Unipotent Hecke algebras of GLn(Fq)

This paper describes a family of Hecke algebras Hμ=EndG(IndUG(ψμ)), where U is the subgroup of unipotent upper-triangular matrices of G=GLn(Fq) and ψμ is a linear character of U. The main results combinatorially index a basis of Hμ, provide a large commutative subalgebra of Hμ, and after describing the combinatorics associated with the representation theory of Hμ, generalize the RSK correspondence that is typically found in the representation theory of the symmetric group.