Fix a flag of subspaces V_{1} in V_{2} in V_{3}, etc. all in C^{n}.

Consider the space of pairs of commuting linear transformations A and B such that:

A preserves the flag (i.e. A(V_{i}) is in V_{i}), and

B *strictly* preserves the flag (i.e. B(V_{i}) is in V_{i-1}).

Does anyone know anything about this space? Is there any literature on it? Is it smooth?

Just as an example: if the flag is trivial, B=0 and A can be anything. So that's smooth.