## Unipotent Hecke algebras: the structure, representation theory, and combinatorics

Last updated: 26 March 2015

## Unipotent Hecke algebras

### The algebras

#### Important subgroups of a Chevalley group

Let $G={G}_{V}$ be a Chevalley group defined with a $𝔤\text{-module}$ $V$ as in Section 2.2.2. Recall that ${R}^{+}$ is the set of positive roots according to some fixed set of simple roots $\left\{{\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{\ell }\right\}$ of $𝔤$ (Section 2.2.1). The group $G$ contains a subgroup $U$ given by $U=⟨xα(t) | α∈R+,t∈𝔽q⟩,$ which decomposes as $U=∏α∈R+Uα, whereUα= ⟨xα(t) | t∈𝔽q⟩,$ with uniqueness of expression for any fixed ordering of the positive roots [Ste1967, Lemma 18]. For each $\alpha \in {R}^{+},$ the map $Uα ⟶∼ 𝔽q+ xα(t) ⟼ t$ is a group homomorphism.

For $\alpha \in R,$ define the maps $sα: 𝔥* ⟶ 𝔥* γ ⟼ γ-γ(Hα)α, (3.1) sα: 𝔥 = Z(𝔤)⊕𝔥s ⟶ 𝔥 H+Hβ ⟼ H+Hβ-β(Hα)Hα, for β∈R. (3.2)$

The Weyl group of $G$ is $W=⟨{s}_{\alpha } | \alpha \in R⟩$ and has a presentation given by $W= ⟨ s1,s2,…,sℓ | si2=1, (sisj)mij =1,1≤i≠j≤ℓ ⟩ ,mij∈ℤ>0, si=sαi.$ If $w={s}_{{i}_{1}}{s}_{{i}_{2}}\cdots {s}_{{i}_{r}}$ with $r$ minimal, then the length of $w$ is $\ell \left(w\right)=r\text{.}$

Let ${𝔥}_{ℤ}$ be as in (2.3). If $q>3,$ then the subgroup $T= ⟨ hH(t) | H∈𝔥ℤ,t∈𝔽q* ⟩$ has its normalizer in $G$ given by $N= ⟨ wα(t),h | α∈R,h∈T,t∈𝔽q* ⟩ ,where wα(t)= xα(t)x-α (-t-1)xα(t).$ If $\alpha \in R,$ then ${h}_{{H}_{\alpha }}\left(t\right)={w}_{\alpha }\left(t\right){w}_{\alpha }{\left(1\right)}^{-1}\text{.}$ Write ${h}_{\alpha }\left(t\right)={h}_{{H}_{\alpha }}\left(t\right)$ and ${h}_{i}\left(t\right)={h}_{{\alpha }_{i}}\left(t\right)\text{.}$

There is a natural surjection from $N$ onto the Weyl group $W$ with kernel $T$ given by $π: N ⟶ W wα(t) ⟼ sα, for α∈R,t∈𝔽q*, h ⟼ 1, for h∈T. (3.3)$ Suppose $v\in N\text{.}$ Then for each minimal expression $π(v)= si1si2 ⋯sir, with ℓ(π(v)) =r,$ there is a unique decomposition of $v$ as $v=v1v2⋯ vrvT, where vk= wik(1) and vT∈T. (3.4)$ Write $ξi=wi(1). (3.5)$

#### Unipotent Hecke algebras

Let $G$ be a finite Chevalley group. Fix a nontrivial homomorphism $\psi :{𝔽}_{q}^{+}\to {ℂ}^{*}\text{.}$ If $μ: R+ ⟶ 𝔽q α ⟼ μα satisfies μα=0 for all α not simple, (3.6)$ then the map $ψμ: U ⟶ ℂ* xα(t) ⟼ ψ(μαt) (3.7)$ is a linear character of $U\text{.}$ In fact, with the exception of a few degenerate special cases of $G$ (which can be avoided if $q>3\text{),}$ all linear characters of $U$ are of this form [Yok1969, Theorem 1].

The unipotent Hecke algebra $ℋ\left(G,U,{\psi }_{\mu }\right)$ is $ℋμ=EndℂG (IndUG(ψμ)). (3.8)$ Using the anti-isomorphism (2.2), view ${ℋ}_{\mu }$ as the subset of $ℂG$ $ℋμ=eμℂG eμ,where eμ=1∣U∣ ∑u∈Uψμ (u-1)u. (3.9)$

#### The choices $\mu$ and $\psi$

This section assumes all linear characters of $U$ are of the form ${\psi }_{\mu }$ for some nontrivial homomorphism$\psi :{𝔽}_{q}^{+}\to {ℂ}^{*}$ and $\mu$ as in (3.6).

The group $T$ normalizes ${U}_{\alpha },$ since $hxα(t)h-1 =xα(α(h)t), for α∈R+,t∈ 𝔽q,h∈T,α(h) ∈𝔽q*.$ Thus, if $\chi :U\to {ℂ}^{*}$ is a linear character, then $hχ: U ⟶ ℂ* xα(t) ⟼ χ(hxα(t)h-1)$ is a linear character for every $h\in T\text{.}$

Suppose $\chi :U\to {ℂ}^{*}$ is a linear character. For every $h\in T,$ $IndUG(χ)= IndUG(hχ).$

Proof.

Recall that $IndUG(hψμ) (g)=1∣U∣ ∑x∈Gxgx-1∈U hψμ(xgx-1)= 1∣U∣ ∑x∈Gxgx-1∈U ψμ(hxgx-1h-1).$ Since $T$ normalizes $U,$ the sum over all $x\in G$ such that $xg{x}^{-1}\in U$ is the same as the sum over $hx\in G$ such that $hxg{x}^{-1}{h}^{-1}\in U\text{.}$ Thus, $IndUG(hψμ) (g)=1∣U∣ ∑x′∈Gx′gx′-1∈U ψμ(x′gx′-1) =IndUG(ψμ)(g).$

$\square$

The type of a linear character $\chi :U\to {ℂ}^{*}$ is the set $J\subseteq \left\{{\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{\ell }\right\}$ such that $χ(xαi(t))≠ 1 for all t∈𝔽qif and only if αi∈J.$ A unipotent Hecke algebra $ℋ\left(G,U,\chi \right)$ has type $J$ if $\chi$ has type $J\text{.}$

