Affine Hecke algebras

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

Last update: 9 January 2013

The affine Hecke algebra: presentations

$C_m$ 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. $\phantom{{𝔥}^{{\alpha }_0^{\vee }},}{𝔥}^{{\alpha }_1^{\vee }},\dots ,{𝔥}^{{\alpha }_l^{\vee }}$ are the walls of $C_0$,
3. ${𝔥}^{{\alpha }_0^{\vee }},{𝔥}^{{\alpha }_1^{\vee }},\dots ,{𝔥}^{{\alpha }_n^{\vee }}$ are the walls of $C_1$,
4. $s_0,\dots ,s_n$ are the reflections in ${𝔥}^{{\alpha }_0^{\vee }},{𝔥}^{{\alpha }_1^{\vee }},\dots ,{𝔥}^{{\alpha }_n^{\vee }}$, and
5. $\Omega =\{g\in W|g{C}_1=C_1\}.$

where $s_i\lambda =\lambda -⟨\lambda ,{\alpha }_i^{\vee }⟩{\alpha }_i$ determines ${\alpha }_i\in {𝔥}_{\mathbb{Z}}.$

The Dynkin diagram, or Coxeter diagram, of $W$ is the graph with $$\text{vertices}{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\text{and}\text{labeled edges}{\alpha }_i\stackrel{m_{ij}}{—}{\alpha }_j,$$ (the graph of the "1-skeleton of $C_1$").

For $w\in W$ define $$\ell \left(w\right)=\left(\text{the number of hyperplanes between}{C}_1\text{and}w{C}_1\right).$$

(Coxeterish presentation) The affine Weyl group $W$ is presented by generators $s_0,s_1,\dots ,s_n$ and $\Omega$ such that $\Omega$ is a subgroup, $$s_i^2=1,{\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{s_i{s}_j{s}_i\cdots }}}={\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{s_j{s}_i{s}_j\cdots }}}\text{for}i\ne j,$$ $$g{s}_i{g}^{-1}=s_{g\left(i\right)},\text{where}g{𝔥}^{{\alpha }_i^{\vee }}={𝔥}^{{\alpha }_{g\left(i\right)}^{\vee }},$$ for $i=0,1,...,n$, and where $\frac{\pi }{m_{ij}}={𝔥}^{{\alpha }_i^{\vee }}\measuredangle {𝔥}^{{\alpha }_j^{\vee }}$.

(Weylish presentation) Let $W$ be the affine Weyl group. Let $q$ be an indeterminate and let $𝕂=\mathbb{Z}[q,q^{-1}]$. The affine Hecke algebra $H$ is presented by generators $T_w$, $w\in W$, and relations $$\begin{array}{cc}{T}_{w_1}{T}_{w_2}=T_{w_1{w}_2},& \text{if}\ell \left(w_1{w}_2\right)=\ell \left(w_1\right)+\ell \left(w_2\right),\\ T_{s_i}{T}_w=(q-q^{-1})T_w+T_{s_{i}w},& \text{if}\ell \left(s_{i}w\right)<\ell \left(w\right)(0\le i\le n)\text{.}\end{array}$$ for $i\in \{0,1,\dots ,n\}$.

(Coxeterish presentation) Let $t^{1/2}$ be an indeterminate and let $𝕂=\mathbb{Z}[t_i^{1/2},t_i^{-1/2}]$. HOW SHOULD WE DEAL WITH THE ISSUE OF MULTIPLE PARAMETERS--PERHAPS AN EXERCISE??
The affine Hecke algebra $H$ is presented by generators $T_{s_0},T_{s_1},\dots ,T_{s_n}$ and $\Omega$ such that $\Omega$ is a subgroup and $$T_{s_i}^2=(t_i^{1/2}-t_i^{-1/2})T_{s_i}+1,{\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{T_{s_i}{T}_{s_j}{T}_{s_i}\cdots }}}={\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{T_{s_j}{T}_{s_i}{T}_{s_j}\cdots }}}\text{for}i\ne j,$$ $$g{T}_{s_i}{g}^{-1}=T_{s_{g\left(i\right)}},\text{where}g{𝔥}^{{\alpha }_i^{\vee }}={𝔥}^{{\alpha }_{g\left(i\right)}^{\vee }},$$ for $i=0,1,...,n$, and where $\frac{\pi }{m_{ij}}={𝔥}^{{\alpha }_i^{\vee }}\measuredangle {𝔥}^{{\alpha }_j^{\vee }}$.

