Conference: The Hitchin system, Langlands duality and mirror symmetry, ICMAT Madrid, April 2023
Minicourse: Mirror symmetry and big algebras
After surveying some mirror symmetry phenomena in the geometry of Langlands dual Hitchin systems, we will model the Hitchin system on cominuscule upward flows by the equivariant cohomology of cominuscule flag varieties like the Grassmannian. Then we will explain how mirror symmetry implies a synthetic BNR correspondence. To generalise to the non-minuscule case we will construct a big commutative subalgebra of the Kirillov algebra to model the mirror of the universal principal bundle in an irreducible representation. We finish with an overview of results and conjectures on involutions on big algebras induced from quasi-split real forms.