Hessenberg varieties, intersections of quadrics, and the Springer correspondence

We continue our study of the Springer correspondence in the case of symmetric spaces initiated in our previous paper. In this paper we introduce a certain class of families of Hessenberg varieties and study their monodromy representations in detail in a special case when the Hessenberg varieties can be expressed in terms of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations