Iwahori-Hecke algebra of a BN-pair ($H_k(G,B)$)

For a finite group $G$ with a BN-pair, the Iwahori-Hecke algebra over $k$ is the endomorphism algebra of the permutation module $k[G/B]$, up to the conventional opposite algebra. Its standard basis is indexed by the Bruhat double cosets.

For a representative ${\dot{x}}_i\in N$, the Hecke parameter isThe Iwahori-Hecke algebra of a BN-pair is obtained from the generic algebra by $a_i\mapsto q_i\cdot 1_k$.