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Let us consider intersections of Springer fibres and Schubert cells in type A.

Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let $$ F_Y = \{ V_0 = 0 \subset V_1 \subset \dots \subset V_n = \mathbb C^n : Y V_i \subset V_i \} $$ be the Springer fibre for $ Y$.

Let $ W_\bullet \in F_Y $ be a point in the Springer fibre.

Let $ w \in S_n $ be a permutation. Let $X_w(W_\bullet) $ be the Schubert cell with respect to $ W_\bullet $, namely those flags whose intersection with $ W_\bullet $ is given by $ w $.

What can we say about the intersection $ X_w(W_\bullet) \cap F_Y $? Is it always an affine space? How does it depend on the choice of $ W_\bullet$?

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