Twisted quadratic foldings of root systems

The present paper is devoted to twisted foldings of root systems that generalize the involutive corresponding automorphisms Dynkin diagrams. A motivating example Lusztig’s projection system typeE8E_8onto subring icosians quaternion algebra, which gives H 4">H4H_4. By using moment graph techniques for any such folding, a map at equivariant cohomology level constructed. It shown this commutes with characteristic classes and Borel maps. Restrictions usual projective homogeneous varieties, group their virtual analogues finite reflection groups are also introduced studied.