Weyl's character formula

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 April 2010

Weyl's character formula

The adjoint representation 𝔤 is a unitary representation of G. So the Weyl group W acts on 𝔥 by unitary operators. So W acts on 𝔱 by orthonormal matrices. Identify 𝔱 and 𝔱*=Hom𝔱ℝ=α:𝔱→ℝ with the innder product, 𝔱→~𝔱*α↦αċ For a root α define α∨=2ααα andHα=x∈𝔱|αx=0. Then the reflection sα in the hyperplane Hα which comes from Zα=ZGTα∘/Tα∘, is sα:𝔱→𝔱λ↦λ-λα∨α.

INSERT DIAGRAM OF HYPERPLANE WITH REFLECTION HERE LATER

So

1. W acts on 𝔱, and
2. 𝔱-⋃α∈RHα=ℝn\⋃α∈RHα is a union of chambers (these are the connected components).

PICTURE OF CHAMBERS AND WEIGHT LATTICE

The Weyl group W permutes these chambers and if we fix a choice of chamber C then we can identify the chambers wC,w∈C. (See Brocker-tom Dieck V (2.3iv) and the Claim at the bottom of p 193.

PICTURE OF CHAMBERS LABELLED BY wC

Let RT=representation ring of  T=Grothendieck ring of representations of  G,  andRG=representation ring of  G.

This means that RG=span-Gλ|λ∈G^ with

1. addition given by Gλ+Gμ=Gλ⊕Gμ, and
2. multiplication given by GλGμ=Gλ⊗Gμ.

Thus, in RG it makes sense to write ∑λ∈G^mλGλinstead of⊕λ∈G^Gλ⊕mλ. Define ℂP=span-eλ|λ∈Pwith multiplicationeλeμ=eλ+μ, for λ,μ∈P. Then ℂP≅RT,sinceRT=span-Xλ|λ∈P. The action of W on RT (see (???)) induces an action of W on ℂP given by weλ=ewλ,forw∈W,λ∈P. Note that εw=det𝔥w=±1 since the action of w on 𝔥 is by an orthogonal matrix. The vector spaces of symmetric and alternating functions are ℂPW=f∈ℂP|wf=f  for all  w∈W, and 𝒜=f∈ℂP|wf==εwf  for all  w∈W, respectively. Note that ℂPW is a ring but 𝒜 is only a vector space.

Define P+=P∩C-andP++=P∩C. The set P+ is the set of dominant weights.

Every W-orbit on P contains a unique element of P+ and so the set of monomial symmetric functions mλ=∑γ∈Wλeγ,λ∈P+, forms a basis of ℂPW. Define aμ=∑w∈Wεwewμ, for μ∈P. Then

1. waμ=εwaμ, for all w∈W and all μ∈P,
2. aμ=0, if μ∈Hα for some α, and
3. aμ|μ∈P++ is a basis of 𝒜.

The fundamental weights ω1,…,ωn in 𝔱 are defined by ωiαj∨=δij, where Hαj are the walls of C. Write α>0ifλα>0for allλ∈C. Then ρ=∑i=1nωi=12∑α>0α, is the element of 𝔱 defined by ραi∨=1,for allα1,…,αn.

The map P+→P++λ↦λ+ρ is a bijection, and ℂPW→𝒜f↦aρf is a vector space isomorphism.

		Proof.
	Since waρf=waρwf=εwaρf, the second map is well defined. Let g=∑λ∈Pgλeλ∈𝒜. Then, for a positive root α, -g=sαg=∑λ∈Pgλesαλ, and so g=∑λ,λα>0gλeλ-ssαλ. Since eλ-esαλ=eλ-α+…+eλ-λα∨αeα-1, the element g is divisible by eα-1. Thus, since all the factors in the product are coprime in ℂP, gis divisible by ∏α>0eα-1=eα∏α>0eα2-e-α2=eρaρ, where the last equality follows from the fact that aρ is divisible by the product ∏α>0eα2-e-α2 and these two expressions have the same top monomial, eρ. Since g∈𝒜 divisble by aρ the map ℂP→𝒜 is invertible. □

Define χλ=aλ+ρaρ,forλ∈P+, so that the χλ|λ∈P+ are the basis of ℂPW obtained by taking the inverse image of the basis aλ+ρ|λ∈P+ of 𝒜. Extend these functions to all of U by setting χλgtg-1=χλt,for allg∈U. Since ∫TXλtXμtdt=δλμ, for λ,μ∈P, then we have ∫Taλ+ρtaμ+ρtdt=δλμW, and thus, by (???), δλμ=∫Gχλgχμgdg,for allλ,μ∈P+. Thus the χλ,λ∈P+ are an orthonormal basis of the set of class functions in CGrep. If Uλ is an irreducible representation of U then TrUλg=∑i=1dMiiλg,whereMijλ=viλgvjλ, for an orthonormal basis v1λ,…,vnλ of Uλ. Then ∫GTrUλgTrUμgdg=δλμ, and so the functions TrUλ are another orthonormal basis of the set of class functions in CGrep. It follows that χλ =±TrUλ.

It only remains to check that the sign is positive to show that the χλ are the irreducible characters of U. This follows from the following computation. χλ1=limt→0χλetρ=limt→0∑w∈WεwXwλ+ρetρ∑w∈WεwXwρetρ=limt→0∑w∈Wεwewλ+ρtρ∑w∈Wεwewρtρ=limt→0∑w∈Wεwetλ+ρw-1ρ∑w∈W εwetρw-1ρ=limt→0aρetλ+ρaρetρ=limt→0∏α>0Xα2-X-α2etλ+ρ∏α>0Xα2-X-α2etρ=limt→0∏α>0etλ+ρα2-e-tλ+ρα2∏α>0etρα2-e-tρα2=limt→0sinhtλ+ρα2sinhtρα2=∏α>0λ+ρα2ρα2=∏α>0λ+ρα∨ρα∨.

Let U be a compact connected Lie group and let T be the maximal torus and L the corresponding lattice.

1. The irreducible representations of U are indexed by dominant integral weights λ∈L+ under the correspondence irreducible representations→1-1P+Vλ↦highest weight of  Vλ
2. The character of Vλ is χλ=∑w∈Wεwewλ+ρ∑w∈Wεwewρ where ρ∈P+ is defined by ραi∨=1 for 1≤i≤n and εw=detw.
3. The dimension of Vλ is dλ=∏α>0λ+ρα∨∏α>0ρα∨.
4. χλ=∑p∈𝒫λep1, where 𝒫λ is the set of all paths obtained by acting on pλ by root operators.

Remark: By part (d) dimVλμ=number of paths in  𝒫λ which end at μ. (For the path model some copying can be done from the Barcelona abstract.)

Remark: Point out that RT=ℤL, where L is the lattice corresponding to T. Also point out that RU=RTW≅ℤLW.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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