Root Reflections

3.1 Roots and Weights

3.1.2 Root Reflections

Lemma 3.1.4.Let G=SO(C^2l,B). The subgroup R_l⊂W_Gis normal, and W_Gis the semidirect product of R_l andπ¯(S_l). The action of W_Gon the coordinate functions inO[H]is by x_i7→ x_σ(i)±1 (i=1, . . . ,l), for every permutationσand choice±1 of exponents with anevennumber of negative exponents.

Proof. By the same argument as in the proof of Lemma 3.1.2 we see that the normal- izer ofHinO(C^2l,B)is given by theHcosets of the elementsβ_Fπ(σ)asσ ranges overS_l andF ranges over all subsets of{1, . . . ,l}. Sinceπ(σ)∈Norm_G(H),we haveβ_Fπ(σ)∈Norm_G(H)if and only if Card(F)is even. ut

3.1 Roots and Weights 133

The action of the Weyl group onh^∗can be expressed in terms of these reflection operators, as follows:

Lemma 3.1.6.Let W =Norm_G(H)/H be the Weyl group of G. Identify W with a subgroup ofGL(h^∗)by the natural action of W onX(H).

1. For everyα∈Φthere exists w∈W such that w acts onh^∗by the reflection s_α . 2. W·∆=Φ .

3. W is generated by the reflections{sα : α∈∆}. 4. If w∈W and wΦ⁺=Φ⁺then w=1.

5. There exists a unique element w0∈W such that w0Φ⁺=−Φ⁺.

Proof. We proceed case by case using the enumeration of∆ from Section 2.4.3 and the description ofW from Section 3.1.1. In all cases we use the characterization (3.5) ofs_α.

Type A (G=SL(l+1,C)): Here W ∼=S_l+1 acts on h^∗ by permutations of ε1, . . . ,εl+1. Letα=εi−εj. Then

hεk,hαi=







1 ifk=i,

−1 ifk=j, 0 otherwise.

This shows that s_αε_k=ε_σ(k), where σ is the transposition (i,j)∈S_l+1. Hence s_α∈W, which proves (1). Property (2) is clear, since the transpositionσ= (i+1,j) carries the simple rootε_i−εi+1toε_i−ε_jfor anyj6=i. Property (3) follows from the fact that any permutation is the product of transpositions(i,i+1). If the permutation σ preservesΦ⁺thenσ(i)<σ(j)for alli<jand henceσ is the identity, which proves (4). To prove (5), letσ∈S_l+1act byσ(1) =l+1,σ(2) =l, . . . ,σ(l+1) =1 and letw0∈W correspond toσ. Thenw0αi=ε_σ(i)−ε_σ(i+1)=−αl+1−i for i= 1, . . . ,l. Hencew₀is the desired element. It is unique, by (4).

The root systems of typesB_l,C_l, andD_l each contain subsystems of typeA_l−1 and the corresponding Weyl groups containS_l. Furthermore, the set of simple roots for these systems is obtained by adjoining one root to the simple roots forA_l−1. So we need consider only the roots and Weyl group elements not in theA_l−1subsystem in these remaining cases. We use the same notation for elements ofW in these cases as in Section 3.1.1.

Type B (G=SO(C^2l+1,B)): Forα=εiwe haves_αεi=−εi, whereass_αεj=εjif i6=j. Sos_αgives the action ofγionh^∗. Whenα=εi+εjwithi6=j, thens_αεi=−εj

ands_αfixesεkfork6=i,j. Hences_αhas the same action onh^∗asγiγjϕ(σ), where σ= (i j), proving (1). Sinceγjtransformsεi−εjintoεi+εjand the transposition σ = (il)interchangesεi andεl, we obtain (2). We know from Lemma 3.1.3 that W contains elements that act on h^∗ by s_α1, . . . ,s_αl−1 and generate a subgroup of W isomorphic toS_l. Combining this with the relationγ_i=ϕ(σ)γlϕ(σ), where σ = (il), we obtain (3). HenceW acts by orthogonal transformations ofh^∗_R. Ifw preservesΦ⁺thenwmust permute the set of short positive roots, and sowε_i=ε_σ(i)

for someσ ∈S_l. Arguing as in the case of typeA_l we then conclude that σ=1.

