Highest Weights

Highest Weight Theory

7.1 Highest Weights

7

As an example, recall that the action ofsu(2)C=sl(2,C)onVn(C²),n ∈Z^≥0, from Equation 6.7 is given by

E·(z^k₁zⁿ₂^−k)= −k z^k₁⁻¹zⁿ₂^−k+1 H·(z^k₁zⁿ₂^−k)=(n−2k)z^k₁z₂^n−k

F·(z^k₁zⁿ₂^−k)=(k−n)z^k₁⁺¹zⁿ₂^−k−1,

and recall that {Vn(C²) | n ∈ Z^≥0} is a complete list of irreducible representa- tions forSU(2). Takingit=diag(θ,−θ),θ ∈ R, there are two roots,±12, where 12(diag(θ,−θ))=2θ. Choosing⁺(sl(2,C))= {12}, it follows thatzⁿ₂is a high- est weight vector ofVn(C²)of weightn₂¹². Notice that the set of dominant analyti- cally integral weights is{n₂¹² |n∈Z^≥0}. Thus there is a one-to-one correspondence between the set of highest weights of irreducible representations ofSU(2)and the set of dominant analytically integral weights. This correspondence will be established for all connected compact groups in Theorem 7.34.

Theorem 7.3.Let G be a connected compact Lie group and V an irreducible repre- sentation of G.

(a)V has a unique highest weight,λ0.

(b)The highest weightλ0is dominant and analytically integral, i.e.,λ0∈ A(T).

(c)Up to nonzero scalar multiplication, there is a unique highest weight vector.

(d)Any weightλ∈(V)is of the form λ=λ0−

αi∈(g_C)

niαi

for ni∈Z^≥0.

(e)Forw ∈ W ,wV_λ = V_wλ, so thatdimV_λ =dimV_wλ. Here W(G)is identified with W((g_C)), as in Theorem 6.43 via theAd-action from Equation 6.35.

(f)Using the norm induced by the Killing form,λ ≤ λ0with equality if and only ifλ=wλ0forw∈W(g_C).

(g)Up to isomorphism, V is uniquely determined byλ0.

Proof. Existence of a highest weightλ0follows from the finite dimensionality ofV and Theorem 6.11. Letv0be a highest weight vector forλ0and inductively define Vn = Vn−1+n⁻Vn−1 whereV0 = Cv0. This defines a filtration on the(n⁻⊕t_C)- invariant subspaceV_∞= ∪nVn ofV. Ifα∈(g_C), then [g_α,n⁻]⊆n⁻⊕t_C. Since g_αV0 = 0, a simple inductive argument shows thatg_αVn ⊆ Vn. In particular, this suffices to demonstrate thatV_∞isg_C-invariant. Irreducibility impliesV =V_∞and part (d) follows.

Ifλ1 is also a highest weight, thenλ1 =λ0−

niαi andλ0 =λ1− miαi

for ni,mi ∈ Z^≥0. Eliminating λ1 andλ0 shows that −

niαi =

miαi. Thus

−ni =mi, so thatni =mi=0 andλ1=λ0. Furthermore, the weight decomposition shows thatV_∞∩V_λ0 =V0=Cv0, so that parts (a) and (c) are complete.

The proof of part (e) is done in the same way as the proof of Theorem 6.36. For part (b), notice thatr_αiλ0is a weight by part (e). Thus

7.1 Highest Weights 153 λ0−2B(λ0, αi)

B(αi, αi)αi =λ0−

αj∈(g_C)

njαj

for nj ∈ Z^≥0. Hence 2^B_B^(λ_(α^0,αi)

i,αi) = ni, so that B(λ0, αi) ≥ 0 andλ0 is dominant.

Theorem 6.27 shows thatλ0(in fact, any weight ofV) is analytically integral.

For part (f), Theorem 6.43 shows that it suffices to takeλdominant by using the Weyl group action. Writeλ=λ0−

niαi. Solving forλ0and using dominance in the second line,

λ0²= λ²+2

αi∈(g_C)

niB(λ, αi)+++

+++

αi∈(g_C)

niαi

++++

+

2

≥ λ²+++

+++

αi∈(g_C)

niαi

++++

+

2

≥ λ².