(Bernstein presentation) The affine Hecke algebra $H$ is presented by generators $T_{s_1},T_{s_2},\dots ,T_{s_n}$ and $X^{\lambda },\lambda \in {𝔥}_{\mathbb{Z}}$, with relations $$T_{s_i}^2=(t_i^{1/2}-t_i^{-1/2})T_{s_i}+1,{\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{T_{s_i}{T}_{s_j}{T}_{s_i}\cdots }}}={\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{T_{s_j}{T}_{s_i}{T}_{s_j}\cdots }}}\text{for}i\ne j,$$ $$X^{\lambda }{X}^{\mu }=X^{\lambda +\mu },\text{and}{T}_{s_i}{X}^{\lambda }=X^{s_i\lambda }{T}_{s_i}+(t_i^{1/2}-t_i^{-1/2})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}},$$

(Intertwiner presentation) The affine Hecke algebra $H$ is presented by generators ${\tau }_1,{\tau }_2,\dots ,{\tau }_n$ and $X^{\lambda },\lambda \in {𝔥}_{\mathbb{Z}}$, with relations $${\tau }_i^2=\frac{(t_i^{1/2}-t_i^{-1/2}{X}^{{\alpha }_i})(t_i^{1/2}-t_i^{-1/2}{X}^{-{\alpha }_i})}{(1-X^{{\alpha }_i})(1-X^{-{\alpha }_i})},{\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{{\tau }_i{\tau }_j{\tau }_i\cdots }}}={\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{{\tau }_j{\tau }_i{\tau }_j\cdots }}}\text{for}i\ne j,$$ $$X^{\lambda }{X}^{\mu }=X^{\lambda +\mu },\text{and}{\tau }_i{X}^{\lambda }=X^{s_i\lambda }{\tau }_i,$$

(Graded presentation) The affine Hecke algebra $H$ is presented by generators $t_{s_1},t_{s_2},\dots ,t_{s_n},$ and $x_{\lambda },\lambda \in {𝔥}_{\mathbb{Z}}$, with relations $$t_{s_i}^2=1,{\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{t_{s_i}{t}_{s_j}{t}_{s_i}\cdots }}}={\displaystyle \underset{m_{ij}\text{factors}}{\underbrace{t_{s_j}{t}_{s_i}{t}_{s_j}\cdots }}}\text{for}i\ne j,$$ $$x_{\lambda }{x}_{\mu }=x_{\lambda +\mu },\text{and}{t}_{s_i}{x}_{\lambda }=x_{s_i\lambda }{t}_{s_i}+⟨\lambda ,{\alpha }_i^{\vee }⟩,$$

(Homogeneous presentation) The affine Hecke algebra $H$ is presented by generators ${\psi }_1,{\psi }_2,\dots ,{\psi }_n$, $x_{\lambda },\lambda \in {𝔥}_{\mathbb{Z}}$, and $e_u$, $u\in \Gamma$ with relations $$KLR\; LIKE\; RELATIONS$$

Normalization. Setting $\stackrel{\sim }{T}=t_i^{1/2}{T}_{s_i}$, the relation $T_{s_i}^2=(t_i^{1/2}-t_i^{-1/2})T_{s_i}+1$ is equivalent to ${\stackrel{\sim }{T}}^2=(t_i-1)\stackrel{\sim }{T}+t_i.$

Remark. The identities $$g=Y^{{\omega }_g^{\vee }}{T}_{w_0{w}_g}^{-1},T_0=Y^{{\phi }^{\vee }}{T}_{s_{\phi }}^{-1},X^{\lambda }=X^{t_{\lambda }},X^{\lambda }{T}_w^{-1}=X^{t_{\lambda }w},$$ help to provide conversions between the Coxeterish and Bernstein presentations. It is also useful to note that $$X^w=T_w,\text{if}w\in W\text{is dominant.}$$

