Weights of Irreducible Representations - Irreducible Representations

3.2 Irreducible Representations

3.2.2 Weights of Irreducible Representations

3.2 Irreducible Representations 153

as an s(α)-module. Now hµ,himust occur as an eigenvalue of h in one of the submodules in (3.27), and hence−hµ,hialso occurs as an eigenvalue by Lemma 2.3.2. It follows that there exists a weight of the formµ+kαinX(λ)with

hµ+kα,hi=−hµ,hi.

Thus 2k=−2hµ,hi. Hences_αµ=µ+kα∈X(λ), which proves (iv).

We can now complete the proof of the theorem. The Weyl groupW is generated by the reflectionss_α withα ∈∆ by Theorem 3.1.9. Since (iv) holds for all simple rootsα, we conclude that the setX(λ)is invariant underW. We already know from Lemma 3.2.2 that dimL^λ(µ)<∞for allµ. Thus the finite-dimensionality ofL^λ is a consequence of the following property:

(v) The cardinality ofX(λ)is finite.

Indeed, letµ∈X(λ). Thenµ∈h^∗

R. By Proposition 3.1.12 there existss∈W such that ξ =sµ is in the positive Weyl chamberC. Since ξ ∈X(λ), we know from Lemma 3.2.2 thatξ =λ−Q, whereQ=β1+···+βr withβi∈Φ⁺. Let(·,·)be the inner product onh^∗

Ras in Remark 3.1.5. Then

(ξ,ξ) = (λ−Q,ξ) = (λ,ξ)−(Q,ξ)

≤(λ,ξ) = (λ,λ−Q)

≤(λ,λ).

Here we have used the inequalities(λ,β_i)≥0 and(ξ,βi)≥0, which hold for elements ofC. SinceW acts by orthogonal transformations, we have thus shown that

(µ,µ)≤(λ,λ). (3.28)

This implies thatX(λ)is contained in the intersection of the ball of radiuskλkwith the weight latticeP(g). This subset ofP(g)is finite, which proves (v). ut

The dual action ofsαonβ ∈h^∗issαβ=β− hβ,hαiα. The Weyl groupW ofgis the finite group generated by the root reflections.

The root lattice and weight lattice are contained in the real vector space h^∗ spanned by the roots, and the inner product(α,β)onh^∗ R

Rdefined by the Killing form is positive definite (see Corollary 2.5.22); for the classical groups this inner product is proportional to the standard inner product with(εi,εj) =δi j. Letkαk²= (α,α)be the associated norm. Since the root reflections are real orthogonal transformations, the Weyl group preserves this inner product and norm.

Proposition 3.2.7.Let(π,V)be a finite-dimensional representation ofgwith weight space decomposition

V= ^M

µ∈X(V)

V(µ).

Forα∈Φ let{eα,fα,hα}be a TDS triple associated withα. Define E=π(eα), F=π(fα), andτα=exp(E)exp(−F)exp(E)∈GL(V). Then

1.τ_απ(Y)τ_α⁻¹=π(s_αY)for Y∈h; 2.ταV(µ) =V(sαµ)for allµ∈h^∗; 3.dimV(µ) =dimV(s·µ)for all s∈W .

Hence the weightsX(V)and the weight multiplicity function m_V(µ) =dimV(µ) are invariant under W .

Proof. (1): From Theorem 2.3.6 and Proposition 2.3.3 we know thatE andF are nilpotent transformations. If X is any nilpotent linear transformation onV, then ad(X)is nilpotent on End(V)and we have

exp(X)Aexp(−X) =exp(adX)A for allA∈End(V). (3.29) (This follows from Lemma 1.6.1 and the fact that the differential of the representation Ad is the representation ad.) ForY ∈hwe have

ad(E)π(Y) =−π(ad(Y)e_α) =−hα,YiE. Hence ad(E)²(π(Y)) =0, and so from (3.29) we obtain

exp(E)π(Y)exp(−E) =π(Y)− hα,YiE. In particular, exp(E)π(hα)exp(−E) =π(hα)−2E. We also have

(adE)F=π(hα) and (adE)²F=−hα,h_αiE=−2E. Hence(adE)³F=0, and so from (3.29) we obtain

exp(E)Fexp(−E) =F+π(hα)−E.

