Springer correspondence for the split symmetric pair in type $A$
  In this paper we establish Springer correspondence for the symmetric pair $(\mathrm{SL}(N),\mathrm{SO}(N))$ 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 $q=-1$. 
Tsao-Hsien Chen, Kari Vilonen, Ting Xue
math.RT math.AG