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

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

Last updated: 26 March 2015

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

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 $\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\ell }\}$ of $𝔤$ (Section 2.2.1). The group $G$ contains a subgroup $U$ given by $$U=⟨x_{\alpha }\left(t\right)\hspace{0.17em}|\hspace{0.17em}\alpha \in R^{+},t\in {𝔽}_q⟩,$$ which decomposes as $$U=\prod _{\alpha \in R^{+}}{U}_{\alpha },\text{where}{U}_{\alpha }=⟨x_{\alpha }\left(t\right)\hspace{0.17em}|\hspace{0.17em}t\in {𝔽}_q⟩,$$ with uniqueness of expression for any fixed ordering of the positive roots [Ste1967, Lemma 18]. For each $\alpha \in R^{+},$ the map $$\begin{array}{ccc}{U}_{\alpha }& \stackrel{\sim }{⟶}& {𝔽}_q^{+}\\ x_{\alpha }\left(t\right)& ⟼& t\end{array}$$ is a group homomorphism.

For $\alpha \in R,$ define the maps $$\begin{array}{cc}{\displaystyle \begin{array}{cccc}{s}_{\alpha }:& {𝔥}^{*}& ⟶& {𝔥}^{*}\\ & \gamma & ⟼& \gamma -\gamma \left(H_{\alpha }\right)\alpha ,\end{array}}& \text{(3.1)}\\ {\displaystyle \begin{array}{cccccc}{s}_{\alpha }:& 𝔥& =& Z\left(𝔤\right)\oplus {𝔥}_s& ⟶& 𝔥\\ & & & H+H_{\beta }& ⟼& H+H_{\beta }-\beta \left(H_{\alpha }\right)H_{\alpha },& \text{for}\hspace{0.17em}\beta \in R\text{.}\end{array}}& \text{(3.2)}\end{array}$$

The Weyl group of $G$ is $W=⟨s_{\alpha }\hspace{0.17em}|\hspace{0.17em}\alpha \in R⟩$ and has a presentation given by $$W=⟨s_1,s_2,\dots ,s_{\ell }\hspace{0.17em}|\hspace{0.17em}{s}_i^2=1,{\left(s_i{s}_j\right)}^{m_{ij}}=1,1\le i\ne j\le \ell ⟩,m_{ij}\in {\mathbb{Z}}_{>0},s_i=s_{{\alpha }_i}\text{.}$$ 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 ${𝔥}_{\mathbb{Z}}$ be as in (2.3). If $q>3,$ then the subgroup $$T=⟨h_H\left(t\right)\hspace{0.17em}|\hspace{0.17em}H\in {𝔥}_{\mathbb{Z}},t\in {𝔽}_q^{*}⟩$$ has its normalizer in $G$ given by $$N=⟨w_{\alpha }\left(t\right),h\hspace{0.17em}|\hspace{0.17em}\alpha \in R,h\in T,t\in {𝔽}_q^{*}⟩,\text{where}\hspace{0.17em}{w}_{\alpha }\left(t\right)=x_{\alpha }\left(t\right)x_{-\alpha }(-t^{-1})x_{\alpha }\left(t\right)\text{.}$$ 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 $$\begin{array}{cc}{\displaystyle \begin{array}{cccc}\pi :& N& ⟶& W\\ & w_{\alpha }\left(t\right)& ⟼& s_{\alpha },& \text{for}\hspace{0.17em}\alpha \in R,t\in {𝔽}_q^{*},\hfill \\ & h& ⟼& 1,& \text{for}\hspace{0.17em}h\in T\text{.}\hfill \end{array}}& \text{(3.3)}\end{array}$$ Suppose $v\in N\text{.}$ Then for each minimal expression $$\pi \left(v\right)=s_{i_1}{s}_{i_2}\cdots s_{i_r},\text{with}\hspace{0.17em}\ell \left(\pi \left(v\right)\right)=r,$$ there is a unique decomposition of $v$ as $$\begin{array}{cc}{\displaystyle v=v_1{v}_2\cdots v_r{v}_T,\text{where}\hspace{0.17em}{v}_k=w_{i_k}\left(1\right)\hspace{0.17em}\text{and}\hspace{0.17em}{v}_T\in T\text{.}}& \text{(3.4)}\end{array}$$ Write $$\begin{array}{cc}{\displaystyle {\xi }_i=w_i\left(1\right)\text{.}}& \text{(3.5)}\end{array}$$

