## The Weyl Character formula

Last update: 8 October 2012

## Symmetric functions

Initial data:

1. $\left({W}_{0},{𝔥}_{ℤ}^{*}\right)$ a finite $ℤ$-reflection group
2. ${𝔥}_{ℤ}^{*}$ is a free $ℤ$-modules
3. ${W}_{0}$ a finite subgroup of $GL\left({𝔥}_{ℤ}^{*}\right)$ generated by reflections

Example: Type $G{L}_{n}$

$𝔥ℤ*=ℤ-span {ε1,…,εn} W0=Snacting by permuting ε1,…,εn.$

The group algebra of ${𝔥}_{ℤ}^{*}$ is

$ℂ[X]=span {Xλ∣λ∈𝔥ℤ*} withXλXμ= Xλ+μ.$

${W}_{0}$ acts on $ℂ\left[X\right]$ by $w{X}^{\lambda }={X}^{w\lambda }\text{.}$

The ring of symmetric functions is

$ℂ[X]W0= { f∈ℂ[X]∣ wf=fforw∈W0 } .$

Example: Type $G{L}_{3}$

Let ${z}_{1}={X}^{{\epsilon }_{1}},$ ${z}_{2}={X}^{{\epsilon }_{2}},$ ${z}_{3}={X}^{{\epsilon }_{3}},$ ${W}_{0}={S}_{3},$ $ℂ\left[X\right]=ℂ\left[{z}_{1}^{±1},{z}_{2}^{±1},{z}_{3}^{±1}\right]$ and

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

where

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

## Weyl characters

Let ${C}_{0}$ be a fundamental region for ${W}_{0}$ acting on ${𝔥}_{ℝ}\text{.}$

$𝔥ℤ+=𝔥ℤ∩ C‾0and 𝔥ℤ++=𝔥ℤ ∩C0$

where ${\stackrel{‾}{C}}_{0}$ is the closure of ${C}_{0}\text{.}$ Then

$𝔥ℤ+ ⟶∼ 𝔥ℤ++ λ⟼ρ+λ as𝔥ℤ+-modules$

Define

$ℂ[X]det= { f∈ℂ[X]∣ wf=det(w)f forw∈W0 } .$

$ℂ{\left[X\right]}^{\text{det}}$ is a free $ℂ{\left[X\right]}^{{W}_{0}}$-module of rank 1

$ℂ[X]W0 ⟶∼ ℂ[X]det as ℂ[X]W0-modules f⟼aρf "naive basis"mλ sλ↤ aλ+ρ "naive basis"$

where ${m}_{\lambda }=\sum _{\gamma \in {W}_{0}\lambda }{X}^{\gamma }$ and ${a}_{\mu }=\sum _{w\in {W}_{0}}\text{det}\phantom{\rule{0.2em}{0ex}}\left(w\right){X}^{W\mu }\text{.}$

The Weyl character is

$sλ= aλ+ρaρ, forλ∈𝔥ℤ+.$

## Weyl's Theorems

Let ${G}^{\vee }\left(ℂ\right)$ be the reductive algebraic group corresponding to $\left({W}_{0},{𝔥}_{ℤ}\right),$ ${T}^{\vee }a$ a maximal torus of ${G}^{\vee }\text{.}$

1. The simple ${T}^{\vee }$-modules ${X}^{\lambda }$ are indexed by $\lambda \in {𝔥}_{ℤ}$
2. The simple ${G}^{\vee }$-modules $L\left(\lambda \right)$ are indexed by $\lambda \in {𝔥}_{ℤ}^{+}$
3. The character of $L\left(\lambda \right)$ is $ResTG(L(λ)) =sλ$
4. ${a}_{\rho }={X}^{\rho }\prod _{\alpha \in {R}^{+}}\left(1-{X}^{-\alpha }\right)$
where ${R}^{+}$ is an index set for the reflections in ${W}_{0}$ such that $sαμ=μ- ⟨μ,α∨⟩α forα∈R+, μ∈𝔥ℤ.$
$C0 s1C0 s2C0 ρ s1s2C0 s2s1C0 s1s2s1C0=s2s1s2C0$

## The affine Hecke algebra $H$

Let ${𝔥}^{{\alpha }_{1}},\dots ,{𝔥}^{{\alpha }_{\ell }}$ be the walls of ${C}_{0},$ ${s}_{1},\dots ,{s}_{\ell }$ the corresponding reflections, so that

$si:𝔥ℤ⟶𝔥ℤ withsiλ=λ- ⟨λ,αi∨⟩ αi.$

The affine Hecke algebra $H$ is generated by

$T1,…,Tℓand Xλ,λ∈𝔥ℤ$

with relations

$TiTjTi… ⏟ mijfactors = TjTiTj… ⏟ mijfactors ,fori≠j withπmij= 𝔥αi≮𝔥αj Ti2= (t12-t-12) Ti+1,fori=1,…,ℓ XλXμ=Xλ+μ, forλ,μ∈𝔥ℤ TiXλ=Xsiλ Ti+ (t12-t-12) Xλ-Xsiλ 1-X-αi$

Define

$Tw=Ti1… Tuℓfor a reduced word w=si1…siℓ .$

Then

${ XλTw∣ λ∈𝔥ℤ,w∈W0 } is a basis ofH$

and

$ℂ[X]=span {Xλ∣λ∈𝔥ℤ} andH0=span {Tw∣w∈W0}$

are subalgebras.

## Geometric Langlands

Let ${1}_{0},{\epsilon }_{0}\in {H}_{0}$ be such that

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

Then

$H⟶H10=ℂ[X]10 h⟼h10 makesℂ[X] anH-module,$

the polynomial representation of $H\text{.}$ Then

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

where

$Mλ=10Xλ 10andAμ =ε0Xμ10,$

${c}_{\lambda }$ is the Kazhdan-Lusztig basis of the spherical Hecke algebra ${1}_{0}H{1}_{0}=\text{?????}\text{Perv}\phantom{\rule{0.2em}{0ex}}\left(G}{K}\right)$ the Grothendieck group of the category of perverse sheaves on the loop Grassmannian $G}{k}\text{.}$

${P}_{\lambda }\left(0,t\right)$ is Macdonald's spherical function for $G\left({ℚ}_{p}\right)}{G\left({ℤ}_{p}\right)}\text{.}$

## 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.