The linear map takinge_αto−f_α, f_α to−e_α, andh_αto−h_αis an automorphism of s(α)(onsl(2,C)this is the mapX7→ −X^t). Hence the calculations just made also

3.2 Irreducible Representations 155

prove that

exp(−F)π(Y)exp(F) =π(Y)− hα,YiF, exp(−F)Eexp(F) =E+π(hα)−F. Combining these relations we obtain

ταπ(Y)τ_α⁻¹=exp(E)exp(−F)π(Y)− hα,YiE exp(F)exp(−E)

=expEπ(Y)− hα,YiE− hα,Yiπ(hα) exp(−E)

=π(Y)− hY,αiπ(hα) =π(sαY).

(2): Let v∈V(µ) andY ∈h. Then by (1) we haveπ(Y)ταv=ταπ(sαY)v= hs_αµ,Yiταv. This shows thatταv∈V(sαµ).

(3): This follows from (2), sinceW is generated by the reflectionss_α. ut Remark 3.2.8.The definition of the linear transformationταcomes from the matrix identity_{1 1}

0 1 1 0

−1 1 1 1 0 1

= _{0 1}

−1 0

inSL(2,C), where the element on the right is a representative for the nontrivial Weyl group element.

Lemma 3.2.9.Let(π,V)be a finite-dimensional representation ofgand letX(V) be the set of weights of V . Ifλ∈X(V)thenλ−kα∈X(V)for all rootsα∈Φand all integers k between0andhλ,h_αi, inclusive, where h_α∈his the coroot toα. Proof. We may suppose that the integerm=hλ,hαi is nonzero. Sinces_α·λ = λ−mα, we have

dimV(λ) =dimV(λ−mα)

by Proposition 3.2.7. Take 06=v∈V(λ). Ifm>0 then from Theorem 2.3.6 and Proposition 2.3.3 we have 06=π(f_α)^kv∈V(λ−kα)fork=0,1, . . . ,m. Ifm<0 then likewise we have 06=π(eα)⁻^kv∈V(λ−kα)fork=0,−1, . . . ,m. This shows

thatλ−kα∈X(V). ut

We say that a subsetΨ⊂P(g)isΦ-saturatedif for allλ ∈Ψ andα ∈Φ, one hasλ−kα∈Ψfor all integerskbetween 0 andhλ,h_αi. In particular, aΦ-saturated set is invariant under the Weyl groupW, sinceW is generated by the reflections s_α(λ) =λ− hλ,h_αiα. An element λ ∈Ψ is calledΦ-extreme if for allα ∈Φ, eitherλ+α∈/Ψorλ−α∈/Ψ.

From Lemma 3.2.9 the set of weights of a finite-dimensional representation ofg isΦ-saturated, and the highest weight isΦ-extreme. Using these notions we now show how to construct the complete set of weights of an irreducible representation starting with the highest weight.

Proposition 3.2.10.Let V be the finite-dimensional irreducibleg-module with highest weightλ.

1.X(V)is the smallestΦ-saturated subset of P(g)containingλ.

2. The orbit ofλ under the Weyl group is the set ofΦ-extreme elements ofX(V).

Proof. (1): LetΨ⁰⊂X(V)be the smallestΦ-saturated subset ofP(g)containing λ. IfΨ⁰⁰=X(V)\Ψ⁰ were nonempty, then it would contain a maximal element µ (relative to the root order). Sinceµ is not the highest weight, there must exist α∈Φ⁺such thatµ+α∈X(V). Let p,qbe the largest integers such thatµ+pα and µ−qα are weights ofV. Then p≥1, q≥0, and sα(λ+pα) =λ−qα by Corollary 2.4.5. Becauseµis maximal inΨ⁰⁰, we haveµ+pα∈Ψ⁰. However,Ψ⁰, beingΦ-saturated, is invariant underW, soµ−qα∈Ψ⁰also. Henceµ+kα∈Ψ⁰ for all integers k∈[−q,p]. In particular, taking k=0 we conclude thatµ∈Ψ⁰, which is a contradiction.

