The Weyl Character formula

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 8 October 2012

Symmetric functions

Initial data:

  1. (W0,𝔥*) a finite -reflection group
  2. 𝔥* is a free -modules
  3. W0 a finite subgroup of GL(𝔥*) generated by reflections

Example: Type GLn

𝔥*=-span {ε1,,εn} W0=Snacting by permuting ε1,,εn.

The group algebra of 𝔥* is

[X]=span {Xλλ𝔥*} withXλXμ= Xλ+μ.

W0 acts on [X] by wXλ=Xwλ.

The ring of symmetric functions is

[X]W0= { f[X] wf=fforwW0 } .

Example: Type GL3

Let z1=Xε1, z2=Xε2, z3=Xε3, W0=S3, [X]= [ z1±1, z2±1, z3±1 ] and

[X]W0= [ z1±1, z2±1, z3±1 ] S3 = [e1,e2,e3±1]


e1=z1+z2+z3, e2=z1z2+z1z3 +z2z3,e3=z1 z2z3.

Weyl characters

Let C0 be a fundamental region for W0 acting on 𝔥.

𝔥+=𝔥 C0and 𝔥++=𝔥 C0

where C0 is the closure of C0. Then

𝔥+ 𝔥++ λρ+λ as𝔥+-modules


[X]det= { f[X] wf=det(w)f forwW0 } .

[X]det is a free [X]W0-module of rank 1

[X]W0 [X]det as [X]W0-modules faρf "naive basis"mλ sλ aλ+ρ "naive basis"

where mλ=γW0λ Xγ and aμ=wW0 det(w)XWμ .

The Weyl character is

sλ= aλ+ρaρ, forλ𝔥+.

Weyl's Theorems

Let G() be the reductive algebraic group corresponding to (W0,𝔥), Ta a maximal torus of G.

  1. The simple T-modules Xλ are indexed by λ𝔥
  2. The simple G-modules L(λ) are indexed by λ𝔥+
  3. The character of L(λ) is ResTG(L(λ)) =sλ
  4. aρ=Xρ αR+ (1-X-α)
    where R+ is an index set for the reflections in W0 such that sαμ=μ- μ,αα forαR+, μ𝔥.
C0 s1C0 s2C0 ρ s1s2C0 s2s1C0 s1s2s1C0=s2s1s2C0

The affine Hecke algebra H

Let 𝔥α1,,𝔥α be the walls of C0, s1,,s the corresponding reflections, so that

si:𝔥𝔥 withsiλ=λ- λ,αi αi.

The affine Hecke algebra H is generated by

T1,,Tand Xλ,λ𝔥

with relations

TiTjTi mijfactors = TjTiTj mijfactors ,forij withπmij= 𝔥αi𝔥αj Ti2= (t12-t-12) Ti+1,fori=1,, XλXμ=Xλ+μ, forλ,μ𝔥 TiXλ=Xsiλ Ti+ (t12-t-12) Xλ-Xsiλ 1-X-αi


Tw=Ti1 Tufor a reduced word w=si1si .


{ XλTw λ𝔥,wW0 } is a basis ofH


[X]=span {Xλλ𝔥} andH0=span {TwwW0}

are subalgebras.

Geometric Langlands

Let 10,ε0H0 be such that

102=10 ε02=ε0 and Ti10= t1210 Tiε0= (-t-12) ε0 fori=1,,


HH10=[X]10 hh10 makes[X] anH-module,

the polynomial representation of H. Then

[X]W0 = Z(H) 10H10 ε0H10 f f f10 Aρf10 sλ aa Cλ aa Aλ+ρ Pλ(0,t) aa Mλ


Mλ=10Xλ 10andAμ =ε0Xμ10,

cλ is the Kazhdan-Lusztig basis of the spherical Hecke algebra 10H10=?????Perv(GK) the Grothendieck group of the category of perverse sheaves on the loop Grassmannian Gk.

Pλ(0,t) is Macdonald's spherical function for G(p) G(p) .

Notes and References

These are a typed version of lecture notes for the third in a series of lectures given at the Brazil Algebra Conference, Salvador, 18 July 2012.

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