Letw0be the product of the cosetsγ_iHfori=1, . . . ,l. Thenw0acts by−Ionh^∗, so it is the desired element.

Type C (G=Sp(C^2l,Ω)): Sinces_2εi=s_εi, we can use Lemma 3.1.2 and the same argument as for typeB, replacingγibyτi. In this case an elementw∈W that pre- servesΦ⁺will permute the set{2ε_i}oflongpositive roots. Againw₀acts by−I.

Type D (G=SO(C^2l,B)): Forα =ε_i+ε_j the reflectionsα has the same action onh^∗asβ_iβ_jπ(σ), whereσ= (i j). This proves (1). We know from Lemma 3.1.4 thatWcontains elements that act onh^∗bysα₁, . . . ,sα_l−1 and generate a subgroup of W isomorphic toS_l. Since we can move the simple rootα_l =ε_l−1+ε_l toα by a permutationσ, we obtain (2).

This same permutation action conjugates the reflections_αl tos_α, so we get (3).

IfwpreservesΦ⁺then for 1≤i<j≤lwe have w(εi+εj) =ε_σ(i)±ε_σ(j)

for someσ∈S_l withσ(i)<σ(j). Henceσ is the identity andwε_i=εifor 1≤ i≤l−1. Sincewcan only change the sign of an even number of theεi, it follows thatwfixesεl, which proves (4). Iflis even, letw₀be the product of all the cosets β_iHfor 1≤i≤l. Thenw₀acts by−I. Iflis odd takew₀to be the product of these cosets for 1≤i≤l−1. In this case we havew₀α_l−1=−α_l,w₀α_l=−α_l−1, and w₀α_i=−α_ifori=1, . . . ,l−2, which shows thatw₀is the desired element. ut Now we consider a semisimple Lie algebragoverC(see Section 2.5); the reader who has omitted Section 2.5 can take gto be a classical semisimple Lie algebra in all that follows. Fix a Cartan subalgebrah⊂gand letΦ be the set of roots of hong(the particular choice ofhis irrelevant, due to Theorem 2.5.28). Choose a setΦ⁺of positive roots and let∆ ⊂Φ⁺ be the simple roots (see Section 2.5.3).

Enumerate∆={α1, . . . ,αl}. We introduce a basis forh^∗whose significance will be more evident later.

Definition 3.1.7.Thefundamental weights(relative to the choice of simple roots∆) are elements{ϖ1, . . . ,ϖ_l} ofh^∗dual to the coroot basis{αˇ1, . . . ,αˇ_l}forh^∗. Thus (ϖi,αˇj) =δi j for i,j=1, . . . ,l.

Forα ∈Φ define the root reflection sα by equation (3.4). Then sα is the or- thogonal reflection in the hyperplaneα^⊥ and acts as a permutation of the setΦ by Theorem 2.5.20 (4). In particular, the reflections in the simple root hyperplanes transform the fundamental weights by

s_αiϖj=ϖj−δi jαi. (3.7) Definition 3.1.8.TheWeyl groupof(g,h)is the groupW =W(g,h)of orthogonal transformations ofh^∗

Rgenerated by the root reflections.

We note thatW is finite, sincew∈W is determined by the corresponding permuta- tion of the finite setΦ. In the case of the Lie algebra of a classical group, Lemma 3.1.6 (3) shows that Definition 3.1.8 is consistent with that of Section 3.1.1.

3.1 Roots and Weights 135

Theorem 3.1.9.LetΦ be the roots,∆ ⊂Φ a set of simple roots, and W the Weyl group for a semisimple Lie algebragand Cartan subalgebrah⊂g. Then all the properties(1)–(5)of Lemma 3.1.6are satisfied.

