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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.
On the other hand, if the flag is complete, then all matrices preserving a complete flag commute, so we're just taking pairs of matrices, one preserving the flag, and one strictly preserving it; that's also smooth.

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

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.
On the other hand, if the flag is complete, then all matrices preserving a complete flag commute, so we're just taking pairs of matrices, one preserving the flag, and one strictly preserving it; that's also smooth.