(a) Fix a nontrivial homomorphism $\psi :{𝔽}_{q}^{+}\to {ℂ}^{*}\text{.}$ If $\psi \prime :{𝔽}_{q}^{+}\to {ℂ}^{*}$ is a homomorphism, then there exists $k\in {𝔽}_{q}$ such that $ψ′(t)= ψ(kt), for all t∈𝔽q+.$ (b) The linear characters $\chi :U\to {ℂ}^{*}$ and ${}^{h}\chi :U\to {ℂ}^{*}$ have the same type for any $h\in T\text{.}$ (c) Let $J\subseteq \left\{{\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{\ell }\right\}$ and ${𝒮}_{J}=\left\{\text{type} J \text{linear characters of} U\right\}\text{.}$ Then the number of $T\text{-orbits}$ of ${𝒮}_{J}$ is $∣ZJ∣(q-1)∣J∣∣T∣, where ZJ= { h∈T | α(h) =1,for α∈J } .$ (d) The number of distinct (nonisomorphic) type $J$ unipotent Hecke algebras is at most $∣ZJ∣(q-1)∣J∣∣T∣.$

Proof.

(a) The map $𝔽q+ ⟶ Hom(𝔽q+,ℂ*) k ⟼ ψk: 𝔽q+ ⟶ ℂ* t ⟼ ψ(kt)$ is a group homomorphism, and the kernel is trivial because $\psi$ is nontrivial. Since $\mid \text{Hom}\left({𝔽}_{q}^{+},ℂ\right)\mid =q,$ the map is also surjective.

(b) Since ${U}_{\alpha }\cong {𝔽}_{q}^{+},$ the restriction of $\chi$ to ${U}_{\alpha }$ gives a linear character of ${𝔽}_{q}^{+}\text{.}$ Part (a) implies that for every $\alpha \in {R}^{+}$ there exists ${k}_{\alpha }\in {𝔽}_{q}$ such that $χ(xα(t))= ψ(kαt)$ Furthermore, if $\chi$ has type $J,$ then ${k}_{\alpha }\ne 0$ if and only if $\alpha \in J\text{.}$ Write $χ=ψ(k1,…,kℓ) where ki=kαi.$ Since $h{x}_{\alpha }\left(t\right){h}^{-1}={x}_{\alpha }\left(\alpha \left(h\right)t\right)$ for $h\in T,$ $hχ= ψ(α1(h)k1,…,αℓ(h)kℓ),$ and $\alpha \left(h\right){k}_{\alpha }\ne 0$ if and only if ${k}_{\alpha }\ne 0$ if and only if $\alpha \in J\text{.}$ Thus, the action of $T$ preserves the type of $\chi \text{.}$

(c) Suppose a group $K$ acts on a set $𝒮\text{.}$ Then by [Isa1994, Theorem 4.18] the number of orbits of the $K$ action is $1∣K∣ ∑g∈KCard {s∈𝒮 | g(s)=s}.$ In this case, $K=T$ acts on the set $𝒮=𝒮J= { ψ(k1,k2,…,kℓ) | ki=0, unless αi∈J } .$ Furthermore, $hψ(k1,k2,…,kℓ) =ψ(k1,k2,…,kℓ) if and only ifαi(h)=1 for all αi∈J.$ Finally, $\mid {𝒮}_{J}\mid ={\left(q-1\right)}^{\mid J\mid }\text{.}$

$\square$

Examples. 1. If $J=\varnothing ,$ then $\mid J\mid =0$ and $\mid {Z}_{J}\mid =\mid T\mid ,$ so there is a unique unipotent Hecke algebra of type $J\text{;}$ in this case, ${\psi }_{\mu }$ is trivial and ${ℋ}_{\mu }$ is the Yokonuma algebra. 2. A character is in general position if it has type $J=\left\{{\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{\ell }\right\}\text{.}$ In this case, ${Z}_{J}$ is the center of $G,$ and $\mid J\mid =\ell ,$ so the number of type $J$ unipotent Hecke algebras is at most $∣Z(G)∣(q-1)ℓ∣T∣ .$

#### A natural basis

The group $G$ has a double-coset decomposition $G=⨆v∈NUvU, [Ste1967, Theorem 4 and B=UT] (3.10)$ and if $Nμ = {v∈N | eμveμ≠0} = { v∈V | u,vuv-1 ∈U implies ψμ(u) =ψμ(vuv-1) } (3.11)$ then the set $\left\{{e}_{\mu }v{e}_{\mu } | v\in {N}_{\mu }\right\}$ is a basis for ${ℋ}_{\mu }$ [CRe1981, Prop. 11.30].

Theorem 3.7 gives a set of relations similar to those of the Yokonuma algebra (Example 1, below) for evaluating the product $(eμueμ) (eμveμ), with u,v∈Nμ$ in any unipotent Hecke algebra ${ℋ}_{\mu }\text{.}$

Examples.

1. The Yokonuma Hecke algebra. If ${\mu }_{\alpha }=0$ for all positive roots $\alpha ,$ then ${\psi }_{\mu }=1$ is the trivial character and ${N}_{1}=N\text{.}$ Let ${T}_{v}={e}_{1}v{e}_{1}$ for $v\in N,$ with ${T}_{i}={T}_{{\xi }_{i}}$ $\text{(}{\xi }_{i}$ as in (3.5)) and ${T}_{H}\left(t\right)={T}_{{h}_{H}\left(t\right)}\text{.}$ If $v\in N$ has a decomposition $v={v}_{1}{v}_{2}\cdots {v}_{r}{V}_{T}$ (as in (3.4)), then $Tv=Ti1Ti2⋯ TirTvT(See Chapter 6).$ Thus, the Yokonuma algebra ${ℋ}_{1}$ has generators ${T}_{i},$ ${T}_{h}$ for $1\le i\le \ell ,$ $h\in T$ (see [Yok1969-2]) with relations, $Ti2 = q-1THαi (-1)+q-1 ∑t∈𝔽q* THαi (t-1)Ti, 1≤i≤ℓ, TiTjTi⋯ ⏟mij terms = TjTiTj⋯ ⏟mij terms , (sish)mij=1, TiTh = TsijTi, h∈T, ThTk = Thk, h,k∈T.$ These relations give an “efficient” way to compute arbitrary products $\left({e}_{1}u{e}_{1}\right)\left({e}_{1}v{e}_{1}\right)$ in ${ℋ}_{1}\text{.}$ There is a surjective map from the Yokonuma algebra onto the Iwahori-Hecke algebra that sends ${T}_{h}↦1$ for all $h\in T\text{.}$ “Setting ${T}_{h}=1\text{”}$ in the Yokonuma algebra relations recovers relations for the Iwahori-Hecke algebra, $Ti2=q-1+ q-1(q-1) Ti, TiTj⋯ ⏟mij terms = TjTi⋯ ⏟mij terms .$ Furthermore, there is a surjective map from the Iwahori Hecke algebra onto the Weyl group given by mapping ${T}_{i}↦{s}_{i}$ and $q↦1\text{.}$ Thus, by “setting ${T}_{i}={s}_{i}$ and $q=1\text{”}$ we retrieve the Coxeter relations of $W,$ $si2=1, sisjsj⋯ ⏟mij terms = sjsisj⋯ ⏟mij terms .$ For a more detailed discussion of the Yokonuma algebra, see Chapter 6.

