Theorem of the Highest Weight

3.2 Irreducible Representations

3.2.1 Theorem of the Highest Weight

Throughout this sectiongwill be a semisimple Lie algebra. Fix a Cartan subalgebra hof gand letΦ be the root system ofgwith respect to h. Choose a set Φ⁺ of positive roots inΦ. Ifα∈Φ⁺then letg_α denote the corresponding root space. As in Section 2.5.3, we definen⁺=^L_α∈Φ+g_α and n⁻=^L_α∈Φ+g_−α. We also write

s(α) =g_−α+ [gα,g_−α] +g_α,

as in Section 2.4.2. Thens(α)is a Lie subalgebra ofgthat has basisx∈g_α,h∈ [gα,g_−α], andy∈g_−αsuch that[x,y] =h,[h,x] =2x, and[h,y] =−2y. Recall that the elementh, called thecoroot associated withα, is unique, and is denoted byh_α. Let(π,V)be a representation ofg(we do not assume that dimV <∞). Ifλ∈h^∗ then we set

V(λ) ={v∈V :π(h)v=hλ,hivfor allh∈h}, as in the finite-dimensional case. We define

X(V) ={λ ∈h^∗:V(λ)6=0}.

We call an element ofX(V)aweightof the representationπand the spaceV(λ)the λ weight spaceofV. We note that ifα∈Φ⁺andx∈g_α thenxV(λ)⊂V(λ+α).

Furthermore, the weight spaces for different weights are linearly independent.

Put a partial order onh^∗by definingµ≺λifµ=λ−β1−···−βrfor some (not necessarily distinct)βi∈Φ⁺and some integerr≥1. We call this theroot order.

From (3.13) we see that on pairsλ,µ withλ−µ∈Q(g)this partial order is the same as the partial order defined by the dual positive Weyl chamber (see the proof of Proposition 3.1.12).

IfV is infinite-dimensional, thenX(V)can be empty. We now introduce a class ofg-modules that have weight-space decompositions with finite-dimensional weight spaces. Ifgis a Lie algebra thenU(g)denotes its universal enveloping algebra; every Lie algebra representation (π,V)of gextends uniquely to an associative algebra representation(π,V)ofU(g)(see Section C.2.1). We use module notation and write π(T)v=T vforT ∈U(g)andv∈V.

Definition 3.2.1.A g-moduleV is a highest-weight representation (relative to a fixed set Φ⁺ of positive roots) if there areλ ∈h^∗and a nonzero vector v₀∈V such that (1) n⁺v₀=0, (2) hv₀=hλ,hiv₀ for allh∈h, and (3) V=U(g)v₀.

Setb=h+n⁺. A vectorv₀satisfying properties (1) and (2) in Definition 3.2.1 is called ab-extremevector. A vectorv₀satisfying property (3) in Definition 3.2.1 is called ag-cyclic vectorfor the representation.

Lemma 3.2.2.Let V be a highest-weight representation ofgas in Definition3.2.1.

Then

V=Cv₀⊕^M

µ≺λ

V(µ), (3.21)

3.2 Irreducible Representations 149

withdimV(µ)<∞for allµ. In particular,dimV(λ) =1andλis the unique max- imal element inX(V), relative to the root order; it is called thehighest weightof V .

Proof. Sinceg=n⁻⊕h⊕n⁺, the Poincar´e–Birkhoff–Witt theorem implies that there is a linear bijectionU(n⁻)^NU(h)^NU(n⁺) //U(g)such that

Y⊗H⊗X7→Y HX forY ∈U(n⁻),H∈U(h), andX∈U(n⁺)

(see Corollary C.2.3). We haveU(h)U(n⁺)v0=Cv0, so it follows thatV=U(n⁻)v0. Thus V is the linear span of v0 together with the elements y1y2···yrv0, for all yj∈g_−β

j withβj∈Φ⁺andr=1,2, . . .. Ifh∈hthen hy₁y₂···y_rv₀=y₁y₂···y_rhv₀+

∑

^r

i=1

y₁···y_i−1[h,y_i]yi+1···y_rv₀

= hλ,hi −

r

i=1

∑

hβi,hi

y1···yrv0.

