Geometric representations of GL(n,R), cellular Hecke algebras and the embedding problem

We study geometric representations of GL(n,R)GL(n,R) for a ring RR. The structure of the associated Hecke algebras is analyzed and shown to be cellular. Multiplicities of the irreducible constituents of these representations are linked to the embedding problem of pairs of RR-modules x⊂yx⊂y.