2. The Gelfand-Graev Hecke algebra. By definition, if ${\mu }_{\alpha }\ne 0$ for all simple roots $\alpha ,$ then ${\psi }_{\mu }$ is in general position. In this case, the Gelfand-Graev character ${\text{Ind}}_{U}^{G}\left({\psi }_{\mu }\right)$ is multiplicity free as a $G\text{-module}$ ([Yok1968],[Ste1967, Theorem 49]). The corresponding Hecke algebra ${ℋ}_{\mu }$ is therefore commutative. However, decomposing the product $\left({e}_{\mu }u{e}_{\mu }\right)\left({e}_{\mu }v{e}_{\mu }\right)$ into basis elements is more challenging than in the Yokonuma case [Cha1976, Cur1988, Rai2002].

### Parabolic subalgebras of ${ℋ}_{\mu }$

Let ${\psi }_{\mu }:U\to G$ be as in (3.7). Fix a subset $J\subseteq \left\{{\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{\ell }\right\}$ such that $if μαi≠0, then αi∈J.$ For example, if ${\psi }_{\mu }$ is in general position, then $J=\left\{{\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{\ell }\right\},$ but if ${\psi }_{\mu }$ is trivial, then $J$ could be any subset.

Let $WJ=⟨si∈W | αi∈J⟩, PJ=⟨U,T,WJ⟩ andRJ=ℤ-span {αi∈J}∩R.$ Then ${P}_{J}$ has subgroups $LJ=⟨T,WJ,Uα | α∈RJ⟩ andUJ=⟨Uα | α∈R+-RJ⟩ (3.12)$ (a Levi subgroup and the unipotent radical of ${P}_{J},$ respectively). Note that $UJLJ=PJ, UJ∩LJ=1,and, in fact, PJ=UJ⋊LJ.$

Define the idempotents of $ℂU,$ $eμJ=1∣LJ∩U∣ ∑u∈LJ∩U ψμ(u-1)u andeJ′= 1∣UJ∣ ∑u∈UJu, (3.13)$ so that ${e}_{\mu }={e}_{\mu J}{e}_{J}^{\prime }\text{.}$

The group homomorphisms $PJ ⟶ LJ lu ⟼ l and PJ ⟶ G lu ⟼ lu for l∈LJ,u∈UJ,$ induce maps $InfLJPJ: {LJ-modules} ⟶ {PJ-modules} M ⟼ eJ′M, IndPJG: {PJ-modules} ⟶ {G-modules} M′ ⟼ ℂG⊗ℂPJM′$ whose composition is the map ${\text{Indf}}_{{L}_{J}}^{G}\text{.}$ Note that in the special case when $ℂ{L}_{J}e$ is an irreducible ${L}_{J}\text{-module}$ with corresponding idempotent $e,$ then $IndfLJG: {LJ-modules} ⟶ {G-modules} ℂLJe ⟼ ℂGeeJ′.$

The map ${\psi }_{\mu }:U\to {ℂ}^{*}$ restricts to a linear character ${\text{Res}}_{U\cap {L}_{J}}^{U}\left({\psi }_{\mu }\right):{L}_{J}\cap U\to {ℂ}^{*}\text{.}$ To make the notation less heavy-handed, write ${\psi }_{\mu }:{L}_{J}\cap U\to {ℂ}^{*},$ for ${\text{Res}}_{U\cap {L}_{J}}^{U}\left({\psi }_{\mu }\right)\text{.}$

Let ${\psi }_{\mu }$ be as in (3.7). Then $IndUG(ψμ)≅ IndfLJG (IndU∩LJLJ(ψμ)).$

Proof.

Recall ${\text{Ind}}_{U}^{G}\left({\psi }_{\mu }\right)\cong ℂG{e}_{\mu }\text{.}$ On the other hand, $IndU∩LJLJ (ψμ)≅ℂ LJeμJimplies IndfLJG (IndU∩LJLJ(ψμ)) ≅ℂGeμJeJ′,$ where ${e}_{\mu J}$ is as in (3.13). But ${e}_{\mu J}{e}_{J}^{\prime }={e}_{\mu },$ so $IndUG(ψμ)≅ ℂGeμ≅ ℂeμJeJ′≅ IndfLJG(IndU∩LJLJ(ψμ)).$

$\square$

The map $θ: EndℂLJ(IndU∩LJLJ(ψμ)) ⟶ ℋμ eμJveμJ ⟼ eμveμ, for v∈LJ∩Nμ,$ is an injective algebra homomorphism.

Proof.

Let $v\in N\cap {L}_{J}\text{.}$ Since ${e}_{\mu J}v{e}_{\mu J}\in ℂ{L}_{J},$ ${e}_{J}^{\prime }\in ℂ{U}_{J}$ and ${L}_{J}\cap {U}_{J}=1,$ $eμJveμJ=0 if and only ifeJ′ eμJveμJ=0.$ Because ${L}_{J}$ normalizes ${U}_{J},$ both ${e}_{\mu J}$ and $v$ commute with ${e}_{J}^{\prime }\text{.}$ Therefore, $eμJveμJ=0 if and only ifeJ′eμJ veJ′eμJ= eμveμ=0,$ and ${eμJveμJ≠0} if and only ifv∈Nμ∩LJ.$ Consequently, $\theta$ is both well-defined and injective.

Consider $θ(eμJueμJ) θ(eμJveμJ)= eμueμ eμveμ= eμJeJ′u eμJeJ′v eJ′eμJ.$ Since $u$ commutes with ${e}_{J}^{\prime },$ $θ(eμJueμJ) θ(eμJveμJ) = eμJeJ′u eμJveJ′ eμJ = eμueμJv eμ = θ(eμJueμJveμJ),$ and so $\theta$ is an algebra homomorphism.

