# Springer fibres for nilpotents of type $(n,n)$; framed tangles

My question is about the paper "Affine tangles and irreducible exotic sheaves" (arxiv.org/abs/0802.1070).

Background: Let $\mathcal{B}, \mathcal{N}$ denote the flag variety and nilpotent cone of $G=SL_{2n}(\mathbb{C})$; let $T^* \mathcal{B} = \tilde{\mathcal{N}}$ with projection map $p:T^*\mathcal{B} \rightarrow \mathcal{B}$. Let $P_k$ be the space of partial flags with the $k$-th flag omitted; let $V_k$ be the tautological vector bundle on $\mathcal{B}$ of dimension $k$, and let $\mathcal{E}_k$ be the pullback under $p$ of the quotient $V_k/V_{k-1}$.

For $z \in \mathcal{N}$, let $B_{z}$ denote the Springer fibre corresponding to $z$ (i.e. the preimage of $z$ under the Springer resolution map $\pi: \tilde{\mathcal{N}} \rightarrow \mathcal{N}$). Let $S_z$ be the Slodowy slice for $z$, and let $U_z = \pi^{-1}(S_z \cap \mathcal{N})$. Let $D_z = D^b_{B_z}(U_z)$ be the derived category of coherent sheaves on $U_z$ supported on $B_z$. If $z = Z_{2n}$ is the nilpotent matrix of size $2n$ with Jordan type $(n,n)$, define $D_{2n} = D_z$.

Also define $P_{k,z}$ to be the subvariety of $P_k$ consisting of all flags $$(V_0 \subset V_1 \subset \cdots \subset V_{k-1} \subset V_{k+1} \subset \cdots \subset V_n)$$ satisfying $\text{dim} V_i = i$, $zV_{i} \subset V_{i-1}$ for $i \neq k,k+1$, and $zV_{k+1} \subset V_{k-1}$.

Q1 Could someone suggest a reference for framed tangles? Specifically, at the start of Section 3.3, I don't understand what "blackboard framing" means, or what "positive/negative twists of the framing of the i-th strand", means.

Q2 My question is how the functor $G_{2n}^k: D_{2m} \rightarrow D_{2n}$ is defined (cf. bottom pg $10$, (1)). It is originally defined on pages $3-4$, as a functor $D^b(T^*P_k) \rightarrow D^b(\tilde{\mathcal{N}})$. Specifically, if $\pi_k: T^*P_k \times_{P_k} \mathcal{B} \rightarrow T^*P_k$ is the projection, and $$i_k: T^*P_k \times_{P_k} \mathcal{B} \rightarrow \tilde{\mathcal{N}}$$ be the embedding; then $G_{2n}^k = \mathcal{E}_k \otimes (i_k)_* p_k^*$. But the domain and image of this map are different, so I'm confused. I thought perhaps Lemma $5$ on pg $10$, that $P_{k, Z_{2n}} \simeq B_{Z_{2n-2}}$ may be useful, but I'm still not quite sure.

• It's often simplest when reading a preprint to consult the author about questions like those raised here. Rina Anno is currently a postdoc at UMass Amherst and in fact has an office just down the hall from the office I share. Her email address is: anno@ [followed by the usual ending as in my address] – Jim Humphreys Nov 19 '11 at 14:45

For Q2: I believe she's restricting the functor to $U_z$ and using the isomorphism you mention, but Jim is right that you might just want to ask Rina.