Exercises

Roots and Associated Structures

6.4 Weyl Group

6.4.5 Exercises

146 6 Roots and Associated Structures

is path connected (Exercise 6.31). Thus there exists a piecewise linear pathγ (t) : [0,1] → itfromH towH that does not intersect. Modifyingγ (t)if necessary (Exercise 6.31), there is a partition{si}_i^N₌₁of [0,1], Weyl chambersCi withC₀=C andCN =wC, and rootsαi, so thatγ (si−1,si)⊆Ci, 1≤i ≤ N, andγ (si)∈h^⊥_αi, 1≤i ≤N−1.

Asγ (t)does not intersect, there is an entire ball,Bi, aroundγ (si)inh^⊥_αi (of codimension 1 init) lying on the boundary of bothCi−1andCi, 1≤i ≤N−1. For small nonzeroε, it follows that Bi+εh_α lies inCi−1 orCi, depending on the sign ofε. Sincerh_αi(Bi +εh_α) = Bi −εh_α and sincerh_αi preserves Weyl chambers, it follows thatrh_αiCi−1=Ci. In particular,rh_α1· · ·rh_αNwC=C.

Now supposew0 ∈ N(T)satisfies Ad(w0)C =C. To finish part (a), it suffices to show thatw0 ∈ T, so thatw0 acts by the identity operator onit. Let =(C) and defineρas in Equation 6.39. By Lemma 6.42, it follows thatw=, so that Ad(w0)ρ=ρ. Thusc_w0e^{i tuρ} =e^{i tuρ},t∈R.

Choose a maximal torus S of ZG(w0)⁰ containing w0. Note that S is also a maximal torus ofGby the Maximal Torus Theorem. Sincee^iRuρ is in turn contained in some (other) maximal torus of ZG(w0)⁰, Corollary 5.10 shows that there exists g∈ZG(w0)⁰, socge^iRuρ ⊆S. LetSbe the maximal torus ofGgiven byS=cg⁻¹S. Thenw0 ∈ S ande^iRuρ ⊆ S(c.f. Exercise 5.12). By definition ofρ and a simple a system, the root space decomposition ofg_Cshowsz_gC(u_ρ)=tsoz_g(i u_ρ)=t. But sinces⊆z_g(i u_ρ)=t, maximality impliess=t, and so S =T. Thusw0 ∈ T, as desired.

Part (b) is done in a similar fashion to part (a). Part (c) is a corollary of the proof

of part (a).

(1)Ifis the union of all intersections of distinct hyperplanes of the formh^⊥_α for α∈(g_C), show(it)\is path connected.

(2)Supposeγ (t): [0,1] → itis a piecewise linear path that does not intersect withγ (0)andγ (1)elements of (different) Weyl chambers. Show there is a piecewise linear pathγ(t): [0,1]→ itthat does not intersect, satisfiesγ(0)=γ (0)and γ(1)=γ (1),γ (si−1,si)⊆Ci, 1≤i ≤ N, andγ (si)∈ h^⊥_αi, 1≤i ≤ N −1, for some partition{si}i^N=0of [0,1], some Weyl chambersCi, and some rootsαi. Exercise 6.32 LetG be compact Lie group with semisimple Lie algebragandta Cartan subalgebra of g. Fix a basis of(it)^∗. With respect to this basis, thelexico- graphic order on(it)^∗ is defined by settingα > β if the first nonzero coordinate (with respect to the given basis) ofα−βis positive.

(1) Let = {α ∈ (g_C) | α > 0 andα = β1 +β2 for anyβi ∈ (g_C)with βi >0}. Showis a simple base of(g_C)with⁺(g_C)= {α ∈(g_C)|α >0} and⁻(g_C)= {α∈(g_C)|α <0}.

(2)Show that all simple systems arise in this fashion.

(3)Show that there is a uniqueδ ∈ ⁺(g_C), so thatδ > β,β ∈ ⁺(g_C)\{δ}. The rootδis called thehighest root. For the classical compact Lie groups, showδis given by the following table:

G SU(n) Sp(n)S O(E_2n)S O(E_2n+1) δ 1−n 21 1+2 1+2. (4)Show thatB(δ, β)≥0 for allβ ∈⁺(g_C).

