Geometric realization of ψ and Δ^ψ via quiver varieties

Develop a geometric interpretation of the anti-involution ψ, the topological coproduct Δ^ψ, and the resulting tensor product within the framework of Nakajima’s morphism to equivariant K-theory of quiver varieties on affine Dynkin diagrams, including their behavior under specialization to torus fixed points.

Background

Nakajima’s construction relates quantum toroidal algebras to equivariant K-theory of quiver varieties, where some aspects of the Drinfeld coproduct have geometric interpretations via specialization to torus fixed points.

The paper’s new horizontal–vertical anti-involution ψ and the twisted topological coproduct Δ^ψ underlie the tensor product and R-matrix structures developed later; a geometric understanding of these within quiver varieties remains unclear and would connect algebraic and geometric frameworks.