Thusy₁y₂···y_rv₀∈V(µ), whereµ=λ−β1−···−βrsatisfiesµ≺λ. This implies (3.21). We have dimg_−βj =1, and for a fixedµ there is only a finite number of choices ofβj∈Φ⁺such thatµ=λ−β1− ··· −βr. Hence dimV(µ)<∞. ut Corollary 3.2.3.Let(π,V)be a nonzero irreducible finite-dimensional representa- tion ofg. There exists a unique dominant integralλ ∈h^∗such thatdimV(λ) =1.

Furthermore, everyµ∈X(V)withµ6=λ satisfiesµ≺λ.

Proof. By Theorem 3.1.16 we know thatX(V)⊂P(g). Letλ be any maximal ele- ment (relative to the root order) in the finite setX(V). Ifα∈Φ⁺andx∈g_α then xV(λ)⊂V(λ+α). ButV(λ+α) =0 by the maximality ofλ. Henceπ(n⁺)V(λ) = 0. Let 06=v₀∈V(λ). ThenL=U(g)v₀6= (0)is ag-invariant subspace, and hence L=V. Now apply Lemma 3.2.2. The fact thatλ is dominant follows from the rep- resentation theory ofsl(2,C)(Theorem 2.3.6 applied to the subalgebras(α)). ut Definition 3.2.4.If(π,V)is a nonzero finite-dimensional irreducible representation ofgthen the elementλ ∈X(V)in Corollary 3.2.3 is called thehighest weight of V. We will denote it byλ_V.

We now use the universal enveloping algebraU(g)to prove that the irreducible highest-weight representations ofg(infinite-dimensional, in general) are in one-to- one correspondence withh^∗via their highest weights. By the universal property of U(g), representations ofgare the same as modules forU(g). We define aU(g)- module structure onU(g)^∗by

g f(u) =f(ug) forg,u∈U(g)and f ∈U(g)^∗. (3.22) Clearlyg f∈U(g)^∗and the mapg,f7→g fis bilinear. Also forg,g⁰∈U(g)we have

g(g⁰f)(u) =g⁰f(ug) = f(ugg⁰) = (gg⁰)f(u),

so definition (3.22) does give a representation ofU(g)onU(g)^∗. Here the algebra U(g)is playing the role previously assigned to the algebraic group G, while the vector spaceU(g)^∗is serving as the replacement for the space of regular functions onG.

Theorem 3.2.5.Letλ∈h^∗.

1. There exists an irreducible highest-weight representation (σ,L^λ) of g with highest weightλ.

2. Let (π,V) be an irreducible highest-weight representation of g with highest weightλ. Then(π,V)is equivalent to(σ,L^λ).

Proof. To prove (1) we useλ to define a particular element ofU(g)^∗as follows.

Sincehis commutative, the algebraU(h)is isomorphic to the symmetric algebra S(h), which in turn is isomorphic to the polynomial functions on h^∗. Using these isomorphisms, we define λ(H), forH ∈h, to be evaluation at λ. This gives an algebra homomorphismH7→λ(H)fromU(h)toC. IfH=h1···hnwithhj∈h, then

λ(H) =hλ,h₁i···hλ,h_ni.

For any Lie algebramthere is a unique algebra homomorphismε:U(m) //C such that

ε(1) =1 and ε(U(m)m) =0.

