Lecture 13: The Weyl Character Formula

Daniel Bump

May 19, 2020

Notation

Most of the setup for the Weyl Character formula is familiarfrom our previous lectures.

Let G be a compact Lie group such as U(n) with maximal torusT and Weyl group W = N(T)/T, root system Φ = Φ+ ∪ Φ−.Embed the weight lattice Λ = X∗(T) into a real vector spaceV = R⊗Z Λ, which we give a W-invariant inner product. Let

Σ = {α1, · · · , αr}

be the simple roots and si ∈ W be the simple reflections.

Expansions of Class functions

Consider a function on T with an expansion

f (t) =∑λ∈Λ

aλtλ.

We ask whether it can be extended to a class function on G.We proved in Lecture 10 that if two elements t and t′ of T areconjugate in G then they are conjugate in N(T). Thus theconjugacy classes of G are in bijection with the orbits of T in theaction of W. Thus the function f (t) on T may be extended to aclass function on G provided aλ = aw(λ) for all w ∈ W.

A symbolic ring for characters

It will be convenient to embed Λ also in its group algebraE = Z[Λ]. Since Λ is written additively, we will use the notationeλ for a weight λ regarded as an element of E . Thus E consistsof the ring of formal expressions

f =∑λ∈Λ

aλeλ

where aλ ∈ Z and the support

supp(f ) = {λ ∈ Λ|aλ 6= 0}

is finite.

The Weyl vector

Sometimes we will make use of the larger ring E2 that is thecharacter ring Z

[12Λ]. For example the Weyl vector

ρ =12

∑α∈Φ+

α

will appear, and this may or may not be an element of Λ. Therings E and E2 are unique factorization domains.

Characters are in EW

We will denote by EW and the subring of sums∑

λ∈Λ aλeλ suchthat aλ = awλ for w ∈ W, and similarly EW

2 . From our previousdiscussion we may associate with such an element a function

f (t) =∑λ∈Λ

aλtλ

and this is a class function. Particularly, suppose f = χ is thecharacter of a representation. Each aλ is the multiplicity of aweight λ (a 1-dimensional representation of T), so aλ ∈ Z andaλ > 0. Only finitely many aλ are nonzero. Finally, since χ is aclass function aw(λ) = aλ for w ∈ W. Thus∑

λ∈Λ

aλeλ ∈ EW .

So we may regard characters as elements of EW .

Coroots applied to weights

PropositionLet α be a root and let λ ∈ Λ be a weight. Then α∨(λ) ∈ Z.

We will prove this under the assumption that V(λ) 6= 0 for someirreducible representation (π,V). It may be deduced from thePeter-Weyl theorem that such a representation exists(Proposition 18.11).

Proof

Let iα : SU(2) −→ G be the embedding discussed in Lecture 12.We consider the iα SU(2)-module generated by V(λ). This is⊕

k∈ZV(λ+ kα)

by out classification of SU(2)-modules. Since

wα = iα

(−1

1

)is in iα SU(2), this is invariant under wα.

Now wα induces the reflection sα which sends λ to λ− α∨(λ)α.Thus −α∨(λ) = k for some k.

The Weyl vector

Recall the Weyl vector

ρ =12

∑α∈Φ+

α,

PropositionWe have α∨i (ρ) = 1 for each simple root α.

Because si sends αi −→ −αi and permutes the remainingpositive roots, we have

si(ρ) = ρ− αi.

Sincesi(ρ) = ρ− α∨i (ρ)αi

we have α∨i (ρ) = 1.

The Weyl denominator

Define the Weyl denominator

∆ = eρ∏α∈Φ+

(1− e−α) =∏α∈Φ+

(eα/2 − e−α/2).

Lemma

w(∆) = (−1)`(w)∆.

It is enough to check this when w = si is a simple reflection.Then si sends αi → −αi and permutes the remaining positiveroots. So it changes the sign of exactly one factor eα/2 − e−α/2.

If α is a positive root we recall that rα ∈ W is the reflectioncorresponding to α. We reserve the notation sα or si = sαi forthe case where α = αi is a simple root.

A divisibility criterion (I)

LemmaIf φ ∈ E and rα(φ) = −φ then φ is divisible (in E) by 1− e−α.

Indeed φ is a linear combination of terms

eλ − esiλ = eλ − eλ−α∨i (λ)α = eλ(1− e−kα)

where k = α∨i (λ) ∈ Z. If k = 0, this vanishes. If k > 0

eλ(1− e−kα) = (1− e−α)(1 + e−α + . . .+ e−(k−1)α).