Remark. In the presence of the relations $$T_{s_i}^2=(t_i^{1/2}-t_i^{-1/2})T_{s_i}+1,\text{and}{X}^{\lambda }{X}^{\mu }=X^{\lambda +\mu },$$ the relation $$T_{s_i}{X}^{\lambda }=X^{s_i\lambda }{T}_{s_i}+(t_i^{1/2}-t_i^{-1/2})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}$$ is equivalent to $$\begin{array}{cc}\begin{array}{ccc}{T}_i{X}^{\mu }=X^{s_i\mu }{T}_i,& & \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=0,\\ T_i{X}^{\mu }{T}_i=X^{s_i\mu },& & \text{if}⟨\mu ,{\alpha }_i^{\vee }⟩=1,\end{array}\text{for}i=1,2,\dots ,n,& \text{(2.23)}\end{array}$$

The conversion between the Bernstein and graded presentations is given by $$X^{\lambda }=e^{x_{\lambda }},t_i^{1/2}=e^{c_i},{\tau }_i=T_{s_i}+\frac{t_i^{-1/2}(1-t_i)}{1-X^{-{\alpha }_i}}=T_{s_i}^{-1}+\frac{t_i^{-1/2}(1-t_i)X^{-{\alpha }_i}}{1-X^{-{\alpha }_i}}$$ and $${\tau }_i=t_{s_i}+\frac{c_{{\alpha }_i}}{{\alpha }_i}$$ THE NORMALIZATION NEEDS TO BE FIXED HERE SO THAT ${\tau }_i^2$ matches both cases.

Bases of $H$

The periodic orientation has

1. (a) If $0\in {𝔥}^{\alpha }$ then $C_0$ is on the positive side of ${𝔥}^{\alpha }$.
2. (b) Parallel hyperplanes have parallel orientation.

For example, when ${𝔤}_0={\mathrm{𝔰𝔩}}_3$,

The alcoves are the triangles and the (centers of) hexagons are the elements of $Q^{\vee }.$

Let $w\in W$. A reduced word for $w$, $$w=g{s}_{i_1}\cdots s_{i_l},g\in \Omega ,i_1,\dots ,i_l\in \{0,1,\dots ,l\}$$ 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 elements of $R\left(w\right)=\{{\beta }_1,...,{\beta }_l\}$ are the elements of ${\tilde{R}}_{\mathrm{re}}^I$ corresponding to the sequence of hyperplanes crossed by the walk.

For a reduced word $w=g{s}_{i_1}\cdots s_{i_{\ell }}$ define $$T_w=g{T}_{i_1}\cdots T_{i_l}\text{and}{X}^w=g{T}_{i_1}^{{ϵ}_1}\cdots T_{i_l}^{{ϵ}_l}$$ where $${ϵ}_j=\{\begin{array}{ll}+1,& \text{if the}{j}^{\mathrm{th}}\text{step of}\stackrel{\rightharpoonup }{w}\text{is}\begin{array}{c} \[ {𝔥}^{{\beta }_j} \] \[ + \] \[ - \]\end{array}\\ -1,& \text{if the}{j}^{\mathrm{th}}\text{step of}\stackrel{\rightharpoonup }{w}\text{is}\begin{array}{c} \[ {𝔥}^{{\beta }_j} \] \[ - \] \[ + \]\end{array}\end{array}\phantom{\}}$$

The affine Hecke algebra $H$ has $𝕂$-bases $$\left\{T_w\right|w\in W\},\{X^w|w\in W\},\left\{T_{w^{-1}}^{-1}{X}^{\lambda }\right|w\in W_0,\lambda \in P\},\text{and}\{X^{\mu }{T}_{v^{-1}}^{-1}|\mu \in P,v\in W_0\}.$$

Conversions between presentations

The conversion between presentations is given by the relations $$\begin{array}{cc}{T}_w=T_g{T}_{s_{i_1}}\dots T_{s_{i_{\ell }}},\text{if}w\in W\text{and}w=g{s}_{i_1}\dots s_{i_{\ell }}\text{is a reduced word,}& \text{(1.22)}\end{array}$$

