Roots with Respect to a Maximal Torus

2.4 The Adjoint Representation

2.4.1 Roots with Respect to a Maximal Torus

Throughout this sectionGwill denote a connected classical group of rankl. ThusG isGL(l,C),SL(l+1,C),Sp(C^2l,Ω),SO(C^2l,B), orSO(C^2l+1,B), where we take asΩ andBthe bilinear forms (2.6) and (2.9). We setg=Lie(G). The subgroupH of diagonal matrices inGis a maximal torus of rankl, and we denote its Lie algebra byh. In this section we will study the regular representationπ ofHon the vector spaceggiven byπ(h)X=hX h⁻¹forh∈HandX∈g.

Let x₁, . . . ,x_l be the coordinate functions on H used in the proof of Theorem

2.1.5. Using these coordinates we obtain an isomorphism between the groupX(H) of rational characters ofHand the additive groupZ^l(see Lemma 2.1.2). Under this isomorphism,λ= [λ₁, . . . ,λ_l]∈Z^lcorresponds to the characterh7→h^λ, where

h^λ=

∏

^l

k=1

x_k(h)^λk, forh∈H. (2.20) Forλ,µ∈Z^landh∈Hwe haveh^λh^µ=h^λ+µ.

For making calculations it is convenient to fix the following bases forh^∗: (a) Let G=GL(l,C). Define hε_i,Ai=a_i for A=diag[a₁, . . . ,a_l]∈h. Then

{ε₁, . . . ,ε_l}is a basis forh^∗.

(b) Let G=SL(l+1,C). Thenhconsists of all diagonal matrices of trace zero.

With an abuse of notation we will continue to denote the restrictions tohof the linear functionals in (a) byεi. The elements ofh^∗can then be written uniquely as∑^l+1_i=1λiεi with λi∈C and ∑^l+1_i=1λi=0. A basis forh^∗is furnished by the functionals

εi− 1

l+1(ε1+···+εl+1) for i=1, . . . ,l.

(c) LetGbeSp(C^2l,Ω)orSO(C^2l,B). For i=1, . . . ,l define hεi,Ai=a_i, where A=diag[a1, . . . ,a_l,−a_l, . . . ,−a₁]∈h.Then{ε1, . . . ,εl}is a basis forh^∗. (d) LetG=SO(C^2l+1,B). For A=diag[a1, . . . ,a_l,0,−a_l, . . . ,−a₁]∈h and i=

1, . . . ,l define hεi,Ai=a_i. Then{ε1, . . . ,εl}is a basis forh^∗.

We define P(G) ={dθ : θ ∈X(H)} ⊂h^∗. With the functionalsε_i defined as above, we have

P(G) =

l M

k=1

Zεk. (2.21)

Indeed, givenλ=λ₁ε₁+···+λ_lε_l withλ_i∈Z, let e^λ denote the rational character ofHdetermined by[λ₁, . . . ,λ_l]∈Z^las in (2.20). Every element ofX(H)is of this form, and we claim that de^λ(A) =hλ,AiforA∈h. To prove this, recall from Section 1.4.3 thatA∈hacts by the vector field

X_A=

∑

^l

i=1hεi,Aix_i ∂

∂x_i

onC[x1,x⁻₁¹, . . . ,x_l,x⁻_l¹]. By definition of the differential of a representation we have de^λ(A) =X_A(x^λ₁¹···x^λ_l^l)(1) =

∑

^l

i=1

λ_ihε_i,Ai=hλ,Ai

as claimed. This proves (2.21). The mapλ7→e^λ is thus an isomorphism between the additive groupP(G)and the character groupX(H), by Lemma 2.1.2. From (2.21) we see thatP(G)is alattice(free abelian subgroup of rankl)inh^∗, which is called the weight latticeof G(the notationP(G)is justified, since all maximal tori are conjugate inG).

We now study the adjoint action ofHandhong. Forα∈P(G)let

2.4 The Adjoint Representation 93

g_α ={X∈g: hX h⁻¹=h^αX for allh∈H}

={X∈g: [A,X] =hα,AiXfor allA∈h}.

(The equivalence of these two formulas forg_αis clear from the discussion above.) Forα=0 we haveg₀=h, by the same argument as in the proof of Theorem 2.1.5.

