5.5. PROBLEMS 95
Figure 5.1: Weights ofπ1,1.
The multiplicity of the 8 weights in the outer layer is1, since these are inW -orbit of the highest weight. The multiplicity of the 4 weights in the inner layer could be1 or higher; we need more work to determine it (cf. Prob-lem 10 in Chapter 6).
3 Letg =sl(3,C)andV = Λ2(S2C3). In this problem, we show thatV is irreducible.
a. Check that the weights ofV and their multiplicities are as follows:
3θi+θj 1 2θi+ 2θj 1 2θi+θj+θk=θi 2 wherei,j,kdenote pairwise different indices.
5.5. PROBLEMS 97
Figure 5.2: Weights ofΛ2(S2C3).
b. Check that the highest weight is3θ1 +θ2 and that e21 ∧(e1e2) is a highest weight vector.
c. Denote byW the irreducible subrepresentation ofV generated bye21∧ (e1e2) and deduce from Lemma 5.4.8 thatW has the same weights asV.
We want to see thatW = V. For that purpose, we need to show that the weight spaces ofV of dimension2are contained inW.
d. Check that e21 ∧ (e2e3) and (e1e2) ∧(e1e3) are linearly independent weight vectors of weight2θ1+θ2+θ3.
e. Recall thate21∧(e1e2)∈W, and explain whye21∧e22 ∈W.
f. ComputeE31(e21∧(e1e2))andE32(e21∧e22)(Eij is the matrix with1in the(i, j)-entry and zero in the other entries).
g. Use part (f) and the fact that the Weyl group acts on the weights pre-serving the multiplicities to conclude that W = V and hence V is irreducible.
4 Letλbe a dominant algebraically integral weight and denote byπλ the irreducible representation with highest weightλ. Show that0occurs as a
weight ofπλif and only ifλis a linear combination of roots.
5 Check that the fundamental weights are as indicated:
a. Bn:̟1 =θ1, . . . , ̟n−1=θ1+· · ·+θn−1,̟n= 12(θ1+· · ·+θn).
b. Cn:̟1 =θ1, . . . , ̟n=θ1+· · ·+θn.
c. Dn:̟1 =θ1, . . . , ̟n−2 =θ1+· · ·+θn−2,̟n−1 = 12(θ1+· · ·+θn−1−θn),
̟n= 12(θ1+· · ·+θn−1+θn).
6 Consider the Lie algebra of typeG2. Letα1 denote the long simple root and letα2 denote the short simple root. Check that̟1 = 2α1 + 3α2 and
̟2 =α1+ 2α2.
7 Letπnben+ 1-dimensional irreducible unitary representation ofSU(n).
Prove the followingClebsch-Gordandecompositions:
πm⊗πn=
minX{m,n} k=0
πm+n−2k,
S2(πn) =
⌊n2⌋
X
k=0
π2n−4k,
Λ2(πn) =
⌊Xn−21⌋ k=0
π2n−4k−2.
The projectionsπm⊗πn→ πm+nandS2(πn) →π2ncan be identified with
“multiplication of polynomials”.
8 Letgbe a complex semisimple Lie algebra. For each dominant integral weight λ, denote by πλ the complex irreducible representation of g with highest weight vector.
a. Prove that if a pure tensor is a highest weight vector ofπλ⊗πµthen it must be the highest weight vector of the Cartan componentπλ+µ. b. Show that the Cartan componentπλ+µindeed occurs with
multiplic-ity one inπλ⊗πµ.
c. Prove that ifπν ⊂πλ⊗πµ, thenν=λ+µ′, whereµ′is a weight ofπµ. 9 Letπbe a complex representation of a Lie groupGon a finite-dimensional vector spaceV. The representationπis said to be ofreal typeif it comes from a representation ofGon a real vector space by extension of scalars, namely, it admits a real form; andπis said to be ofquaternionic typeif it comes from a representation on a quaternionic vector space by restriction of scalars, that is,V is a quaternionic vector space viewed as complex and π(g)is a quaternionic linear transformation for allg∈G. Ifπ is neither of real type nor of quaternionic type, one can say it is ofcomplex type.
