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(Vi) is in Vi), and
B strictly preserves the flag (i.e. B(Vi) is in Vi-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.