Let $w_0$ be the longest element of $W$ and let $w_i$ be the longest element of the subgroup $W_{{\omega }_i}=\{w\in W\mid w{\omega }_i={\omega }_i\}\text{.}$ Let ${\phi }^{\vee }=c_1{\alpha }_1^{\vee }+\dots +c_n{\alpha }_n^{\vee }\text{.}$ Then let

$$\begin{array}{cc}\Omega =\{g\in \stackrel{\sim }{W}\mid \ell \left(g\right)=0\}=\left\{1\right\}\cup \{g_i\mid c_i=1\},\text{where}{g}_i=t_{{\omega }_i}{w}_i{w}_0,& \text{(1.19)}\end{array}$$

(see [Bou1981, VI § no. 3 Prop. 6]). Each element $g\in \Omega$ sends the alcove $A$ to itself and thus permutes the walls $H_{{\alpha }_0},H_{{\alpha }_1},\dots ,H_{{\alpha }_n}$ of $A\text{.}$ Denote the resulting permutation of $\{0,1,\dots ,n\}$ also by $g\text{.}$ Then

$$\begin{array}{cc}g{s}_i{g}^{-1}=s_{g\left(i\right)},\text{for}0\le i\le n,& \text{(1.20)}\end{array}$$

With notations as in (1.10-1.20) the conversion between the two presentations is given by the relations $$\begin{array}{cc}\begin{array}{cc}{T}_w=T_{i_1}\dots T_{i_p},& \text{if}w\in W_{\text{aff}}\text{and}w=s_{i_1}\dots s_{i_p}\text{is a reduced word,}\\ T_{g_i}=x^{{\omega }_i}{T}_{w_0{w}_i}^{-1},& \text{for}{g}_i\in \Omega \text{as in (1.19),}\\ x^{\lambda }=T_{t_{\mu }}{T}_{t_{\nu }}^{-1},& \text{if}\lambda =\mu -\nu \text{with}\mu ,\nu \in P^{+},\\ T_{s_0}=T_{s_{\varphi }}{x}^{-\varphi },& \text{where}\varphi \text{is the highest short root of}R\text{.}\end{array}& \text{(1.22)}\end{array}$$

For $w\in W$ and $\lambda \in P$ define elements $$\begin{array}{rcl}{T}_{w^{-1}}^{-1}& =& \left(image\; in\tilde{H}of\; a\; minimal\; length\; alcove\; walk\; fromAtowA\right),\\ X^{\lambda }& =& \left(image\; in\tilde{H}of\; a\; minimal\; length\; alcove\; walk\; fromAto{t}_{\lambda }A\right).\end{array}$$ $$T_{w^{-1}}^{-1}=\begin{array}{c} \[ A \] \[ wA \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \]\end{array}{X}^{\lambda }=\begin{array}{c} \[ WA \] \[ \lambda +WA \]\end{array}$$ The following proposition shows that the alcove walk definition of the affine Hecke algebra coincides with the standard definition by generators and relations (see [IM] and [LU]). A consequence of the proposition is that $$\begin{array}{ll}\begin{array}{rcl}\text{the <em>finite Hecke algebra</em>,}& H=\mathrm{span}\left\{T_{w^{-1}}^{-1}\right|w\in W\},& and\\ \text{the Laurent polynomial ring,}& 𝕂\left[P\right]=\mathrm{span}\left\{X^{\lambda }\right|\lambda \in P\},& \end{array}& (AHA\; 6)\end{array}$$ are subalgebras of $\tilde{H}.$

Let $g\in \Omega ,$ $\lambda ,\mu \in P,$$w\in W$ and $1\le i\le n.$ Let $\phi$ be the element of $R^{+}$ such that $H_{{\alpha }_0}=H_{\phi ,1}$ is the wall of $A$ which is not a wall of $C$ and let $s_{\phi }$ be the reflection in $H_{\phi }.$ Let $w_0$ be the longest element of $W.$ The following identities hold in $\tilde{H}.$