The case k < 0 is almost the same.

A divisibility criterion (II)

Proposition

If w(φ) = (−1)`(w)φ for all w ∈ W then φ is divisible in E2 by

∆ = eρ∏α∈Φ+

(1− e−α).

We express the hypothesis that w(φ) = (−1)`(w)φ by saying thatφ is antisymmetric or alternating.

To prove this, by the Lemma, it is divisible by each of the factors(1− e−α), and of course eρ is a unit in E . The factors 1− e−α

are coprime in the unique factorization domain E2, and eρ is aunit. The statement follows.

The partial order

We will make use of a partial order on the weight lattice, inwhich λ < µ if λ− µ is a linear combination of the positive rootswith nonnegative integer coefficients. This means that λ− µlies in the root lattice Λroot

LemmaIf λ is dominant then

w(λ) 4 λ

for all w ∈ W.

Proof

It is easy to see that, for x ∈ V, x < 0 if and only if 〈x, v〉 > 0 forall v ∈ C◦+. So if λ ∈ C+ and λw(λ), then there exists v ∈ C◦+such that 〈λ− w(λ), v〉 < 0. We choose w to maximize 〈w(λ), v〉.Since w(λ) 6= λ and λ ∈ C+, it follows that w(λ) 6∈ C+. Therefore,there exists α ∈ Σ such that 〈w(λ), α〉 < 0, or equivalently,α∨(w(λ)) < 0. Now

〈sαw(λ), v〉 = 〈w(λ)− α∨(w(λ))α, v〉

= 〈w(λ), v〉 − 2〈w(λ), α〉〈α, α〉

〈α, v〉 > 〈w(λ), v〉.

The maximality of 〈w(λ), v〉 is contradicted.

Definition of χλ

Now let λ be a dominant weight. Define

χλ =

∑w∈W(−1)`(w)ew(λ+ρ)

∆.

Proposition

χλ is an element of EW , hence represents a class function on G.

The expression ∑w∈W

(−1)`(w)ew(λ+ρ)

is alternating, so it is divisible by ∆ in E2. This proves that thequotient is in E2, and also since both the numerator anddenominator are alternating, it is W-invariant.

Proof, continued

To see that χ is in E divide numerator and denominator by eρ:

χλ =

∑w∈W(−1)`(w)ew(λ+ρ)−ρ∏

α∈Φ+(1− e−α)

then both the numerator and denominator are in E .

We have proved that χλ ∈ EW .

Support properties of χλ

Eventually we will prove that χλ is the character of anirreducible representation, and moreover every irreduciblecharacter is a χλ.

Let us writeχλ =

∑µ∈supp(χλ)

aµeµ.

PropositionIf µ ∈ supp(χλ) then µ 4 λ. The function aµ is W-invariant andaλ = 1.

Because of this, λ is the highest weight of χλ, the weight that ismaximal with respect to the partial order 4.

Example

Before we prove the support properties, here is an example, theirreducible representation of U(3) with highest weight (3, 1, 0).

λ = (3, 1, 0)

λ− α2λ− 2α1

The shaded region is the positive Weyl chamber. The theregion within the dashed lines is µ such that µ 4 λ.

A completion of E

We have proved that χλ ∈ EW , so aλ is W-invariant. We provethe assertion that µ 4 λ for µ ∈ supp(χλ). We work in a ring Ethat is a completion of E . In the ring E we allow sums∑

aλeλ

with an infinite number of terms. However we require that thereexist a finite number of vectors vector vi ∈ V (depending on theelement) such that aλ = 0 unless λ 4 vi for some i.

Proof

Let us restate:

PropositionIf µ ∈ supp(χλ) then µ 4 λ. The function aµ is W-invariant andaλ = 1.

In the ring E we may write χλ as∑w∈W(−1)`(w)ew(λ+ρ)−ρ∏

α∈Φ+(1− e−α)=

∑(−1)`(w)ew(λ+ρ)−ρ

∏α∈Φ+

(1 + e−α + e−2α + · · · ).

From this we see that if µ ∈ supp(χλ) then µ 4 λ and aλ = 1.We have already proved the W-invariance.

Introducing the Weyl denomonator formula

The Weyl denominator formula is a generalization of theVandermonde determinant identity

det(zr−1−ji )i,j =

∣∣∣∣∣∣∣∣∣zr−1

1 zr−12 · · · zr−1

r...

......

z1 z2 · · · zr

1 1 · · · 1

∣∣∣∣∣∣∣∣∣ =∏i<j

(zi − zj).

