Suppose $G$ is a finite group with $BN$-pair, and $(W,S)$ is its Coxeter system.
Iwahori's theorem that the corresponding Hecke algebra $\mathcal{H}$ has a standard basis $\{a_w:w\in W\}$ where $a_w=|B|^{-1}\sum_{x\in BwB}x$, satisfying relations $$ a_{s_i}a_w=a_{s_iw}\qquad\text{if } \ell(sw)>\ell(w) $$ and $$ a_{s_i}a_w=q_ia_{s_iw}+(q_i-1)a_w\qquad\text{if }\ell(s_iw)<\ell(w) $$ where $q_i$ is the index parameter $q_i=[B:s_iBs_i\cap B]$.
It's known that if $s_i$ and $s_j$ are conjugate in $W$, then $q_i=q_j$. Why is this? In Curtis and Reiner's Methods of Representation Theory, they say is is a Corollary of Iwahori's theorem, but I don't see it. I was able to prove that if $s_i=s_ks_js_k$, then $q_i=q_j$, but observing that $\ell(s_ks_j)=\ell(s_js_k)=2$, and then left or multiplying by $s_k$ again must decrease the length. Using the relations I could write $a_{s_i}$ in terms of $a_{s_k}$ and $a_{s_j}$, and then apply the index homomorphism.
I would like to induct on the length of $w$ in the case $s_i=ws_jw^{-1}$, but I'm struggling to set it up correctly since if $w=s_rw'$, with $\ell(w')<\ell(w)$, then $s_i=s_rw's_jw'^{-1}s_r$, the problem is $w's_jw'^{-1}$ might not have length one, so it's not an element of $S$.