5.5. PROBLEMS 99 a. Prove thatπis of real type (resp. quaternionic type) if and only there exists a conjugate linear map ǫ : V → V that commutes with the action ofGand such thatǫ2 =I (resp.ǫ2 =−I). (Hint: ǫis complex conjugation, resp. multiplication byj).
b. AssumeGis compact. Prove thatπis of real type (resp. quaternionic type) if and only there exists aG-invariant nondegenerate symmetric (resp. skew-symmetric) bilinear form onV. (Hint: ConsiderB(u, v) = hu, ǫvi, whereh,iis aG-invariant Hermitian product onV andǫis an in part (a). Conversely, ifB exists and one definesǫby this formula, one can checkǫ2is a definite, Hermitian operator onV, which can be then renormalized.)
c. Check thatπ is equivalent to its contragredient representation onV∗ if and only ifπis of real or quaternionic type.
d. In caseV is irreducible, show thatπmust be exactly of one of those three types: real, quaternionic or complex.
C H A P T E R 6
The Weyl formulae
In this chapter and in the following, our approach will be more in an an-alytic vein and focused more on the group level. We will recover and/or complete some results from earlier chapters.
We know from Chapter 1 that, for a complex (unitary) representationπ of a compact Lie groupG, the character function
χπ :G→C, χπ(g) = traceπ(g)
completely determines the equivalence class of the representation. The character function, in its turn, is Ad-invariant, so it is determined by its values on a maximal torusT ofG. Now
χπ(expH) =X
µ
mµeµ(H)
whereH belongs the Lie algebrat ofT, µruns through the weights ofπ, andmµdenotes the multiplicity ofµ.
The character function is thus a complete invariant of the representa-tion, but how to compute it? How to determine the mutiplicities of the weights? Even for a given π, this in general is labourious and indeed a daunting task. Well, of course we can restrict to irreducible representations, and from Chapter 5, we know that an irreducible π is completely deter-mined by its highest weightµπ, which (say, in caseGis simply-connected) can be any dominant weight; and the multiplicity ofµπ is always 1; and the multiplicities of the weights are invariant under the action of Weyl group1. The other weights ofπ can be obtained fromπ by taking the in-tegral weights in the convex hull of theW-orbit ofµπ, which are congruent toµπ modulo the root lattice. Still, this does not tell much about the actual values of the multiplicities.
The main result in this chapter is the Weyl Character Formula. The WCF gives not only the multiplicities, but the full character of an irre-ducible representationπ in terms of its highest weightµπ. There are a few
1Ref?
101
other formulae for the character and for the multiplicities, as well general-izations and extensions of WCF (the most notable perhaps being Freuden-thal’s, Kostant’s, Steinberg’s, Harishchandras’s, Demazure’s, Weyl-Kac’s, etc.); however, WCF remains the most “classical” one, and a very beauti-ful and most usebeauti-ful theoretical statement, although perhaps not the most suitable formula for practical computations.
In the caseGis simply-connected, the formula can be stated as χλ(expH) = AltWe(λ+δ)(H)
AltWeδ(H) ,
whereλ=µπis the highest weight,δis half the sum of the positive roots, AltW = X
w∈W
sgn(W)w
is the antisymmetrization operator, based onW, acting on characters of the maximal torus. The difficulty in using this formula is of course in com-puting the quotient between the linear combinations of exponentials. It is rather surprising that the numerator and the denominator of the formula involve a much smaller number of terms than its quotient (the character function itself).2
The basic idea of Hermann Weyl’s proof relies on an integration for-mula, bearing his own name, which (together with the fact that the L2 -norm overGof the character of an irreducible representation is1) implies that theL2-norm overT of the function given byχλ multiplied by the the denominator of WCF is equal to|W|; this function is a linear combination of torus characters, which are themselves orthonormal, and thus it follows that the function involves exactly|W|of such torus characters; on the other hand, we know thateλ+δ must be a torus character appearing in the func-tion, only once, and cannot cancel out; since its W-translates must also appear, each with multiplicity one, and there are exactly|W|of them, it fol-lows that these are all torus characters that contribute to the function, and hence we get the numerator of WCF.
