Geometry and Topology without figures

Let $G$ be a split reductive group over a finite field $k = \mathbb{F}_q$. Let $T \subset B \subset G$ be a choice of maximal and Borel. One of the fundamental object in representation theory is the Hecke algebra $$\mathcal{H}(k) = \operatorname{Fun}_{B(k) \times B(k)}(G(k),\mathbb{C})$$ consisting of complex valued functions on $G(k)$ invariant on both sides by $B(k)$.  This is an algebra with the convolution product $$(f_1 \star f_2)(g) = \frac{1}{\left| B(k) \right|} \sum_{h_1,h_2 \in G(k), h_1h_2=g} f_1(h_1)f_2(h_2).$$ It is a routine check that character functions on double cosets $B(k)wB(k)$ with $w \in W$ the Weyl group, form a $\mathbb{Z}$-basis of the Hecke algebra.  Let $S \subset W$ be the set of simple reflections, then Iwahori showed that $\mathcal{H}(k)$ admits a canonical basis $t_s$ (with $s \in S$) such that $$\begin{align*} t_s^2 & = (q-1)t_s + q t_1 \\ \underset{m_{su}}{\underbrace{t_s t_u \cdots}} & = \underset{m_{su}}{\underbrace{t_u t_s \cdot...