Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type A
Ivan Mirkovic, Maxim Vybornov
TL;DR
This work establishes explicit isomorphisms among three central geometric lenses for type A representations: Nakajima’s framed quiver varieties, nilpotent/matrix conjugacy data, and Beilinson-Drinfeld loop Grassmannians. A key innovation is the regular transverse normal slice T to nilpotent orbits, which embeds into BD-Grassmannians and yields a concrete, computable bridge via explicit formulas; this enables a decomposition of the loop Grassmannian into quiver varieties and a compactification of Nakajima varieties. The Isomorphism Theorem ties these settings through phi, tilde-phi, psi, and tilde-psi, with deformations along the central parameter c linking nilpotent orbits to general conjugacy classes and BD fibers to BD-Grassmannians. Applications include geometric SKew and Symmetric GL dualities, realized through multiplicity spaces and the Geometric Satake correspondence, illustrating deep ties between quiver variety geometry and representation theory. The Appendix provides an explicit formula for the isomorphism ψ ∘ φ, derived from an alternative moduli of bundles perspective, reinforcing the canonical nature of the correspondence and enabling practical computations in the cohomological framework of loop Grassmannians.
Abstract
In type A we find equivalences of geometries arising in three settings: Nakajima's (``framed'') quiver varieties, conjugacy classes of matrices and loop Grassmannians. These are now all given by explicit formulas. In particular, we embedd the framed quiver varieties into Beilinson-Drinfeld Grassmannians. This provides a compactification of Nakajima varieties and a decomposition of affine Grassmannians into Nakajima varieties. As an application we provide a geometric version of symmetric and skew $(GL(m), GL(n))$ dualities.