Multiplying the basis on the tree
The same model $\SL_2(q)$, now with one operation added: convolution. Every product $T_xT_y$ is a count of edges, and two counts at a single vertex give the quadratic relation. The finite Hecke algebra, the idempotent $\eK$ and the two characters follow.
The model with convolution: products are edge counts Proved here
(the running example, with convolution)
$\SL_2(q)$ as on Part I, with $\cH=\Cc(I\backslash\GF/I)$, the Haar measure normalized by $\vol(I)=1$, the convolution of Hecke algebras §2, and $T_w=\one_{IwI}$ for $w\in\Waff=\langle s_0,s_1\rangle$.
(structure constants on the tree) Proved here
For $x,y\in\Waff$ and $g\in\GF$, with $\delta$ the relative position of edges from Part I §3,
Proof
By Hecke algebras, Lemma 2.2(ii), $(T_x\ast T_y)(g)=\#\{uI\in IxI/I:\ u^{-1}g\in IyI\}$. The coset $uI$ is the edge $c=ue_0$; $u\in IxI$ means $\delta(e_0,c)=x$; and $u^{-1}g\in IyI$ means $\delta(ue_0,ge_0)=y$.
The mechanism: two counts at one vertex give the quadratic relation Proved here
(quadratic relation for $\SL_2$) Proved here
Proof
- Support. $Is_iI\cdot Is_iI=I\sqcup Is_iI$ (Part I, Lemma 4.1(iii)), so $T_{s_i}^2=a\,T_e+b\,T_{s_i}$ for some integers $a,b\ge0$.
- Coefficient of $T_e$: take $g=e$ in . The edges $c$ with $\delta(e_0,c)=s_i$ and $\delta(c,e_0)=s_i$ are the edges at the type-$i$ vertex of $e_0$ other than $e_0$: there are $q$ of them. So $a=q$.
- Coefficient of $T_{s_i}$: take $ge_0$ one of those $q$ edges. The edges $c$ that are $s_i$-adjacent to both $e_0$ and $ge_0$ are the edges at the same vertex other than $e_0$ and $ge_0$: there are $(q+1)-2=q-1$ of them. So $b=q-1$.
Cross-check with the index character
$\ind$ is an algebra homomorphism with $\ind(T_e)=1$ and $\ind(T_{s_i})=q$ (Hecke algebras, Lemma 3.2). So $q^2=\ind(T_{s_i}^2)=a+bq$, which with $a=q$ gives $b=q-1$ without the third count. The Concept uses this shortcut for every $G$.
Worked computation: products with a simple reflection Proved here
(the Iwahori–Matsumoto relations for $\SL_2$) Proved here
For $i\in\{0,1\}$ and $w\in\Waff$:
Proof
- Case $\len(s_iw)=\len(w)+1$, so the alternating word of $s_iw$ is $s_i$ followed by the word of $w$. By Part I, Lemma 3.1, $Is_iwI/I$ is the set of edges reached from $e_0$ by a path of type $(i,\text{word of }w)$, and $Is_iI\cdot IwI=Is_iwI$. So $T_{s_i}\ast T_w=c\,T_{s_iw}$. For $ge_0\in Is_iwI/I$, an edge $c$ with $\delta(e_0,c)=s_i$ and $\delta(c,ge_0)=w$ is an edge at the type-$i$ vertex of $e_0$ from which $ge_0$ is reached by a path of type the word of $w$. Since the tree has no cycles, $c$ is the second edge of the unique path from $e_0$ to $ge_0$, so $c=1$.
- Case $\len(s_iw)=\len(w)-1$: put $w'=s_iw$, so $w=s_iw'$ with $\len(w)=\len(w')+1$ and $T_w=T_{s_i}T_{w'}$ by step 1. Then $T_{s_i}T_w=T_{s_i}^2T_{w'}=qT_{w'}+(q-1)T_{s_i}T_{w'}=qT_{s_iw}+(q-1)T_w$ by .