Proof. Property (1) is true by definition ofW. LetW₀⊂W be the subgroup gener- ated by the reflections{s_α1, . . . ,s_αl}. The following geometric property is basic to all arguments involving root reflections:

Ifβ∈Φ⁺\ {α_k}, then sα_kβ ∈Φ⁺\ {α_k}. (3.8) (Thus the reflection in a simple root hyperplane sends the simple root to its negative and permutes the other positive roots.) To prove (3.8), letβ =∑iciαi. There is an index j6=ksuch thatcj>0, whereas insα_kβ the coefficients of the simple roots other thanα_kare the same as those ofβ. Henceallthe coefficients ofs_αkβ must be nonnegative.

Proof of (2): Letβ ∈Φ⁺\∆. We shall prove by induction on ht(β)that β ∈ W₀·∆. We can writeβ =∑ic_iα_i withc_i≥0 andc_i6=0 for at least two indicesi.

Furthermore, there must exist an indexksuch that(β,α_k)>0, since otherwise we would have(β,β) =∑_ic_i(β,α_i)≤0,forcingβ=0. We haves_αkβ∈Φ⁺by (3.8), and we claim that

ht(sα_kβ)<ht(β). (3.9)

Indeed,sα_kβ=β−d_kα_k, whered_k=2(β,α_k)/(α_k,α_k)>0 by (3.6). Thus insα_kβ the coefficient ofα_kisc_k−d_k<c_k. This proves (3.9)

By inductions_αkβ∈W₀·∆, henceβ∈W₀·∆. Thus we can writeβ =wα_jfor somew∈W₀. Hence

−β=w(−αj) =w s_αjαj

∈W₀∆. This completes the proof thatΦ=W₀·∆, which implies (2).

Proof of (3): Letβ∈Φ. Then by (2) there existw∈W₀and an indexisuch that β =wα_i. Hence forγ∈h^∗

Rwe have s_βγ =γ− 2(wαi,γ)

(wα_i,wαi)wα_i=w

w⁻¹γ−2(αi,w⁻¹γ) (α_i,α_i) α_i

= ws_αiw⁻¹ γ.

This calculation shows thats_β =wsα_iw⁻¹∈W₀, proving (3).

Proof of (4): Letw∈W and supposewΦ⁺=Φ⁺. Assume for the sake of con- tradiction thatw6=1. Then by (3) and (3.8) we can writew=s₁···s_r, wheres_j is the reflection relative to a simple rootαi_jandr≥2. Among such presentations ofw we choose one with the smallest value ofr. We have

s₁···s_r−1αir=−s₁···s_r−1s_rαir=−wα_ir ∈ −Φ⁺. Sinceα_ir∈Φ⁺, there must exist an index 1≤ j<rsuch that

s_jsj+1···s_r−1α_ir∈ −Φ⁺ and sj+1···s_r−1α_ir ∈Φ⁺.

(If j=r−1 then the productsj+1···s_r−1equals 1.) Hence by (3.8) we know that sj+1···sr−1αir =αi_j . (3.10) Setw₁=sj+1. . .sr−1∈W. Then (3.10) implies thatw₁s_r(w₁)⁻¹=s_j(as in the proof of (3)). We now use this relation to writewas a product ofr−2 simple reflections:

w=s1···s_j−1 w1s_r(w1)⁻¹

w1s_r=s1···s_j−1sj+1···s_r−1

(since(sr)²=1). This contradicts the minimality ofr, and hence we conclude that w=1, proving (4).

Proof of (5): Leth∈h^∗

Rbe a regular element. Define ρ=ϖ1+···+ϖ_l and choosew∈W to maximize(s(h),ρ). We claim that

(w(h),αj)>0 for j=1, . . . ,l. (3.11) To prove this, note thats_αjρ=ρ−αjfor j=1, . . . ,lby (3.7). Thus

(w(h),ρ)≥(sαjw(h),ρ) = (w(h),ρ)−(w(h),αˇj)(αj,ρ).

Hence (w(h),αj)≥0, since (αj,ρ) = (αj,αj)(αˇj,ρ)/2= (αj,αj)/2>0. But (w(h),α_j) = (h,w⁻¹α_j)6=0, sincehis regular. Thus we have proved (3.11).

