Induced Representations

Highest Weight Theory

7.4 Borel–Weil Theorem

7.4.1 Induced Representations

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.

7.4 Borel–Weil Theorem 177 (d)V is ahomogeneousvector bundle overM for the Lie groupGif(i)the action ofGonV preserves fibers;(ii)the resulting action ofGonM is transitive; and(iii) eachg∈GmapsVxtoVgxlinearly forx∈M.

(e)IfVis a homogeneous vector bundle over M, the vector space(M,V)carries an action ofGgiven by

(gs)(x)=g(s(g⁻¹x)) fors∈(M,V).

(f)Two homogeneous vector bundlesVandVoverM forGareequivalentif there is a diffeomorphismϕ:V→V, so thatπ◦ϕ =ϕ◦π.

Note it suffices to study manifolds of the formM=G/H,Ha closed subgroup ofG, when studying homogenous vector bundles.

Definition 7.43.LetGbe a Lie group andH a closed subgroup ofG. Given a rep- resentationV ofH, define the homogeneous vector bundleG×HV overG/H by

G×HV =(G×V) /^∼, where^∼is the equivalence relation given by

(gh, v)^∼(g,hv)

forg∈G,h∈ H, andv∈V. The projection mapπ :G×HV →G/His given by π(g, v)=g H and theG-action is given byg(g, v)=(gg, v)forg∈G.

It is necessary to verify thatG×HVis indeed a homogeneous vector bundle over G/H. SinceH is a regular submanifold, this is a straightforward argument and left as an exercise (Exercise 7.27).

Theorem 7.44.Let G be a Lie group and H a closed subgroup of G. There is a bijection between equivalence classes of homogenous vector bundlesVon G/H and representations of H .

Proof. The correspondence mapsVtoVe H. By definitionVe H is a representation of H. Conversely, given a representationV ofH, the vector bundleG×HV inverts the

correspondence.

Definition 7.45.LetGbe a Lie group andHa closed subgroup ofG. Given a repre- sentation(π,V)ofH, define the smooth (continuous)induced representationofG by

Ind^G_H(V)=Ind^G_H(π)= {smooth (continuous) f :G→V | f(gh)=h⁻¹f(g)}

with action(g₁f)(g₂)= f(g₁⁻¹g₂)forgi ∈G.

Theorem 7.46.Let G be a Lie group, H a closed subgroup of G, and V a represen- tation of H . There is a linear G-intertwining bijection between(G/H,G×HV) andInd^G_H(V).

Proof. Identify(G×HV)e H withV by mapping(h, v)∈ (G×H V)e H toh⁻¹v ∈ V. Given s ∈ (G/H,G ×H V), let fs ∈ Ind^G_H(V) be defined by fs(g) = g⁻¹(s(g H)). Conversely, given f ∈ Ind^G_H(V), letsf ∈ (G/H,G×H V)be de- fined bysf(g H)=(g, f(g)). It is easy to use the definitions to see these maps are well defined, inverses, andG-intertwining (Exercise 7.28).

Theorem 7.47 (Frobenius Reciprocity).Let G be a Lie group and H a closed sub- group of G. If V is a representation of H and a W is a representation of G, then as vector spaces

HomG(W,Ind^G_H(V))∼=HomH(W|H,V).

Proof. Map T ∈ HomG(W,Ind^G_H(V)) to ST ∈ HomH(W|H,V) by ST(w) = (T(w))(e)forw∈ W and mapS ∈HomH(W|H,V)toTS ∈ HomG(W,Ind^G_H(V)) by(TS(w))(g) = S(g⁻¹w). Verifying these maps are well defined and inverses is

straightforward (Exercise 7.28).

In the special case of H = {e} and V = C, the continuous version gives (G/H,G×HV)∼= Ind^G_H(V)= C(G). In this setting, Frobenius Reciprocity al- ready appeared in Lemma 3.23.