Unipotent Hecke algebras

Let $G$ be a finite Chevalley group. Fix a nontrivial homomorphism $\psi :{𝔽}_q^{+}\to {ℂ}^{*}\text{.}$ If $$\begin{array}{cc}{\displaystyle \begin{array}{cccc}\mu :& R^{+}& ⟶& {𝔽}_q\\ & \alpha & ⟼& {\mu }_{\alpha }\end{array}\text{satisfies}\hspace{0.17em}{\mu }_{\alpha }=0\hspace{0.17em}\text{for all}\hspace{0.17em}\alpha \hspace{0.17em}\text{not simple,}}& \text{(3.6)}\end{array}$$ then the map $$\begin{array}{cc}{\displaystyle \begin{array}{cccc}{\psi }_{\mu }:& U& ⟶& {ℂ}^{*}\\ & x_{\alpha }\left(t\right)& ⟼& \psi \left({\mu }_{\alpha }t\right)\end{array}}& \text{(3.7)}\end{array}$$ 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 $\mathscr{H}(G,U,{\psi }_{\mu })$ is $$\begin{array}{cc}{\displaystyle {\mathscr{H}}_{\mu }={\text{End}}_{ℂG}\left({\text{Ind}}_U^G\left({\psi }_{\mu }\right)\right)\text{.}}& \text{(3.8)}\end{array}$$ Using the anti-isomorphism (2.2), view ${\mathscr{H}}_{\mu }$ as the subset of $ℂG$ $$\begin{array}{cc}{\displaystyle {\mathscr{H}}_{\mu }=e_{\mu }ℂG{e}_{\mu },\text{where}{e}_{\mu }=\frac{1}{\mid U\mid }\sum _{u\in U}{\psi }_{\mu }\left(u^{-1}\right)u\text{.}}& \text{(3.9)}\end{array}$$

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 $$h{x}_{\alpha }\left(t\right)h^{-1}=x_{\alpha }\left(\alpha \left(h\right)t\right),\text{for}\hspace{0.17em}\alpha \in R^{+},t\in {𝔽}_q,h\in T,\alpha \left(h\right)\in {𝔽}_q^{*}\text{.}$$ Thus, if $\chi :U\to {ℂ}^{*}$ is a linear character, then $$\begin{array}{cccc}{}^h\chi :& U& ⟶& {ℂ}^{*}\\ & x_{\alpha }\left(t\right)& ⟼& \chi \left(h{x}_{\alpha }\left(t\right)h^{-1}\right)\end{array}$$ is a linear character for every $h\in T\text{.}$

Suppose $\chi :U\to {ℂ}^{*}$ is a linear character. For every $h\in T,$ $${\text{Ind}}_U^G\left(\chi \right)={\text{Ind}}_U^G\left({}^h\chi \right)\text{.}$$

The type of a linear character $\chi :U\to {ℂ}^{*}$ is the set $J\subseteq \{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\ell }\}$ such that $$\chi \left(x_{{\alpha }_i}\left(t\right)\right)\ne 1\hspace{0.17em}\text{for all}\hspace{0.17em}t\in {𝔽}_q\text{if and only if}{\alpha }_i\in J\text{.}$$ A unipotent Hecke algebra $\mathscr{H}(G,U,\chi )$ 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 $$\psi \prime \left(t\right)=\psi \left(kt\right),\text{for all}\hspace{0.17em}t\in {𝔽}_q^{+}\text{.}$$ (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 $J\subseteq \{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\ell }\}$ and ${𝒮}_J=\left\{\text{type}\hspace{0.17em}J\hspace{0.17em}\text{linear characters of}\hspace{0.17em}U\right\}\text{.}$ Then the number of $T\text{-orbits}$ of ${𝒮}_J$ is $$\frac{\mid Z_J\mid {(q-1)}^{\mid J\mid }}{\mid T\mid },\text{where}\hspace{0.17em}{Z}_J=\{h\in T\hspace{0.17em}|\hspace{0.17em}\alpha \left(h\right)=1,\text{for}\hspace{0.17em}\alpha \in J\}\text{.}$$ (d) The number of distinct (nonisomorphic) type $J$ unipotent Hecke algebras is at most $$\frac{\mid Z_J\mid {(q-1)}^{\mid J\mid }}{\mid T\mid }\text{.}$$

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 ${\mathscr{H}}_{\mu }$ is the Yokonuma algebra. 2. A character is in general position if it has type $J=\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\ell }\}\text{.}$ 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 $$\frac{\mid Z\left(G\right)\mid {(q-1)}^{\ell }}{\mid T\mid }\text{.}$$

A natural basis

The group $G$ has a double-coset decomposition $$\begin{array}{cc}{\displaystyle G=\underset{v\in N}{⨆}UvU,\text{[Ste1967, Theorem 4 and}\hspace{0.17em}B=UT\text{]}}& \text{(3.10)}\end{array}$$ and if $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle N_{\mu }}& =& {\displaystyle \{v\in N\hspace{0.17em}|\hspace{0.17em}{e}_{\mu }v{e}_{\mu }\ne 0\}}\\ & =& {\displaystyle \{v\in V\hspace{0.17em}|\hspace{0.17em}u,vu{v}^{-1}\in U\hspace{0.17em}\text{implies}\hspace{0.17em}{\psi }_{\mu }\left(u\right)={\psi }_{\mu }\left(vu{v}^{-1}\right)\}}\end{array}}& \text{(3.11)}\end{array}$$ then the set $\left\{e_{\mu }v{e}_{\mu }\hspace{0.17em}\right|\hspace{0.17em}v\in N_{\mu }\}$ is a basis for ${\mathscr{H}}_{\mu }$ [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 $$\left(e_{\mu }u{e}_{\mu }\right)\left(e_{\mu }v{e}_{\mu }\right),\text{with}\hspace{0.17em}u,v\in N_{\mu }$$ in any unipotent Hecke algebra ${\mathscr{H}}_{\mu }\text{.}$

