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

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 05 April 2012

Initial data

The inital data is

1. a finite $\mathbb{Z}-$reflection group, $(W_1,{𝔥}_{\mathbb{Z}}).$

This means

1. ${𝔥}_{\mathbb{Z}}$ is a free $\mathbb{Z}-$module (has $\mathbb{Z}-$basis $\{{\omega }_1,...,{\omega }_l\}$)
2. $W_0$ is a finite subgroup of $GL\left({𝔥}_{\mathbb{Z}}\right)$ generated by reflections.

A reflection is a matrix conjugate (in $GL\left({𝔥}_{ℂ}\right)$) to $$\left(\begin{array}{cccc}\xi & & & \\ & 1& & 0\\ & & \ddots & \\ & 0& & 1\end{array}\right)with\xi \ne 1.$$

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 $S{L}_3\left(ℂ\right).$ $W_0=\{1,s_1,s_2,s_1{s}_2,s_2{s}_1,s_1{s}_2{s}_2\}$ contains 3 reflection, $s_1, s_2$ and $s_1{s}_2{s}_1=s_{\theta },$ the reflections in ${𝔥}^{{\alpha }_1^{\vee }}, {𝔥}^{{\alpha }_2^{\vee }}, {𝔥}^{{\theta }^{\vee }},$ respectively.

The affine Weyl group (semidirect product presentation)

$$W=\left\{t_w{X}^{\lambda } \right| w\in W_0, \lambda \in {𝔥}_{\mathbb{Z}}\}$$ with $$t_u{t}_v=t_{uv},X^{\lambda }{X}^{\mu }=X^{\lambda +\mu }and{t}_w{X}^{\lambda }=X^{w\lambda }{t}_w$$ for $u,v,w\in W_0$ and $\lambda ,\mu \in {𝔥}_{\mathbb{Z}}.$ Then $W$ acts on $$𝔥={𝔥}_{ℝ}\oplus ℝ{\Lambda }_0({𝔥}_{ℝ}=ℝ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}=ℝ-\mathrm{span}\{{\omega }_1,...,{\omega }_l\})$$ by $$t_w{X}^{\lambda }(\mu +m{\Lambda }_0)=\left(\begin{array}{cccc}& & & \\ & t_w& & \lambda \\ & & & \\ 0& \cdots & 0& 1\end{array}\right)\left(\begin{array}{c}\\ \mu \\ \\ m\end{array}\right)=\left(\begin{array}{c}\\ t_w\mu +m\lambda \\ \\ m\end{array}\right)=t_w\mu +m\lambda +m{\Lambda }_0.$$

In our $S{L}_3\left(ℂ\right)$ example: $${𝔥}_1={𝔥}_{ℝ}+{\Lambda }_0\begin{array}{c}\end{array}$$

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.
5. $\Omega =\{g\in W | g{C}_1=C_1\}.$

$W$ is presented by generators $s_0,s_1,...,s_l$ and $\Omega$ such that $\Omega$ is a subgroup, $$\begin{array}{lcl}{s}_i^2=1,& & for\; i=0,1,...,l,\\ {\displaystyle \underset{m_{ij} \; factors}{\underbrace{s_i{s}_j{s}_i\cdots }}}={\displaystyle \underset{m_{ij} \; factors}{\underbrace{s_j{s}_i{s}_j\cdots }}}& & for\; i\ne j, \; where\; \frac{\pi }{m_{ij}}={𝔥}^{{\alpha }_i^{\vee }}\measuredangle {𝔥}^{{\alpha }_j^{\vee }}\\ g{s}_i{g}^{-1}=s_{g\left(i\right)},& & where\; g{𝔥}^{{\alpha }_i^{\vee }}={𝔥}^{{\alpha }_{g\left(i\right)}^{\vee }}.\end{array}$$

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

1. vertices ${\alpha }_0^{\vee },...,{\alpha }_l^{\vee },$
2. and edges ${\alpha }_i^{\vee } \stackrel{m_{ij}}{—} {\alpha }_j^{\vee },$

(the graph of the "1-skeleton of $C_1$").

The affine Hecke algebra (Bernstein presentation)

$H$ is generated by $T_1,...,T_l$ and $X^{\lambda },\lambda \in {𝔥}_{\mathbb{Z}}$ with $$\begin{array}{lcl}{T}_i^2=1 \; for\; i=1,2,...,l,& & X^{\lambda }{X}^{\mu }=X^{\lambda +\mu },\\ {\displaystyle \underset{m_{ij} \; factors}{\underbrace{T_i{T}_j{T}_i\cdots }}}={\displaystyle \underset{m_{ij} \; factors}{\underbrace{T_j{T}_i{T}_j\cdots }}}& & for\; i\ne j,\\ T_i{X}^{\lambda }=X^{s_i\lambda }+(q-q^{-1})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}},& & \end{array}$$ where $s_i\lambda =\lambda -⟨\lambda ,{\alpha }_i^{\vee }⟩{\alpha }_i$ determines ${\alpha }_i\in {𝔥}_{\mathbb{Z}}.$