WhenZ∈U(m)is written in terms of a P–B–W basis,ε(Z)is the coefficient of 1. Combining these homomorphisms (whenm=n^±) with the linear isomorphism U(g)∼=U(n⁻)⊗U(h)⊗U(n⁺)already employed in the proof of Lemma 3.2.2, we can construct a unique element f_λ∈U(g)^∗that satisfies

f_λ(Y HX) =ε(Y)ε(X)λ(H) for Y ∈U(n⁻),H∈U(h), and X∈U(n⁺). Ifh∈h, thenY HX h=Y HhX+Y HZwithZ= [X,h]∈n⁺U(n⁺). Sinceε(Z) =0 andλ(Hh) =hλ,hiλ(H), we obtain

h f_λ(Y HX) =f_λ(Y HhX) +f_λ(Y HZ) =hλ,hif_λ(Y HX). (3.23) Furthermore, ifx∈n⁺thenε(x) =0, so we also have

x f_λ(Y HX) =f_λ(Y HX x) =ε(Y)λ(H)ε(X)ε(x) =0. (3.24) DefineL^λ ={g f_λ : g∈U(g)}to be theU(g)-cyclic subspace generated by f_λ relative to the action (3.22) ofU(g)onU(g)^∗, and letσbe the representation ofgon L^λ obtained by restriction of this action. Then (3.23) and (3.24) show that(σ,L^λ) is a highest-weight representation with highest weightλ. SinceHX∈U(h)U(n⁺) acts on f_λ by the scalarλ(H)ε(X), we have

3.2 Irreducible Representations 151

L^λ ={Y f_λ :Y∈U(n⁻)}. (3.25) We now prove thatL^λ is irreducible. Suppose 06=M⊂L^λ is ag-invariant sub- space. By Lemma 3.2.2 we see thatMhas a weight-space decomposition underh:

M=^M

µλ

M∩L^λ(µ).

If µλ then there are only finitely many ν∈h^∗such that µνλ. (This is clear by writingλ−µ,ν−µ, and λ−ν as nonnegative integer combinations of the simple roots.) Hence there exists a weight ofM, sayµ, that is maximal. Take 06= f ∈M(µ). Then n⁺f =0 by maximality of µ, and henceX f =ε(X)f for allX∈U(n⁺). By (3.25) we know that f =Y₀f_λ for someY₀∈U(n⁻). Thus for Y ∈U(n⁻),H∈U(h), andX∈U(n⁺)we can evaluate

f(Y HX) =X f(Y H) =ε(X)f(Y H) =ε(X)f_λ(Y HY0).

ButHY₀=Y₀H+[H,Y0], and[H,Y0]∈n⁻U(n⁻). Thus by definition of f_λwe obtain f(Y HX) =ε(X)ε(Y)ε(Y0)λ(H) =ε(Y0)f_λ(Y HX).

Thusε(Y₀)6=0 and f =ε(Y₀)f_λ. This implies thatM=L^λ, and henceL^λ is irre- ducible, completing the proof of (1).

To prove (2), we note by Lemma 3.2.2 thatV has a weight-space decomposition and dim(V/n⁻V) =1. Letp:V //V/n⁻V be the natural projection. Then the re- striction ofptoV(λ)is bijective. Fix nonzero elementsv₀∈V(λ)andu₀∈V/n⁻V such that p(v0) =u₀. The representationπ extends canonically to a representa- tion ofU(g)onV. ForX∈U(n⁺)we haveX v₀=ε(X)v0, forH∈U(h)we have Hv₀=λ(H)v0, while forY ∈U(n⁻)we haveY v₀≡ε(Y)v0 modn⁻V. Thus

p(Y HX v0) =ε(X)λ(H)p(Y v0) =ε(Y)ε(X)λ(H)u0=f_λ(Y HX)u0. (3.26) For anyg∈U(g)andv∈V, the vectorp(gv)is a scalar multiple ofu₀. If we fix v∈V, then we obtain a linear functionalT(v)∈U(g)^∗such that

p(gv) =T(v)(g)u₀ for allg∈U(g)

(since the mapg7→p(gv)is linear). The mapv7→T(v)defines a linear transforma- tionT :V //U(g)^∗, and we calculated in (3.26) thatT(v0) = f_λ. Ifx∈gthen

T(xv)(g)u0=p(gxv) =T(v)(gx)u0= (xT(v))(g)u0.