$\square$

Write $ℒJ=θ (EndℂLJ(IndU∩LJLJ(ψμ))) ⊆ℋμ (3.14)$ The ${ℒ}_{J}$ are “parabolic” subalgebras of ${ℋ}_{\mu }$ in that they have a similar relationship to the representation theory of ${ℋ}_{\mu }$ as parabolic subgroups ${P}_{J}$ have with the representation theory of $G\text{.}$

#### Weight space decompositions for ${ℋ}_{\mu }\text{-modules}$

An important special case of Theorem 3.4 is when $J=Jμ= {αi∈{α1,α2,…,αℓ} | μαi≠0},$ so that ${\psi }_{\mu }$ has type ${J}_{\mu }\text{.}$ Write ${L}_{\mu }={L}_{{J}_{\mu }},$ ${W}_{\mu }={W}_{{J}_{\mu }},$ etc.

The algebra ${ℒ}_{\mu }$ is a nonzero commutative subalgebra of ${ℋ}_{\mu }\text{.}$

Proof.

As a character of $U\cap {L}_{\mu },$ ${\psi }_{\mu }$ is in general position, so ${\text{Ind}}_{{L}_{\mu }\cap U}^{{L}_{\mu }}\left({\psi }_{\mu }\right)$ is a Gelfand-Graev module and ${ℒ}_{\mu }$ is a Gelfand-Graev Hecke algebra (and therefore commutative).

$\square$

Since ${ℒ}_{\mu }$ is commutative, all the irreducible modules are one-dimensional; let ${\stackrel{ˆ}{ℒ}}_{\mu }$ be an indexing set for the irreducible modules of ${ℒ}_{\mu }\text{.}$ Suppose $V$ is an ${ℋ}_{\mu }\text{-module.}$ As an ${ℒ}_{\mu }\text{-module,}$ $V≅⨁γ∈ℒˆμ VγwhereVγ= {v∈V | xv=γ(x)v,x∈ℒμ}.$ If $\gamma \in {\stackrel{ˆ}{ℒ}}_{\mu },$ then ${V}_{\gamma }$ is the $\gamma \text{-weight}$ space of $V,$ and $V$ has a weight $\gamma$ if ${V}_{\gamma }\ne 0\text{.}$ See Chapter 6 for an application of this decomposition.

Examples 1. In the Yokonuma algebra ${\psi }_{\mu }=1,$ ${J}_{1}=\varnothing$ and ${ℒ}_{1}={e}_{1}ℂT{e}_{1}\text{.}$ 2. In the Gelfand-Graev Hecke algebra case, ${J}_{\mu }=\left\{{\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{\ell }\right\}$ and ${ℒ}_{\mu }={ℋ}_{\mu }$ (confirming, but not proving, that ${ℋ}_{\mu }$ is commutative).

### Multiplication of basis elements

This section examines the decomposition of products in terms of the natural basis $(eμueμ) (eμveμ)= ∑v′∈Nμ cuvv′ (eμv′eμ).$ In particular, Theorem 3.7, below, gives a set of braid-like relations (similar to those of the Yokonuma algebra) for manipulating the products, and Corollary 3.9 gives a recursive formula for computing these products.

#### Chevalley group relations

The relations governing the interaction between the subgroups $N,$ $U,$ and $T$ will be critical in describing the Hecke algebra multiplication in the following section. They can all be found in [Ste1967].

The subgroup $U=⟨xα(t) | α∈R+,t∈𝔽q⟩$ has generators $\left\{{x}_{\alpha }\left(t\right) | \alpha \in {R}^{+},t\in {𝔽}_{q}\right\},$ with relations $xα(a) xβ(b) xα(a)-1 xβ(b)-1 = ∏γ=iα+jβ∈R+i,j∈ℤ>0 xγ(zγaibj), (U1) xα(a) xα(b) = xα(a+b), (U2)$ where ${z}_{\gamma }\in ℤ$ depends on $i,j,\alpha ,\beta ,$ but not on $a,b\in {𝔽}_{q}$ [Ste1967]. The ${z}_{\gamma }$ have been explicitly computed for various types in [Dem1965, Ste1967] (See also Appendix A).

The subgroup $N$ has generators $\left\{{\xi }_{i},{h}_{H}\left(t\right) | i=1,2,\dots ,\ell ,H\in {𝔥}_{ℤ},t\in {𝔽}_{q}^{*}\right\},$ with relations $ξi2 = hi(-1), (N1) ξiξjξiξj⋯ ⏟mij terms = ξjξiξjξi⋯ ⏟mij terms , where (sisj)mij =1 in W, (N2) ξihH(t) = hsi(H) (t)ξi, (N3) hH(a) hH(b) = hH(ab), (N4) hH(a) hH′(b) = hH′(b) hH(a), for H,H′∈𝔥, (N5) hH(a) hH′(a) = hH+H′(a), for H,H′∈𝔥, (N6) hH1(t1) hH2(t2)⋯ hHk(tk)=1, if t1λj(H1)⋯ tkλj(Hk)=1 for all 1≤j≤n, (N7)$ where ${\lambda }_{j}:𝔥\to ℂ$ depends on $V$ as in (2.6).

The double-coset decomposition of $G$ (3.10) implies $G=⟨U,N⟩\text{.}$ Thus, $G$ is generated by $\left\{{x}_{\alpha }\left(a\right),{\xi }_{i},{h}_{H}\left(b\right) | \alpha \in {R}^{+},a\in {𝔽}_{q},i=1,2,\dots ,\ell ,H\in {𝔥}_{ℤ},b\in {𝔽}_{q}^{*}\right\}$ with relations (U1)-(N7) and $ξixα(t) xi-1 = xsi(α) (ciαt), where ciα= ±1 (ciαi =-1) (UN1) hxα(b)h-1 = xα(α(h)b), for h∈T, (UN2) ξixi(t) ξi = xi(t-1) hi(t-1) ξixi(t-1), where xi(t)=xαi (t) and t≠0, (UN3)$ where for $\alpha \in R$ and ${h}_{H}\left(t\right)\in T,$ $α(hH(t))= tα(H). (3.15)$

Fix a ${\psi }_{\mu }:U\to {ℂ}^{*}$ as in (3.7). For $k\in {𝔽}_{q},$ let $eα(k)=1q ∑t∈𝔽qψ (-μαkt)xα (t)with the convention eα=eα(1). (3.16)$ Note that for any given ordering of the positive roots, the decomposition $U=∏α∈R+Uα implieseμ=∏α∈R+ eα. (3.17)$ In particular, given any $\alpha \in {R}^{+},$ we may choose the ordering of the positive roots to have ${e}_{\alpha }$ appear either first or last. Therefore, since ${e}_{\alpha }$ is an idempotent, $eμeα=eμ= eαeμ. (3.18)$

If $w={s}_{{i}_{1}}{s}_{{i}_{2}}\cdots {s}_{{i}_{r}}\in W$ with $r$ minimal, then let $Rw = {α∈R+ | w(α)∈R-} = { αir, sir (αir-1), …,sirsir-1 ⋯si2(αi1) } , (3.19)$ where the second equality is in [Bou2002, VI.1, Corollary 2 of Proposition 17].