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 $$T_v=T_{i_1}{T}_{i_2}\cdots T_{i_r}{T}_{v_T}\text{(See Chapter 6).}$$ Thus, the Yokonuma algebra ${\mathscr{H}}_1$ has generators $T_i,$ $T_h$ for $1\le i\le \ell ,$ $h\in T$ (see [Yok1969-2]) with relations, $$\begin{array}{rcll}{\displaystyle T_i^2}& =& {\displaystyle q^{-1}{T}_{H_{{\alpha }_i}}(-1)+q^{-1}\sum _{t\in {𝔽}_q^{*}}{T}_{H_{{\alpha }_i}}\left(t^{-1}\right)T_i,}& {\displaystyle 1\le i\le \ell ,}\\ {\displaystyle \underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{T_i{T}_j{T}_i\cdots }}& =& {\displaystyle \underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{T_j{T}_i{T}_j\cdots },}& {\displaystyle {\left(s_i{s}_h\right)}^{m_{ij}}=1,}\\ {\displaystyle T_i{T}_h}& =& {\displaystyle T_{s_{i}j}{T}_i,}& {\displaystyle h\in T,}\\ {\displaystyle T_h{T}_k}& =& {\displaystyle T_{hk},}& {\displaystyle h,k\in T\text{.}}\end{array}$$ 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 ${\mathscr{H}}_1\text{.}$ There is a surjective map from the Yokonuma algebra onto the Iwahori-Hecke algebra that sends $T_h\mapsto 1$ for all $h\in T\text{.}$ “Setting $T_h=1\text{”}$ in the Yokonuma algebra relations recovers relations for the Iwahori-Hecke algebra, $$T_i^2=q^{-1}+q^{-1}(q-1)T_i,\underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{T_i{T}_j\cdots }=\underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{T_j{T}_i\cdots }\text{.}$$ Furthermore, there is a surjective map from the Iwahori Hecke algebra onto the Weyl group given by mapping $T_i\mapsto s_i$ and $q\mapsto 1\text{.}$ Thus, by “setting $T_i=s_i$ and $q=1\text{”}$ we retrieve the Coxeter relations of $W,$ $$s_i^2=1,\underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{s_i{s}_j{s}_j\cdots }=\underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{s_j{s}_i{s}_j\cdots }\text{.}$$ 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 ${\mathscr{H}}_{\mu }$ 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 ${\mathscr{H}}_{\mu }$

Let ${\psi }_{\mu }:U\to G$ be as in (3.7). Fix a subset $J\subseteq \{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\ell }\}$ such that $$\text{if}\hspace{0.17em}{\mu }_{{\alpha }_i}\ne 0\text{, then}\hspace{0.17em}{\alpha }_i\in J\text{.}$$ For example, if ${\psi }_{\mu }$ is in general position, then $J=\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\ell }\},$ but if ${\psi }_{\mu }$ is trivial, then $J$ could be any subset.

Let $$W_J=⟨s_i\in W\hspace{0.17em}|\hspace{0.17em}{\alpha }_i\in J⟩,P_J=⟨U,T,W_J⟩\text{and}{R}_J=\mathbb{Z}\text{-span}\{{\alpha }_i\in J\}\cap R\text{.}$$ Then $P_J$ has subgroups $$\begin{array}{cc}{\displaystyle L_J=⟨T,W_J,U_{\alpha }\hspace{0.17em}|\hspace{0.17em}\alpha \in R_J⟩\text{and}{U}_J=⟨U_{\alpha }\hspace{0.17em}|\hspace{0.17em}\alpha \in R^{+}-R_J⟩}& \text{(3.12)}\end{array}$$ (a Levi subgroup and the unipotent radical of $P_J,$ respectively). Note that $$U_J{L}_J=P_J,U_J\cap L_J=1,\text{and, in fact,}{P}_J=U_J⋊L_J\text{.}$$

Define the idempotents of $ℂU,$ $$\begin{array}{cc}{\displaystyle e_{\mu J}=\frac{1}{\mid L_J\cap U\mid }\sum _{u\in L_J\cap U}{\psi }_{\mu }\left(u^{-1}\right)u\text{and}{e}_J^{\prime }=\frac{1}{\mid U_J\mid }\sum _{u\in U_J}u,}& \text{(3.13)}\end{array}$$ so that $e_{\mu }=e_{\mu J}{e}_J^{\prime }\text{.}$

The group homomorphisms $$\begin{array}{ccc}{P}_J& ⟶& L_J\\ lu⟼& \text{<mi>l</mi>}\end{array}\text{and}\begin{array}{ccc}{P}_J& ⟶& G\\ lu& ⟼& lu\end{array}\text{for}\hspace{0.17em}l\in L_J,u\in U_J,$$ induce maps $$\begin{array}{cccc}{\text{Inf}}_{L_J}^{P_J}:& \left\{L_J\text{-modules}\right\}& ⟶& \left\{P_J\text{-modules}\right\}\\ & M& ⟼& e_J^{\prime }M,\\ {\text{Ind}}_{P_J}^G:& \left\{P_J\text{-modules}\right\}& ⟶& \left\{G\text{-modules}\right\}\\ & M\prime & ⟼& ℂG{\otimes }_{ℂP_J}M\prime \end{array}$$ whose composition is the map ${\text{Indf}}_{L_J}^G\text{.}$ Note that in the special case when $ℂL_{J}e$ is an irreducible $L_J\text{-module}$ with corresponding idempotent $e,$ then $$\begin{array}{cccc}{\text{Indf}}_{L_J}^G:& \left\{L_J\text{-modules}\right\}& ⟶& \left\{G\text{-modules}\right\}\\ & ℂL_{J}e& ⟼& ℂGe{e}_J^{\prime }\text{.}\end{array}$$

