Dominant Weights - Roots and Weights - Graduate Texts in Mathematics 255

3.1 Roots and Weights

3.1.4 Dominant Weights

3.1 Roots and Weights 141

weights is contained in the positive Weyl chamber (a cone of opening 60^◦), and the action of the Weyl group is generated by the reflections in the dashed lines that bound the chamber. The only root that is a dominant weight isα1+α2=ϖ1+ϖ2, and it is regular.

Fig. 3.2 Roots and dominant weights forSp(2,C).

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α1

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−2α◦1−α2

−α1◦−α2

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Type C₂:In this caseP(g)⊂Rα1+Rα2, but now(α1,α1) =2,(α2,α2) =4, and (α1,α2) =−2. Hence the angle between the two simple roots is 135^◦. The roots, fundamental weights, and some of the dominant weights forSp(2,C)are shown in Figure 3.2. Here there is a square symmetry of the set of roots and the weight lattice.

The set of dominant weights is contained in the positive Weyl chamber (a cone of opening 45^◦), and the action of the Weyl group is generated by the reflections in the dashed lines that bound the chamber. The only roots that are dominant weights are α₁+α₂and 2α₁+α2.

For the orthogonal groups the situation is a bit more subtle.

Proposition 3.1.19.When G=SO(2l+1,C), then P++(G)consists of all weights n1ϖ1+···+n_l₋₁ϖ_l₋₁+n_l(2ϖ_l)with ni∈Nfor i=1, . . . ,l. When G=SO(2l,C) with l≥2, then P₊₊(G)consists of all weights

n₁ϖ1+···+n_l−₂ϖ_l−2+n_l−₁(2ϖ_l−1) +n_l(2ϖ_l) +n_l+1(ϖ_l−1+ϖ_l) (3.16) with n_i∈Nfor i=1, . . . ,l+1.

Proof. In both cases we haveP(G) =∑^l_i=1Zεi. Thus the first assertion is obvious from the formulas forϖi. Now assume G=SO(2l,C). Then every weight of the form (3.16) is inP₊₊(G). Conversely, suppose thatλ=k₁ϖ1+···+klϖl∈P₊₊(g).

Then the coefficients ofεl−1andεlinλ are(kl+k_l₋₁)/2 and(kl−k_l₋₁)/2, respec- tively. Thusλ∈P(G)if and only ifk_l+k_l₋₁=2pandk_l−k_l₋₁=2qwithp,q∈Z. Supposeλ ∈P₊₊(G). Sincek_i≥0 for alli, we havep≥0 and−p≤q≤p. Set n_i=k_ifori=1, . . . ,l−2. Ifq≥0 setn_l₋₁=0,n_l=q, andn_l+1=p−q. Ifq≤0 set

3.1 Roots and Weights 143

n_l₋₁=−q,n_l=0, andn_l+1=p+q. Thenλ is given by (3.16) with all coefficients

n_i∈N. ut

The roots, fundamental weights, and some of the dominant weights forso(4,C) are shown in Figure 3.3. The dominant weights ofSO(4,C)are indicated by?, and the dominant weights ofso(4,C)that are not weights ofSO(4,C)are indicated by

•. In this case the two simple roots are perpendicular and the Dynkin diagram is disconnected. The root system and the weight lattice are the product of two copies of thesl(2,C)system (this occurs only in rank two), corresponding to the isomor- phismso(4,C)∼=sl(2,C)⊕sl(2,C). The set of dominant weights is contained in the positive Weyl chamber, which is a cone of opening 90^◦. The dominant weights forSO(4,C)are the nonnegative integer combinations of 2ϖ₁, 2ϖ₂, andϖ1+ϖ2in this case.

The root systems of typesB₂ andC₂ are isomorphic; to obtain the roots, fundamental weights, and dominant weights forso(5,C), we just interchange the sub- scripts 1 and 2 in Figure 3.2.

Fig. 3.3 Roots and dominant

weights forg=so(4,C). ^...

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−◦α1

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⋆ ρ

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P++

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The definition of dominant weight depends on a choice of the system Φ⁺ of positive roots. We now prove that any weight can be transformed into a unique dominant weight by the action of the Weyl group. This means that the dominant weights give a cross section for the orbits of the Weyl group on the weight lattice.

Proposition 3.1.20.For everyλ ∈P(g)there isµ∈P++(g)and s∈W such that λ=s·µ. The weightµis uniquely determined byλ. Ifµis regular, then s is uniquely determined byλ and hence the orbit W·µhas|W|elements.

For each type of classical group the dominant weights are given in terms of the weights{εi}as follows:

1. Let G=GL(n,C)orSL(n,C). Then P++(g)consists of all weights

µ=k₁ε1+···+knεnwith k₁≥k₂≥ ··· ≥k_nand k_i−k_i+1∈Z. (3.17) 2. Let G=SO(2l+1,C). Then P++(g)consists of all weights

µ=k1ε1+···+k_lε_lwith k1≥k2≥ ··· ≥k_l≥0. (3.18)

Here2k_iand k_i−k_jare integers for all i,j.

