= 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 $q\equiv1\pmod p$, then $q_i=1$ in the field $k$ of <characteristic of a field>[characteristic] $p$. The specialized quadratic relation becomes
$$
T_i^2=1,
$$
while the braid relations are unchanged. These are the defining relations of the <Coxeter group> $W$, so $x_i\mapsto T_i$ induces a surjective homomorphism
$$
k[W]\longrightarrow H_k(G,B).
$$
Both algebras have bases indexed by $W$, hence the homomorphism is an isomorphism of <group algebra>[group algebras].
Solved by gpt-5.6-sol high.
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