In particular, taking h=−h₀, where h₀ is a regular element definingΦ⁺ (see Section 2.5.3), we obtain an elementw∈Wsuch that(w(h₀),α)<0 for allα∈Φ⁺. Setw₀=w⁻¹. Thenw₀Φ⁺=−Φ⁺. Ifw₁∈W is another element sendingΦ⁺to

−Φ⁺thenw0(w1)⁻¹preserves the setΦ⁺, and hence must be the identity, by (4).

Thusw0is unique. ut

Remark 3.1.10.The proof of Theorem 3.1.9 has introduced new arguments that will be used in the next proposition and in later sections. Forga classical Lie algebra the proof also furnishes a more geometric and less computational explanation for the validity of Lemma 3.1.6.

Definition 3.1.11.LetC={µ∈h^∗

R:(µ,αi)≥0 fori=1, . . . ,l}. ThenCis a closed convex cone in the Euclidean space h^∗

R that is called the positive Weyl chamber (relative to the choiceΦ⁺of positive roots).

Ifµ=∑^l_i=1c_iϖi, thenc_i= (µ,αˇi) =2(µ,αi)/(αi,αi). Hence

µ∈C if and only ifc_i≥0 fori=1, . . . ,l. (3.12) We shall also need thedual cone

3.1 Roots and Weights 137

C^∗={λ ∈h^∗_R :(λ,ϖi)≥0 fori=1, . . . ,l}. Ifλ=∑^l_i=1diαi, thendi=2(λ,ϖi)/(αi,αi). Hence

λ∈C^∗ if and only ifdi≥0 fori=1, . . . ,l. (3.13) We now prove that the positive Weyl chamber is afundamental domainfor the action ofW onh^∗

R.

Proposition 3.1.12.Ifλ∈h^∗

Rthen there existµ∈C and w∈W such that w·λ=µ.

The elementµis unique, and ifλ is regular then w is also unique.

Proof. (For a case-by-case proof of this result for the classical Lie algebras, see Proposition 3.1.20.) We use the dual coneC^∗to define a partial order onh^∗_R: say thatλµifµ−λ∈C^∗. This partial order is compatible with addition and multi- plication by positive scalars. For a fixedλthe set

{w·λ :w∈W andλ w·λ}

is nonempty and finite. Letµbe a maximal element in this set (relative to the partial order). Sincesα_iµ=µ−(µ,αˇi)αiandµis maximal, the inequality(µ,αˇi)<0 is impossible by (3.13). Thusµ∈C.

To prove uniqueness ofµ, we may assume thatλ,µ∈C\ {0}andw·λ=µfor somew∈W. We will use the same type of argument as in the proof of Theorem 3.1.9 (4). Writew=s₁···s_r, wheres_jis the reflection relative to a simple rootαi_j. Assumew6=1. Then there exists an indexksuch thatwα_k∈ −Φ⁺(since otherwise w∆⊂Φ⁺, which would implywΦ⁺=Φ⁺, contradicting Theorem 3.1.9 (4)). Thus there exists an index 1≤j≤rsuch that

s_js_j+1···s_rα_k∈ −Φ⁺ and s_j+1···s_rα_k∈Φ⁺.

(Ifj=rthen the products_j+1···s_requals 1.) Hence by (3.8) we haves_j+1···s_rαk= αi_j. Setw₁=s_j+1···s_r∈W. Thenw₁s_αk(w1)⁻¹=s_j. We can use this relation to writewas

w=s₁···s_j−1 w₁s_αk(w1)⁻¹

w₁=s₁···s_j−1s_j+1···s_rs_αk.

Hencew₁=ws_αk=s₁···s_j−1s_j+1···s_ris a product ofr−1 simple reflections. Since λ,µ∈Candwα_k∈ −Φ⁺, we have

0≥(µ,wα_k) = (w⁻¹µ,α_k) = (λ,α_k)≥0.

Hence(λ,α_k) =0; thussα_kλ =λ andw1λ =µ. Ifw16=1 we can continue this shortening process until we reach the identity. This proves thatλ =µ and thatw is the product of simple reflections that fixλ. In particular, ifλ is regular, then no

simple reflection fixes it, sow=1 in this case. ut