Fourier-transform Springer correspondence for the split symmetric pair in type A

Let $\lambda$ be a partition of $N$. For a partition with at least one odd part, let $\mathcal O_\lambda$ be the corresponding nilpotent $K$-orbit, let $\mathcal E$ range over irreducible $K$-equivariant local systems on it, and let $\Sigma_\lambda$ be the set of triples defined above. Let $\mathfrak F$ denote the Fourier transform, and let $\operatorname{IC}$ and $\mathcal T$ denote the intersection cohomology complexes and the corresponding local systems used in the statement.

Fourier-transform Springer correspondence. If $\lambda$ has at least one odd part, then $\mathfrak F$ induces a bijection

$$\mathfrak F:\\{\operatorname{IC}(\mathcal O_\lambda,\mathcal E)\mid \mathcal E\text{ irreducible }K\text{-equivariant local system on }\mathcal O_\lambda\text{ up to isomorphism}\\} \xrightarrow{\sim} \\{\operatorname{IC}(\mathfrak g_1^{|\nu|},\mathcal T(\nu;\mu^1,\mu^2))\mid (\nu;\mu^1,\mu^2)\in\Sigma_\lambda\\}.$$

Moreover, it sends the constant local system to the unique triple $(\nu_0;\mu^1_0,\mu^2_0)\in\Sigma_\lambda$ satisfying

$$|\nu_0|=\max\\{ |\nu|: (\nu;\mu^1,\mu^2)\in\Sigma_\lambda\\},$$

and such that the parts of $\mu^1_0$ and $\mu^2_0$ have opposite parity. If all parts of $\lambda$ are even, the analogous bijection holds between the irreducible $K$-equivariant local systems on $\mathcal O_\lambda^\omega$, for $\omega=\mathrm{I},\mathrm{II}$, and the two displayed families of intersection cohomology complexes indexed by $\Sigma_\lambda^\omega$; moreover, the constant local system is sent to $\operatorname{IC}(\mathfrak g_1^{n,\omega},\mathcal T(\nu_0;\emptyset,\emptyset))$ with $|\nu_0|=n$ and $(\nu_0;\emptyset,\emptyset)\in\Sigma_\lambda^\omega$. This identifies the Fourier transform with the Springer correspondence for the split symmetric pair in type $A$, parametrizing all relevant irreducible equivariant local systems by the stated data.