1. $X^{\lambda }{X}^{\mu }=X^{\lambda +\mu }=X^{\mu }{X}^{\lambda }.$
2. $T_{s_i}{T}_w=\{\begin{array}{ll}{T}_{s_{i}w},& ifl\left(s_{i}w\right)>l\left(w\right),\\ T_{s_{i}w}+(q-q^{-1})T_w,& ifl\left(s_{i}w\right)<l\left(w\right).\end{array}\phantom{\}}$
3. If $⟨\lambda ,{\alpha }_i^{\vee }⟩=0$ then $T_{s_i}{X}^{\lambda }=X^{\lambda }{T}_{s_i}.$
4. If $⟨\lambda ,{\alpha }_i^{\vee }⟩=1$ then $T_{s_i}{X}^{s_i\lambda }{T}_{s_i}=X^{\lambda }.$
5. $T_{s_i}{X}^{\lambda }=X^{s_i\lambda }{T}_{s_i}+(q-q^{-1})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}.$
6. $T_{s_0}{T}_{s_{\phi }}=X^{\phi }.$
7. $X^{{\omega }_i}=g{T}_{w_0{w}_i},$ where the action of $g$ on $A$ sends the origin to ${\omega }_i$ and $w_i$ is the longest element of the stabilizer $W_{{\omega }_i}$ of ${\omega }_i$ in $W.$

		Proof.
	Use notations for alcove walks as in (AHA 4). If $p_{\lambda }$ is a minimal length walk from $A$ to $t_{\lambda }A$ and $p_{\mu }$ is a minimal length walk from $A$ to $t_{\mu }A$ then $$p_{\lambda }{p}_{\mu }\text{and}{p}_{\mu }{p}_{\lambda } \; are\; both\; nonfolded\; walks\; from\; A \; to\; t_{\lambda +\mu }A.$$ Thus the images of $p_{\lambda }{p}_{\mu }$ and $p_{\mu }{p}_{\lambda }$ are equal in $H$. If $l\left(w{s}_i\right)>l\left(w\right)$ and $p_w$ is a minimal length walk from $A$ to $wA$ then $$p_{w{s}_i}=p_w{c}_i^{-} \; is\; a\; minimal\; length\; walk\; from\; A \; to\; w{s}_{i}A,$$ and so $T_{s_i{w}^{-1}}^{-1}=T_{w{s}_i^{-1}}^{-1}=T_{w^{-1}}^{-1}{T}_{s_i}^{-1}={\left(T_{s_i}{T}_{w^{-1}}\right)}^{-1}$ in $\tilde{H}.$ Taking inverses gives the first result, and the second follows by switching $w$ and $w{s}_i$ and using the relation $T_{s_i}^{-1}=T_{s_i}-(q-q^{-1})$ which follows from (AHA 2) and (AHA 5). Let $p_{\lambda }$ be a minimal length alcove walk from $A$ to $t_{\lambda }A.$ If $⟨\lambda ,{\alpha }_i^{\vee }⟩=0$ then $H_{{\alpha }_i}$ is a wall of $t_{\lambda }A$ and $s_i\lambda =\lambda$ and $$c_i^{-}{p}_{\lambda }{c}_i^{+}is\; a\; nonfolded\; walk\; from\; A \; to\; t_{\lambda }A.$$ Thus $T_{s_i}^{-1}{X}^{\lambda }{T}_{s_i}=X^{\lambda }=X^{s_i\lambda }$ in $\tilde{H}.$ $$\begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ H_{{\alpha }_0} \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \]\end{array}$$ Let $p_{\lambda }$ be a minimal length walk from $A$ to $t_{\lambda }A$. If $⟨\lambda ,{\alpha }_i^{\vee }⟩=1$ then there is a minimal length walk from $A$ to $t_{\lambda }A$ of the form $p_{\lambda }=p_{t_{\lambda }{s}_i}{c}_i^{+}$ where $p_{t_{\lambda }{s}_i}$ is a minimal length walk from $A$ to $t_{\lambda }{s}_{i}A$. Then $$c_i^{-}{p}_{t_{\lambda }{s}_i} \; is\; a\; minimal\; length\; walk\; from\; A \; to\; t_{s_i\lambda }A.$$ Thus $T_{s_i}^{-1}\left(X^{\lambda }{T}_{s_i^{-1}}\right)=X^{s_i\lambda }$ in $H$. $$\begin{array}{c} \[ H_{{\alpha }_1} \] \[ H_{{\alpha }_1+{\alpha }_2} \] \[ H_{{\alpha }_2} \] \[ H_{{\alpha }_1+2{\alpha }_2} \] \[ H_{{\alpha }_0} \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 0 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 1 \] \[ 2 \] \[ 1 \] \[ 2 \]\end{array}$$ Note that (c) and (d) are special cases of (e). If the statement of (e) holds for $\lambda$ then, by multiplying on the left by $X^{-s_i\lambda }$ and on the right by $X^{-\lambda }$, it holds for $-\lambda$. If the statement (e) holds for $\lambda$ and $\mu$ then it holds for $\lambda +\mu$ since $$\begin{array}{rcl}{T}_{s_i}{X}^{\lambda }{X}^{\mu }& =& (X^{s_i\lambda }{T}_{s_i}+(q-q^{-1})\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}})X^{\mu }\\ & =& X^{s_i\lambda }(X^{s_i\mu }{T}_{s_i}+(q-q^{-1})\frac{X^{\mu }-X^{s_i\mu }}{1-X^{-{\alpha }_i}})+(q-q^{-1})\left(\frac{X^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}\right)X^{\mu }\\ & =& X^{s_i(\lambda +\mu )}{T}_{s_i}+(q+q^{-1})\frac{X^{\lambda +\mu }-X^{s_i(\lambda +\mu )}}{1-X^{-{\alpha }_i}}.\end{array}$$ Thus, to prove (e) it is sufficient to verify (c) and (d), which has already been done. Let $p_{s_{\phi }}$ be a minimal length walk from $s_{\phi }A$ to $A,$ then $$p_{\phi }=c_0^{+}{p}_{s_{\phi }} \; is\; a\; minimal\; length\; walk\; from\; A \; to\; t_{\phi }A.$$ Thus $T_0{T}_{s_{\phi }}=X^{\phi }$ in $\tilde{H}.$ If $p_{w_0{w}_i}$ is a minimal length walk from $w_i{w}_{0}A$ to $A$ then $$p_{{\omega }_i}=g{p}_{w_0{w}_i} \; is\; a\; minimal\; length\; walk\; from\; A \; to\; t_{{\omega }_i}A.$$ Thus $X^{{\omega }_i}=g{T}_{w_0{w}_i}$ in $\tilde{H}.$ For example, in type $C_2,$ $w_0=s_2{s}_1{s}_2{s}_1$ and there is one element $g$ in $\Omega$ such that $g\ne 1$ for which $g{\omega }_2=0$ and $w_2=s_1$ so that $w_0{w}_2=s_2{s}_1{s}_2.$ $\square$

