the following really should be a comment, but i don't have enough points for that yet.  not all pairs of matrices preserving a complete flag commute.  consider the complete flag determined by the basis e_1,...,e_n.  let A = diag(a_1,...,a_n) and B(e_i) = e_{i+1} where e_{n+1} := 0.  then AB(e_1) = a_2 e_2 and BA(e_1) = a_1 e_2.  if $a_1 \neq a_2$ then these matrices do not commute.  

for the case where n = 2 and the flag is complete the variety in question is isomorphic to A^2 x Spec C[x,y]/(xy).