Lie algebra is called reductive. The previous result of Freudenthal and deVries

has been generalized by Kostant from a semi-simple Lie algebra to all reductive

Lie algebras: Suppose that g is merely reductive, and that we have chosen a

symmtric bilinear form on g which is invariant under the adjoint representation,

and denote the associated Casimir element by Casg . We claim that (7.28)

generalizes to

1

tr ad(Casg ) = (ρ, ρ). (7.29)

24

(Notice that if g is semisimple and we take our symmetric bilinear form to be

the Killing form ( , )κ (7.29) becomes (7.28).) To prove (7.29) observe that

both sides decompose into sums as we decompose g into as sum of its center

and its simple ideals, since this must be an orthogonal decomposition for our

invariant scalar product. The contribution of the center is zero on both sides,

so we are reduced to proving (7.29) for a simple algebra. Then our symmetric

biinear form ( , ) must be a scalar multiple of the Killing form:

( , ) = c2 ( , )κ

for some non-zero scalar c. If z1 , . . . , zN is an orthonormal basis of g for ( , )κ

then z1 /c, . . . , zN /c is an orthonormal basis for ( , ). Thus

1

Casκ .

Casg = 2

c

So

1 11

tr ad(Casκ ) = 2 dim g.

tr ad(Casg ) =

2

c c 24

—

But on h we have the dual relation

1

(ρ, ρ) = 2 (ρ, ρ)κ .

c

Combining the last two equations shows that (7.29) becomes (7.28).

Notice that the same proof shows that we can generalize (7.8) as

χ» (Cas) = (» + ρ, » + ρ) ’ (ρ, ρ) (7.30)

valid for any reductive Lie algebra equipped with a symmetric bilinear form

invariant under the adjoint representation.

7.9. FUNDAMENTAL REPRESENTATIONS. 131

7.9 Fundamental representations.

We let ωi denote the weight which satis¬es

ωi (hj ) = δij

so that the ωi form an integral basis of L and are dominant. We call these

the basic weights.If (V, ρ) and (W, σ) are two ¬nite dimensional irreducible

representations with highest weights » and σ, then V — W, ρ — σ contains the

irreducible representation with highest weight » + µ, and highest weight vector

v» — wµ , the tensor product of the highest weight vectors in V and W . Tak-

ing this “highest” component in the tensor product is known as the Cartan

product of the two irreducible representations.

Let (Vi , ρi ) be the irreducible representations corresponding to the basic

weight ωi . Then every ¬nite dimensional irreducible representation of g can be

obtained by Cartan products from these, and for that reason they are called the

fundamental representations.

For the case of An = sl(n + 1) we have already veri¬ed that the fundamental

representations are §k (V ) where V = Cn+1 and where the basic weights are

ωi = L1 + · · · + Li

We now sketch the results for the other classical simple algebras, leaving the

details as an exercise in the use of the Weyl dimension formula.

For Cn = sp(2n) it is immediate to check that these same expressions give the

basic weights. However while V = C2n = §1 (V ) is irreducible, the higher order

exterior powers are not: Indeed, the symplectic form „¦ ∈ §2 (V — ) is preserved,

and hence so is the the map

§j (V ) ’ §j’2 (V )

given by contraction by „¦. It is easy to check that the image of this map

is surjective (for j = 2, . . . , n). the kernel is thus an invariant subspace of

dimension

2n 2n

’

2j ’ 2

j

and a (not completely trivial) application of the Weyl dimension formula will

show that these are indeed the dimensions of the irreducible representations

with highest weight ωj . Thus these kernels are the fundamental representations

of Cn . Here are some of the details:

We have

ρ = ω1 + · · · + ωn = (n ’ i + 1)Li .

The most general dominant weight is of the form

ki ωi = a1 L1 + · · · + an Ln

132 CHAPTER 7. CYCLIC HIGHEST WEIGHT MODULES.

where

a1 = k1 + · · · + kn , a2 = k2 + · · · + kn , · · · an = kn

where the ki are non-negative integers. So we can equally well use any decreasing

sequence a1 ≥ a2 ≥ · · · ≥ an ≥ 0 of integers to parameterize the irreducible

representations. We have

(ρ, Li ’ Lj ) = j ’ i, (ρ, Li + Lj ) = 2n + 2 ’ i ’ j.

Multiplying these all together gives the denominator in the Weyl dimension

formula.

Similarly the numerator becomes

(li ’ lj ) (li + lj )

i<j i¤j.

where

li := ai + n ’ i + 1.

If we set mi := n ’ i + 1 then we can write the Weyl dimension formula as

2 2

li ’ lj li

dim V (a1 , . . . , an ) = ,

m2 ’ m2 mi

i j

i<j i

where for the case i = j we have taken out a common factor of 2n from the

numerator and the denominator.

An easy induction shows that

(m2 ’ m2 ) mi = (2n ’ 1)!(2n ’ 3)! · · · 1!.

i j

i<j i

so if we set

ri = l i ’ 1 = a + n ’ i

then

’ rj )(ri + rj + 2)

i<j (ri i (ri + 1)

dim V (a1 , . . . , an ) = .

(2n ’ 1)!(2n ’ 3)! · · · 1!

For example, suppose we want to compute the dimension of the fundamental

representation corresponding to »2 = L1 + L2 so a1 = a2 = 1, ai = 0, i > 2. In

applying the preceding formula, all of the terms with 2 < i are the same as for

the trivial representation, as is r1 ’ r2 . The ratios of the remaining factors to

those of the trivial representation are

n n n

j j j

· =

j ’ 1 j=3 j ’ 2 j=3 j ’ 2

j=3

coming from the ri ’ rj terms, i = 1, 2. Similarly the ri + rj terms give a factor

n

2n + 2 ’ j

2n + 1

2n ’ 1 j=3 2n ’ j

7.10. EQUAL RANK SUBGROUPS. 133

and the terms r1 + 1, r2 + 1 contribute a factor

n+1

.