This paper describes a family of Hecke algebras; H_mu=End_G(Ind_U^G(psi_mu)), where U is the subgroup of unipotent; upper-triangular matrices of G=GL_n(F_q) and psi_mu is a linear character of; U. The main results combinatorially index a basis of H_mu, provide a large; commutative subalgebra of H_mu, and after describing the combinatorics; associated with the representation theory of H_mu, generalize the RSK; correspondence that is typically found in the representation theory of the; symmetric group.