The affine Hecke algebra (Coxeterish presentation)

$H$ is presented by generators $T_0,T_1,...,T_l$ and $\Omega$ with $$\begin{array}{lcl}{T}_i^2=(q-q^{-1})T_i+1,& & for\; i=0,1,...,l,\\ {\displaystyle \underset{m_{ij} \; factors}{\underbrace{T_i{T}_j{T}_i\cdots }}}={\displaystyle \underset{m_{ij} \; factors}{\underbrace{T_j{T}_i{T}_j\cdots }}}& & for\; i\ne j,\\ gh=hgand& & g{T}_i{g}^{-1}=T_{g\left(i\right)}\end{array}$$ for $g,h\in \Omega .$

Bases of $H$

Let $w\in W.$ A reduced word for $w,$ $$w=g{s}_{i_1}\cdots s_{i_l},g\in \Omega ,i_1,...,i_l\in \{0,1,...,l\}$$ is a minimal length sequence $$\stackrel{\rightharpoonup }{w}=(g,C_1\begin{array}{c} \[ {𝔥}^{{\beta }_1} \]\end{array}{s}_{i_1}{C}_1\begin{array}{c} \[ {𝔥}^{{\beta }_2} \]\end{array}{s}_{i_1}{s}_{i_2}{C}_1\begin{array}{c}\end{array}\cdots \begin{array}{c} \[ {𝔥}^{{\beta }_l} \]\end{array}{s}_{i_1}\cdots s_{i_l}{C}_1).$$ 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 $$T_w=g{T}_{i_1}\cdots T_{i_l}and{X}^w=g{T}_{i_1}^{{ϵ}_1}\cdots T_{i_l}^{{ϵ}_l}$$ where $${ϵ}_j=\{\begin{array}{ll}+1,& if\; the\; j^{\mathrm{th}} \; step\; of\stackrel{\rightharpoonup }{w} \; is\begin{array}{c} \[ {𝔥}^{{\beta }_j} \] \[ + \] \[ - \]\end{array}\\ -1,& if\; the\; j^{\mathrm{th}} \; step\; of\; \stackrel{\rightharpoonup }{w} \; is\begin{array}{c} \[ {𝔥}^{{\beta }_j} \] \[ - \] \[ + \]\end{array}\end{array}\phantom{\}}$$ Then $$\begin{array}{lcl}\left\{T_w \right| w\in W\},& & \left\{X^w \right| w\in W\}\\ \left\{T_v{X}^{\mu } \right| v\in W_0, \lambda \in {𝔥}_{\mathbb{Z}}\},& & \left\{X^{\lambda }{T}_w \right| \lambda \in {𝔥}_{\mathbb{Z}}, w\in W_0\}\end{array}$$ are bases of $H.$

Note: $$X^{\lambda }=X^{X^{\lambda }}and{X}^w\stackrel{?}{=}{T}_v{X}^{\mu } \; if\; w=v{t}_{\mu }.$$

The path model (the algebra of paths)

Four kinds of steps $$\begin{array}{c} \[ j \] \[ v \] \[ v{s}_j \] \[ - \] \[ + \]\end{array}\begin{array}{c} \[ j \] \[ v \] \[ v{s}_j \] \[ - \] \[ + \]\end{array}\begin{array}{c} \[ j \] \[ v \] \[ v{s}_j \] \[ - \] \[ + \]\end{array}\begin{array}{c} \[ j \] \[ v \] \[ v{s}_j \] \[ - \] \[ + \]\end{array}$$ 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 $$\begin{array}{c} \[ - \] \[ + \]\end{array}= \begin{array}{c} \[ - \] \[ + \]\end{array}+ \begin{array}{c} \[ - \] \[ + \]\end{array}and\begin{array}{c} \[ - \] \[ + \]\end{array}= \begin{array}{c} \[ - \] \[ + \]\end{array}+ \begin{array}{c} \[ - \] \[ + \]\end{array}$$ to straighten any sequence of steps to a linear combination of alcove walks.

1. Fix $\lambda \in {𝔥}_{\mathbb{Z}}$ 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 $$T_{w^{-1}}^{-1}{X}^{\lambda }=\sum _p{(-1)}^{\#\; neg\; folds\; of\; p}{(q-q^{-1})}^{\#\; of\; folds\; of\; p}{X}^{\mathrm{wt}\left(p\right)}{T}_{\phi {\left(p\right)}^{-1}}^{-1}$$ where the sum is over all alcove walks $$c_{i_1}^{-}\cdots c_{i_r}^{-}{p}_{j_1}\cdots p_{j_s}where\; p_{j_k} \; is\; c_{j_k}^{+} \; or\; c_{j_k}^{-} \; or\; f_{j_k}^{{ϵ}_k}$$ and $$\mathrm{end}\left(p\right)=\mathrm{wt}\left(p\right)+\phi \left(p\right)C_1$$ is the ending alcove of $p.$

Notes and References

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

References

References?

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