Before the Concept · II of II

Hecke algebras of double cosets

For a compact open subgroup $J$ of $\GF$, the functions on $J\backslash\GF/J$ form an algebra under convolution. This page defines it, proves the three general facts the Concept needs (structure constants are counts, the integral is a character, the index is the integral), and shows the finite Hecke algebra inside the Iwahori one.

Table 1. The objects of this page and their types.
object	type	defined in
$J$	a compact open subgroup of $\GF$ (on this site $J=I$ or $J=K$)	§1
$dg$, $\vol$	the Haar measure on $\GF$ with $\vol(J)=1$	§1
$\Cc(J\backslash\GF/J)$	a $\C$-vector space with basis the indicator functions of double cosets	§1
$\ast:\cH_J\times\cH_J\to\cH_J$	convolution	§2
$\ind:\cH_J\to\C$	the integral, an algebra homomorphism	§3
$\Cc(J\backslash Y)$	a left $\cH_J$-module, for a $\GF$-set $Y$	§4
$\Hzero\subset\cH$	the finite Hecke algebra inside the Iwahori one	§5

The Hecke algebra of a compact open subgroup Definition

(Hecke algebra) [HKP10, §1.2], [IM65, §3.1]

Let $J\subset\GF$ be a compact open subgroup and $dg$ the Haar measure on $\GF$ with $\vol(J)=1$. Put

$$\cH_J=\Cc(J\backslash\GF/J)=\{f:\GF\to\C\ \text{ with compact support, }\ f(j_1gj_2)=f(g)\ \text{ for }j_1,j_2\in J\}.$$

It has the basis $\one_{JxJ}$, one for each double coset; on this site $\cH=\cH_I$ and $\HK=\cH_K$.

Each double coset $JxJ$ is compact and open, so it is a finite union of left cosets $yJ$; the number of them is finite because $JxJ$ is compact and the $yJ$ are open. A function with compact support that is $J$-invariant on both sides is therefore a finite linear combination of the $\one_{JxJ}$, which are linearly independent since the double cosets are disjoint.

In the running example $J=I$ for $\SL_2$: the double cosets are the $I$-orbits on edges of the tree, one for each word in $s_0,s_1$ (SL₂ I §3). $J=K$: the double cosets $Kt^{\mu}K$, $\mu\ge0$, are the $K$-orbits on vertices, one for each distance from the base vertex.

Convolution Proved here

(convolution) [HKP10, §1.2, footnote 1]

For $f\in\cH_J$ and a left $J$-invariant function $\varphi$ on a $\GF$-set (for instance $\varphi\in\cH_J$),

$$(f\ast\varphi)(x)=\int_{\GF}f(y)\,\varphi(y^{-1}x)\,dy=\sum_{y\in\GF/J}f(y)\,\varphi(y^{-1}x).$$

($\cH_J$ is an algebra; structure constants are counts) Proved here

(i)The sum is finite and well defined, $f\ast\varphi$ is left $J$-invariant, and $\supp(f\ast\varphi)\subseteq\supp(f)\cdot\supp(\varphi)$.

(ii)$\ast$ makes $\cH_J$ an associative algebra with unit $\one_J$, and $\one_{JxJ}\ast\one_{JyJ}=\sum_z c_{x,y}^{z}\,\one_{JzJ}$ with

$$c_{x,y}^{z}=\#\{\,uJ\in JxJ/J:\ u^{-1}z\in JyJ\,\}\ \in\ \N,$$

a number that depends only on the double coset of $z$, and is zero unless $JzJ\subseteq JxJ\cdot JyJ$.

Proof

The summand depends only on $yJ$: $f(yj)=f(y)$ and $\varphi(j^{-1}y^{-1}x)=\varphi(y^{-1}x)$. Only the finitely many cosets $yJ\subseteq\supp(f)$ contribute. For $j\in J$, $(f\ast\varphi)(jx)=\sum_y f(y)\varphi(y^{-1}jx)=\sum_{y'}f(jy')\varphi(y'^{-1}x)$ with $y'=j^{-1}y$, and $f(jy')=f(y')$. If $(f\ast\varphi)(x)\neq0$ then $x=y\cdot(y^{-1}x)$ with $y\in\supp f$ and $y^{-1}x\in\supp\varphi$.
Associativity is Fubini for the sum over $\GF/J\times\GF/J$; $(\one_J\ast\varphi)(x)=\sum_{y\in J/J}\varphi(y^{-1}x)=\varphi(x)$, and likewise on the right.
$(\one_{JxJ}\ast\one_{JyJ})(z)=\sum_{u\in\GF/J}\one_{JxJ}(u)\one_{JyJ}(u^{-1}z)$ is the count $c^{z}_{x,y}$. It is right $J$-invariant in $z$ by (i) applied on the other side (the same computation with $x\mapsto xj$), so it is constant on $JzJ$; it vanishes unless $z\in JxJ\cdot JyJ$.

This is the multiplication law of [IM65, p. 276], where $c^{z}_{x,y}$ is written $m^{\mu}_{\sigma,\tau}=|B\backslash B\omega(\sigma)^{-1}B\omega(\mu)\cap B\omega(\tau)B|$.