The sets $$\begin{array}{ll}\left\{T_{w^{-1}}^{-1}{X}^{\lambda }\right|w\in W_0,\lambda \in P\}\text{and}\left\{X^{\mu }{T}_{v^{-1}}^{-1}\right|\mu \in P,v\in W_0\}& (AHA\; 7)\end{array}$$ are bases of $H$.

If $p$ is an alcove walk then the weight of $p$ and the final direction of $p$ are $$\begin{array}{ll}\mathrm{wt}\left(p\right)\in P\text{and}\phi \left(p\right)\in W_{0}such\; thatp\text{ends in the alcove}\mathrm{wt}\left(p\right)+\phi \left(p\right)A.& (AHA\; 8)\end{array}$$ Let $$\begin{array}{ll}\begin{array}{rcl}{f}^{-}\left(p\right)& =& \left(number\; of\; negative\; folds\; ofp\right),\\ f^{+}\left(p\right)& =& \left(number\; of\; positive\; folds\; ofp\right),and\\ f\left(p\right)& =& \left(total\; number\; of\; folds\; ofp\right).\end{array}& (AHA\; 9)\end{array}$$ The following theorem provides a combinatorial formulation of the transition matrix between the bases in (AHA 7). It is a $q-$version of the main result of [LP] and an extension of Corollary 6.1 of [Sc].

Use notations as in (AHA 4). Let $\lambda \in P$ and $w\in W.$ Fix a minimal length walk $p_w=c_{i_1}^{-}{c}_{i_2}^{-}\cdots c_{i_r}^{-}$ from $A$ to $wA$ and a minimal length walk $p_{\lambda }=c_{j_1}^{{\epsilon }_1}\cdots c_{j_s}^{{\epsilon }_s}$ from $A$ to $t_{\lambda }A.$ Then, with notations as in (AHA 8) and (AHA 9), $$T_{w^{-1}}^{-1}{X}^{\lambda }=\sum _p{(-1)}^{f^{-}\left(p\right)}{(q-q^{-1})}^{f\left(p\right)}{X}^{\mathrm{wt}\left(p\right)}{T}_{\phi {\left(p\right)}^{-1}}^{-1},$$ where the sum is over all alcove walks $p=c_{i_1}^{-}\cdots c_{i_r}^{-}{p}_{j_1}\cdots p_{j_s}$ such that $p_{j_k}$ is either $c_{j_k}^{{\epsilon }_k}, c_{j_k}^{-{\epsilon }_k}$ or $f_{j_k}^{{\epsilon }_k}.$

		Proof.
	The product $p_w{p}_{\lambda }=c_{i_1}^{-}{c}_{i_2}^{-}\cdots c_{i_r}^{-}{c}_{j_1}^{{\epsilon }_1}\cdots c_{j_s}^{{\epsilon }_s}$ may not necessarily be a walk, but its straightening produces a sum of walks, and this decomposition gives the formula in the statement. $\square$

The initial direction $\iota \left(p\right)$ and the final direction $\phi \left(p\right)$ of an alcove walk $p$ appear naturally in Theorem 2.2. These statistics also appear in the Pieri-Chevalley formula in the K-theory of the flag variety (see [PR], [GR], [Br] and [LP]).

In Theorem 2.2, for certain $\lambda$ the walk $p_{\lambda }$ may be chosen so that all the terms in the expansion of $T_{w^{-1}}^{-1}{X}^{\lambda }$ have the same sign. For example, if $\lambda$ is dominant, then $p_{\lambda }$ can be taken with all ${\epsilon }_k=+,$ in which case all folds which appear in the straightening of $p_w{p}_{\lambda }$ will be positive folds and so all terms in the expansion will be positive. If $\lambda$ is antidominant then $p_{\lambda }$ can be taken with all ${\epsilon }_k=-$ and all terms in the expansion will be negative. This fact gives positivity results for products in the cohomology and the K-theory of the flag variety (see [PR], [Br]).

The affine Hecke algebra $\tilde{H}$ has basis $\left\{X^{\lambda }{T}_{w^{-1}}^{-1}\right|\lambda \in P,w\in W\}$ in bijection with the alcoves in $\Omega \times {𝔥}_{ℝ}^{*},$ where $X^{\lambda }{T}_{w^{-1}}^{-1}$ is the image in $\tilde{H}$ of a minimal length alcove walk from $A$ to the alcove $\lambda +wA.$ Changing the orientation of the walls of the alcoves chances the resulting basis in the affine Hecke algebra $\tilde{H}.$ The orientation in (AHA 1) is the one such that $$\begin{array}{ll}the\; most\; negative\; point\; is\; -\infty \rho , \; deep\; in\; the\; chamber\; w_{0}C.& (AHA\; 10)\end{array}$$ Another standard orientation is where $$\begin{array}{ll}the\; most\; negative\; point\; is\; the\; center\; of\; the\; fundamental\; alcove\; A.& (AHA\; 11)\end{array}$$ Using the orientation of the walls given by (AHA 11) produces the basis commonly denoted $\left\{T_w\right|w\in \tilde{W}\}$ by taking $T_w$ to be the image in $\tilde{H}$ of a minimal length alcove walk from $A$ to $w^{-1}A.$ Since $T_i^{-1}=T_i-(q-q^{-1})$ the transition matrix between the basis $\left\{X^{\lambda }{T}_{w^{-1}}^{-1}\right|\lambda \in P,w\in W\}$ and the basis $\left\{T_w\right|w\in \tilde{W}\}$ is triangular.

Convolution algebra presentation

Let ${𝔽}_q$ be a finite field with $q$ elements, The Weyl group of $G$ is $W_0=N/T,$ where$ N=\{g\in G|gT{g}^{-1}=T\}$is\; the\; normalizer\; of$ T$in$ G$.$$

(a) Let $w\in W_0$. Then $$BwB\cdot B{s}_{j}B=\{\begin{array}{cc}Bw{s}_{j}B,\hfill & \text{if}w{s}_j>w,\hfill \\ BwB\cup Bw{s}_{j}B,\hfill & \text{if}w{s}_j<w,\hfill \end{array}$$ (b) Bruhat decomposition: $$G=\underset{w\in W_0}{⨆}BwB.$$ (c) The characteristic functions $\left\{T_w\right|w\in W\}$ of the double cosets $BwB$ are a basis of the Hecke algebra $H=C\left(B\backslash G/B\right)$ and $$T_w{T}_{s_j}=\{\begin{array}{ll}{T}_{w{s}_j},& \text{if}w{s}_j>w,\\ q{T}_{w{s}_j}+(q-1)T_w,& \text{if}w{s}_j<w.\end{array}$$

For the moment, we refer to affflags1.14.07.pdf for the proof.

Leftover junk

Let $w\in W$ and let $w=s_{i_1}^{\vee }\dots s_{i_{\ell }}^{\vee }$ be a reduced word for $w.$ For $k=1,\dots ,\ell$ let

$$\begin{array}{cc}{\beta }_k^{\vee }=s_{i_{\ell }}^{\vee }{s}_{i_{\ell -1}}^{\vee }\dots s_{i_{k+1}}^{\vee }{s}_{i_k}^{\vee }\text{and}{t}_{{\beta }_k^{\vee }}=t_{i_k},& \text{(2.35)}\end{array}$$

so that the sequence ${\beta }_{\ell }^{\vee },{\beta }_{\ell -1}^{\vee },\dots ,{\beta }_1^{\vee }$ is the sequence of labels of the hyperplanes crossed by the walk $w^{-1}=s_{i_{\ell }}^{\vee }{s}_{i_{\ell -1}}^{\vee }\dots s_{i_1}^{\vee }\text{.}$ For example, in Type $A_2,$ with $w=s_2^{\vee }{s}_0^{\vee }{s}_1^{\vee }{s}_2^{\vee }{s}_1^{\vee }{s}_0^{\vee }{s}_2^{\vee }{s}_1^{\vee }$ the picture is

$$\[ 1 \] \[ w-1 \] \[ {𝔥}^{{\beta }_7^{\vee }} \] \[ {𝔥}^{{\beta }_5^{\vee }} \] \[ {𝔥}^{{\beta }_8^{\vee }} \] \[ {𝔥}^{{\beta }_6^{\vee }} \] \[ {𝔥}^{{\beta }_4^{\vee }} \] \[ {𝔥}^{{\beta }_2^{\vee }} \] \[ {𝔥}^{{\beta }_3^{\vee }} \] \[ {𝔥}^{{\beta }_1^{\vee }} \]$$

Let $v\in W^{\vee }\text{.}$ An alcove walk of type $i_1,\dots ,i_{\ell }$ beginning at $v$ is a sequence of steps, where a step of type $j$ is

$$\begin{array}{cccc} \[ z \] \[ z{s}_j \] \[ - \] \[ + \]& \[ z \] \[ z{s}_j \] \[ - \] \[ + \]& \[ z \] \[ z{s}_j \] \[ - \] \[ + \]& \[ z \] \[ z{s}_j \] \[ - \] \[ + \]\\ \text{positive}j\text{–crossing}& \text{negative}j\text{–crossing}& \text{positive}j\text{–fold}& \text{negative}j\text{–fold}\end{array}$$

Let $ℬ(v,\overrightarrow{w})$ be the set of alcove walks of type $\overrightarrow{w}=(i_1,\dots ,i_{\ell })$ beginning at $v.$ For a walk $p\in ℬ(v,\overrightarrow{w})$ let

$$\begin{array}{cc}\begin{array}{ccc}{f}^{+}\left(p\right)& =& \{k\mid \text{the}k\text{th step of}p\text{is a positive fold}\},\\ f^{-}\left(p\right)& =& \{k\mid \text{the}k\text{th step of}p\text{is a negative fold}\},\end{array}& \text{(2.36)}\end{array}$$

and

$$\begin{array}{cc}\text{end}\left(p\right)=\text{endpoint of}p\text{(an element of}W\text{).}& \text{(2.37)}\end{array}$$

Notes and References

These notes are intended to supplement various lecture series given by Arun Ram. These important facts about Iwahori-Hecke algebras are found in Bourbaki ????. The original papers are [Iw] Iwahori ????, and [IM] Iwahori-Matsumoto ????. One can also see Steinberg Lecture notes ?????.

References

References?

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