abstract: We develop a Springer theory in the case of symmetric spaces using Fourier transform. We discuss applications in computing cohomology of Hessenberg varieties. Examples of such varieties include classical objects in algebraic geometry: Jacobians and moduli spaces of vector bundles on curves with extra structure, Fano varieties of k-planes in the intersection of two quadrics, etc. This is joint work with Tsao-hsien Chen and Kari Vilonen.