## Presenting the Affine Hecke algebra: Iwahori and Bernstein Presentations and the Path model

Last update: 05 April 2012

## Initial data

The inital data is

1. a finite $ℤ-$reflection group, $\left({W}_{1},{𝔥}_{ℤ}\right).$
This means
1. ${𝔥}_{ℤ}$ is a free $ℤ-$module (has $ℤ-$basis $\left\{{\omega }_{1},...,{\omega }_{l}\right\}$)
2. ${W}_{0}$ is a finite subgroup of $GL\left({𝔥}_{ℤ}\right)$ generated by reflections.
A reflection is a matrix conjugate (in $GL\left({𝔥}_{ℂ}\right)$) to $ξ 1 0 ⋱ 0 1 with ξ≠1.$

HW: Show that if $s$ is a reflection in a finite subgroup of ${\mathrm{GL}}_{n}\left(ℂ\right)$ then $\xi$ is a root of unity?

Our favourite example: Type $S{L}_{3}\left(ℂ\right).$ ${W}_{0}=\left\{1,{s}_{1},{s}_{2},{s}_{1}{s}_{2},{s}_{2}{s}_{1},{s}_{1}{s}_{2}{s}_{2}\right\}$ contains 3 reflection, and ${s}_{1}{s}_{2}{s}_{1}={s}_{\theta },$ the reflections in respectively.

## The affine Weyl group (semidirect product presentation)

with $tutv = tuv, XλXμ = Xλ+μ and twXλ = Xwλtw$ for $u,v,w\in {W}_{0}$ and $\lambda ,\mu \in {𝔥}_{ℤ}.$ Then $W$ acts on $𝔥 = 𝔥ℝ⊕ℝΛ0 ( 𝔥ℝ = ℝ⊗ℤ𝔥ℤ = ℝ-span{ω1,...,ωl} )$ by $twXλ (μ+mΛ0) = tw λ 0 ⋯ 0 1 μ m = twμ+mλ m = twμ + mλ + mΛ0.$

In our $S{L}_{3}\left(ℂ\right)$ example: $𝔥1 = 𝔥ℝ+Λ0$

## Coxeter generators

${C}_{m}$ are fundamental regions for $W$ acting on ${𝔥}_{m}$ such that

1. $\stackrel{_}{{C}_{0}}\supseteq \cdots \supseteq \stackrel{_}{{C}_{2}}\supseteq \stackrel{_}{{C}_{1}}\supseteq 0.$
2. ${𝔥}^{{\alpha }_{1}^{\vee }},...,{𝔥}^{{\alpha }_{l}^{\vee }}$ are the walls of ${C}_{0}.$
3. ${𝔥}^{{\alpha }_{0}^{\vee }},...,{𝔥}^{{\alpha }_{l}^{\vee }}$ are the walls of ${C}_{1},$
4. ${s}_{0},...,{s}_{l}$ the corresponding reflections.

$W$ is presented by generators ${s}_{0},{s}_{1},...,{s}_{l}$ and $\Omega$ such that $\Omega$ is a subgroup,

The Dynkin, or Coxeter diagram of $W$ is the graph with

1. vertices ${\alpha }_{0}^{\vee },...,{\alpha }_{l}^{\vee },$
2. and edges
(the graph of the "1-skeleton of ${C}_{1}$").

## The affine Hecke algebra (Bernstein presentation)

$H$ is generated by ${T}_{1},...,{T}_{l}$ and ${X}^{\lambda },\lambda \in {𝔥}_{ℤ}$ with where ${s}_{i}\lambda =\lambda -⟨\lambda ,{\alpha }_{i}^{\vee }⟩{\alpha }_{i}$ determines ${\alpha }_{i}\in {𝔥}_{ℤ}.$

## The affine Hecke algebra (Coxeterish presentation)

$H$ is presented by generators ${T}_{0},{T}_{1},...,{T}_{l}$ and $\Omega$ with for $g,h\in \Omega .$

## Bases of $H$

Let $w\in W.$ A reduced word for $w,$ $w = gsi1⋯ sil, g∈Ω, i1,...,il ∈ {0,1,...,l}$ is a minimal length sequence $w⇀ = ( g, C1 𝔥β1 si1C1 𝔥β2 si1 si2C1 ⋯ 𝔥βl si1⋯ sil C1 ).$ The periodic orientation is:

1. If $0\in {𝔥}^{\alpha }$ then ${C}_{0}$ is on the positive side of ${𝔥}^{\alpha }.$
2. Parallel hyperplanes have parallel orientation.
For a reduced work $w=g{s}_{{i}_{1}}\cdots {s}_{{i}_{l}}$ define $Tw = gTi1 ⋯ Til and Xw = gTi1ϵ1 ⋯ Tilϵl$ where Then are bases of $H.$

Note:

## The path model (the algebra of paths)

Four kinds of steps $j v vsj - + j v vsj - + j v vsj - + j v vsj - +$ An alcove walk is a sequence of steps such that

1. the tail of the first step is in ${C}_{1}$
2. at every step, the head of each arrow is in the same alcove as the tail of the next.
Use the relations to straighten any sequence of steps to a linear combination of alcove walks.

1. Fix $\lambda \in {𝔥}_{ℤ}$ and $w\in {W}_{0}.$
2. Fix a minimal length walk ${c}_{{i}_{1}}^{-}\cdots {c}_{{i}_{r}}^{-}$ from ${C}_{1}$ to $w{C}_{1}$
3. and a minimal length walk ${c}_{{j}_{1}}^{{ϵ}_{1}}\cdots {c}_{{j}_{s}}^{{ϵ}_{s}}$ from ${C}_{1}$ to $\lambda +{C}_{1}.$
Then where the sum is over all alcove walks and $end(p) = wt(p) + φ(p)C1$ is the ending alcove of $p.$

## Notes and References

These notes are from lecture notes of Arun Ram (CBMS Lecture 1).

References?