= Solution
A <BN-pair> consists of subgroups $B,N\leq G$ for which $G=\langle B,N\rangle$, the subgroup $H=B\cap N$ is normal in $N$, the quotient $W=N/H$ is generated by a distinguished set $S$ of involutions, and the Bruhat multiplication and nondegeneracy axioms hold. The quotient is the associated <Weyl group>, and the axioms give the <Bruhat decomposition of a BN-pair>
$$
G=\bigsqcup_{w\in W}B\dot wB.
$$
The <Iwahori-Hecke algebra of a BN-pair> may be defined, up to the usual opposite-algebra convention, by
$$
H_k(G,B)=\operatorname{End}_{kG}(k[G/B]).
$$
Its standard basis $(T_w)_{w\in W}$ is indexed by the Bruhat double cosets. For a simple generator $x_i$ represented by $\dot x_i\in N$, set
$$
q_i=[B:B\cap\dot x_iB\dot x_i^{-1}].
$$
The double-coset multiplication rule is the generic rule from part a with $a_i$ specialized to $q_i\cdot1_k$. Thus $H_k(G,B)$ is a <specialization of an algebra>[specialization] of the generic algebra.
Solved by gpt-5.6-sol high.
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