(2): SinceW·Φ=Φ, the set ofΦ-extreme elements ofX(V)is invariant under W. Thus it suffices to show that ifµ∈P₊₊(g)∩X(V)isΦ-extreme then µ=λ. Take a simple rootαiand corresponding reflections_i. Let the nonnegative integers p,qbe as in (1) relative toµandα_i. Sinces_i(µ+pα_i) =µ−qα_iwe haveq−p= hµ,H_ii. But sinceµis dominant,hµ,H_ii ≥0. Henceq≥p. Ifp≥1 thenq≥1 and µ±α_i∈X(V). This would contradict the assumption thatµisΦ-extreme. Hence p=0 andµ+α_i∈/X(V)fori=1, . . . ,l. We conclude thatµis a maximal element ofX(V). But we have already shown in Corollary 3.2.3 thatλis the unique maximal

weight. ut

On the set of dominant weights of a representation, the root order≺ has the following inductive property:

Proposition 3.2.11.Let V be any finite-dimensional representation ofg. Suppose µ∈P₊₊(g),ν∈X(V), andµ≺ν. Thenµ∈X(V).

Proof. By assumption,ν=µ+β, whereβ =∑^l_i=1niαi∈Q+. We proceed by induction on ht(β) =∑ni, the result being true ifβ=0. Ifβ6=0 then

0<(β,β) =

∑

ⁿⁱ^(β^,αⁱ⁾^.

Thus there exists an indexisuch thatn_i≥1 and(β,αi)>0. For this value ofiwe havehβ,H_ii ≥1. Sincehµ,H_ji ≥0 for all j, it follows thathν,H_ii ≥1. ButX(V) isΦ-saturated, soν⁰=ν−αi∈X(V). Setβ⁰=β−αi.Thenβ⁰∈Q₊, ht(β⁰) = ht(β)−1, andµ=ν⁰−β⁰. By induction,µ∈X(V). ut Corollary 3.2.12.Let L^λ be the finite-dimensional irreducibleg-module with highest weightλ. ThenX(L^λ)∩P₊₊(g)consists of allµ∈P₊₊(g)such thatµλ.

Corollary 3.2.12 and inequality (3.28) give an explicit algorithm for finding the weights ofL^λ. Take allβ ∈Q+such thatkλ−βk ≤ kλk(there are only finitely many) and write µ =λ−β in terms of the basis of fundamental weights. If all the coefficients are nonnegative, thenµis a weight ofL^λ. This gives all the dominant weights, and the Weyl group orbits of these weights make up the entire set of weights.

3.2 Irreducible Representations 157

Example

Consider the representation of sl(3,C) with highest weight λ =ϖ1+2ϖ2. The weights of this representation (as determined by the algorithm just described) are shown in Figure 3.4.

◦

α1

◦

α2

.. .. .. .. .. .. ..

. .. . . .. .

. .. . . .

⋆ λ=̟1+2̟2

⋆

s2(λ)

⋆

s1(λ)

⋆

s2s1(λ)

⋆

s1s2(λ)

⋆

s2s1s2(λ)

λ−⋄∗α2

⋄∗

λ−α1−α•2

•

............................................................................................................................... ...

......

.......

...

......

...

......

.......

...

......

...

......

.......

...

.......

...

......

.......

.....

... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... .. ........ ..

.. .. .. .

.. .. ..

.. .. .. ..

.. .. .. .. .

.. .. .. .. ..

.. .. .. .. .. .

.. .. .. .. .. ..

.. .. .. .. .. .. .

.. .. .. .. .. .. .. .

.. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .

.. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .

.. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .

.. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .

.. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .

.. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .

.. .. .. .. .. .. ..

.. .. .. .. .. .. .

.. .. .. .. .. ..

.. .. .. .. .. .

Fig. 3.4 Weights of the representationL^ϖ¹^+2ϖ²ofsl(3,C).

It is clear from the picture thatλ−α2andλ−α1−α2are the only dominant weights inλ−Q₊. The highest weight is regular, and its orbit under the Weyl group W =S₃ (indicated by ?) has |W|=6 elements (in Figure 5.1, s₁ ands₂ denote the reflections for the simple rootsα1andα2, respectively). Each of the weights λ−α2 and λ−α1−α2 is fixed by one of the simple reflections, and theirW- orbits (indicated by∗and•, respectively) have three elements. The weights? in theouter shellhave multiplicity one, by Proposition 3.2.7 and Corollary 3.2.3. The weights marked by∗have multiplicity one, whereas the weights marked by•have multiplicity two, so dimV^µ =15 (we shall see how to obtain these multiplicities in Section 8.1.2).

Dalam dokumen Graduate Texts in Mathematics 255 (Halaman 172-176)