7.4.2 Complex Structure onG/T

Definition 7.48.LetGbe a compact connected Lie group with maximal torusT. (a)Choosing a faithful representation, assumeG ⊆U(n)for somen. By Theorem 4.14 there exists a unique connected Lie subgroup ofG L(n,C)with Lie algebrag_C. WriteG_Cfor this subgroup and call it thecomplexificationofG.

(b)Fix⁺(g_C)a system of positive roots and recalln⁺=

α∈⁺(g_C)g_α. The corre- spondingBorel subalgebraisb=t_C⊕n⁺.

(c)Let N,B, and Abe the unique connected Lie subgroups ofG L(n,C)with Lie algebrasn⁺,b, anda=it, respectively.

For example, ifG=U(n)with the usual positive root system,G_C=G L(n,C), Nis the subgroup of upper triangular matrices with 1’s on the diagonal,Bis the sub- group of all upper triangular matrices, and Ais the subgroup of diagonal matrices with entries inR^>0. Although not obvious from Definition 7.48,G_Cis in fact unique up to isomorphism when G is compact. More generally for certain types of non- compact groups, complexifications may not be unique or even exist (e.g., [61], VII

§1). In any case, what is important for the following theory is thatG_Cis a complex manifold.

Lemma 7.49.Let G be a compact connected Lie group with maximal torus T . (a)The mapexp :n⁺→N is a bijection.

(b)The mapexp :a→ A is a bijection.

(c)N , B, A, and AN are closed subgroups.

(d)The map from T×a×n⁺to B sending(t,X,H)→te^Xe^His a diffeomorphism.

7.4 Borel–Weil Theorem 179 Proof. SinceT consists of commuting unitary matrices, we may assumeT is con- tained in the set of diagonal matrices of G L(n,C). By using the Weyl group of G L(n,C), we may further assumeu_ρ=diag(c1, . . . ,cn)withci ≥ci+1. Therefore ifX ∈g_α,α∈⁺(g_C), withX =

i,jki,jEi,j, then

i,j

(ci−cj)ki,jEi,j =[u_ρ,X]=α(u_ρ)X =

i,j

B(α, ρ)ki,jEi,j.

SinceB(α, ρ) >0, it follows thatki,j =0 wheneverci−cj ≤0. In turn, this shows that Xis strictly upper triangular.

It is well known and easy to see that the set of nilpotent matrices are in bijection with the set of unipotent matrices by the polynomial mapM →e^Mwith polynomial inverse M → ln(I +(M−I))=

k (−1)^k+1

k (M −I)^k. In particular ifX,Y ∈n⁺, there is a unique strictly upper triangularZ ∈gl(n,C), so thate^Xe^Y =e^Z.

Dynkin’s formula is usually only applicable to smallXandY. However,⁺(g_C) is finite, so [X_n⁽ⁱⁿ⁾, . . . ,X⁽₁ⁱ¹⁾] is 0 for sufficiently large ij for Xj ∈ n⁺. Thus all the sums in the proof of Dynkin’s formula are finite and the formula for Z is a polynomial inXandY. Coupled with the already mentioned polynomial formula for Z, Dynkin’s Formula therefore actually holds for all X,Y ∈n⁺. As a consequence, Z ∈n⁺and expn⁺is a subgroup. SinceNis generated by expn⁺, part (a) is finished.

The group N is closed since exp :n⁺ → N is a bijection and the exponential map restricted to the strictly upper triangular matrices has a continuous inverse.

Part (b) and the fact that A is closed in G_C follows from the fact that a is Abelian and real valued. Next note that AN is a subgroup. This follows from the two observations that (an)(an) = (aa)((ca⁻¹n)n),a,a ∈ A and n,n ∈ N, and that ce^He^X = exp

e^ad(H)X

, H ∈ a and X ∈ n⁺. Since the map from b = t⊕a⊕n⁺ → G_Cgiven by(H1,H2,X) → e^H1e^H2e^X is a local diffeomor- phism near 0, products of the formtan,t ∈ T,a ∈ A, andn ∈ N, generateB. Just as withAN,T AN is a subgroup, so thatB=T e^aeⁿ⁺. It is an elementary fact from linear algebra that this decomposition is unique and the proof is complete.

