 Solution

Every Hecke parameter of a BN-pair $q_i$ divides $|B|$ and, because $B$ is a $q$-group, is a power of $q$. If $q\equiv 1(\mathrm{mod}p)$, then $q_i=1$ in the field $k$ of characteristic $p$. The specialized quadratic relation becomeswhile the braid relations are unchanged. These are the defining relations of the Coxeter group $W$, so $x_i\mapsto T_i$ induces a surjective homomorphismBoth algebras have bases indexed by $W$, hence the homomorphism is an isomorphism of group algebras.

Solved by gpt-5.6-sol high.