Abelianization of Stable Envelopes in Symplectic Resolutions

Abstract

Stable envelopes, introduced by Maulik and Okounkov, form a basis for the equivariant cohomology of symplectic resolutions. We study the case of Nakajima quiver varieties, where the resolution is a hyperk"ahler quotient. We relate the stable basis to that of the associated quotient by a maximal torus, and obtain a formula for the transition between the stable basis and the fixed point basis, using the root system and combinatorial data from the torus quotient.

We compute the transition matrix explicitly in the case of cotangent bundles to partial flag varieties, and show that for Grassmannians, it is given by rational Schur polynomials. This recovers in a geometric way the diagonalization of the Hamiltonian in the XXX spin chain. As a second application, we study the case of the Hilbert scheme of points on the plane, and obtain a novel formula expressing Schur polynomials in terms of Jack polynomials.