Fundamental Group

Highest Weight Theory

7.3 Weyl Character Formula

7.3.6 Fundamental Group

Here we finish the proof of Theorem 6.30. This is especially important in light of the Highest Weight Classification. Of special note, it shows that whenGis a simply connected compact Lie group with semisimple Lie algebra, then the irreducible rep- resentations are parametrized by the set of dominant algebraic weights, P. In turn, this also classifies the irreducible representations ofg(Theorem 4.16). At the oppo- site end of the spectrum, Theorem 6.30 shows that the irreducible representations of Ad(G)∼=G/Z(G)(Lemma 5.11) are parametrized by the dominant elements of the root lattice,R. The most general group lies between these two extremes.

Lemma 7.35.Let G be a compact connected Lie group with maximal torus T . Let G^sing =G\G^reg. Then G^singis a closed subset withcodimG^sing≥3in G.

Proof. It follows from Theorem 7.7 thatG^sing is closed and the mapψ : G/T × T^sing → G^sing is surjective. Moreover t ∈ T^sing if and only if there exists α ∈ ⁺(g_C), soξα(t)=1 so thatT^sing = ∪∈⁺(g_C)kerξα. As a Lie subgroup ofT, kerξα

is a closed subgroup of codimension 1. LetU_α= {gtg⁻¹|g∈Gandt ∈kerξα}, so thatG^sing = ∪∈⁺(g_C)U_α.

Recall thatzg(t)= {X ∈g|Ad(t)X = X}(Exercise 4.22). Since Ad(t)acts on g_αasξ_α(t), it follows thatg∩(g_−α⊕g_α)⊆z_g(t)whent ∈ kerξ_α. Now choose a standard embeddingϕ_α:SU(2)→Gcorresponding toαand letV_αbe the compact

7.3 Weyl Character Formula 171 manifoldV_α =G/(ϕα(SU(2))T)×kerξα. Observe that dimV_α =dimG−3 and thatψmapsV_αontoU_α. Therefore the precise version of this lemma is thatG^singis a finite union of closed images of compact manifolds each of which has codimension

3 with respect toG.

Thinking of a homotopy of loops as a two-dimensional surface, Lemma 7.35 cou- pled with standard transversality theorems ([42]), show that loops inGwith a base point inG^regcan be homotoped to loops inG^reg. As a corollary, it is straightforward to see that

π1(G)∼=π1(G^reg).

LetGbe a compact Lie group with maximal torusT. Recall from Theorem 7.7 that e^H ∈ T^reg if and only if H ∈ {H ∈ t | α(H) /∈ 2πiZfor all roots α}. The connected regions of{H ∈ t | α(H) /∈ 2πiZfor all roots α}are convex and are given a special name.

Definition 7.36.LetGbe a compact Lie group with maximal torusT. The connected components of{H ∈t|α(H) /∈2πiZfor all rootsα}are calledalcoves.

Lemma 7.37.Let G be a compact connected Lie group with maximal torus T and fix a base t0=e^H0∈T^regwith H0∈t.

(a)Any continuous loopγ : [0,1]→G^regwithγ (0)=t0can be written as γ (s)=cg_se^H(s)

with g0 =e, H(0) = H0, and the maps s → gsT ∈ G/T and s → H(s) ∈ t^reg continuous. In that case, g1∈ N(T)and

H(1)=Ad(g1)⁻¹H0+X_γ

for some X_γ ∈ 2πikerE. The element X_γ is independent of the homotopy class of γ.

(b)Write A0for the alcove containing H0. Keeping the same base t0, the map π1(G^reg)→ A₀∩ {wH₀+Z |w∈W((g_C)^∨)and Z∈2πi A(T)^∗} induced byγ → X_γ is well defined and bijective.

Proof. Using the Maximal Torus Theorem, writeγ (s)=cg_sτ(s)withτ(s)∈ T^reg, τ(0)=t0, andg0 =e. In fact, sinceψ:G/T ×T^reg→G^regis a covering, the lifts s→τ(s)∈T^regands→gsT ∈G/T are uniquely determined by these conditions and continuity. Since exp : t^reg → T^reg is also a local diffeomorphism (Theorem 5.14), there exists a unique continuous lifts→ H(s)∈t^regofτ, soH(0)=H0and γ (s)=cg_se^H(s).

