Equivariant structures in mirror symmetry

When a variety \(X\) is equipped with the action of an algebraic group \(G\), it is natural to study the \(G\)-equivariant vector bundles or coherent sheaves on \(X\). When \(X\) furthermore has a mirror partner \(Y\), one can ask for the corresponding notion of equivariance in the symplectic geometry of \(Y\). The infinitesimal notion (equivariance for a single vector field) was introduced by Seidel and Solomon (GAFA 22 no. 2), and it involves identifying a vector field with a particular element in symplectic cohomology. I will describe the analogous situation for a Lie algebra of vector fields, and discuss the application of this theory to mirror symmetry of flag varieties. In this situation, we expect to find a close connection to the canonical bases of Gross-Hacking-Keel. This talk is based on joint work with Yanki Lekili and Nick Sheridan.