STRATIFICATION OF A SINGULAR COMPONENT OF A SPRINGER FIBER OVER x ² = 0 OF THE SIMPLEST TYPE

Let x ∈ M_n(K) satisfy x ² = 0. Let F_x be the Springer fiber over x. The components of F_x are labeled by standard Young tableaux with two columns. For a Young tableau T with two columns, one can define a numerical invariant ρ(T). The component F_T is singular if and only if ρ(T) ≥ 2. In this paper, we construct the stratification of F_T for T with ρ(T) = 2, which reflects also the inner structure of the singular locus, and show that such components have equivalent stratifications if and only if they are isomorphic as algebraic varieties and provide the combinatorial procedure which partitions the set of such components into the classes of isomorphic components. We explain why the stratification of F_T for T with ρ(T) ≥ 3 is very complex and does not reflect the inner structure of the singular locus.