The point of the next theorem is thatG/T has aG-invariant complex structure inherited from the fact thatG_C/Bis a complex manifold. This will allow us to study holomorphic sections onG/T.

Theorem 7.50.Let G be a compact connected Lie group with maximal torus T . The inclusion G→G_Cinduces a diffeomorphism

G/T ∼=G_C/B.

Proof. Recall thatg= {X+θX |X ∈g_C}, so thatg/tandg_C/bare both spanned by the projections of{X_α+θX_α| X_α∈g_α,α∈⁺(g_C)}. In particular, the differential of the mapG→G_C/Bis surjective. Thus the image ofGcontains a neighborhood ofe BinG_C/B. As left multiplication bygandg⁻¹,g ∈G, is continuous, the image of G is open inG_C/B. Compactness ofG shows that the image is closed so that connectedness shows the mapG→G_C/Bis surjective.

It remains to see thatG∩ B = T. Let g ∈ G∩ B. Clearly Ad(g)preserves g∩b=t, so thatg∈ N(T). Writingwfor the corresponding element ofW((g_C)), the fact thatg∈ Bimplies thatwpreserves⁺(g_C). In turn, this meanswpreserves the corresponding Weyl chamber. Since Theorem 6.43 shows thatW((g_C))acts simply transitively on the Weyl chambers,w=Iandg∈T. 7.4.3 Holomorphic Functions

Definition 7.51.LetGbe a compact Lie group with maximal torusT. Forλ∈ A(T), writeC_λfor the one-dimensional representation ofT given byξ_λand writeL_λfor theline bundle

L_λ=G×T C_λ.

By Frobenius Reciprocity,(G/T,L_λ)is a huge representation ofG. However by restricting our attention to holomorphic sections, we will obtain a representation of manageable size.

Definition 7.52.LetGbe a compact connected Lie group with maximal torusT and λ∈ A(T).

(a)Extendξλ:T →Cto a homomorphismξ_λ^C:B→Cby ξ_λ^C(te^{i H}e^X)=ξλ(t)e^iλ(H) fort ∈T,H ∈t, andX ∈n⁺.

(b)LetL^C_λ =G_C×BC_λwhereC_λis the one-dimensional representation ofBgiven byξ_λ^C.

Lemma 7.53.Let G be a compact connected Lie group with maximal torus T and λ ∈ A(T). Then(G/T,L_λ) ∼=(G_C/B,L^C_λ)andInd^G_T(ξλ)∼= Ind^G_B^C(ξ_λ^C)as G- representations.

Proof. Since the mapG → G_C/B induces an isomorphismG/T ∼= G_C/B, any h ∈ G_C can be written ash = gbforg ∈ G andb ∈ B. Moreover, ifh = gb, g∈Gandb∈B, then there ist∈T sog=gtandb=t⁻¹b.

On the level of induced representations, map f ∈Ind^G_T(ξλ)toFf ∈ Ind^G_B^C(ξ_λ^C) by Ff(gb)= f(g)ξ_−λ^C(b)forg ∈ Gandb ∈ B and mapF ∈ Ind^G_B^C(ξ_λ^C)to fF ∈ Ind^G_T(ξ_λ)by fF(g)= F(g). It is straightforward to verify that these maps are well defined,G-intertwining, and inverse to each other (Exercise 7.31).

Definition 7.54.LetGbe a compact connected Lie group with maximal torusT and λ∈ A(T).

(a)A sections ∈(G/T,L_λ)is said to beholomorphicif the corresponding func- tion F ∈Ind^G_B^C(ξ_λ^C), c.f. Theorem 7.46 and Lemma 7.53, is a holomorphic function onG_C, i.e., if

d F(i X)=i d F(X)

at eachg∈G_Cand for allX ∈Tg(G_C)whered F(X)=X(ReF)+i X(ImF). (b)Writehol(G/T,L_λ)for the set of all holomorphic sections.