3. Let G=Sp(l,C). Then P++(g)consists of allµsatisfying(3.18)with k_iintegers for all i.

4. Let G=SO(2l,C), l≥2. Then P₊₊(g)consists of all weights

µ=k₁ε1+···+klεlwith k₁≥ ··· ≥k_l₋₁≥ |k_l|. (3.19) Here2k_iand k_i−k_jare integers for all i,j.

The weightµis regular when all inequalities in(3.17),(3.18), or(3.19)are strict.

Proof. For a general reductive Lie algebra, the first part of the proposition follows from Proposition 3.1.12. We will prove (1)–(4) for the classical Lie algebras by explicit calculation.

(1): The Weyl group isW=S_n, acting onh^∗by permutingε₁, . . . ,ε_n. Ifλ∈P(g), then by a suitable permutationswe can makeλ =s·µ, whereµ=∑ⁿ_i=1k_iε_iwith k1≥ ··· ≥k_n. Clearlyµis uniquely determined byλ. Since

hµ,Hii=ki−ki+1, (3.20)

the integrality and regularity conditions forµare clear. Whenµis regular, the coefficientsk₁, . . . ,k_nare all distinct, soλ is fixed by no nontrivial permutation. Hence sis unique in this case.

(2): The Weyl group acts by all permutations and sign changes ofε₁, . . . ,ε_l. Given λ =∑^l_i=1λ_iε_i∈P(g), we can thus finds∈W such thatλ=s·µ, whereµsatisfies (3.18) and is uniquely determined byλ. Equations (3.20) hold fori=1, . . . ,l−1 in this case, andhµ,H_li=2k_l. The integrality conditions onk_iand the condition forµ to be regular are now clear. Evidently ifµis regular ands∈W fixesµthens=1.

Conversely, everyµ∈P++(g)is of the form

µ=n1ϖ1+···+n_lϖ_l, ni∈N.

A calculation shows thatµsatisfies (3.18) withk_i=n_i+···+nl−1+ (1/2)nl. (3): The Weyl groupW is the same as in (2), so the dominant weights are given by (3.18). In this casehµ,H_li=k_l, so the coefficientsk_iare all integers in this case.

The regularity condition is obtained as in (2).

(4): The Weyl group acts by all permutations ofε1, . . . ,ε_land all changes of aneven number of signs. So by the action ofW we can always make at leastl−1 of the coefficients ofλ positive. Then we can permute theεisuch thatµ=s·λ satisfies (3.19). In this case (3.20) holds fori=1, . . . ,l−1, and hµ,Hli=k_l₋₁+k_l. The integrality condition and the condition for regularity are clear. Just as in (2) we see that ifµis regular then the only element ofW that fixesµis the identity.

Conversely, ifµ∈P₊₊(g)thenµis given as

µ=n₁ϖ1+···+n_lϖl, n_i∈N.

3.1 Roots and Weights 145

Let k_i=n_i+···+n_l₋₂+ (1/2)(n_l₋₁+n_l) for i=1, . . . ,l−2. If we set k_l₋₁= (1/2)(n_l+n_l₋₁)andk_l= (1/2)(n_l−n_l₋₁), thenµsatisfies (3.19). ut There is a particular dominant weight that plays an important role in representa- tion theory and which already appeared in the proof of Theorem 3.1.9. Recall that the choice of positive roots gives a triangular decompositiong=n⁻+h+n⁺. Lemma 3.1.21.Defineρ∈h^∗by

hρ,Yi=1

2tr(ad(Y)|n⁺) =1

∑

α∈Φ⁺

hα,Yi

for Y∈h. Thenρ=ϖ1+···+ϖl. Henceρis a dominant regular weight.

Proof. Letsi∈W be the reflection in the rootαi. By (3.8) we have s_i(ρ) =−1

2αi+1

∑

β∈Φ⁺\{α_i}

β=ρ−αi.

But we also haves_i(ρ) =ρ− hρ,H_iiα_i by the definition ofs_i. Hencehρ,H_ii=1

andρ=ϖ1+···+ϖ_l. ut

Let gbe a classical Lie algebra. From the equationshρ,Hii=1 it is easy to calculate thatρis given as follows:

ρ= l

2ε1+l−2

2 ε2+··· −l−2 2 εl−l

2εl+1, (TypeA_l)

ρ= l−1

ε1+ l−3

ε2+···+3

2ε_l−1+1

2ε_l, (TypeB_l) ρ=lε₁+ (l−1)ε2+···+2ε_l₋₁+εl, (TypeC_l) ρ= (l−1)ε₁+ (l−2)ε₂+···+2ε_l₋₂+ε_l₋₁. (TypeD_l) See Figures 3.1, 3.2, and 3.3 for the rank-two examples.

Dalam dokumen Graduate Texts in Mathematics 255 (Halaman 160-164)