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 $${\text{Ind}}_U^G\left({\psi }_{\mu }\right)\cong {\text{Indf}}_{L_J}^G\left({\text{Ind}}_{U\cap L_J}^{L_J}\left({\psi }_{\mu }\right)\right)\text{.}$$

The map $$\begin{array}{cccc}\theta :& {\text{End}}_{ℂL_J}\left({\text{Ind}}_{U\cap L_J}^{L_J}\left({\psi }_{\mu }\right)\right)& ⟶& {\mathscr{H}}_{\mu }\\ & e_{\mu J}v{e}_{\mu J}& ⟼& e_{\mu }v{e}_{\mu },& \text{for}\hspace{0.17em}v\in L_J\cap N_{\mu },\end{array}$$ is an injective algebra homomorphism.

Write $$\begin{array}{cc}{\displaystyle {ℒ}_J=\theta \left({\text{End}}_{ℂL_J}\left({\text{Ind}}_{U\cap L_J}^{L_J}\left({\psi }_{\mu }\right)\right)\right)\subseteq {\mathscr{H}}_{\mu }}& \text{(3.14)}\end{array}$$ The ${ℒ}_J$ are “parabolic” subalgebras of ${\mathscr{H}}_{\mu }$ in that they have a similar relationship to the representation theory of ${\mathscr{H}}_{\mu }$ as parabolic subgroups $P_J$ have with the representation theory of $G\text{.}$

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

An important special case of Theorem 3.4 is when $$J=J_{\mu }=\{{\alpha }_i\in \{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\ell }\}\hspace{0.17em}|\hspace{0.17em}{\mu }_{{\alpha }_i}\ne 0\},$$ 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 ${\mathscr{H}}_{\mu }\text{.}$

Since ${ℒ}_{\mu }$ is commutative, all the irreducible modules are one-dimensional; let ${\widehat{ℒ}}_{\mu }$ be an indexing set for the irreducible modules of ${ℒ}_{\mu }\text{.}$ Suppose $V$ is an ${\mathscr{H}}_{\mu }\text{-module.}$ As an ${ℒ}_{\mu }\text{-module,}$ $$V\cong \underset{\gamma \in {\widehat{ℒ}}_{\mu }}{⨁}{V}_{\gamma }\text{where}{V}_{\gamma }=\{v\in V\hspace{0.17em}|\hspace{0.17em}xv=\gamma \left(x\right)v,x\in {ℒ}_{\mu }\}\text{.}$$ If $\gamma \in {\widehat{ℒ}}_{\mu },$ 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 ${ℒ}_1=e_1ℂT{e}_1\text{.}$ 2. In the Gelfand-Graev Hecke algebra case, $J_{\mu }=\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\ell }\}$ and ${ℒ}_{\mu }={\mathscr{H}}_{\mu }$ (confirming, but not proving, that ${\mathscr{H}}_{\mu }$ is commutative).