Throughout this chapter we fix a compact connected Lie group G of dimensionnand rankkand a maximal torusTofG. Also, let∆be the root system of gc with respect to tc and fix an ordering of the roots. Then we have the (complex root) decomposition
gc =tc+X
α∈∆
gcα
2Note that this formula can also be expressed asχλ· Aδ=Aλ+δ, whereAµ= AltWeµ; this identity can be interpreted in the group algebra of the algebraically integral weights, written multiplicatively as formal linear combinations ofeµ.
6.1. THE ANALYTIC WEYL GROUP 103 and the real root decomposition
g=t+ X
α∈∆+
(gcα+gc−α)∩g.
For brevity, writeg(±α):= (gcα+gc−α)∩g.
6.1 The analytic Weyl group
The Weyl group W was defined in Chapter 4 as the group generated by the reflections{sα}associated to the rootsα∈∆. We can realizeW on the group level as follows.
The normalizer ofT inGis the closed subgroup NG(T) ={g∈G|gT g−1=g}.
Of course, NG(T) acts by inner automorphisms on T. The kernel of this action is the centralizer
ZG(T) ={g∈G|gtg−1=tfor allt∈T}.
6.1.1 Lemma The centralizer of any torusSinGis the union of all maximal tori containingS; in particular,ZG(S)is connected.
It follows from the Lemma 6.1.1 thatZG(T) = T. The analytically de-fined Weyl group is dede-fined as
WG =NG(T)/T.
6.1.2 Proposition LetGbe a compact connected semisimple Lie group with Lie algebrag, and fix a maximal torusT with Lie algebrat, so thath=tc is a CSA of gc. Then the analytic Weyl groupWG = NG(T)/T is canonically isomorphic to the algebraic Weyl groupWgdefined in Chapter 4 (as the group generated by the reflections on the singular hyperplanes).
Proof. Let α ∈ ∆+ and consider the reflectionsα. We will exhibit an elementnα ∈NG(T)such thatAdnα =sαonh.
Recall the Lie algebrag[α] =CH¯α+CEα+CFα∼=sl(2,C). The element Xα := 21(Eα−Fα)belongs to a compact real formsu(2). Note thatadXαH= 0ifH ∈kerαandad2XαH¯α =−H¯α. ForH ∈hR=it, we compute
Adexp(tXα)H =etadXα
= X∞ k=0
tk
k!adkXαH,
which yieldsH, ifH ∈kerα, and Adexp(tXα)H¯α =
X∞ m=0
t2m
(2m)!(−1)mH¯α+ X∞ m=0
t2m+1
(2m+ 1)!(−1)madXαH¯α
= costH¯α+ sintadXαH¯α
=−H¯α
ift=π. Hence we can takenα:= exp(πXα).
Note that WG permutes the roots. Indeed, for n ∈ NG(T) = NG(t), α∈∆andEα∈(gc)α, we have
[H,AdnEα] = Adn[Adn−1H, Eα]
=α(Adn−1H)AdnEα
= (Adtnα)(H) AdnEα,
showing thatAdnEα is a root vector for the rootAdtnα. It follows thatWG acts on the set of Weyl chambers. SinceWg actssimplytransitively on the set of Weyl chambers, we deduce thatWG∼=Wg. 6.1.3 Corollary The multiplicities of the weights of a representation of a complex semisimple Lie algebragare invariant under the Weyl group.
6.1.4 Proposition The inclusionT → Ginduces a homeomorphism between or-bit spacesh:T /W →G/Ad.
Proof. h is well-defined since W is a subquotient ofG. It is surjective by the Maximal Torus Theorem 4.1.3. Let us prove injectivity. Supposes, t∈TareG-conjugate,s=gtg−1for someg∈G. ThenZG(s)0is a compact connected Lie group and T, gT g−1 are two maximal tori in it so, again by 4.1.3, there is z ∈ ZG(s)0 such thatT = z(gT g−1)z−1 = (zg)T(zg)−1. Nown:=zg∈NG(T)andntn−1 =s. Finally,his continuous andT /W is
compact, sohis a homeomorphism.