Asγ is a loop, γ (0) = γ (1), soe^H0 = cg₁e^H(1). Becausee^H0 ande^H(1) are regular,T =ZG(e^H0)⁰=cg₁ZG(e^H(1))⁰=cg₁T, so thatg₁∈ N(T)is a Weyl group element. Writingw =Ad(g₁), it follows that H₀ ≡wH(1)modulo 2πikerE, the

kernel of exp : t → T. Therefore write H(1) = w⁻¹H₀ +X_γ for some X_γ ∈ 2πikerE.

To see thatX_γ is independent of the homotopy class ofγ, supposeγ: [0,1]→ G^reg withγ(0)=t₀is another loop and thatγ (s,t)is a homotopy betweenγ and γ. Thusγ (s,0) = γ (s),γ (s,1) = γ(s), andγ (0,t) = γ (1,t) = t0. Using the same arguments as above and similar notational conventions, writeγ(s)=cgse^H(s) and H(1) = w⁻¹H0+X_γ. Similarly, write γ (s,t) = cg_s,te^H(s,t) and H(1,s) = ws⁻¹H₀+X_γ(s). Notice thatw0 =w,w1 =w, X_γ(0)= X_γ, and X_γ(1)= X_γ. Since ws and X_γ(s)vary continuously with s and since W(T)and 2πikerE are discrete,wsandX_γ(s)are constant. This finishes part (a).

For part (b), first note that continuity of H(s)implies that H(1)is still in A0, so that the map is well defined. To see surjectivity, fix H ∈ A0 ∩ {wH0 + Z | w ∈ W((g_C)^∨)and Z ∈ 2πi A(T)^∗}and write H = w⁻¹H0 +Z for w ∈ W((g_C)^∨)and Z ∈ 2πikerE. Choose a continuous paths → g_s ∈ G, so that g₀ = eand Ad(g₁) = w. Let H(s) = H0+s(H−H0)∈ A0and consider the curveγ(s)=cg_se^H(s). Sinceγ(0)=t0andγ(1)= e^wH = e^H0+Z = t0,γis a loop with base pointt0. By construction,X_γ = H, as desired. To see injectivity, observe that ifX_γ =X_γwithγ (s)=cg_se^H(s)andγ(s)=cg_se^H(s), thenγ (s,t)= cg_se^{(1−t)H(s)+t H(s)}is a homotopy between the two.

Lemma 7.38.Let G be a compact connected Lie group with maximal torus T . (a)Each homotopy class in G with base e can be represented by a loop of the form

γ (s)=e^{s Xγ}

for some X_γ ∈2πikerE, i.e., for some Xγ in the kernel ofexp :t→T . The surjec- tive map from2πikerEtoπ1(G)induced by X_γ →γis a homomorphism.

(b) Fix an alcove A₀ and H₀ ∈ A₀. The above map restricts to a bijection on {Z ∈2πikerE|wH₀+Z ∈ A₀for somew∈W((g_C)^∨)}.

Proof. Lemma 7.37 shows that each homotopy class inGwith baset0can be repre- sented by a curve of the formγ (s)=cg_se^H(s)withH(1)=Ad(g1)⁻¹H0+X_γ for someX_γ ∈ 2πikerE. Using the homotopyγ (s,t)=cg_se^{(1−t)H(s)+t[H0+s(H(1)−H0)]}, we may assumeH(s)is of the formH(s)=H0+s

Ad(g1)⁻¹H0+X_γ −H0

. Translating back to the identity, it follows that each homotopy class inG with baseecan be represented by a curve of the form

γ (s)=e^−H0cgse^H0+s(^{Ad(g1)−1H0+Xγ−H0}).

Using the homotopyγ (s,t) = e^{−t H0}cg_se^{t H0+s}(^{tAd(g1)−1H0+Xγ−t H0}), we may assume γ (s) = cg_se^{s Xγ}. Finally, using the homotopy γ (s,t) = cg_ste^{s Xγ}, we may assume γ (s)=e^{s Xγ}. Verifying that the mapγ → X_γ is a homomorphism is straightforward and left as an exercise (Exercise 7.24). Part (b) follows from Lemma 7.37.

Note that a corollary of Lemma 7.38 shows that the inclusion map T → G induces a surjectionπ1(T)→π1(G).

7.3 Weyl Character Formula 173 Definition 7.39.Let G be a compact connected Lie group with maximal torus T. Theaffine Weyl groupis the group generated by the transformations oftof the form H →wH+Z forw∈W((g_C)^∨)andZ ∈2πi R^∨.

Lemma 7.40.Let G be a compact connected Lie group with maximal torus T . (a) The affine Weyl group is generated by the reflections across the hyperplanes α⁻¹(2πi n)forα∈(g_C)and n∈Z.