Let $v\in N,$ and let $w=\pi \left(v\right)$ (with $\pi :N\to W$ as in (3.3)). $ξieα (k)ξi-1 = esiα (ciαk), for α∈R+, 1≤i≤n-1, (E1) veαv-1 = ewα, for α∉Rw, v∈Nμ, (E2) heα(k) h-1 = eα(kα(h)-1), for h∈T, (E3) eμxα(t) = ψ(μαt)eμ= xα(t)eμ. (E4)$

Proof.

(E1) Using relation (UN1), we have $ξieα(k) ξi-1 = 1q∑t∈𝔽q ψ(-μαkt) ξixα(t) ξi-1 = 1q∑t∈𝔽q ψ(-μαkt) xsiα (ciαt) = 1q∑t′∈𝔽q ψ(-μαciαkt′) xsiα(t′) = esiα(ciαk).$ (E2) Suppose $\alpha \notin {R}_{w}\text{.}$ Since $v\in {N}_{\mu },$ $ψ(μαt) = ψμ(xα(t)) = ψμ(vxα(t)v-1) = ψμ(xwα(kt)) (by (UN1)) = ψ(μwαkt), for some k∈ℤ≠0.$ In particular, ${\mu }_{\alpha }=k{\mu }_{w\alpha }$ and we may conclude $veαv-1= 1q∑t∈𝔽q ψ(-μαt) xwα(kt)= 1q∑t′∈𝔽q ψ(-μαk-1t′) xwα(t′)= ewα.$ (E3) Since $h{x}_{\alpha }\left(t\right){h}^{-1}={x}_{\alpha }\left(\alpha \left(h\right)t\right),$ we have $heα(k)h-1= 1q∑t∈𝔽qψ (-kt)xα(α(h)t) =∑t∈𝔽qψ (-ktα(h)-1) xα(t)=eα (kα(h)-1).$ (E4) Note that $eαxα(t) = 1q∑a∈𝔽q ψ(-μαa) xα(a)xα(t) = 1q∑a∈𝔽q ψ(-μαa)xα (a+t) = 1q∑a′∈𝔽q ψ(-μα(a′-t)) xα(a′) = 1q∑a′∈𝔽q ψ(-μαa′)ψ (μαt)xα(a′) = ψ(μαt)eα.$ Therefore, by (3.18), ${e}_{\mu }{x}_{\alpha }\left(t\right)={e}_{\mu }{e}_{\alpha }{x}_{\alpha }\left(t\right)=\psi \left({\mu }_{\alpha }t\right){e}_{\mu }\text{.}$

$\square$

#### Local Hecke algebra relations

Let $u={u}_{1}{u}_{2}\cdots {u}_{r}{u}_{T}\in N$ decompose according to ${s}_{{i}_{1}}{s}_{{i}_{2}}\cdots {s}_{{i}_{r}}\in W\text{.}$ For $1\le k\le r$ define constants ${c}_{k}=±1$ and roots ${\beta }_{k}\in {R}^{+}$ by the equation $xβk(ckt)= (uk+1⋯ur)-1 xαik(t) (uk+1⋯ur). (3.20)$ Note that ${R}_{\pi \left(u\right)}=\left\{{\beta }_{1},{\beta }_{2},\dots ,{\beta }_{r}\right\}$ (see (3.19)). Define ${f}_{u}\in {𝔽}_{q}\left[{y}_{1},{y}_{2},\dots ,{y}_{r}\right]$ by $fu=- μβ1c1β1(uT)y1- μβ2c2β2(uT)y2-⋯- μβrcrβr(uT)yr (3.21)$ and for $k=1,2,\dots ,r,$ let $uk(t)=ξik (t),where ξi(t)=ξi xi(t). (3.22)$

Let $u={u}_{1}{u}_{2}\cdots {u}_{r}{u}_{T},$ $v={v}_{1}{v}_{2}\cdots {v}_{s}{v}_{T}\in {N}_{\mu }$ decompose according to ${s}_{{i}_{1}}{s}_{{i}_{2}}\cdots {s}_{{i}_{r}}\in W$ and ${s}_{{j}_{1}}{s}_{{j}_{2}}\cdots {s}_{{j}_{s}}\in W,$ respectively, as in (3.4). Then (a)  $(eμueμ) (eμveμ)= 1qr∑t∈𝔽qr (ψ∘fμ)(t) eμ(u1(t1)u2(t2)⋯ur(tr)) (v1v2⋯vs) heμ,$ where $h={v}_{T}{v}^{-1}{u}_{T}v\in T\text{.}$ (b) The following local relations suffice to compute the product $\left({e}_{\mu }u{e}_{\mu }\right)\left({e}_{\mu }v{e}_{\mu }\right)\text{.}$ $∑t∈𝔽q (ψ∘f)(t) ξi(t)ξi = (ψ∘f)(0) ξi2+ ∑t∈𝔽q* (ψ∘f)(t) xi(t-1) ξihi(t) xi(t-1), (ℋ1) ξixα(t) = xsi(α) (ciαt)ξi, (ℋ2) xα(t)h = hxα(α(h)-1t), (ℋ3) eμxα(t) = ψ(μαt)eμ= xα(t)eμ, (ℋ4) (ψ∘f)(t) (ψ∘g)(t) = (ψ∘(f+g))(t), for f,g∈𝔽q [y1±1,…,yr±1] ,t∈𝔽qr, (ℋ5) hα(t)ξi = ξihsi(α) (t), (ℋ6) ξi(a)xα(b) = ∏γ=mαi+nα∈R+m,n≥0 xsiγ (cisi(γ)zγambn) ξi(a),where α≠αi, (ℋ7) ∑a∈𝔽qΦ (a)ξi(a) xi(b) = ∑a∈𝔽qΦ (a-b)ξi(a), for some map Φ: 𝔽q→ℂG, (ℋ8) hα(a) hα(b) = hα(ab), (ℋ9) ξiξjξiξj⋯ ⏟mij terms = ξjξiξjξi⋯ ⏟mij terms , where mij is the order of sisj in W. (ℋ10)$

Proof.

