Springer correspondence for the split symmetric pair in type $A$

In this paper we establish Springer correspondence for the symmetric pair $(\mathrm{S}\mathrm{L}(N),\mathrm{S}\mathrm{O}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain results on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.