(5)ForG =S O(E2n+1),n ≥2, show that there is another root besidesδsatisfying the condition in part (4).

Exercise 6.33 Let G be compact Lie group with semisimple Lie algebra g and t a Cartan subalgebra of g. Fix a simple system = {αi} of (g_C). For any β ∈ ⁺(g_C), show thatβ can be written as β = αi1 +αi2 + · · · +αiN, where αi₁+αi₂+ · · · +αi_k ∈⁺(g_C)for 1≤k≤N.

Exercise 6.34 LetGbe compact Lie group with a Cartan subalgebrat. Fix a simple systemof(g_C).

(1) Forα ∈ andβ ∈ ⁺(g_C)\{α}, writer_αβ = β −2^B_B^(β,α)_(α,α)αto show that β ∈⁺(g_C). Conclude thatr_α(⁺(g_C)\{α})=⁺(g_C)\{α}.

(2)Let

ρ=1 2

β∈⁺(g_C)

β

and conclude from part (1) thatr_αρ=ρ−α. Use the definition ofr_αto show that ρ=ρ.

Exercise 6.35 LetG be compact Lie group with semisimple Lie algebragandta Cartan subalgebra ofg. Fix a simple system = {αi}of(g_C)and letW((g_C))

148 6 Roots and Associated Structures be the subgroup ofW((g_C))generated{r_αi}.

(1)Given anyβ ∈(g_C), choosex∈(±β)^⊥not lying on any other root hyperplane.

For all sufficiently smallε >0, show thatx+εβlies in a Weyl chamberCand that β ∈=(C).

(2) Writeρfor the element of(it)^∗ satisfying 2^B_B^(ρ_(α^,αi)

i,αi) = 1 from Equation 6.39 (c.f. Exercise 6.34) and choosew∈W((g_C))so thatB(wρ, ρ)is maximal. By examiningB(rαiwρ, ρ), show thatwρ ∈C(). Conclude thatwβ∈. (3)Show thatr_β=w⁻¹r_wβw. Conclude thatW((g_C))=W((g_C)).

Exercise 6.36 LetG be compact Lie group with semisimple Lie algebragandta Cartan subalgebra ofg. Fix a simple system= {αi}of(g_C)and recall Exercise 6.35. Forw∈W((g_C)), letn(w)={β ∈⁺(g_C)|wβ∈⁻(g_C)}. Forw= I, writew=r_α1· · ·r_αN withNas small as possible. Thenr_α1· · ·r_αN is called areduced expression forw. Thelengthofw, with respect to, is defined byl(w)= N. For w=I,l(I)=0.

(1)Use Exercise 6.34 to show n(wr_αi)=

n(w)−1 ifwαi ∈⁻(g_C) n(w)+1 ifwαi ∈⁺(g_C).

Conclude thatn(w)≤l(w).

(2)Use Theorem 6.43 and induction on the length to show thatn(w)=l(w).

Exercise 6.37 (Chevalley’s Lemma) Let G be compact Lie group with semisim- ple Lie algebra g andt a Cartan subalgebra of g. Fix λ ∈ (it)^∗ and let W_λ = {w ∈ W((g_C)) | wλ = λ}. Choose a Weyl chamberC, so thatλ ∈ C and let

=(C).

(1)Ifβ ∈(g_C)withB(λ, β) >0, show thatβ∈⁺(g_C).

(2)Ifα∈andw∈W_λwithwα∈⁻(g_C), showB(λ, α)≤0.

(3) Chevalley’s Lemma states W_λ is generated by W(λ) = {r_α | B(λ, α) = 0}.

Use Exercise 6.36 to prove this result. To this end, argue by contradiction and let w∈W_λ\ W(λ)be of minimal length.

(4)Show that the only reflections inW((g_C))are of the formr_αforα∈W(g_C).