(a) Order the positive roots so that by (3.17) $eμueμveμ = eμu (∏α∉Rweα) eβ1eβ2⋯ eβrveμ (definition βk) = eμ(∏α∉Rwewα) ueβ1eβ2⋯ eβrveμ (Lemma 3.6, E2) = eμueβ1 eβ2⋯eβr veμ(Lemma 3.6, E4) = eμu1u2⋯ uruTeβ1 eβ2⋯eβr veμ = eμu1u2⋯ur eβ1(μβ1β1(uT)) eβ2(μβ2β2(uT))⋯ eβr(μβrβr(uT)) uTveμ(Lemma 3.6, E3) = eμ u1eαi1(uβ1c1β1(uT)) u2eαi2(uβ2c2β2(uT))⋯ ureαir(uβrcrβr(uT)) uTveμ(Lemma 3.6, E1) = eμ u1eαi1(uβ1c1β1(uT)) u2eαi2(uβ2c2β2(uT))⋯ ureαir(uβrcrβr(uT)) v1⋯vsvT v-1uTveμ = eμqr ∑t1,…,tr∈𝔽q ψ(-μβ1c1t1β1(uT)) u1(t1)⋯ ψ(-μβrcrtrβr(uT)) ur(tr) v1⋯vsheμ (definition eα) = 1qr∑t∈𝔽qr (ψ∘fu)(t) eμu1(t1)⋯ ur(tr)v1⋯ vsheμ,(by (ℋ5))$ where $h={v}_{T}{v}^{-1}{u}_{T}v\in T,$ as desired.

(b) First, note that these relations are in fact correct (though not necessarily sufficient): $\left(ℋ1\right)$ comes from (UN3); $\left(ℋ2\right)$ comes from (UN1); $\left(ℋ3\right)$ comes from (UN2); $\left(ℋ4\right)$ comes from (E4); $\left(ℋ5\right)$ comes from the multiplicativity of $\psi \text{;}$ $\left(ℋ6\right)$ comes from (N3); $\left(ℋ7\right)$ comes from (U1) and (UN1); $\left(ℋ8\right)$ comes from (U2); $\left(ℋ9\right)$ is (N4); and $\left(ℋ10\right)$ is (N2). It therefore remains to show sufficiency.

By (a) we may write $(eμueμ) (eμveμ) =1qr ∑t∈𝔽qr (ψ∘f)(t) eμu1(t1) ⋯ur(tr) v1⋯vsheμ$ for some $f\in {𝔽}_{q}\left[{y}_{1},\dots ,{y}_{r}\right]$ and $h\in T\text{.}$ Say ${t}_{k}$ is resolved if the only part of the sum depending on ${t}_{k}$ is $\left(\psi \circ f\right)\text{.}$ The product is reduced when all the ${t}_{k}$ are resolved. I will show how to resolve ${t}_{r}$ and the result will follow by induction.

Use relation $\left(ℋ2\right)$ to define the constant $d$ and the root $\gamma \in R$ by $(v1v2⋯vs)-1 xαir(t) (v1v2⋯vs)= xγ(dt) (where ℓ(π(v)) =s). (3.23)$ There are two possible situations:

Case 1. $\gamma \in {R}^{+},$

Case 2. $\gamma \in {R}^{-}\text{.}$

In Case 1, $(eμueμ) (eμveμ) = 1qr∑t∈𝔽qr (ψ∘f)(t) eμu1(t1)⋯ urxir(tr)→ v1⋯vsheμ (by (a)) = 1qr∑t∈𝔽qr (ψ∘f)(t) eμu1(t1)⋯ ur-1(tr-1) urv1⋯vs xγ(dtr)→ heμ(by (ℋ2)) = 1qr∑t∈𝔽qr (ψ∘f)(t) eμu1(t1)⋯ ur-1(tr-1) urv1⋯vsh xγ(dγ(h)-1tr)eμ_ (by (ℋ3)) = 1qr∑t∈𝔽qr (ψ∘f)(t) eμu1(t1)⋯ ur-1(tr-1) urv1⋯vsh ψ(μγdγ(h)-1tr)_ eμ (by (ℋ4)) = 1qr∑t∈𝔽qr (ψ∘f)(t) eμu1(t1)⋯ ur-1(tr-1) urv1⋯vsh (ψ∘μγdγ(h)-1yr)(tr)← eμ = 1qr∑t∈𝔽qr (ψ∘g)(t) eμu1(t1)⋯ ur-1(tr-1) urv1⋯vsheμ, (by (ℋ5))$ where $g=f+{\mu }_{\gamma }d\gamma {\left(h\right)}^{-1}{y}_{r}\text{.}$ We have resolved ${t}_{r}$ in Case 1.