(b)The affine Weyl group acts simply transitively on the set of alcoves.

Proof. Recall that h_α ∈ R^∨ and notice the reflection across the hyperplane α⁻¹(2πi n)is given byrh_α,n(H)=rh_αH +2πi hα (Exercise 7.25). Since the Weyl group is generated by the reflectionsrh_α, part (a) is finished. The proof of part (b) is very similar to Theorem 6.43 and the details are left as an exercise (Exercise 7.26).

Theorem 7.41.Let G be a connected compact Lie group with semisimple Lie alge- bra and maximal torus T . Thenπ1(G)∼=kerE/R^∨∼= P/A(T).

Proof. By Lemma 7.38, it suffices to show that the loop γ (s) = e^{s Xγ}, X_γ ∈ 2πikerE, is trivial if and only if X_γ ∈ 2πi R^∨. For this, first consider the stan- dardsu(2)-triple corresponding to α ∈ (g_C) and letϕ_α : SU(2) → G be the corresponding embedding. The loop γ_α(s) = e^{2πi shα} is the image underϕ_α of the loops→diag(e^{2πi s},e^{−2πi s})inSU(2). AsSU(2)is simply connected,γ_αis trivial.

Thus there is a well-defined surjective map 2πikerE/2πi R^∨→π1(G).

It remains to see that it is injective. Fix an alcove A0 and H0 ∈ A0. Since 2πikerE ⊆ 2πi P^∨, A0 − X_γ is another alcove. By Lemma 7.40, there is a w ∈ W((g_C)^∨) and H ∈ 2πi R^∨, so that wH0 + H ∈ A0 − X_γ. Thus wH0 +(Xγ +H) ∈ A0. Because the loops → e^{s H} is trivial, we may use a ho- motopy onγ and assume H = 0, so thatwH0+X_γ ∈ A0. But as H0+0 ∈ A0, Lemma 7.38 shows thatγ must be homotopic to the trivial loops→e^s0. 7.3.7 Exercises

Exercise 7.12 Show that the functione^ρdescends to the maximal torus forSU(n), S O(2n), andSp(2n), but not forS O(2n+1).

Exercise 7.13 LetGbe a compact Lie group with a maximal torusT. Letu_ρ ∈it, so thatρ(H)=B(H,u_ρ)forH ∈t. Show thati tu_ρ ∈for small positivet.

Exercise 7.14 Show that the dominant analytically integral weights of SU(3)are all expressions of the formλ = nπ1+mπ2 forn,m ∈ Z^≥0 whereπ1, π2 are the fundamental weightsπ1= ²_31,2+¹_32,3andπ2= ¹_31,2+²_32,3. Conclude that

dimV(λ)= (n+1)(m+1)(n+m+2)

2 .

Exercise 7.15 LetGbe a compact Lie group with semisimplegand a maximal torus T. The set of dominant weight vectors are of the formλ=

iniπi where{πi}are the fundamental weights andni ∈Z^≥0. Verify the following calculations.

(1) ForG=SU(n),

dimV(λ)= 8

1≤i<j≤n

1+ni+ · · · +nj−1

j−i

.

(2) ForG=Sp(n), dimV(λ)= 8

1≤i<j≤m

1+ni+ · · · +nj−1

j−i

· 8

1≤i<j≤m

1+ni+ · · · +nj−1+2

nj+ · · · +nm−1

2n+2−i−j

· 8

1≤i≤m

1+ni+ · · · +nm−1+nm

n+1−i

.

(3) ForG=Spin_2m+1(R), dimV(λ)= 8

1≤i<j≤m

1+ni+ · · · +nj−1

j−i

· 8

1≤i<j≤m

1+ni+ · · · +nj−1+2

nj+ · · ·nm−1

+nm

2m+1−i−j

· 8

1≤i≤m

1+2(ni+ · · · +nm−1)+nm

2n+1−2i

.

(4) ForG=Spin_2m(R), dimV(λ)= 8

1≤i<j≤m

1+ni+nj−1

j−i

· 8

1≤i<j≤m

1+ni+ · · · +nj−1+2

nj+ · · · +nm−1 +nm

2m−i−j

. Exercise 7.16 For each groupGbelow, show that the listed representation(s)V of Ghas minimal dimension among nontrivial irreducible representations.

(1) ForG=SU(n),V is the standard representation onCⁿor its dual.

(2) ForG=Sp(n),V is the standard representation onC²ⁿ.

