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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.

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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.

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