(5)IfW_λ= {I}, show that there existsα∈(g_C)soλ∈α^⊥.

Exercise 6.38 LetG be compact Lie group with semisimple Lie algebragandta Cartan subalgebra ofg. Fix a simple system= {αi}of(g_C). Forw∈W((g_C)), let sgn(w)=(−1)^l(w)(c.f. Exercise 6.36). Show that sgn(w)=det(w).

Exercise 6.39 LetG be compact Lie group with semisimple Lie algebragandta Cartan subalgebra ofg. Fix a Weyl chamberCandH∈(it)^∗.

(1)SupposeH ∈C∩wCforw∈W((g_C)). Show thatwH =H.

(2)LetH ∈(it)^∗be arbitrary. Show thatCis afundamental chamberfor the action ofW((g_C)), i.e., show that the Weyl group orbit ofH intersectsC in exactly one point.

Exercise 6.40 LetG be compact Lie group with semisimple Lie algebragandta Cartan subalgebra ofg. Fix a simple systemof(g_C).

(1)Show that there is a uniquew0∈W((g_C)), so thatw0= −.

(2)Show thatw0= −I ∈W((g_C))forGequal toSU(2),S O(E_2n+1),Sp(n), and S O(E4n).

(3) Show thatw0 = −I, so−I ∈/ W((g_C))forG equal to SU(n)(n ≥ 3) and S O(E4n+2).

Exercise 6.41 LetGbe compact Lie group with simple Lie algebragandta Cartan subalgebra ofg. Fix a simple system= {αi}of(g_C).

(1)Ifαi andαj are joined by a single edge in the Dynkin diagram, show that there existsw∈W((g_C)), so thatωαi =αj.

(2) If G is a classical compact Lie group, i.e., G is SU(n), Sp(n), S O(E2n), or S O(E2n+1), show the set of roots of a fixed length constitutes a single Weyl group orbit.

Exercise 6.42 LetG be compact Lie group with semisimple Lie algebragandta Cartan subalgebra ofg. Fix a simple system = {αi}of(g_C)and letπi ∈ (it)^∗ be defined by the relation 2^B_B^(π_(α^i,αj)

j,αj)=δi,j.

(1) Show that{αi}is aZ-basis for the root lattice R and{πi}is aZ-basis for the weight latticeP.

(2)Show the matrix

B(αi, αj)

is positive definite. Conclude det

2_B^B_(α^(αi,αj)

j,αj) >0.

(3)It is well known from the study of free Abelian groups ([3]) that there exists a Z-basis{λi}of P andki ∈Z, so that{kiλi}is a basis forR. Thus there is a change of basis matrix from the basis{λi}to{πi}with integral entries and determinant±1.

Show that |P/R| = det

2^B_B(α^(αi,αj)

j,αj) . The matrix

2^B_B(α^(αi,αj)

j,αj) is called the Cartan matrixofg_C.

Exercise 6.43 Let G be a compact Lie group with semisimple Lie algebra gand t a Cartan subalgebra of g. Fix a simple system = {αi} of (g_C). For each β ∈(g_C), choose a standardsl(2,C)-triple associated toβ,{Eβ,H_β,F_β}. Leth =

β∈⁺(g_C)H_β and definek_αi ∈ Z>0, soh =

αi∈k_αiH_αi. Sete =

i

6k_αiE_αi, f =

ik_αiF_αi, ands=span_C{e,h,f}.

(1)Show _B^B₍⁽_h^h,hαi)

αi,h_αi) =1 (c.f. Exercise 6.34).

(2)Show thats∼=sl(2,C). The subalgebrasis called theprincipal three-dimensional subalgebraofg_C.

Exercise 6.44 LetGbe a compact Lie group with semisimple Lie algebragand let T be a maximal torus ofG. Fix a Weyl chamberCofitand letNG(C)= {g ∈G| Ad(g)C = C}. Show that the inclusion map of NG(C) → Ginduces an isomor- phismNG(C)/T ∼=G⁰/G.

7