Twisted quadratic foldings of root systems

The present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig's projection of the root system of type $E_8$ onto the subring of icosians of the quaternion algebra, which gives the root system of type $H_4$.
By using moment graph techniques for any such folding, a map at the equivariant cohomology level is constructed. It is shown that this map commutes with characteristic classes and Borel maps. Restrictions of this map to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups are also introduced and studied.