Ifα6=0 andg_α6=0 thenαis called arootofHongandg_αis called aroot space.

Ifαis a root then a nonzero element ofg_αis called aroot vectorforα. We call the setΦof roots theroot systemofg. Its definition requires fixing a choice of maximal torus, so we writeΦ=Φ(g,h)when we want to make this choice explicit. Applying Proposition 2.1.3, we have theroot space decomposition

g=h⊕^M

α∈Φ

g_α. (2.22)

Theorem 2.4.1.Let G⊂GL(n,C)be a connected classical group, and let H⊂G be a maximal torus with Lie algebrah. LetΦ⊂h^∗be the root system ofg.

1.dimg_α=1for allα∈Φ.

2. Ifα∈Φ and cα∈Φfor some c∈Cthen c=±1.

3. The symmetric bilinear form(X,Y) =tr_Cⁿ(XY)ongis invariant:

([X,Y],Z) =−(Y,[X,Z]) for X,Y,Z∈g. 4. Letα,β ∈Φandα6=−β. Then(h,g_α) =0and(gα,g_β) =0. 5. The form(X,Y)ongis nondegenerate.

Proof of (1): We shall calculate the roots and root vectors for each type of clas- sical group. We take the Lie algebras in the matrix form of Section 2.1.2. In this realization the algebras are invariant under the transpose. ForA∈handX ∈gwe have[A,X]^t=−[A,X^t]. Hence ifX is a root vector for the rootα, thenX^t is a root vector for−α.

Type A:LetGbeGL(n,C)orSL(n,C). ForA=diag[a1, . . . ,a_n]∈hwe have [A,e_{i j}] = (ai−a_j)ei j=hεi−εj,Aie_{i j}.

Since the set{e_{i j}: 1≤i,j≤n,i6=j}is a basis ofgmoduloh, the roots are {±(ε_i−ε_j): 1≤i<j≤n},

each with multiplicity 1. The root spaceg_λ isCe_{i j}forλ=ε_i−ε_j.

Type C:LetG=Sp(C^2l,Ω). Label the basis forC^2l as e_±1, . . . ,e_±l, wheree_−i= e_2l+1−i. Lete_i,jbe the matrix that takes the basis vectore_jtoe_iand annihilatese_kfor k6=j(hereiandjrange over±1, . . . ,±l). SetX_εi−εj=e_i,j−e_−j,−ifor 1≤i,j≤l, i6=j. ThenX_εi−εj∈gand

[A,X_εi−εj] =hεi−εj,AiX_εi−εj, (2.23)

forA∈h. Henceε_i−ε_j is a root. These roots are associated with the embedding gl(l,C) //ggiven byY 7→h

Y 0

0−s_lY^ts_l

i

forY ∈gl(l,C), wheres_l is defined in (2.5). Set X_εi+εj =e_i,−j+e_j,−i, X_−εi−εj =e_−j,i+e_−i,jfor 1≤i<j≤l, and set X_2εi =e_i,−ifor 1≤i≤l. These matrices are ing, and

[A,X_±(εi+εj)] =±hεi+εj,AiX_±(εi+εj)

forA∈h. Hence±(εi+εj)are roots for 1≤i≤j≤l. From the block matrix form (2.8) ofgwe see that

{X_{±(εi−εj)},X_±(εi+εj) : 1≤i<j≤l} ∪ {X_±2εi : 1≤i≤l}

is a basis forgmoduloh. This shows that the roots have multiplicity one and are

±(εi−εj) and ±(εi+εj) for 1≤i<j≤l, ±2ε_k for 1≤k≤l. Type D: LetG=SO(C^2l,B). Label the basis for C^2l and defineX_εi−εj as in the case ofSp(C^2l,Ω). ThenXε_i−ε_j ∈gand (2.23) holds forA∈h, soε_i−ε_jis a root.

These roots arise from the same embeddinggl(l,C) //gas in the symplectic case. Set Xε_i+ε_j=e_i,−j−e_j,−i and X₋ε_i−ε_j=e_−j,i−e_−i,j for 1≤i<j≤l. Then X_±(εi+εj)∈gand

[A,X_±(ε

i+ε_j)] =±hεi+εj,AiX_±(ε

i+ε_j)

forA∈h. Thus±(εi+εj)is a root. From the block matrix form (2.7) forgwe see that

{X_{±(εi−εj)} : 1≤i<j≤l} ∪ {X_±(εi+εj) : 1≤i<j≤l}

is a basis forgmoduloh. This shows that the roots have multiplicity one and are

±(ε_i−ε_j)and±(ε_i+ε_j)for 1≤i<j≤l.

