  In this paper we introduce a certain class of families of Hessenberg
varieties arising from Springer theory for symmetric spaces. We study the
geometry of those Hessenberg varieties and investigate their monodromy
representations in detail using the geometry 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. The results of the paper refine (part of)
the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in
[CVX2].