In the case of equality, it follows that

αi∈(g_C)niαi =0, so thatni =0 andλ=λ0. For part (g), supposeVis an irreducible representation ofGwith highest weight λ0 and corresponding highest weight vector v₀. Let W = V ⊕ V and define Wn = Wn−1 +n⁻Wn−1, where W0 = C(v0, v₀). As above, W_∞ = ∪nWn is a subrepresentation of V ⊕V. IfU is a nonzero subrepresentation ofW_∞, thenU has a highest weight vector,(u0,u₀). In turn, this means thatu0 andu₀ are highest weight vectors ofV andV, respectively. Part (a) then shows thatC(u0,u₀)=W0. ThusU =W_∞andW_∞is irreducible. Projection onto each coordinate establishes

theG-intertwining mapV ∼=V.

The above theorem shows that highest weights completely classify irreducible representations. It only remains to parametrize all possible highest weights of irre- ducible representations. This will be done in §7.3.5 where we will see there is a bijection between the set of dominant analytically integral weights and irreducible representations ofG.

Definition 7.4.LetG be connected and letV be an irreducible representation ofG with highest weightλ. AsVis uniquely determined byλ, writeV(λ)forV and write χλfor its character.

Lemma 7.5.Let G be connected. If V(λ)is an irreducible representation of G, then V(λ)^∗ ∼= V(−w0λ), where w0 ∈ W((g_C)) is the unique element mapping the positive Weyl chamber to the negative Weyl chamber (c.f. Exercise 6.40).

Proof. SinceV(λ)is irreducible, the character theory of Theorems 3.5 and 3.7 show that V(λ)^∗ is irreducible. It therefore suffices to show that the highest weight of V(λ)^∗is−w0λ.

Fix aG-invariant inner product,(·,·), onV(λ), so thatV(λ)^∗= {µv|v∈V(λ)}, whereµv(v) = (v, v)forv ∈ V(λ). By the invariance of the form,gµv = µgv

forg ∈ G, so that Xµv =µXv for X ∈ g. Since(·,·)is Hermitian, it follows that Zµv =µθ(Z)vforZ ∈g_C.

Letvλbe a highest weight vector forV(λ). IdentifyingW(G)withW((g_C)^∨) andW((g_C))as in Theorem 6.43 via the Ad-action of Equation 6.35, it follows from Theorem 7.3 that w0v_λ is a weight vector of weight w0λ (called the lowest weight vector). Asθ(Y)= −Y forY ∈itand since weights are real valued onit, it follows thatµ_w0vλis a weight vector of weight−w0λ.

It remains to see that n⁻w0v_λ = 0 since Lemma 6.14 shows θn⁺ = n⁻. By construction,w0⁺(g_C)=⁻(g_C)andw₀²=I, so that Ad(w0)n⁻=n⁺. Thus

n⁻w0vλ=w0

Ad(w₀⁻¹)n⁻

vλ=w0n⁺vλ=0

and the proof is complete.

7.1.1 Exercises

Exercise 7.1 Consider the representation of SU(n)onpCⁿ. For T equal to the usual set of diagonal elements, show that a basis of weight vectors is given by vectors of the formel1∧ · · · ∧elp with weightp

i=1li. Verify that onlye₁∧ · · · ∧epis a highest weight to conclude thatp

Cⁿis an irreducible representation ofSU(n)with highest weightp

i=1i.

Exercise 7.2 Recall that Vp(Rⁿ), the space of complex-valued polynomials onRⁿ homogeneous of degree p, andHp(Rⁿ), the harmonic polynomials, are representa- tions ofS O(n). LetT be the standard maximal torus given in §5.1.2.3 and §5.1.2.4, lethj = E_2j−1,2j −E₂j,2j−1 ∈ t, 1 ≤ k ≤ m ≡ ;_n

2

<, i.e.,hj is an embedding of 0 1

−1 0

, and letj ∈t^∗be defined byj(hj)= −iδj,j(c.f. Exercise 6.14).

(1)Show thathjacts onVp(Rⁿ)by the operator−x2j∂x_2j−1+x2j−1∂x_2j.

(2)Forn=2m+1, conclude that a basis of weight vectors is given by polynomials of the form

(x₁+i x₂)^j1· · ·(x_2m−1+i x_2m)^jm(x₁−i x₂)^k1· · ·(x_2m−1−i x_2m)^km x^l_2m⁰ ₊₁, l₀+

i ji+

iki = p, each with weight

i(ki−ji)i.

