Twisted quadratic foldings of root systems

TL;DR

This paper explores twisted foldings of root systems, extending involutive foldings, and constructs equivariant cohomology maps that relate to various algebraic and geometric structures.

Contribution

It introduces a general framework for twisted foldings of root systems and develops equivariant cohomology maps that unify several algebraic and geometric contexts.

Findings

Constructed equivariant cohomology maps for twisted foldings.

Showed these maps commute with characteristic classes and Borel maps.

Analyzed restrictions to cohomology of projective varieties and reflection groups.

Abstract

In the present paper we study twisted foldings of root systems which generalize usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Our motivating example is the Lusztig 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 $H_4$. Using moment graph techniques for any such folding we construct a map at the equivariant cohomology level. We show that this map commutes with characteristic classes and Borel maps. We also introduce and study its restrictions to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups.