## Affine Hecke algebras

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. $\Omega =\left\{g\in W\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}g{C}_{1}={C}_{1}\right\}.$
where ${s}_{i}\lambda =\lambda -⟨\lambda ,{\alpha }_{i}^{\vee }⟩{\alpha }_{i}$ determines ${\alpha }_{i}\in {𝔥}_{ℤ}.$

The Dynkin diagram, or Coxeter diagram, of $W$ is the graph with $vertices α0, α1, …, αn and labeled edges αi —mij αj,$ (the graph of the "1-skeleton of ${C}_{1}$").

For $w\in W$ define $ℓ(w) = ( the number of hyperplanes between C1 and wC1).$

(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, $si2 = 1, sisjsi⋯ ⏟ mij factors = sjsisj⋯ ⏟ mij factors for i≠j,$ $gsi g-1 = sg(i) , where g𝔥 αi∨ = 𝔥α g(i)∨ ,$ for $i=0,1,...,n$, and where $\frac{\pi }{{m}_{ij}}={𝔥}^{{\alpha }_{i}^{\vee }}\measuredangle \phantom{\rule{0.5em}{0ex}}{𝔥}^{{\alpha }_{j}^{\vee }}$.

(Weylish presentation) Let $W$ be the affine Weyl group. Let $q$ be an indeterminate and let $𝕂=ℤ\left[q,{q}^{-1}\right]$. The affine Hecke algebra $H$ is presented by generators ${T}_{w}$, $w\in W$, and relations $Tw1 Tw2= Tw1w2, ifℓ(w1w2) =ℓ(w1)+ ℓ(w2), TsiTw= (q-q-1)Tw+ Tsiw, ifℓ(siw)< ℓ(w) (0≤i≤n).$ for $i\in \left\{0,1,\dots ,n\right\}$.

(Coxeterish presentation) Let ${t}^{1/2}$ be an indeterminate and let $𝕂=ℤ\left[{t}_{i}^{1/2},{t}_{i}^{-1/2}\right]$. 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 $Tsi2 = ( ti1/2 - ti-1/2 ) Tsi+1 , Tsi Tsj Tsi⋯ ⏟ mij factors = Tsj Tsi Tsj⋯ ⏟ mij factors for i≠j,$ $g Tsi g-1 = T sg(i) , where g𝔥 αi∨ = 𝔥α g(i)∨ ,$ for $i=0,1,...,n$, and where $\frac{\pi }{{m}_{ij}}={𝔥}^{{\alpha }_{i}^{\vee }}\measuredangle \phantom{\rule{0.5em}{0ex}}{𝔥}^{{\alpha }_{j}^{\vee }}$.

(Bernstein presentation) The affine Hecke algebra $H$ is presented by generators ${T}_{{s}_{1}},{T}_{{s}_{2}},\dots ,{T}_{{s}_{n}}$ and ${X}^{\lambda },\lambda \in {𝔥}_{ℤ}$, with relations $Tsi2 = ( ti1/2 - ti-1/2 ) Tsi+1 , Tsi Tsj Tsi⋯ ⏟ mij factors = Tsj Tsi Tsj⋯ ⏟ mij factors for i≠j,$ $XλXμ = Xλ+μ, and Tsi Xλ = Xsiλ Tsi + ( ti1/2 - ti-1/2 ) Xλ-Xsiλ 1-X-αi ,$

(Intertwiner presentation) The affine Hecke algebra $H$ is presented by generators ${\tau }_{1},{\tau }_{2},\dots ,{\tau }_{n}$ and ${X}^{\lambda },\lambda \in {𝔥}_{ℤ}$, with relations $τi2 = ( ti1/2 - ti-1/2 Xαi ) ( ti1/2 - ti-1/2 X-αi ) (1- Xαi ) (1- X-αi ) , τi τj τi⋯ ⏟ mij factors = τj τi τj⋯ ⏟ mij factors for i≠j,$ $XλXμ = Xλ+μ, and τiXλ = Xsiλ τi ,$

(Graded presentation) The affine Hecke algebra $H$ is presented by generators ${t}_{{s}_{1}},{t}_{{s}_{2}},\dots ,{t}_{{s}_{n}},$ and ${x}_{\lambda },\lambda \in {𝔥}_{ℤ}$, with relations $tsi2 =1 , tsi tsj tsi ⋯ ⏟ mij factors = tsj tsi tsj ⋯ ⏟ mij factors for i≠j,$ $xλxμ = xλ+μ, and tsi xλ = xsiλ tsi + ⟨λ,αi ∨⟩ ,$

(Homogeneous presentation) The affine Hecke algebra $H$ is presented by generators ${\psi }_{1},{\psi }_{2},\dots ,{\psi }_{n}$, ${x}_{\lambda },\lambda \in {𝔥}_{ℤ}$, 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 ${T}_{{s}_{i}}^{2}=\left({t}_{i}^{1/2}-{t}_{i}^{-1/2}\right){T}_{{s}_{i}}+1$ is equivalent to ${\stackrel{\sim }{T}}^{2}=\left({t}_{i}-1\right)\stackrel{\sim }{T}+{t}_{i}.$

Remark. The identities $g= Yωg∨ Tw0 wg-1, T0= Yφ∨ Tsφ-1 , Xλ= Xtλ, Xλ Tw-1 = Xtλw ,$ help to provide conversions between the Coxeterish and Bernstein presentations. It is also useful to note that $Xw=Tw, if w∈W is dominant.$

Remark. In the presence of the relations $Tsi2 = ( ti1/2 - ti-1/2 ) Tsi+1 , and XλXμ = Xλ+μ,$ the relation $Tsi Xλ = Xsiλ Tsi + ( ti1/2 - ti-1/2 ) Xλ-Xsiλ 1-X-αi$ is equivalent to $TiXμ= Xsiμ Ti , if ⟨μ, αi∨⟩ =0, TiXμ Ti =Xsiμ , if ⟨μ, αi∨⟩ =1 , for i=1,2, …,n, (2.23)$

The conversion between the Bernstein and graded presentations is given by $Xλ = exλ, ti1/2 = eci, τi = Tsi + ti-1/2 (1-ti) 1- X-αi = Tsi-1 + ti-1/2 (1-ti) X-αi 1- X-αi$ and $τi = tsi + cαi αi$ 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 $w$, $w= gsi1⋯ sil, g∈Ω, i1,…, il∈ {0,1,…, l}$ is a minimal length sequence $w⇀ = ( g, C1 𝔥β1 si1C1 𝔥β2 si1 si2C1 ⋯ 𝔥βl si1⋯ sil C1 ).$

The elements of $R\left(w\right)=\left\{{\beta }_{1},...,{\beta }_{l}\right\}$ are the elements of ${\stackrel{˜}{R}}_{\mathrm{re}}^{I}$ corresponding to the sequence of hyperplanes crossed by the walk.

For a reduced word $w=g{s}_{{i}_{1}}\cdots {s}_{{i}_{\ell }}$ define $Tw = gTi1 ⋯ Til and Xw = gTi1ϵ1 ⋯ Tilϵl$ where $ϵj = { +1, if the jth step of w⇀ is 𝔥βj + - -1, if the jth step of w⇀ is 𝔥βj - + }$

The affine Hecke algebra $H$ has $𝕂$-bases ${ Tw | w∈W} , {Xw | w∈W} , { Tw-1 -1Xλ | w∈W0, λ∈P }, and { Xμ Tv-1-1 | μ∈P, v∈W0 } .$

## Conversions between presentations

The conversion between presentations is given by the relations $Tw= Tg Tsi1… Tsiℓ, if w∈W and w= gsi1… siℓ is a reduced word, (1.22)$

Let ${w}_{0}$ be the longest element of $W$ and let ${w}_{i}$ be the longest element of the subgroup ${W}_{{\omega }_{i}}=\left\{w\in W\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w{\omega }_{i}={\omega }_{i}\right\}\text{.}$ Let ${\phi }^{\vee }={c}_{1}{\alpha }_{1}^{\vee }+\dots +{c}_{n}{\alpha }_{n}^{\vee }\text{.}$ Then let

$Ω= { g∈W∼∣ℓ (g)=0 } ={1}∪ {gi∣ci=1}, wheregi=tωi wiw0, (1.19)$

(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 $\left\{0,1,\dots ,n\right\}$ also by $g\text{.}$ Then

$gsig-1= sg(i), for0≤i≤n, (1.20)$

With notations as in (1.10-1.20) the conversion between the two presentations is given by the relations $Tw=Ti1… Tip, ifw∈Waff andw=si1… sipis a reduced word, Tgi= xωi Tw0wi-1, forgi∈Ω as in (1.19), xλ=Ttμ Ttν-1, ifλ=μ-νwith μ,ν∈P+, Ts0=Tsϕ x-ϕ, whereϕis the highest short root ofR . (1.22)$

For $w\in W$ and $\lambda \in P$ define elements $Tw-1-1 = ( image in H˜ of a minimal length alcove walk from A to wA ), Xλ = ( image in H˜ of a minimal length alcove walk from A to tλA ).$ $Tw-1-1 = A wA 2 1 2 1 2 1 2 1 0 0 0 0 0 0 0 0 Xλ = WA λ+WA$ 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 are subalgebras of $\stackrel{˜}{H}.$

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 $\stackrel{˜}{H}.$

1. ${X}^{\lambda }{X}^{\mu }={X}^{\lambda +\mu }={X}^{\mu }{X}^{\lambda }.$
2. ${T}_{{s}_{i}}{T}_{w}=\left\{\begin{array}{ll}{T}_{{s}_{i}w},& if\phantom{\rule{.5em}{0ex}}l\left({s}_{i}w\right)>l\left(w\right),\\ {T}_{{s}_{i}w}+\left(q-{q}^{-1}\right){T}_{w},& if\phantom{\rule{.5em}{0ex}}l\left({s}_{i}w\right)
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. ${T}_{{s}_{i}}{X}^{\lambda }={X}^{{s}_{i}\lambda }{T}_{{s}_{i}}+\left(q-{q}^{-1}\right)\frac{{X}^{\lambda }-{X}^{{s}_{i}\lambda }}{1-{X}^{-{\alpha }_{i}}}.$
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 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 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 $\stackrel{˜}{H}.$ Taking inverses gives the first result, and the second follows by switching $w$ and $w{s}_{i}$ and using the relation ${T}_{{s}_{i}}^{-1}={T}_{{s}_{i}}-\left(q-{q}^{-1}\right)$ 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 Thus ${T}_{{s}_{i}}^{-1}{X}^{\lambda }{T}_{{s}_{i}}={X}^{\lambda }={X}^{{s}_{i}\lambda }$ in $\stackrel{˜}{H}.$ $Hα1 Hα1+α2 Hα2 H α1+2α2 Hα0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2$ 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 Thus ${T}_{{s}_{i}}^{-1}\left({X}^{\lambda }{T}_{{s}_{i}^{-1}}\right)={X}^{{s}_{i}\lambda }$ in $H$. $Hα1 Hα1+α2 Hα2 H α1+2α2 Hα0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2$ 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 $Tsi XλXμ = ( Xsiλ Tsi + (q-q-1) Xλ- Xsiλ 1- X-αi ) Xμ = Xsiλ ( Xsiμ Tsi + (q-q-1) Xμ- Xsiμ 1- X-αi ) + (q-q-1) ( Xλ-Xsiλ 1-X-αi ) Xμ = Xsi(λ+μ) Tsi + (q+q-1) Xλ+μ -Xsi (λ+μ) 1-X-αi .$ 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 Thus ${T}_{0}{T}_{{s}_{\phi }}={X}^{\phi }$ in $\stackrel{˜}{H}.$ If ${p}_{{w}_{0}{w}_{i}}$ is a minimal length walk from ${w}_{i}{w}_{0}A$ to $A$ then Thus ${X}^{{\omega }_{i}}=g{T}_{{w}_{0}{w}_{i}}$ in $\stackrel{˜}{H}.$ 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 ${ Tw-1 -1Xλ | w∈W0, λ∈P } and { Xμ Tv-1-1 | μ∈P, v∈W0 } (AHA 7)$ are bases of $H$.

