# Intersection of Springer fibre and Schubert cell

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 $W_\bullet$?