In Case 2, $\gamma \in {R}^{-},$ so we can no longer move ${x}_{{i}_{r}}\left({t}_{r}\right)$ past the ${v}_{j}\text{.}$ Instead, $(eμueμ) (eμveμ) = eμqr ∑t∈𝔽qr (ψ∘f)(t)_ u1(t1)⋯ ur-1(tr-1) urxir(tr) urur-1v1⋯ vsheμ = eμqr ∑t′∈𝔽qr-1 u1(t1)⋯ ur-1(tr-1) ∑tr∈𝔽q (ψ∘f)(t′,tr) urxir(tr)ur_ ur-1v1⋯ vsheμ = eμqr ∑t′∈𝔽qr-1 u1(t1)⋯ ur-1(tr-1) (ψ∘f)(t′,0) ur2ur-1_ v1⋯vsheμ (by (ℋ1)) + eμqr ∑t′∈𝔽qr-1 u1(t1)⋯ ur-1(tr-1) ∑tr∈𝔽q* (ψ∘f)(t′,tr) xir(tr-1) hir(tr-1) urxir(tr-1)→ ur-1 v1⋯vsheμ = eμqr ∑t′∈𝔽qr-1 u1(t1)⋯ ur-1(tr-1) (ψ∘f)(t′,0)← ur v1⋯vsheμ + eμqr ∑t′∈𝔽qr-1 u1(t1)⋯ ur-1(tr-1) ∑tr∈𝔽q* (ψ∘g)(t′,tr) xir(tr-1) hir(-tr)→ v1⋯vsheμ, (by (ℋ2,ℋ3,ℋ4))$ where $g=f-{\mu }_{\gamma }d\gamma {\left(h\right)}^{-1}{y}_{r}^{-1}$ (same as in the analogous steps in Case 1). $= eμqr ∑t′∈𝔽qr-1 (ψ∘f)(t′,0) u1(t1)⋯ ur-1(tr-1) ur v1⋯vsheμ + eμqr ∑t′∈𝔽qr-1 u1(t1)⋯ ur-1(tr-1) ∑tr∈𝔽q* (ψ∘g)(t′,tr)xir(tr-1)← v1⋯vshγ (-tr)heμ, (by (ℋ6)) = eμqr ∑t′∈𝔽qr-1 (ψ∘f)(t′,0) u1(t1)⋯ ur-1(tr-1) ur v1⋯vsheμ + eμqr ∑t′∈𝔽qr-1tr∈𝔽q* (ψ∘φ(g)) (t′,tr) (∏β∈R+xβ(aβ(t,tr))←) u1(t1)⋯ ur-1(tr-1) v1⋯vsh′eμ (by (ℋ7,ℋ8))$ where $\phi :{𝔽}_{q}\left[{y}_{1},\dots ,{y}_{r}\right]\to {𝔽}_{q}\left[{y}_{1},\dots ,{y}_{r}\right]$ catalogues the substitutions to $g$ due to $\left(ℋ8\right),$ the ${a}_{\beta }\left({y}_{1},{y}_{2},\dots ,{y}_{r}^{-1}\right)\in {𝔽}_{q}\left[{y}_{1},\dots ,{y}_{r-1},{y}_{r}\right]$ are determined by repeated applications of $\left(ℋ7\right)$ and $\left(ℋ8\right),$ and $h\prime ={h}_{\gamma }\left(-{t}_{r}\right)h\in T\text{.}$ $= 1qr ∑t′∈𝔽qr-1 (ψ∘f) (t′,0) eμ u1(t1)⋯ ur-1(tr-1) urv1⋯vsheμ +1qr∑t′∈𝔽qr-1tr∈𝔽q* (ψ∘φ(g)) (t′,tr)eμ (∏β∈R+ψ(μβaβ(t,tr))←) u1(t1)⋯ ur-1(tr-1) v1⋯vsh′eμ (by (ℋ4)) = 1qr ∑t′∈𝔽qr-1 (ψ∘f) (t′,0) eμ u1(t1)⋯ ur-1(tr-1) urv1⋯vsheμ +1qr∑t′∈𝔽qr-1tr∈𝔽q* (ψ∘g2) (t′,tr)eμ u1(t1)⋯ ur-1(tr-1) v1⋯vsh′eμ (by (ℋ5))$ where ${g}_{2}=\phi \left(g\right)+\sum _{\beta \in {R}^{+}}{\mu }_{\beta }{a}_{\beta }\left({y}_{1},\dots ,{y}_{r-1},{y}_{r}^{-1}\right)\text{.}$ Now ${t}_{r}$ is resolved for Case 2, as desired.

$\square$

(Resolving ${t}_{k}\text{).}$ Let $u={u}_{1}{u}_{2}\cdots {u}_{k}\in N$ decompose according to ${s}_{{i}_{1}}{s}_{{i}_{2}}\cdots {s}_{{i}_{k}}$ (with ${u}_{T}=1\text{).}$ Suppose $v\in N$ and $f\in {𝔽}_{q}\left[{y}_{1},{y}_{2},\dots ,{y}_{k}\right]\text{.}$ Define $\gamma \in R$ and $d\in ℂ$ by the equation ${v}^{-1}{x}_{{i}_{k}}\left(t\right)v={x}_{\gamma }\left(dt\right)\text{.}$ Then  Case 1 If $\ell \left(\pi \left({u}_{k}v\right)\right)>\ell \left(\pi \left(v\right)\right),$ then $∑t∈𝔽qk (ψ∘f)(t) eμu1(t1)⋯ uk(tk)veμ= ∑t∈𝔽qk (ψ∘(f+μγdcγyk_)) (t)eμu1(t1) ⋯uk-1(tk-1) ukv_eμ.$ Case 2 If $\ell \left(\pi \left({u}_{k}v\right)\right)<\ell \left(\pi \left(v\right)\right),$ then $∑t∈𝔽qk (ψ∘f)(t) eμu1(t1)⋯ uk(tk)veμ = ∑t∈𝔽qk,tk=0_ (ψ∘f)(t)eμ u1(t1)⋯ uk-1(tk-1) ukv_eμ + ∑t∈𝔽qk,tk∈𝔽q*_ (ψ∘(φk(f)-μγdyk-1_)) (t)eμ u1(t1)⋯ uk-1(tk-1) hik(tk-1)v_eμ,$ where ${\phi }^{k}:{𝔽}_{q}\left[{y}_{q}^{±1},\dots ,{y}_{k}^{±1}\right]\to {𝔽}_{q}\left[{y}_{1}^{±1},\dots ,{y}_{k}^{±1}\right]$ is given by $∑t∈𝔽qktk∈𝔽q* (ψ∘f)(t)eμ u1(t1)⋯ uk-1(tk-1) xik(tk-1)←= ∑t∈𝔽qktk∈𝔽q* (ψ∘φk(f)_) (t)eμu1(t1) ⋯uk-1(tk-1).$

Proof.

This Lemma puts $v$ in the place of ${u}_{T}v$ in the proof of Theorem 3.7, (b), and summarizes the steps taken in Case 1 and Case 2.