Multiplication of basis elements

This section examines the decomposition of products in terms of the natural basis $$\left(e_{\mu }u{e}_{\mu }\right)\left(e_{\mu }v{e}_{\mu }\right)=\sum _{v\prime \in N_{\mu }}{c}_{uv}^{v\prime }\left(e_{\mu }v\prime e_{\mu }\right)\text{.}$$ 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 $$U=⟨x_{\alpha }\left(t\right)\hspace{0.17em}|\hspace{0.17em}\alpha \in R^{+},t\in {𝔽}_q⟩$$ has generators $\left\{x_{\alpha }\left(t\right)\hspace{0.17em}\right|\hspace{0.17em}\alpha \in R^{+},t\in {𝔽}_q\},$ with relations $$\begin{array}{rcll}{\displaystyle x_{\alpha }\left(a\right)x_{\beta }\left(b\right)x_{\alpha }{\left(a\right)}^{-1}{x}_{\beta }{\left(b\right)}^{-1}}& =& {\displaystyle \prod _{\underset{i,j\in {\mathbb{Z}}_{>0}}{\gamma =i\alpha +j\beta \in R^{+}}}{x}_{\gamma }\left(z_{\gamma }{a}^i{b}^j\right),}& \text{(U1)}\\ {\displaystyle x_{\alpha }\left(a\right)x_{\alpha }\left(b\right)}& =& {\displaystyle x_{\alpha }(a+b),}& \text{(U2)}\end{array}$$ where $z_{\gamma }\in \mathbb{Z}$ 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 $\{{\xi }_i,h_H\left(t\right)\hspace{0.17em}|\hspace{0.17em}i=1,2,\dots ,\ell ,H\in {𝔥}_{\mathbb{Z}},t\in {𝔽}_q^{*}\},$ with relations $$\begin{array}{rclll}{\displaystyle {\xi }_i^2}& =& {\displaystyle h_i(-1),}& & \text{(N1)}\\ {\displaystyle \underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{{\xi }_i{\xi }_j{\xi }_i{\xi }_j\cdots }}& =& {\displaystyle \underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{{\xi }_j{\xi }_i{\xi }_j{\xi }_i\cdots },}& {\displaystyle \text{where}\hspace{0.17em}{\left(s_i{s}_j\right)}^{m_{ij}}=1\hspace{0.17em}\text{in}\hspace{0.17em}W,}& \text{(N2)}\\ {\displaystyle {\xi }_i{h}_H\left(t\right)}& =& {\displaystyle h_{s_i\left(H\right)}\left(t\right){\xi }_i,}& & \text{(N3)}\\ {\displaystyle h_H\left(a\right)h_H\left(b\right)}& =& {\displaystyle h_H\left(ab\right),}& & \text{(N4)}\\ {\displaystyle h_H\left(a\right)h_{H\prime }\left(b\right)}& =& {\displaystyle h_{H\prime }\left(b\right)h_H\left(a\right),}& {\displaystyle \text{for}\hspace{0.17em}H,H\prime \in 𝔥,}& \text{(N5)}\\ {\displaystyle h_H\left(a\right)h_{H\prime }\left(a\right)}& =& {\displaystyle h_{H+H\prime }\left(a\right),}& {\displaystyle \text{for}\hspace{0.17em}H,H\prime \in 𝔥,}& \text{(N6)}\\ \multicolumn{3}{c}{{\displaystyle h_{H_1}\left(t_1\right)h_{H_2}\left(t_2\right)\cdots h_{H_k}\left(t_k\right)=1,}}& {\displaystyle \text{if}\hspace{0.17em}{t}_1^{{\lambda }_j\left(H_1\right)}\cdots t_k^{{\lambda }_j\left(H_k\right)}=1\hspace{0.17em}\text{for all}\hspace{0.17em}1\le j\le n,}& \text{(N7)}\end{array}$$ 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 $\{x_{\alpha }\left(a\right),{\xi }_i,h_H\left(b\right)\hspace{0.17em}|\hspace{0.17em}\alpha \in R^{+},a\in {𝔽}_q,i=1,2,\dots ,\ell ,H\in {𝔥}_{\mathbb{Z}},b\in {𝔽}_q^{*}\}$ with relations (U1)-(N7) and $$\begin{array}{rclll}{\displaystyle {\xi }_i{x}_{\alpha }\left(t\right)x_i^{-1}}& =& {\displaystyle x_{s_i\left(\alpha \right)}\left(c_{i\alpha }t\right),}& {\displaystyle \text{where}\hspace{0.17em}{c}_{i\alpha }=\pm 1\hspace{0.17em}\text{(}{c}_{i{\alpha }_i}=-1\text{)}}& \text{(UN1)}\\ {\displaystyle h{x}_{\alpha }\left(b\right)h^{-1}}& =& {\displaystyle x_{\alpha }\left(\alpha \left(h\right)b\right),}& {\displaystyle \text{for}\hspace{0.17em}h\in T,}& \text{(UN2)}\\ {\displaystyle {\xi }_i{x}_i\left(t\right){\xi }_i}& =& {\displaystyle x_i\left(t^{-1}\right)h_i\left(t^{-1}\right){\xi }_i{x}_i\left(t^{-1}\right),}& {\displaystyle \text{where}\hspace{0.17em}{x}_i\left(t\right)=x_{{\alpha }_i}\left(t\right)\hspace{0.17em}\text{and}\hspace{0.17em}t\ne 0,}& \text{(UN3)}\end{array}$$ where for $\alpha \in R$ and $h_H\left(t\right)\in T,$ $$\begin{array}{cc}{\displaystyle \alpha \left(h_H\left(t\right)\right)=t^{\alpha \left(H\right)}\text{.}}& \text{(3.15)}\end{array}$$

Fix a ${\psi }_{\mu }:U\to {ℂ}^{*}$ as in (3.7). For $k\in {𝔽}_q,$ let $$\begin{array}{cc}{\displaystyle e_{\alpha }\left(k\right)=\frac{1}{q}\sum _{t\in {𝔽}_q}\psi (-{\mu }_{\alpha }kt)x_{\alpha }\left(t\right)\text{with the convention}{e}_{\alpha }=e_{\alpha }\left(1\right)\text{.}}& \text{(3.16)}\end{array}$$ Note that for any given ordering of the positive roots, the decomposition $$\begin{array}{cc}{\displaystyle U=\prod _{\alpha \in R^{+}}{U}_{\alpha }\text{implies}{e}_{\mu }=\prod _{\alpha \in R^{+}}{e}_{\alpha }\text{.}}& \text{(3.17)}\end{array}$$ 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, $$\begin{array}{cc}{\displaystyle e_{\mu }{e}_{\alpha }=e_{\mu }=e_{\alpha }{e}_{\mu }\text{.}}& \text{(3.18)}\end{array}$$

