an irreducible Cartan matrix via the

classi¬cation of the possible Dynkin diagrams. The four major series in our clas-

si¬cation correspond to the classical simple algebras we introduced in Chapter

III. The remaining ¬ve cases also correspond to simple algebras - the “excep-

tional algebras”. Each deserves a discussion on its own. However a theorem of

Serre guarantees that starting with any Cartan matrix, there is a corresponding

semi-simple Lie algebra. Any root system gives rise to a Cartan matrix. So even

before studying each of the simple algebras in detail, we know in advance that