(3) ForG =Spin_2m+1(R)withm ≥2,V =C^2m+1and the action comes from the covering Spin_2m+1(R)→S O(2m+1).

(4) ForG = Spin_2m(R)withm > 4, V = C^2m and the action comes from the covering Spin_2m(R)→ S O(2m).

7.3 Weyl Character Formula 175 Exercise 7.17 LetG be a compact Lie group with a maximal torusT. SupposeV is a representation of G that possesses a highest weight of weightλ. If dimV = dimV(λ), show thatV ∼=V(λ)and, in particular, irreducible.

Exercise 7.18 Use Exercise 7.17 and the Weyl Dimension Formula to show that the following representationV ofGis irreducible:

(1) G=SU(n)withV =pCⁿ(c.f. Exercise 7.1).

(2) G=S O(n)withV =Hm(Rⁿ)(c.f.Exercise 7.2).

(3) G=S O(2n+1)withV =pC²ⁿ⁺¹, 1≤ p ≤n(c.f. Exercise 7.3).

(4) G=S O(2n)withV =pC²ⁿ, 1≤ p<n(c.f.Exercise 7.3).

(5) G=SU(n)withV =Vp,0(Cⁿ)(c.f. Exercise 7.5).

(6) G=SU(n)withV =V_0,q(Cⁿ)(c.f. Exercise 7.5).

(7) G=SU(n)withV =Hp,q(Cⁿ)(c.f. Exercise 7.5).

(8) G=Spin_2m+1(R)withV =S(c.f. Exercise 7.6).

(9) G=Spin_2m(R)withV =S^±(c.f. Exercise 7.6).

Exercise 7.19 Letλbe a dominant analytically integral weight ofU(n)and write λ=λ11+· · ·+λnn,λj∈Zwithλ1≥ · · · ≥λn. ForH=diag(H₁, . . . ,Hn)∈t, show that the Weyl Character Formula can be written as

χ_λ(e^H)=det

e(^λj+j−1)^Hk det

e^(j−1)Hk .

Exercise 7.20 LetGbe a compact connected Lie group with maximal torusT. (1) IfGis not Abelian, show that the dimensions of the irreducible representations ofGare unbounded.

(2) If gis semisimple, show that there are at most a finite number of irreducible representations of any given dimension.

Exercise 7.21 LetGbe a compact connected Lie group with maximal torusT. For λ∈(it)^∗, theKostant partition functionevaluated atλ,P(λ), is the number of ways of writingλ=

α∈⁺(g_C)m_ααwithm_α ∈Z^≥0. (1) As a formal sum of functions ont, show that

8

α∈⁺(g_C)

1+e^−α+e^−2α+ · · ·

=

λ

P(λ)e^−λ

to conclude that

1=

λ

P(λ)e^−λ 8

α∈⁺(g_C)

1−e^−α . For what values ofH ∈tcan this expression be evaluated?

(2) Themultiplicity,m_µ,of µin V(λ)is the dimension of theµ-weight space in V(λ). Thusχ_λ=

µm_µξ_µ. Use the Weyl Character Formula, part (1), and gather terms to show thatm_µis given by the expression

m_µ=

w∈W((g_C))

det(w)P(w(λ+ρ)−(µ+ρ)). This formula is called theKostant Multiplicity Formula.

(3) ForG=SU(3), calculate the weight multiplicities forV(_1,2+3_2,3).

Exercise 7.22 LetGbe a compact connected Lie group with maximal torusT. The multiplicity,m_µ,of V(µ)in V(λ)⊗V(λ)is the number of timesV(µ)appears as a summand inV(λ)⊗V(λ). Thusχ_λχ_λ =

µm_µχ_µ. Use part (1) of Exercise 7.21 and compare dominant terms to showm_µis given by the expression

m_µ=

w,w∈W((g_C))

det(ww)P

w(λ+ρ)+w(λ+ρ)−(µ+2ρ) . This formula is calledSteinberg’s Formula.

Exercise 7.23 LetGbe a compact connected Lie group with maximal torusT and α∈(g_C). Show that kerξαinT may be disconnected.

Exercise 7.24 Show that the mapγ → X_γ from Lemma 7.38 is a homomorphism.

Exercise 7.25 Let G be a compact connected Lie group with maximal torus T. Show that the reflection across the hyperplaneα⁻¹(2πi n)is given by the formula rh_α,n(H)=rh_αH+2πi nh_αforH ∈t.

Exercise 7.26 LetGbe a compact connected Lie group with maximal torusT. Show that the affine Weyl group acts simply transitively on the set of alcoves.