(3)Forn =2m, conclude that a basis of weight vectors is given by polynomials of the form

(x1+i x2)^j1· · ·(xn−1+i xn)^jm(x1−i x2)^k1· · ·(xn−1−i xn)^km,

i ji+

iki =p, each with weight

i(ki− ji)i.

(4)Using the root system ofso(n,C)and Theorem 2.33, conclude that the weight vector(x₁−i x₂)^p of weight p1 must be the highest weight vector ofHp(Rⁿ)for n ≥3.

(5) Using Lemma 2.27, show that a basis of highest weight vectors for Vp(Rⁿ)is given by the vectors(x₁−i x₂)^p−2jx^2jof weight(p−2j)1, 1≤ j ≤m.

Exercise 7.3 Consider the representation ofS O(n)onp

Cⁿ and continue the no- tation from Exercise 7.2.

7.1 Highest Weights 155 (1)Forn=2m+1, examine the wedge product of elements of the forme₂j−1±i e₂j

as well as e_2m+1 to find a basis of weight vectors (the weights will be of the form

±j1· · · ±jr with 1 ≤ j₁ < . . . < jr ≤ p). For p ≤ m, show that only one is a highest weight vector and conclude thatpCⁿ is irreducible with highest weight p

i=1i.

(2)Forn=2m, examine the wedge product of elements of the forme2j−1±i e2j to find a basis of weight vectors. Forp<m, show that only one is a highest weight vec- tor and conclude thatpCⁿis irreducible with highest weightp

i=1i. Forp=m, show that there are exactly two highest weights and that they arem−1

i=1 i±m. In this case, conclude thatmCⁿis the direct sum of two irreducible representations.

Exercise 7.4 Let G be a compact Lie group, T a maximal torus, and ⁺(g_C) a system of positive roots with respect tot_Cwith corresponding simple system(g_C).

(1)IfV(λ)andV(λ)are irreducible representations ofG, show that the weights of V(λ)⊗V(λ)are of the formµ+µ, whereµis a weight ofV(λ)andµis a weight ofV(λ).

(2)By looking at highest weight vectors, showV(λ+λ)appears exactly once as a summand inV(λ)⊗V(λ).

(3)SupposeV(ν)is an irreducible summand ofV(λ)⊗V(λ)and write the highest weight vector ofV(ν)in terms of the weights ofV(λ)⊗V(λ). By considering a term in which the contribution fromV(λ)is as large as possible, show thatν=λ+µfor a weightµofV(λ).

Exercise 7.5 Recall thatVp,q(Cⁿ)from Exercise 2.35 is a representations ofSU(n) on the set of complex polynomials homogeneous of degree pinz1, . . . ,zn and ho- mogeneous of degreeq inz1, . . . ,znand thatHp,q(Cⁿ)is an irreducible subrepre- sentation.

(1) If H = diag(t₁, . . . ,tn) with

jtj = 0, show that H acts on Vp,q(Cⁿ) as

jtj(−zj∂zj +zj∂zj).

(2)Conclude thatz^k₁¹· · ·z^k_nⁿz₁^l1· · ·znl_n

,

jkj =pand

jlj =q, is a weight vec- tor of weight

j(lj−kj)j.

(3)Show that−pnis a highest weight ofVp,0(Cⁿ).

(4)Show thatq1is a highest weight ofV_0,q(Cⁿ).

(5)Show thatq1−pnis the highest weight ofHp,q(Cⁿ).

Exercise 7.6 Since Spin_n(R)is the simply connected cover ofS O(n),n ≥ 3, the Lie algebra of Spin_n(R)can be identified withso(n)(a maximal torus for Spin_n(R) is given in Exercise 5.5).

(1)Forn=2m+1, show that the weights of the spin representationSare all weights of the form ¹₂(±1· · · ±m)and that the highest weight is ¹₂(1+ · · · +m). (2) Forn = 2m, show that the weights of the half-spin representation S⁺ are all weights of the form ¹₂(±1· · · ±m)with an even number of minus signs and that the highest weight is ¹₂(1+ · · · +m−1+m).

(3) Forn = 2m, show that the weights of the half-spin representation S⁻ are all weights of the form ¹₂(±1· · · ±m)with an odd number of minus signs and that the highest weight is ¹₂(1+ · · · +m−1−m).