In the running example For $J=I$ in $\SL_2$ the count $c^{z}_{x,y}$ is a count of edges in the tree: edges $c$ with $\delta(e_0,c)=x$ and $\delta(c,ze_0)=y$, where $\delta$ is the relative position of two edges (SL₂ II §1).

The index of a double coset and the integral Proved here

(index) [IM65, §3.1]

$$\ind(x)=|JxJ/J|=[J:J\cap x^{-1}Jx]=\int_{\GF}\one_{JxJ}\,dg,\qquad \ind\Big(\sum_x\lambda_x\one_{JxJ}\Big)=\sum_x\lambda_x\ind(x).$$

(the integral is a character) [IM65, p. 276] Proved here

$$\ind(f\ast g)=\ind(f)\,\ind(g)\qquad(f,g\in\cH_J),$$

so $\ind:\cH_J\to\C$ is an algebra homomorphism. In particular $\ind(x)\,\ind(y)=\sum_z c^{z}_{x,y}\,\ind(z)$.

Proof

$\int(f\ast g)(x)\,dx=\int\!\!\int f(y)g(y^{-1}x)\,dy\,dx$. For fixed $y$, substitute $x=yx'$: the Haar measure is left invariant, so the inner integral is $\int g(x')dx'=\ind(g)$.
Hence $\int f\ast g=\ind(f)\ind(g)$. The three expressions for $\ind(x)$ agree because $JxJ=\bigsqcup_{u}uJ$ over $u\in JxJ/J$, each $uJ$ has volume $1$, and $u\mapsto u$ identifies $JxJ/J$ with $J/(J\cap xJx^{-1})$ via $j\mapsto jxJ$.

Only left invariance of $dg$ is used. ($\GF$ is unimodular, but this is not needed.)

This character is the tool that turns a support statement into an equation: if $\one_{JxJ}\ast\one_{JyJ}=c\,\one_{JzJ}$ for a single $z$, then $c=\ind(x)\ind(y)/\ind(z)$. The Concept uses it in exactly this way (the Concept §4).

In the running example $\ind(\one_I)=1$ and $\ind(\one_{Is_1I})=q$: the double coset $Is_1I$ is the union of the $q$ cosets $x_\alpha(\nu)\dot s_1I$ (SL₂ I §4). So $\ind(\one_{Is_1I}\ast\one_{Is_1I})=q^2$.

Modules of invariant functions Definition

The same formula with $\varphi$ a left $J$-invariant compactly supported function on a $\GF$-set $Y$ with open stabilizers makes $\Cc(J\backslash Y)$ a left $\cH_J$-module: $f\ast(g\ast\varphi)=(f\ast g)\ast\varphi$ by the same Fubini argument. Two cases matter on these pages and in the proof workspace:

$Y=\GF/J'$ for a second compact open subgroup $J'$: then $\Cc(J\backslash\GF/J')$ is an $(\cH_J,\cH_{J'})$-bimodule. For $J=I$, $J'=K$ this is the module of Iwahori–spherical translates.
$Y=X(F)$ for a $G$-variety $X$: $\Cc(J\backslash X(F))$ is the module whose generation the proof workspace studies [Draft].

In the running example $Y=\GF/K$ is the set of vertices of type $0$ in the tree, and $\Cc(I\backslash\GF/K)$ has basis the indicator functions of the $I$-orbits on those vertices.

The finite Hecke algebra inside the Iwahori one Proved (cited)

($\Hzero\subset\cH$) [HKP10, §1.7] Proved here

The functions supported on $K$ form a subalgebra

$$\Hzero=\Cc(I\backslash K/I)\ \subset\ \cH,\qquad \Hzero\simeq\C\big[B(\Fq)\backslash G(\Fq)/B(\Fq)\big],$$

with basis $T_w=\one_{IwI}$, $w\in W$, and $\one_K=\sum_{w\in W}T_w$.

Proof

$K$ is a group, so $\supp(f\ast g)\subseteq K$ when both supports lie in $K$ ((i)). Pull back the Bruhat decomposition $G(\Fq)=\bigsqcup_{w\in W}B(\Fq)wB(\Fq)$ along $\ev$: $K=\bigsqcup_{w\in W}IwI$, which gives the basis and the formula for $\one_K$. Functions on $I\backslash K/I$ are functions on $B(\Fq)\backslash G(\Fq)/B(\Fq)$ because $\ker\ev\subseteq I$, and the convolutions agree since $\vol(I)=1$ corresponds to counting cosets of $B(\Fq)$.

This is Iwahori's 1964 algebra (History) sitting inside the affine one. The Concept shows that its relations are the same quadratic relations, now for all $s\in\Saff$ rather than $s\in S$.

In the running example $\Hzero=\C T_e\oplus\C T_{s_1}$ and $\one_K=T_e+T_{s_1}$; $\ind(\one_K)=1+q=[K:I]$.

How this page meets the Concept Context

With $J=I$, the basis of $\cH$ is $\{T_w\}_{w\in\Wt}$ by the Iwahori–Bruhat decomposition (Groups §6). The Concept computes the structure constants $c^{z}_{s,w}$ for $s\in\Saff$: (ii) says which $z$ can occur, and fixes the coefficients once the index $\ind(w)=q^{\len(w)}$ is known.