If $w=s_{i_1}{s}_{i_2}\cdots s_{i_r}\in W$ with $r$ minimal, then let $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle R_w}& =& {\displaystyle \{\alpha \in R^{+}\hspace{0.17em}|\hspace{0.17em}w\left(\alpha \right)\in R^{-}\}}\\ & =& {\displaystyle \{{\alpha }_{i_r},s_{i_r}\left({\alpha }_{i_{r-1}}\right),\dots ,s_{i_r}{s}_{i_{r-1}}\cdots s_{i_2}\left({\alpha }_{i_1}\right)\},}\end{array}}& \text{(3.19)}\end{array}$$ 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)). $$\begin{array}{rclll}{\displaystyle {\xi }_i{e}_{\alpha }\left(k\right){\xi }_i^{-1}}& =& {\displaystyle e_{s_i\alpha }\left(c_{i\alpha }k\right),}& {\displaystyle \text{for}\hspace{0.17em}\alpha \in R^{+},1\le i\le n-1,}& \text{(E1)}\\ {\displaystyle v{e}_{\alpha }{v}^{-1}}& =& {\displaystyle e_{w\alpha },}& {\displaystyle \text{for}\hspace{0.17em}\alpha \notin R_w,v\in N_{\mu },}& \text{(E2)}\\ {\displaystyle h{e}_{\alpha }\left(k\right)h^{-1}}& =& {\displaystyle e_{\alpha }\left(k\alpha {\left(h\right)}^{-1}\right),}& {\displaystyle \text{for}\hspace{0.17em}h\in T,}& \text{(E3)}\\ {\displaystyle e_{\mu }{x}_{\alpha }\left(t\right)}& =& {\displaystyle \psi \left({\mu }_{\alpha }t\right)e_{\mu }=x_{\alpha }\left(t\right)e_{\mu }\text{.}}& {\displaystyle }& \text{(E4)}\end{array}$$

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 $c_k=\pm 1$ and roots ${\beta }_k\in R^{+}$ by the equation $$\begin{array}{cc}{\displaystyle x_{{\beta }_k}\left(c_{k}t\right)={\left(u_{k+1}\cdots u_r\right)}^{-1}{x}_{{\alpha }_{i_k}}\left(t\right)\left(u_{k+1}\cdots u_r\right)\text{.}}& \text{(3.20)}\end{array}$$ Note that $R_{\pi \left(u\right)}=\{{\beta }_1,{\beta }_2,\dots ,{\beta }_r\}$ (see (3.19)). Define $f_u\in {𝔽}_q[y_1,y_2,\dots ,y_r]$ by $$\begin{array}{cc}{\displaystyle f_u=-\frac{{\mu }_{{\beta }_1}{c}_1}{{\beta }_1\left(u_T\right)}{y}_1-\frac{{\mu }_{{\beta }_2}{c}_2}{{\beta }_2\left(u_T\right)}{y}_2-\cdots -\frac{{\mu }_{{\beta }_r}{c}_r}{{\beta }_r\left(u_T\right)}{y}_r}& \text{(3.21)}\end{array}$$ and for $k=1,2,\dots ,r,$ let $$\begin{array}{cc}{\displaystyle u_k\left(t\right)={\xi }_{i_k}\left(t\right),\text{where}{\xi }_i\left(t\right)={\xi }_i{x}_i\left(t\right)\text{.}}& \text{(3.22)}\end{array}$$

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) $\hspace{0.17em}$ $$\left(e_{\mu }u{e}_{\mu }\right)\left(e_{\mu }v{e}_{\mu }\right)=\frac{1}{q^r}\sum _{t\in {𝔽}_q^r}(\psi \circ f_{\mu })\left(t\right)e_{\mu }\left(u_1\left(t_1\right)u_2\left(t_2\right)\cdots u_r\left(t_r\right)\right)\left(v_1{v}_2\cdots v_s\right)h{e}_{\mu },$$ where $h=v_T{v}^{-1}{u}_{T}v\in T\text{.}$ (b) The following local relations suffice to compute the product $\left(e_{\mu }u{e}_{\mu }\right)\left(e_{\mu }v{e}_{\mu }\right)\text{.}$ $$\begin{array}{rcll}{\displaystyle \sum _{t\in {𝔽}_q}(\psi \circ f)\left(t\right){\xi }_i\left(t\right){\xi }_i}& =& {\displaystyle (\psi \circ f)\left(0\right){\xi }_i^2+\sum _{t\in {𝔽}_q^{*}}(\psi \circ f)\left(t\right)x_i\left(t^{-1}\right){\xi }_i{h}_i\left(t\right)x_i\left(t^{-1}\right),}& \left(\mathscr{H}1\right)\\ {\displaystyle {\xi }_i{x}_{\alpha }\left(t\right)}& =& {\displaystyle x_{s_i\left(\alpha \right)}\left(c_{i\alpha }t\right){\xi }_i,}& \left(\mathscr{H}2\right)\\ {\displaystyle x_{\alpha }\left(t\right)h}& =& {\displaystyle h{x}_{\alpha }\left(\alpha {\left(h\right)}^{-1}t\right),}& \left(\mathscr{H}3\right)\\ {\displaystyle e_{\mu }{x}_{\alpha }\left(t\right)}& =& {\displaystyle \psi \left({\mu }_{\alpha }t\right)e_{\mu }=x_{\alpha }\left(t\right)e_{\mu },}& \left(\mathscr{H}4\right)\\ {\displaystyle (\psi \circ f)\left(t\right)(\psi \circ g)\left(t\right)}& =& {\displaystyle (\psi \circ (f+g))\left(t\right),\text{for}\hspace{0.17em}f,g\in {𝔽}_q[y_1^{\pm 1},\dots ,y_r^{\pm 1}],t\in {𝔽}_q^r,}& \left(\mathscr{H}5\right)\\ {\displaystyle h_{\alpha }\left(t\right){\xi }_i}& =& {\displaystyle {\xi }_i{h}_{s_i\left(\alpha \right)}\left(t\right),}& \left(\mathscr{H}6\right)\\ {\displaystyle {\xi }_i\left(a\right)x_{\alpha }\left(b\right)}& =& {\displaystyle \prod _{\underset{m,n\ge 0}{\gamma =m{\alpha }_i+n\alpha \in R^{+}}}{x}_{s_i\gamma }\left(c_{i{s}_i\left(\gamma \right)}{z}_{\gamma }{a}^m{b}^n\right){\xi }_i\left(a\right),\text{where}\hspace{0.17em}\alpha \ne {\alpha }_i,}& \left(\mathscr{H}7\right)\\ {\displaystyle \sum _{a\in {𝔽}_q}\mathrm{\Phi }\left(a\right){\xi }_i\left(a\right)x_i\left(b\right)}& =& {\displaystyle \sum _{a\in {𝔽}_q}\mathrm{\Phi }(a-b){\xi }_i\left(a\right),\text{for some map}\hspace{0.17em}\mathrm{\Phi }:{𝔽}_q\to ℂG,}& \left(\mathscr{H}8\right)\\ {\displaystyle h_{\alpha }\left(a\right)h_{\alpha }\left(b\right)}& =& {\displaystyle h_{\alpha }\left(ab\right),}& \left(\mathscr{H}9\right)\\ {\displaystyle \underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{{\xi }_i{\xi }_j{\xi }_i{\xi }_j\cdots }}& =& {\displaystyle \underset{\underset{m_{ij}\hspace{0.17em}\text{terms}}{⏟}}{{\xi }_j{\xi }_i{\xi }_j{\xi }_i\cdots },\text{where}\hspace{0.17em}{m}_{ij}\hspace{0.17em}\text{is the order of}\hspace{0.17em}{s}_i{s}_j\hspace{0.17em}\text{in}\hspace{0.17em}W\text{.}}& \left(\mathscr{H}10\right)\end{array}$$