| product | result | $\ind$ |
|---|---|---|
| $T_{s_1}T_{s_0}$ | $T_{s_1s_0}$ | $q^2$ |
| $T_{s_1}T_{s_1}$ | $qT_e+(q-1)T_{s_1}$ | $q+(q-1)q=q^2$ |
| $T_{s_1}T_{s_1s_0}$ | $qT_{s_0}+(q-1)T_{s_1s_0}$ | $q^2+(q-1)q^2=q^3$ |
| $T_{s_0}T_{s_1s_0}$ | $T_{s_0s_1s_0}$ | $q^3$ |
| $T_{s_1s_0}T_{s_0s_1}$ | $T_{s_1}\big(qT_e+(q-1)T_{s_0}\big)T_{s_1}=q\,T_{s_1}^2+(q-1)T_{s_1s_0s_1}$ | $q^4$ |
So the algebra is generated by $T_{s_0}$ and $T_{s_1}$, every $T_w$ is the product of the $T_{s_i}$ along the alternating word of $w$, and the only relations are the two quadratic ones: $\Waff$ is the infinite dihedral group, which has no braid relation. The Concept records this as the presentation [IM65, Thm 3.5].
The finite Hecke algebra, $\one_K$ and the idempotent $\eK$ Proved here
($\one_K$ and $\eK$ for $\SL_2$) Proved here
$$K=I\sqcup Is_1I,\qquad \one_K=T_e+T_{s_1},\qquad [K:I]=q+1,$$ $$\one_K\ast\one_K=(q+1)\,\one_K,\qquad \eK:=\frac{\one_K}{q+1},\qquad\eK\ast\eK=\eK .$$$\eK$ is the spherical idempotent: $\eK\ast\varphi=\varphi$ for every left $K$-invariant $\varphi$, and $\eK\cH\eK$ is the spherical Hecke algebra $\HK$ (source page §4). It is not the unit of $\cH$: the unit is $T_e=\one_I$. This distinction is what the proof workspace's Part II is about [Draft].
Worked computation: the index character and the sign character Proved here
(the two one-dimensional modules of $\cH$ for $\SL_2$) Proved here
There are algebra homomorphisms $\ind,\eps:\cH\to\C$ with
Proof
- $\cH$ is generated by $T_{s_0},T_{s_1}$ subject only to the quadratic relations ( and the presentation [IM65, Thm 3.5]). An assignment $T_{s_i}\mapsto\lambda_i$ extends to an algebra homomorphism iff $\lambda_i^2=q+(q-1)\lambda_i$, that is iff $\lambda_i\in\{q,-1\}$.
- $\lambda_0=\lambda_1=q$ gives $\ind$ (it is the integral by Hecke algebras, Lemma 3.2, and $\ind(T_w)=q^{\len(w)}$ by Part I, Lemma 3.1); $\lambda_0=\lambda_1=-1$ gives $\eps$, and $\eps(T_w)=(-1)^{\len(w)}$ because $T_w$ is the product of $\len(w)$ generators.
- $\one_K=T_e+T_{s_1}$: $\ind(\one_K)=1+q$, $\eps(\one_K)=1-1=0$.
The two mixed characters
$\lambda_0=q,\lambda_1=-1$ and $\lambda_0=-1,\lambda_1=q$ are also characters of the affine Hecke algebra of $\SL_2$: there are four one-dimensional modules. The mixed ones do not extend to $\PGL_2$, where $T_\sigma$ conjugates $T_{s_0}$ into $T_{s_1}$ (Part III).
Dictionary to the proof workspace Context
| workspace | here | proved |
|---|---|---|
| $\mathcal H_I$, $T_w=\one_{IwI}$ | $\cH$, $T_w$ | the Concept §3 |
| $\one_K=\sum_{w\in W}T_w$, $[K:I]=\sum_{w\in W}q^{\len(w)}$ | general $G$: the Concept §5 | |
| $e_K=[K:I]^{-1}\one_K$, idempotent | $\eK$ | |
| sign character $\eps$, $\eps(\one_K)=0$ | general $G$: the Concept §6 | |
| $\mathcal H_{I_H}\ast e_{K_H}\ast\mathcal H_{I_H}\subsetneq\mathcal H_{I_H}$ | the ideal of $\eK$ lies in $\ker\eps\not\ni T_e$ | Iwahori level, the group case |