The SL2(Q3) Iwahori convolution corner has affine parameters 3

The determinant equation makes $G$ a closed subspace of ${\mathbb{Q}}_3^4$, hence it is LCH by [F4]; matrix operations and inversion are continuous. The determinant and congruence conditions make $I$ closed in the compact space ${\mathbb{Z}}_3^4$, hence compact. It is open in $G$ because its entry conditions $a,b,d\in {\mathbb{Z}}_3$ and $c\in \pi {\mathbb{Z}}_3$ are open in the matrix topology. By [F5] choose left Haar measure. Its value on $I$ is finite and positive: if it were zero, every compact support of a continuous test function would have a finite cover by translates of the open subgroup $I$, making every Haar integral zero. Rescale to $\mu (I)=1$. This is the only use of arbitrary choice beyond the countable compactness input in [F4]. For $m\ge 1$, let $K_m=\{g\in {\mathrm{SL}}_2({\mathbb{Z}}_3):g\equiv I_2(\mathrm{m}\mathrm{o}\mathrm{d}{\pi }^m)\}$. These are kernels of reduction homomorphisms, hence subgroups. Since ${\mathbb{Z}}_3$ is compact open, ${\mathrm{SL}}_2({\mathbb{Z}}_3)$ is compact open in $G$; the entrywise congruence conditions make each $K_m$ compact open, and the matrix topology shows that $(K_m)$ is a neighborhood basis at the identity. For a locally constant compactly supported integrand, each point of its compact support has a right $K_m$-coset on which it is constant; a finite subcover and the nested subgroup basis give a finite coset partition on which it is constant. Thus convolution integrals and their associativity calculation reduce to finite sums; the change of variables $y=zu$ uses left invariance of $\mu$. Compact support and local constancy are preserved.

Averaging on the left and right by $e$ makes a function invariant under both $I$ actions. Indeed $e\ast f(g)={\int }_{I}f(i^{-1}g)d\mu (i)$ and $f\ast e(g)={\int }_{I}f(gi)d\mu (i)$. The restriction of $\mu$ to $I$ is a left Haar probability on the compact Hausdorff group $I$, so [F7] makes it right invariant. Thus $e\ast e=e$, and every bi-invariant $f$ has $e\ast f=f=f\ast e$. This proves the corner equality. The double cosets are open, and a compact support meets only finitely many of them, so their characteristic functions form a vector-space basis.

The displayed $s_1$ fixes $[L_0]$ and sends the base edge through that type-$0$ vertex; $s_0=D{s}_1{D}^{-1}$ with $D=\mathrm{diag}(1,\pi )$ fixes $[L_1]$ and changes the base edge through type $1$. Their squares are $-I\in I$. The three right cosets in each simple double coset are explicitly $I{s}_{1}I={\coprod }_{b\in {\mathbb{F}}_3}\left(\begin{array}{cc}1& b\\ 0& 1\end{array}\right)s_{1}I,I{s}_{0}I={\coprod }_{b\in {\mathbb{F}}_3}\left(\begin{array}{cc}1& 0\\ \pi b& 1\end{array}\right)s_{0}I\mathrm{.}$ For the first, reduce the action on the four lines at $[L_0]$: $I$ fixes the line $e_1$ and upper unipotents translate the other three. For the second, use the basis $e_1,\pi e_2$ at $[L_1]$, where lower unipotents translate the three lines other than the backward one. These matrices have entries in $I$, their three images are distinct, and [F1] shows they exhaust the orbit.

Under $G\mathrm{/}I$ identified with type-ordered edges, the operator given by $1_{I{s}_{i}I}$ sums over the three other edges at the endpoint of the indicated type. Two such moves at the same vertex return to the original edge in three ways and reach any specified other edge in two ways. Equivalently the finite right-coset formula for convolution gives $1_{I{s}_{i}I}\ast 1_{I{s}_{i}I}=3e+21_{I{s}_{i}I}$. A product with alternating endpoint types follows the unique shortest gallery in the tree. There is exactly one intermediate gallery to any specified final edge of that type word, so the product of the simple characteristic functions is the characteristic function of its entire double coset, with coefficient one. By [F1] these exhaust all double cosets. The infinite dihedral group has no finite braid relation between the two generators, so the quadratic identities give an algebra map from [F2] onto the corner. The basis of [F3] maps bijectively to the distinct double-coset characteristic functions, proving injectivity as well as surjectivity. Rescaling its quadratic identities yields the displayed symmetric normalization.

If $V$ is smooth, each vector $v$ is fixed by an open subgroup, which contains some $K_m$ by step 1.1. For compactly supported locally constant $f$, shrink $m$ further so that $f$ is constant on every right $K_m$-coset meeting its support; compactness gives a finite such partition. Since $\pi (gk)v=\pi (g)v$ for $k\in K_m$, the integral ${\int }_{G}f(g)\pi (g)vd\mu (g)$ is a finite sum. Refinement preserves the sum by additivity of Haar measure, so it is well defined. Finite-sum convolution and the representation law prove multiplicativity. Averaging by $e$ is the projection onto $V^I$ because left translation by $I$ preserves its integral and averaging fixes every $I$-fixed vector. Hence the corner acts on $V^I$ with identity $e$, including $V^I=0$. Finally [F6] applies to $3>1$, giving the precise parameter boundary. No equivalence assertion is used in this construction.