Request for info on the space of commuting matrices preserving a flag.

Question

Fix a flag of subspaces V₁ in V₂ in V₃, etc. all in Cⁿ.

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.

Answer 1

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.