# Homeomorphisms of Springer fibers

Let $$V$$ be a complex $$n$$-dimensional vector space and denote by $${\cal F}$$ its space of complete flags. Let $$g \in Gl(V)$$ be unipotent and consider the Springer fiber $${\cal F}_g$$ of its fixed points in $${\cal F}$$.

My question is: (1) Suppose $$h \in Gl(V)$$ is unipotent and such that $${\cal F}_h$$ is homeomorphic to $${\cal F}_g$$ (in the category of topological spaces, that is, continuous map with continuous inverse). Can we conclude that $$h$$ is conjugate to $$g$$ in $$Gl(V)$$?

Or, equivalently. Since conjugation induces an homeomorphism of $${\cal F}$$ into itself, one can suppose that both $$g = I + N$$ and $$h = I + M$$ are such that $$N$$, $$M$$ are nilpotent matrices in Jordan canonical form. Thus I am asking: (2) does $${\cal F}_h$$ homeomorphic to $${\cal F}_g$$ implies that the sizes of the Jordan blocks of $$N$$ and $$M$$ are the same?

One more equivalence. I know that the sizes of the Jordan blocks of $$N$$ (i.e. the partition of $$n$$, or Young diagram of $$N$$) are related to the topology of $${\cal F}_g$$ in many indirect ways. What I do not know is: (3) are they homeomorphism invariants? If so, how to obtain them from the topological data of $${\cal F}_g$$?

An observation follows. If one uses partial flags, of course the answer to my question is negative, but it seems that one still has some invariants preserved by homeomorphisms. An extreme case is the projective space $${\cal P}(V)$$ of $$V$$: in this case the fixed points of $$g$$ are given by its eigenvectors, which are given by the projectivization $${\cal P}({\rm Ker} N)$$ of $${\rm Ker} N$$ in $${\cal P}(V)$$. An homeomorphism between $${\cal P}({\rm Ker} N)$$ and $${\cal P}({\rm Ker} M)$$ implies that $$\dim {\rm Ker} N = \dim {\rm Ker} M$$ so that $$g$$ and $$h$$ have the same number of Jordan blocks. Thus, the question is if on the complete flag manifold one gets the whole set of conjugation invariants of $$N$$ (the number of Jordan blocks and their sizes) from the topology of its Springer fiber.

At last, if (1) is false, does it become true if one supposes a good enough homeomorphism? If needed, one can suppose that the homeomorphism between $${\cal F}_h$$ and $${\cal F}_g$$ is obtained by the restriction of an homeomorphism of $${\cal F}$$ into itself. Furthermore, if needed, one can suppose that the homeomorphism of $${\cal F}$$ into itself is induced by an homeomorphism of $$V$$ into itself that maps $$0$$ to $$0$$: this would project to a homeomorphism of any partial flag manifold.

PS - A homeomorphism whose induced map in top cohomology intertwines with the Springer representation of the symmetric group $$S_n$$ would imply, by representation theory of $$S_n$$, that the diagram of $$N$$ and $$M$$ coincide. But I do not know which kind of homeomorphisms of $${\cal F}$$ induce such intertwining maps. Also, the homeomorphisms I am interested are in the category of topological spaces. So, at first, I would like to avoid this line of reasoning.

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