If $p$ is an alcove walk then the weight of $p$ and the final direction of $p$ are $wt(p) ∈P and φ(p) ∈ W0 such that p ends in the alcove wt(p)+ φ(p)A. (AHA 8)$ Let $f-(p) = ( number of negative folds of p ), f+(p) = ( number of positive folds of p ), and f(p) = ( total number of folds of p ). (AHA 9)$ 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), $Tw-1-1Xλ = ∑p (-1)f-(p) (q-q-1)f(p) Xwt(p) Tφ(p)-1-1,$ 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 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 $\stackrel{˜}{H}$ has basis $\left\{{X}^{\lambda }{T}_{{w}^{-1}}^{-1}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\lambda \in P,w\in W\right\}$ in bijection with the alcoves in $\Omega ×{𝔥}_{ℝ}^{*},$ where ${X}^{\lambda }{T}_{{w}^{-1}}^{-1}$ is the image in $\stackrel{˜}{H}$ 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 $\stackrel{˜}{H}.$ The orientation in (AHA 1) is the one such that Another standard orientation is where Using the orientation of the walls given by (AHA 11) produces the basis commonly denoted $\left\{{T}_{w}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}w\in \stackrel{˜}{W}\right\}$ by taking ${T}_{w}$ to be the image in $\stackrel{˜}{H}$ of a minimal length alcove walk from $A$ to ${w}^{-1}A.$ Since ${T}_{i}^{-1}={T}_{i}-\left(q-{q}^{-1}\right)$ the transition matrix between the basis $\left\{{X}^{\lambda }{T}_{{w}^{-1}}^{-1}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\lambda \in P,w\in W\right\}$ and the basis $\left\{{T}_{w}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}w\in \stackrel{˜}{W}\right\}$ is triangular.

## Convolution algebra presentation

Let ${𝔽}_{q}$ be a finite field with $q$ elements, $G =G( 𝔽q) a finite Chevalley group over 𝔽q ∪| Ba Borel subgroup ∪| T a maximal torus.$ The Weyl group of $G$ is ${W}_{0}=N/T,whereN=\left\{g\in G\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}gT{g}^{-1}=T\right\}is the normalizer ofTinG.$

(a) Let $w\in {W}_{0}$. Then $BwB ⋅ BsjB = { BwsjB, if wsj> w, BwB ∪ BwsjB, if wsj< w,$ (b) Bruhat decomposition: $G= ⨆w∈W0 BwB.$ (c) The characteristic functions $\left\{{T}_{w}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}w\in W\right\}$ of the double cosets $BwB$ are a basis of the Hecke algebra $H=C\left(B\G/B\right)$ and $Tw Tsj = { Twsj, ifwsj >w, q Twsj + (q-1)Tw, ifwsj

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

$βk∨= siℓ∨ siℓ-1∨ … sik+1∨ sik∨and tβk∨= tik, (2.35)$

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

$1 w-1 𝔥β7∨ 𝔥β5∨ 𝔥β8∨ 𝔥β6∨ 𝔥β4∨ 𝔥β2∨ 𝔥β3∨ 𝔥β1∨$

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

$z zsj - + z zsj - + z zsj - + z zsj - + positivej–crossing negativej–crossing positivej–fold negativej–fold$

Let $ℬ\left(v,\stackrel{\to }{w}\right)$ be the set of alcove walks of type $\stackrel{\to }{w}=\left({i}_{1},\dots ,{i}_{\ell }\right)$ beginning at $v.$ For a walk $p\in ℬ\left(v,\stackrel{\to }{w}\right)$ let

$f+(p) = { k∣the kth step ofp is a positive fold } , f-(p) = { k∣the kth step ofp is a negative fold } , (2.36)$

and

$end(p)= endpoint ofp(an element ofW ). (2.37)$

## 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?