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

Last update: 05 April 2012

## Symmetric functions

The initial data is $\left({W}_{0},{𝔥}_{ℤ}\right),$ a finite $ℤ-$reflection group, i.e.

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

Example.

1. ${𝔥}_{ℤ}=\mathrm{span}\left\{{\epsilon }_{1},...,{\epsilon }_{n}\right\}$ with
2. ${W}_{0}={S}_{n}$ acting by permuting ${\epsilon }_{1},...,{\epsilon }_{n}.$
The group algebra of ${𝔥}_{ℤ}$ is ${W}_{0}$ acts on $ℂ\left[X\right]$ by $w{X}^{\lambda }={X}^{w\lambda }.$

The ring of symmetric functions is

Example: Type $G{L}_{3}$.
Let ${z}_{1}={X}^{{\epsilon }_{1}},{z}_{2}={X}^{{\epsilon }_{2}},{z}_{3}={X}^{{\epsilon }_{3}}.$ Then $W0 = S3, ℂ[X] = ℂ[z1±1,z2±1,z3±1] and ℂ[X]W0 = ℂ[z1±1,z2±1,z3±1]S3 = ℂ[e1,e2,e3±1],$ where $e1 = z1+z2+z3, e2 = z1z2 + z1z3 + z2z3, e3 = z1z2z3.$

## Weyl characters

Let ${C}_{0}$ be a fundamental region for ${W}_{0}$ acting on ${𝔥}_{ℝ}=ℝ{\otimes }_{ℤ}{𝔥}_{ℤ}.$ Let $𝔥ℤ+ = 𝔥ℤ∩ C0_ and 𝔥ℤ++ = 𝔥ℤ∩C0$ where $\stackrel{_}{{C}_{0}}$ is the closure of ${C}_{0}.$ Then

Define

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

where ${m}_{\lambda }=\sum _{\gamma \in {W}_{0}\lambda }{X}^{\gamma }$ and ${a}_{\mu }=\sum _{w\in {W}_{0}}\mathrm{det}\left(w\right){X}^{w\mu }.$

The Weyl character is

## The affine Hecke algebra $H$

Let

1. ${𝔥}^{{\alpha }_{1}},...,{𝔥}^{{\alpha }_{l}}$ be the walls of ${C}_{0},$
2. ${s}_{1},...,{s}_{l}$ the corresponding reflections,
so that ${s}_{i}:{𝔥}_{ℤ}\to {𝔥}_{ℤ}$ is given by

The affine Hecke algebra $H$ is generated by with relations Define $Tw = Ti1⋯ Til for a reduced word w = si1 ⋯ sil.$ Then and are subalgebras.

## Bernstein-Satake-Lusztig isomorphisms

Let ${𝟙}_{0},{\epsilon }_{0}\in {H}_{0}$ be such that Then $H → H𝟙0 = ℂ[X]𝟙0 h ↦ h𝟙0$ makes $ℂ\left[X\right]$ into an $H-$module (the polynomial representation). Then where $Mλ = 𝟙0 Xλ 𝟙0 and Aμ = ε0 Xμ 𝟙0.$ ${C}_{\lambda }$ is the Kazhdan-Lusztig basis of the spherical Hecke algebra ${𝟙}_{0}H{𝟙}_{0}={K}_{0}\left(\mathrm{Perv}\left(G/K\right)\right)$ the Grothendieck group of the category $\mathrm{Perv}\left(G/K\right)$ of perverse sheaves on the loop Grassmannian $G/K.$
${P}_{\lambda }\left(0,t\right)$ is Macdonald's spherical function for $G\left({ℚ}_{p}\right)/G\left({ℤ}_{p}\right).$

## Weyl's Theorems

Let $G\left(ℂ\right)$ be the reductive algebraic group corresponding to $\left({W}_{0},{𝔥}_{ℤ}\right).$

1. The simple $T-$modules ${X}^{\lambda }$ are indexed by ${𝔥}_{ℤ}^{*}.$
2. The simple $G-$modules $L\left(\lambda \right)$ are indexed by $\lambda \in {\left({𝔥}_{ℤ}^{*}\right)}^{+}.$
3. The character of $L\left(\lambda \right)$ is $ResTG(L(λ)) = sλ.$
4. $aρ = Xρ ∏α∈R+ (1-X-α)$ where ${R}^{+}$ is an index set for the reflections ${s}_{\alpha }\in {W}_{0},$ so that $sαμ = μ- ⟨μ,α∨⟩α.$

## $q-$Weyl denominator

$Aρ = ∏α∈R+ ( t12 Xα2 - t-12 X-α2 )$

## 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).

References?