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).

for the case where n = 2 and the complete flag the variety in question is isomorphic to A^2 x Spec C[x,y]/(xy). so the space is not nec. smooth.