HenceT(xv) =xT(v). SinceVis irreducible, we haveV=U(g)v0, and soT(V) = U(g)f_λ=L^λ. ThusT is ag-homomorphism fromVontoL^λ.

We claim that Ker(T) =0. Indeed, ifT(v) =0 then p(gv) =0 for allg∈U(g).

HenceU(g)v⊂n⁻V. Sincen⁻V 6=V, we see thatU(g)vis a proper g-invariant

subspace ofV. ButV is irreducible, soU(g)v=0, and hencev=0. ThusT gives

an isomorphism betweenVandL^λ. ut

The vector spaceL^λ in Theorem 3.2.5 is infinite-dimensional, in general. From Corollary 3.2.3 a necessary condition for L^λ to be finite-dimensional is thatλ be dominant integral. We now show that this condition is sufficient.

Theorem 3.2.6.Let λ ∈h^∗ be dominant integral. Then the irreducible highest weight representation L^λ is finite-dimensional.

Proof. We writeX(λ)for the set of weights ofL^λ. Sinceλ is integral, we know from Lemma 3.2.2 thatX(λ)⊂P(g)⊂h^∗

R. Fix a highest-weight vector 06=v₀∈ L^λ(λ).

Letα∈∆be a simple root. We will show thatX(λ)is invariant under the action of the reflections_α onh^∗_R. The argument proceeds in several steps. Fix a TDS basis x∈g_α,y∈g_−α, andh= [x,y]∈h for the subalgebra s(α). Setn=hα,hiand v_j=y^jv0. Thenn∈N, sinceλ is dominant integral.

(i) Ifβ is a simple root andz∈g_β, thenzvn+1=0 .

To prove (i), suppose first thatβ6=α. Thenβ−αis not a root, since the coefficients of the simple roots are of opposite signs. Hence [z,y] =0, so in this casezv_j= y^jzv₀=0 . Now suppose thatβ=α. Then we may assume thatz=x. We know that xv_j=j(n+1−j)v_j−1by equation (2.16). Thusxvn+1=0.

(ii) The subspaceU(s(α))v0is finite-dimensional.

We have n⁺v_n+1=0 by (i), since the subspacesg_β withβ ∈∆ generaten⁺ by Theorem 2.5.24. Furthermorev_n+1∈L^λ(µ)withµ=λ−(n+1)α. Hence theg- submoduleZ=U(g)vn+1 is a highest-weight module with highest weightµ. By Lemma 3.2.2 every weightγofZsatisfiesγµ. Sinceµ≺α, it follows thatZis a properg-submodule ofL^λ. ButL^λis irreducible, soZ=0. The subspaceU(s(α))v₀ is spanned by{v_j: j∈N}, since it is a highest-weights(α)module. Butv_j∈Zfor

j≥n+1, sov_j=0 for j≥n+1. HenceU(s(α))v₀=Span{v₀, . . . ,v_n}. (iii) Letv∈L^λ be arbitrary and letF=U(s(α))v. Then dimF<∞.

SinceL^λ =U(g)v0, there is an integer jsuch thatv∈U_j(g)v0, whereU_j(g)is the subspace ofU(g)spanned by products of j or fewer elements ofg. By Leibniz’s rule[g,U_j(g)]⊂U_j(g). HenceF⊂U_j(g)U(s(α))v₀. SinceU_j(g)andU(s(α))v₀ are finite-dimensional, this proves (iii).

(iv) s_αX(λ)⊂X(λ).

Letµ∈X(λ)and let 06=v∈L^λ(µ). Since[h,s(α)]⊂s(α), the spaceF=U(s(α))v is invariant underhand the weights ofhonFare of the formµ+kα, wherek∈Z. We know thatF is finite-dimensional by (iii), so Theorem 2.3.6 implies thatF is equivalent to

F^(k1)L···^LF^(kr) (3.27)

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 ele- ments 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