The Weyl character formula from the affine Hecke algebra point of view

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

Last update: 05 April 2012

Symmetric functions

The initial data is (W0,𝔥), a finite -reflection group, i.e.

  1. 𝔥 is a free -module,
  2. W0 a finite subgroup of GL(𝔥) generated by reflections.


  1. 𝔥 = span{ε1,...,εn} with
  2. W0 = Sn acting by permuting ε1,...,εn.
The group algebra of 𝔥 is [X] = span{Xλ  |  λ𝔥} with XλXμ = Xλ+μ . W0 acts on [X] by wXλ = Xwλ.

The ring of symmetric functions is [X]W0 = {f[X]  |  wf=f}.

Example: Type GL3.
Let z1 = Xε1, z2 = Xε2, z3 = Xε3. Then W0 = S3, [X] = [z1±1,z2±1,z3±1] and [X]W0 = [z1±1,z2±1,z3±1]S3 = [e1,e2,e3±1], where e1 = z1+z2+z3, e2 = z1z2 + z1z3 + z2z3, e3 = z1z2z3.

Weyl characters

Let C0 be a fundamental region for W0 acting on 𝔥 = 𝔥. Let 𝔥+ = 𝔥 C0_ and 𝔥++ = 𝔥C0 where C0_ is the closure of C0. Then 𝔥+ 𝔥++ λ ρ+λ as   𝔥+- modules.

Define [X]det = {f[X]  |  wf = det(w)f,   for   wW0}.

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

[X]W0 [X]det   as   [X]W0   modules f aρf sλ aλ+ρ   "naive basis" "naive basis" mλ where mλ = γW0λ Xγ and aμ = wW0 det(w) Xwμ.

The Weyl character is sλ = aλ+ρ aρ , for   λ𝔥+.

The affine Hecke algebra H


  1. 𝔥α1 ,..., 𝔥αl be the walls of C0,
  2. s1,...,sl the corresponding reflections,
so that si: 𝔥𝔥 is given by siλ = λ- λ,αi αi, for   i=1,...,l.

The affine Hecke algebra H is generated by T1,...,Tl and Xλ,  λ𝔥 with relations Ti2 = (t12-t-12)Ti+1, for   i=1,...,l, TiTjTi mij   factors = TjTiTj mij   factors , for   ij   with   πmij = 𝔥αi 𝔥αj, XλXμ = Xλ+μ , for   λ,μ𝔥, TiXλ = Xsiλ Ti + ( t12 - t-12 ) Xλ-Xsiλ 1-X-αi . Define Tw = Ti1 Til for a reduced word w = si1 sil. Then {XλTw  |  λ𝔥,  wW0} is a basis of  H and [X] = span{Xλ  |  λ𝔥} and H0 = span{Tw  |  wW0} are subalgebras.

Bernstein-Satake-Lusztig isomorphisms

Let 𝟙0,ε0H0 be such that 𝟙02 = 𝟙0 ε02 = ε0 and Ti𝟙0 = t12𝟙0 Tiε0 = (-t-12) ε0 for   i=1,...,l. Then H H𝟙0 = [X]𝟙0 h h𝟙0 makes [X] into an H-module (the polynomial representation). Then Bernstein Satake Lusztig [X]W0 = Z(H) 𝟙0H𝟙0 ε0H𝟙0 f f f𝟙0 Aρf𝟙0 sλ Cλ Aλ+ρ }   "naive bases" Pλ(0,t) Mλ where Mλ = 𝟙0 Xλ 𝟙0 and Aμ = ε0 Xμ 𝟙0. Cλ is the Kazhdan-Lusztig basis of the spherical Hecke algebra 𝟙0H𝟙0 = K0 (Perv(G/K)) the Grothendieck group of the category Perv(G/K) of perverse sheaves on the loop Grassmannian G/K.
Pλ(0,t) is Macdonald's spherical function for G(p) / G(p) .

Weyl's Theorems

Let G() be the reductive algebraic group corresponding to (W0,𝔥).

  1. The simple T-modules Xλ are indexed by 𝔥*.
  2. The simple G-modules L(λ) are indexed by λ(𝔥*)+.
  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 sαW0, so that sαμ = μ- μ,αα.

q-Weyl denominator

Aρ = αR+ ( t12 Xα2 - t-12 X-α2 )

Notes and References

These notes are from lecture notes of Arun Ram (CBMS Lecture 2). It was also a lecture for the 2011 semester 2 Representation Theory course at the University of Melbourne (05/08/2011).



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