### Springer fiber components in the two columns case for types $A$ and $D$ are normal

We study the singularities of the irreducible components of the Springer fiber over a nilpotent element $N$ with ${N}^{2}=0$ in a Lie algebra of type $A$ or $D$ (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.