Type B:LetG=SO(C^2l+1,B). We embedSO(C^2l,B)intoGby equation (2.14).

Since H⊂SO(C^2l,B)⊂G via this embedding, the roots ±εi±εj of ad(h) on so(C^2l,B)also occur for the adjoint action ofhong. We label the basis forC^2l+1 as {e_−l, . . . ,e₋₁,e₀,e₁, . . . ,e_l}, where e₀=e_l+1ande_−i=e_2l+2−i. Lete_i,j be the matrix that takes the basis vectore_jtoe_iand annihilatese_kfork6= j(hereiand j range over 0,±1, . . . ,±l). Then the corresponding root vectors from type D are

X_εi−εj=e_i,j−e_−j,−i, X_εj−εi =e_j,i−e_−i,−j, Xε_i+ε_j =e_i,−j−e_j,−i, X₋ε_i−ε_j =e_−j,i−e_−i,j, for 1≤i<j≤l. Define

Xε_i=ei,0−e0,−i, X₋ε_i =e0,i−e₋i,0,

2.4 The Adjoint Representation 95

for 1≤i≤l. Then X_±ε_i ∈g and [A,X_±ε_i] =±hε_i,AiXε_i for A∈h. From the block matrix form (2.10) for g we see that {X_±ε_i : 1≤i≤l} is a basis for g moduloso(C^2l,B). Hence the results above forso(C^2l,B)imply that the roots of so(C^2l+1,B)have multiplicity one and are

±(εi−εj) and ±(εi+εj) for 1≤i< j≤l, ±ε_k for 1≤k≤l. Proof of (2): This is clear from the calculations above.

Proof of (3): LetX,Y,Z∈g. Since tr(AB) =tr(BA), we have ([X,Y],Z) =tr(XY Z−Y X Z) =tr(Y ZX−Y X Z)

=−tr(Y[X,Z]) =−(Y,[X,Z]). Proof of (4): LetX∈g_α,Y ∈g_β, andA∈h. Then

0= ([A,X],Y) + (X,[A,Y]) =hα+β,Ai(X,Y).

Sinceα+β 6=0 we can takeAsuch thathα+β,Ai 6=0. Hence(X,Y) =0 in this case. The same argument, but withY∈h, shows that(h,g_α) =0.

Proof of (5): By (4), we only need to show that the restrictions of the trace form toh×hand tog_α×g_−α are nondegenerate for allα∈Φ. SupposeX,Y∈h. Then

tr(XY) =

(∑ⁿ_i=1ε_i(X)ε_i(Y) if G=GL(n,C) or G=SL(n,C),

2∑^l_i=1εi(X)εi(Y) otherwise. (2.24)

From this it is clear that the restriction of the trace form toh×his nondegenerate.

Forα∈Φwe defineX_α∈g_αfor types A, B, C, and D in terms of the elementary matricese_i,jas above. ThenX_αX_−αis given as follows (the case ofGL(n,C)is the same as type A):

Type A: Xε_i−ε_jXε_j−ε_i =e_i,ifor 1≤i< j≤l+1.

Type B: X_εi−εjX_εj−εi=e_i,i+e_−j,−jandX_εi+ε_jX_−εj−εi=e_i,i+e_j,jfor 1≤i<j≤l.

AlsoX_εiX_−εi=e_i,i+e_0,0for 1≤i≤l.

Type C: Xε_i−ε_jXε_j−ε_i=e_i,i+e_−j,−jfor 1≤i<j≤landXε_i+ε_jX₋ε_j−ε_i=e_i,i+e_j,j for 1≤i≤ j≤l.

Type D: Xε_i−ε_jXε_j−ε_i=e_i,i+e_−j,−jandXε_i+ε_jX₋ε_j−ε_i=e_i,i+e_j,jfor 1≤i<j≤l.

From these formulas it is evident that tr(XαX₋α)6=0 for allα∈Φ. ut