(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 $f\in {𝔽}_q[y_1,y_2,\dots ,y_k]\text{.}$ 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 $$$$ Case 1 If $\ell \left(\pi \left(u_{k}v\right)\right)>\ell \left(\pi \left(v\right)\right),$ then $$\sum _{t\in {𝔽}_q^k}(\psi \circ f)\left(t\right)e_{\mu }{u}_1\left(t_1\right)\cdots u_k\left(t_k\right)v{e}_{\mu }=\sum _{t\in {𝔽}_q^k}(\psi \circ \left(\underset{\_}{f+{\mu }_{\gamma }d{c}_{\gamma }{y}_k}\right))\left(t\right)e_{\mu }{u}_1\left(t_1\right)\cdots u_{k-1}\left(t_{k-1}\right)\underset{\_}{u_{k}v}{e}_{\mu }\text{.}$$ Case 2 If $\ell \left(\pi \left(u_{k}v\right)\right)<\ell \left(\pi \left(v\right)\right),$ then $$\begin{array}{rcl}{\displaystyle \sum _{t\in {𝔽}_q^k}(\psi \circ f)\left(t\right)e_{\mu }{u}_1\left(t_1\right)\cdots u_k\left(t_k\right)v{e}_{\mu }}& =& {\displaystyle \sum _{\underset{\underset{\_}{t_k=0}}{t\in {𝔽}_q^k,}}(\psi \circ f)\left(t\right)e_{\mu }{u}_1\left(t_1\right)\cdots u_{k-1}\left(t_{k-1}\right)\underset{\_}{u_{k}v}{e}_{\mu }}\\ & & {\displaystyle +\sum _{\underset{\underset{\_}{t_k\in {𝔽}_q^{*}}}{t\in {𝔽}_q^k,}}(\psi \circ \left(\underset{\_}{{\phi }_k\left(f\right)-{\mu }_{\gamma }d{y}_k^{-1}}\right))\left(t\right)e_{\mu }{u}_1\left(t_1\right)\cdots u_{k-1}\left(t_{k-1}\right)\underset{\_}{h_{i_k}\left(t_k^{-1}\right)v}{e}_{\mu },}\end{array}$$ where ${\phi }^k:{𝔽}_q[y_q^{\pm 1},\dots ,y_k^{\pm 1}]\to {𝔽}_q[y_1^{\pm 1},\dots ,y_k^{\pm 1}]$ is given by $$\sum _{\underset{t_k\in {𝔽}_q^{*}}{t\in {𝔽}_q^k}}(\psi \circ f)\left(t\right)e_{\mu }{u}_1\left(t_1\right)\cdots u_{k-1}\left(t_{k-1}\right)\underleftarrow{x_{i_k}\left(t_k^{-1}\right)}=\sum _{\underset{t_k\in {𝔽}_q^{*}}{t\in {𝔽}_q^k}}(\psi \circ \underset{\_}{{\phi }_k\left(f\right)})\left(t\right)e_{\mu }{u}_1\left(t_1\right)\cdots u_{k-1}\left(t_{k-1}\right)\text{.}$$

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 $\tau =({\tau }_1,{\tau }_2,\dots ,{\tau }_{r-k})$ be such that ${\tau }_i\in \{+0,-0,1\},$ 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 $\tau =(-0,1,+0,+0,1,1),$ then ${\ell }^{\vee }\left(\tau \right)=4\text{.}$ For $i\in \{+0,-0,1\},$ let $$(i,\tau )=(i,{\tau }_1,{\tau }_2,\cdots ,{\tau }_{r-k})\text{.}$$ By convention, if ${\ell }^{\vee }\left(\tau \right)=r,$ then $\tau =\varnothing \text{.}$