$\square$

#### Global Hecke algebra relations

Fix $u={u}_{1}{u}_{2}\cdots {u}_{r}{u}_{T}\in {N}_{\mu },$ decomposed according ${s}_{{i}_{1}}{s}_{{i}_{2}}\cdots {s}_{{i}_{r}}\in W$ (see (3.4)). Suppose $v\prime \in {N}_{\mu }$ and let $v={u}_{T}v\prime \text{.}$

For $0\le k\le r,$ let $\tau =\left({\tau }_{1},{\tau }_{2},\dots ,{\tau }_{r-k}\right)$ be such that ${\tau }_{i}\in \left\{+0,-0,1\right\},$ where $+0,$ $-0,$ and $1$ are symbols. If $\tau$ has $r-k$ elements, then the colength of $\tau$ is ${\ell }^{\vee }\left(\tau \right)=k\text{.}$ For example, if $r=10$ and $\tau =\left(-0,1,+0,+0,1,1\right),$ then ${\ell }^{\vee }\left(\tau \right)=4\text{.}$ For $i\in \left\{+0,-0,1\right\},$ let $(i,τ)= (i,τ1,τ2,⋯,τr-k).$ By convention, if ${\ell }^{\vee }\left(\tau \right)=r,$ then $\tau =\varnothing \text{.}$

Suppose ${\ell }^{\vee }\left(\tau \right)=k\text{.}$ Define $Ξτ(u,v)= 1qr∑t∈𝔽qτ (ψ∘fτ)(t) eμu1(t1)⋯ uk(tk)vτ (t)eμ, (3.24)$ where $𝔽qτ = { t∈𝔽qr | for k with $+0=-0=0\in ℤ,$ $1=1\in ℤ$ in (3.26); and ${f}^{\tau }$ is defined recursively by $f∅=fu = -μβ1c1β1(uT)y1 -μβ2c2β2(uT)y2 -⋯ -μβrcrβr(uT)yr ,(as in (3.21)), (3.27) f(i,τ) = { fτ+μγτ dτyk, if i=±0, φk(fτ)- μγτdτ yk-1, if i=1, (3.28)$ where ${\left({v}^{\tau }\right)}^{-1}{x}_{{\alpha }_{{i}_{k}}}\left(t\right){v}^{\tau }={x}_{{\gamma }_{\tau }}\left({d}_{\tau }t\right)$ and the map ${\phi }_{k}$ is as in Lemma 3.8, Case 2.

Remarks. 1. By (3.24) and Theorem 3.7 (a), ${\mathrm{\Xi }}^{\varnothing }\left(u,v\right)=\left({e}_{\mu }u{e}_{\mu }\right)\left({e}_{\mu }v\prime {e}_{\mu }\right)$ (recall, $v={u}_{T}v\prime \text{).}$ 2. If ${\ell }^{\vee }\left(\tau \right)=0$ so that $\tau$ is a string of length $r,$ then (a) ${\mathrm{\Xi }}^{\tau }\left(u,v\right)=\frac{1}{{q}^{r}}\sum _{t\in {𝔽}_{q}^{\tau }}\left(\psi \circ {f}^{\tau }\right)\left(t\right){e}_{\mu }{v}^{\tau }{e}_{\mu }$ has no remaining factors of the form ${u}_{k}\left({t}_{k}\right),$ (b) ${\mathrm{\Xi }}^{\tau }\left(u,v\right)=0$ unless ${v}^{\tau }\left(t\right)\in {N}_{\mu }$ for some $t\in {𝔽}_{q}\text{.}$

The following corollary gives relations for expanding ${\mathrm{\Xi }}^{\tau }\left(u,v\right)$ (beginning with ${\mathrm{\Xi }}^{\varnothing }\left(u,v\right)\text{)}$ as a sum of terms of the form ${\mathrm{\Xi }}^{\tau \prime }$ with ${\ell }^{\vee }\left(\tau \prime \right)={\ell }^{\vee }\left(\tau \right)-1\text{.}$ When each term has colength $0$ (or length $r\text{),}$ then we will have decomposed the product $\left({e}_{\mu }u{e}_{\mu }\right)\left({e}_{\mu }v\prime {e}_{\mu }\right)$ in terms of the basis elements of ${ℋ}_{\mu }\text{.}$

In summary, while we compute ${f}^{\tau }$ recursively by removing elements from $\tau ,$ we compute the product $\left({e}_{\mu }u{e}_{\mu }\right)\left({e}_{\mu }v\prime {e}_{\mu }\right)$ by progressively adding elements to $\tau \text{.}$

(The Global Alternative). Let $u,v\prime \in {N}_{\mu }$ such that $u={u}_{1}{u}_{2}\cdots {u}_{r}{u}_{T}$ decomposes according to a minimal expression in $W\text{.}$ Let $v={u}_{T}v\prime \text{.}$ Then (a) $\left({e}_{\mu }u{e}_{\mu }\right)\left({e}_{\mu }v\prime {e}_{\mu }\right)={\mathrm{\Xi }}^{\varnothing }\left(u,v\right)\text{.}$ (b) If ${\ell }^{\vee }\left(\tau \right)=k,$ then $Ξτ(u,v)= { Ξ(+0,τ) (u,v), if ℓ(π(ukvτ)) >ℓ(π(vτ)), Ξ(-0,τ) (u,v)+ Ξ(1,τ) (u,v), if ℓ(π(ukvτ)) <ℓ(π(vτ)).$

Proof.

(a) follows from Remark 1.

(b) Suppose ${\ell }^{\vee }\left(\tau \right)=k\text{.}$ Note that $Ξτ(u,v) = 1qr∑t∈𝔽qτ (ψ∘fτ) (t)eμu1(t1) ⋯uk(tk)vτeμ = 1qr∑t″∈(𝔽qr-k)τ ∑t′∈𝔽qk (ψ∘fτ) (t′,t″)eμu1 (t1)⋯uk (tk)vτeμ$ where ${\left({𝔽}_{q}^{r-k}\right)}^{\tau }=\left\{\left({t}_{k+1},\dots ,{t}_{r}\right)\in {𝔽}_{q}^{r-k} | \text{restrictions according to} \tau \right\}$ (as in (3.25)). Apply Lemma 3.8 to the inside sum with $f≔{f}^{\tau },$ $v≔{v}^{\tau }\text{.}$ Note that the Lemma relations imply ${t′∈𝔽qk} becomes { {t′∈𝔽qk}, if in Case 1, {t′∈𝔽qk | tk=0}, if in Case 2, first sum, {t′∈𝔽qk | tk∈𝔽q*}, if in Case 2, second sum, fτ becomes { f(+0,τ), if in Case 1, f(-0,τ), if in Case 2, first sum, f(1,τ), if in Case 2, second sum, vτ becomes { v(+0,τ), if in Case 1, v(-0,τ), if in Case 2, first sum, v(1,τ), if in Case 2, second sum.$ Thus $Ξτ(u,v)= { Ξ(+0,τ) (u,v), if Case 1, Ξ(-0,τ) (u,v)+ Ξ(1,τ) (u,v), if Case 2,$ as desired.

$\square$

## Notes and References

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

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