Homology representations of GL(n,q) from Grassmannians in cross-characteristics

Let F be a finite field of characteristic q0 and let F be a field of characteristic p0 with 0 < p0 6 q0. We describe a class of homological representations of GL(n,F) over F which are obtained from the permutation action of GL(n,F) on the Grassmannians of F n . The key to these homology representations is a branching rule which allows us to determine irreducibility. It also provides explicit dimension and character formulae for the representations in question. Phenomena reminiscent of the geometry of the homological Steinberg and Lusztig representations for GL(n,F) can be observed.