In its most general form, due to Kac, similar identities for infinitedimensional Lie algebras include identities such as the Jacobitriple product identity:

∞∑n=−∞

(−1)nqnzn2=

∞∏m=1

(1− z2m)(1− qz2m−1)(1− q−1z2m−1).

It general, it relates an alternating sum over the Weyl group to aproduct over the positive roots.

The Weyl denominator formula

Theorem (The Weyl denominator formula)The following two expressions for ∆ are equal:

eρ∏α∈Φ+

(1− e−α) =∑w∈W

(−1)`(w)ew(ρ).

Indeed, take λ = 0. We claim χ0 = 1. Indeed, if µ ∈ supp(χλ)then µ 4 0. If µ 6= 0 then there is a dominant weight ν in theW-orbit of µ and aν = aµ 6= 0, but µ 6= 0 so it is impossible thatµ 4 0, which is a contradiction. We see that µ = 0 is the onlyweight of χ0 and its multiplicity a0 is 1, that is, χ0 = 1.

Equating the numerator and denominator gives the Weyldenominator formula.

The Weyl integration formula

We recall that∫G

f (g) dg =1|W|

∫T

f (t) det(Ip − Ad(t−1)|p)dt

Here we recall p is the sum of the root spaces Xα for all roots.

Proposition

det(I − Ad(t−1)|p) = |∆|2.

Indeed Ad(t) has Xα as an eigenspace, so

det(I − Ad(t−1)|p) =∏α

(1− t−α).

Interpreting ∆ as a function on T means

∆ = tρ∏α∈Φ+

(1− t−α).

Proof, continued

Therefore

∆∆ = tρ∏α∈Φ+

(1− t−α)t−ρ∏α∈Φ+

(1− tα) =∏α∈Φ

(1− t−α).

Thus the Weyl integration formula can be written∫G

f (g) dg =1|W|

∫T

f (t)∆∆ dt.

Orthogonality

PropositionIf λ, µ are dominant weights then

〈χλ, χµ〉 = δλ,µ.

The inner product is∫Gχλ(g)χµ(g) dg =

1W

∫Tχλ(t)χµ(t)∆∆ dt.

The factor ∆∆ cancels the denominators in the definitions ofχλ, χµ and we obtain

1|W|

∫T

∑w∈W

(−1)`(w)tw(λ+ρ)∑

w′∈W

(−1)`(w′)t−w′(µ−ρ) dt.

Proof

Now consider a term ∫T

tw(λ+ρ)−w′(µ+ρ) dt.

This vanishes unless w(λ+ ρ)− w′(µ+ ρ) = 0. Because λ+ ρand µ+ ρ are in the interior of the positive Weyl chamber, thetwo factors w(λ+ ρ) and w′(µ+ ρ) will lie in the interiors ofdistinct Weyl chambers unless w = w′. Then the term cannotvanish unless λ = µ. If λ = µ there are |W| distinct choices forw = w′ so |W| terms equal 1, but we are dividing by 1 and so therequired orthogonality is proved.

The χλ are a basis

Proposition

The χλ are a Z-basis of EW .

If λ is a dominant weight let mλ be the sum of the elements ofthe W-orbit ew(λ). Since every W-orbit meets the positive Weylchamber in a unique dominant weight, it is obvious that the mλ

are a Z-basis of EW . Now

χλ =∑µ4λ

aµ,λmµ

where the coefficients aµ,λ are integers and aλ,λ = 1. Since thetransition matrix (aµ,λ) is triangular with respect to the partialorder 4 it is invertible, so the χλ are a Z-basis also.

The Weyl character formula

Theorem (Weyl)The χλ defined by

χλ =

∑w∈W(−1)`(w)ew(λ+ρ)

eρ∏α∈Φ+(1− e−α)

are precisely the irreducible characters of G.

First, let χ be an irreducible character. We may regard χ as anelement of EW and hence expand it in terms of the χλ:

χ =∑λ

cλχλ.

The coefficients are integers.

Proof (concluded)

Now1 = 〈χ, χ〉 =

∑|cλ|2.

This implies that exactly one of the integers cλ is nonzero, andthat coefficient is ±1. Therefore χ = ±χλ for some λ. Wecannot have χ = −χλ since the multiplicity of the weight λcannot be negative; in −χλ that coefficient would be −1. Thisproves that χ = χλ.

Therefore the characters of irreducible representations areamong the orthogonal functions χλ. There cannot be a χλ thatis not a character of an irreducible representation, since such aχλ would be a class function that is orthogonal to theirreducible characters; no such function exists as aconsequence of the Peter-Weyl theorem.