7.4 Borel–Weil Theorem 181 Since the differentiald Fis alwaysR-linear, the condition of being holomorphic is equivalent to saying thatd FisC-linear. Written in local coordinates, this condition gives rise to the standard Cauchy–Riemann equations (Exercise 7.32).

Definition 7.55.Let G be a connected (linear) Lie group with maximal torus T. WriteC^∞(GC)for the set of smooth functions onG_Cand use similar notation for G.

(a)ForZ ∈g_CandF ∈C^∞(G_C), let [dr(Z)F](h)= d

dtF(he^{t Z})|t=0

forh∈G_C. ForX ∈gand f ∈C^∞(G), let [dr(X)f](g)= d

dt f(ge^{t X})|t=0

forg∈G.

(b)ForZ =X+i Y withX,Y ∈g, let

dr_C(Z)=dr(X)+i dr(Y).

Note thatdr_C(Z)is a well-defined operator onC^∞(G)but thatdr(Z) is not (except whenZ ∈g).

Lemma 7.56.Let G be a compact connected Lie group with maximal torus T ,λ∈ A(T), F∈Ind^G_B^C(ξ_λ^C), and f =F|Gthe corresponding function inInd^G_T(ξλ).

(a)Then F is holomorphic if and only if

dr_C(Z)F =0 for Z ∈n⁺.

(b)Equivalently, F is holomorphic if and only if dr_C(Z)f =0 for Z ∈n⁺.

Proof. Sincedlg :Te(GC)→ Tg(GC)is an isomorphism, F is holomorphic if and only if

d F(dlg(i Z))=i d F(dlgZ) (7.57)

for allg ∈G_CandX ∈g_Cwhere, by definition, d F(dlgZ)= d

dtF(ge^{t Z})|t=0=[dr(Z)F](g).

If Z ∈ n⁺, thene^{t Z} ∈ N, so that F(ge^{t Z}) = F(g). Thus for Z ∈ n⁺, Equation 7.57 is automatic since both sides are 0. IfZ ∈t_C,F(ge^{t Z})=F(g)e^−tλ(Z). Thus for Z ∈t_C, Equation 7.57 also holds since both sides are−iλ(Z)F(g).

Sinceg_C = n⁻⊕t_C⊕n⁺, part (a) will be finished by showing Equation 7.57 holds for Z ∈ n⁻. However, Equation 7.57 is equivalent to requiringdr(i Z)F = i dr(Z)Fwhich in turn is equivalent to requiringdr(Z)F =dr_C(Z)F. If Z ∈ n⁻, thenθZ ∈n⁺andZ +θZ ∈g. Thus

dr(Z)F =dr(Z+θZ)F−dr(θZ)F =dr_C(Z+θZ)F =dr_C(Z)+dr_C(θZ), so thatdr(Z)F =dr_C(Z)Fif and only ifdr_C(θZ)=0, as desired.

For part (b), first, assume F is holomorphic. Since f = F|G, it follows that dr_C(n⁺)f = 0. Conversely, suppose dr_C(n⁺)f = 0. Restricting the above argu- ments fromG_CtoGshowsdr(Z)F|g =dr_C(Z)F|g forg ∈GandZ ∈g_C. Hence ifX ∈g,

(dr(X)F)(gb)= d

dtF(gbe^{t X})|t=0= d

dtF(ge^tAd(b)Xb)|t=0

=ξ−λ(b) d

dtF(ge^tAd(b)X)|t=0

=ξ_−λ(b) (dr(Ad(b)X)F)(g)=ξ_−λ(b) (dr_C(Ad(b)X)F)(g) forg∈Gandb∈ B. Thus if Z =X+i Y ∈n⁺withX,Y ∈g, note Ad(b)Z ∈n⁺ and calculate

(dr_C(Z)F)(gb)=(dr(X)F)(gb)+i(dr(Y)F)(gb)

=ξ_−λ(b)[(dr_C(Ad(b)X)F)(g)+(dr_C(iAd(b)Y)F)(g)]

=ξ−λ(b) (drC(Ad(b)Z)F)(g)=0,

as desired.