For the real Grassmannian Gr$(N,k)$ we have the wellknown isomorphism $$ \text{Gr}(N,k) = O(N)/(O(k) \times O(Nk)) $$ For the complex case, we have $$ \text{Gr}(N,k) = U(N)/(U(k) \times U(Nk)) $$ I would like to know if anything like this holds in the finite field setting, ie can the finite field Grassmannians be described as a homogeneous space of an algebraic group over a finite field, or something like this?

There is a description of these homogeneous spaces over finite fields in terms of graphs in "DistanceRegular Graphs" by Brouwer, Cohen and Neumaier (Springer, 1989) and in "Algebraic combinatorics I: Association schemes" by E.Bannai and T.Ito (Benjamin/Cummings, 1984). 

