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In this paper, we study the affine Springer fiber $\mathcal{F} l_N$ in type$A$ for rectangular type semisimple nil-element $N$ and calculate the relativeposition between irreducible components. In particular, we use the affinematrix ball construction to show the relative position map is compatible withthe Kazhdan-Lusztig cell structure, generalizing the work of Steinberg and vanLeeuwen.

Affine Springer Fibers and the Affine Matrix Ball Construction for Rectangular Type Nilpotents