Suppose ${\ell }^{\vee }\left(\tau \right)=k\text{.}$ Define $$\begin{array}{cc}{\displaystyle {\mathrm{\Xi }}^{\tau }(u,v)=\frac{1}{q^r}\sum _{t\in {𝔽}_q^{\tau }}(\psi \circ f^{\tau })\left(t\right)e_{\mu }{u}_1\left(t_1\right)\cdots u_k\left(t_k\right)v^{\tau }\left(t\right)e_{\mu },}& \text{(3.24)}\end{array}$$ where $$\begin{array}{rcll}{\displaystyle {𝔽}_q^{\tau }}& =& {\displaystyle \{t\in {𝔽}_q^r\hspace{0.17em}|\hspace{0.17em}\text{for}\hspace{0.17em}k<i\le r,\begin{array}{l}\text{if}\hspace{0.17em}{\tau }_{i-k}=+0\text{, then}\hspace{0.17em}{t}_i\in {𝔽}_q,\\ \text{if}\hspace{0.17em}{\tau }_{i-k}=-0\text{, then}\hspace{0.17em}{t}_i=0,\\ \text{if}\hspace{0.17em}{\tau }_{i-k}=1\text{, then}\hspace{0.17em}{t}_i\in {𝔽}_q^{*}\text{.}\end{array}\};}& \text{(3.25)}\\ {\displaystyle v^{\tau }\left(t\right)}& =& {\displaystyle h_{i_{k+1}}{\left(t_{k+1}\right)}^{{\tau }_1}{u}_{k+1}^{1-{\tau }_1}\cdots h_{i_r}{\left(t_r\right)}^{{\tau }_{r-k}}{u}_r^{1-{\tau }_{r-k}}v,}& \text{(3.26)}\end{array}$$ with $+0=-0=0\in \mathbb{Z},$ $1=1\in \mathbb{Z}$ in (3.26); and $f^{\tau }$ is defined recursively by $$\begin{array}{rcll}{\displaystyle f^{\varnothing }=f_u}& =& {\displaystyle -\frac{{\mu }_{{\beta }_1}{c}_1}{{\beta }_1\left(u_T\right)}{y}_1-\frac{{\mu }_{{\beta }_2}{c}_2}{{\beta }_2\left(u_T\right)}{y}_2-\cdots -\frac{{\mu }_{{\beta }_r}{c}_r}{{\beta }_r\left(u_T\right)}{y}_r,\text{(as in (3.21)),}}& \text{(3.27)}\\ {\displaystyle f^{(i,\tau )}}& =& {\displaystyle \{\begin{array}{ll}{f}^{\tau }+{\mu }_{{\gamma }_{\tau }}{d}_{\tau }{y}_k,& \text{if}\hspace{0.17em}i=\pm 0,\\ {\phi }_k\left(f^{\tau }\right)-{\mu }_{{\gamma }_{\tau }}{d}_{\tau }{y}_k^{-1},& \text{if}\hspace{0.17em}i=1,\end{array}}& \text{(3.28)}\end{array}$$ 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), ${\mathrm{\Xi }}^{\varnothing }(u,v)=\left(e_{\mu }u{e}_{\mu }\right)\left(e_{\mu }v\prime e_{\mu }\right)$ (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) ${\mathrm{\Xi }}^{\tau }(u,v)=\frac{1}{q^r}\sum _{t\in {𝔽}_q^{\tau }}(\psi \circ f^{\tau })\left(t\right)e_{\mu }{v}^{\tau }{e}_{\mu }$ has no remaining factors of the form $u_k\left(t_k\right),$ (b) ${\mathrm{\Xi }}^{\tau }(u,v)=0$ unless $v^{\tau }\left(t\right)\in N_{\mu }$ for some $t\in {𝔽}_q\text{.}$

The following corollary gives relations for expanding ${\mathrm{\Xi }}^{\tau }(u,v)$ (beginning with ${\mathrm{\Xi }}^{\varnothing }(u,v)\text{)}$ 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 ${\mathscr{H}}_{\mu }\text{.}$

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) $\left(e_{\mu }u{e}_{\mu }\right)\left(e_{\mu }v\prime e_{\mu }\right)={\mathrm{\Xi }}^{\varnothing }(u,v)\text{.}$ (b) If ${\ell }^{\vee }\left(\tau \right)=k,$ then $${\mathrm{\Xi }}^{\tau }(u,v)=\{\begin{array}{ll}{\mathrm{\Xi }}^{(+0,\tau )}(u,v),& \text{if}\hspace{0.17em}\ell \left(\pi \left(u_k{v}^{\tau }\right)\right)>\ell \left(\pi \left(v^{\tau }\right)\right),\\ {\mathrm{\Xi }}^{(-0,\tau )}(u,v)+{\mathrm{\Xi }}^{(1,\tau )}(u,v),& \text{if}\hspace{0.17em}\ell \left(\pi \left(u_k{v}^{\tau }\right)\right)<\ell \left(\pi \left(v^{\tau }\right)\right)\text{.}\end{array}$$

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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