Representation theory, equivariant topology and Langlands duality via fixed point schemes

Objective

We propose a three-way dictionary between representation theory of complex semisimple Lie groups, equivariant topology of affine Schubert varieties and mirror symmetry for Langlands dual Hitchin systems via the language of fixed point schemes. This unified framework will yield new insights and advances within these fields, with applications extending to other areas of mathematics and quantum physics. We introduce big algebras, which are commutative algebras attached to representations of complex semisimple Lie groups. We view them as commutative avatars of the representation because they bring together a wealth of sophisticated information, including a novel ring structure on multiplicity spaces, the weight diagram and crystal structure. We geometrize the study of equivariant cohomologies of various varieties with group action by representing them as rings of functions of certain fixed point schemes of the group action. Adopting this perspective, we propose novel scheme-theoretic counterparts to several fundamental constructions in algebraic topology, such as equivariant integration, Hodge and Lefschetz theory. For affine Schubert varieties, these counterparts offer an alternative topological approach to understanding big algebras. Due to our central observation that the Hitchin integrable system on various Lagrangians can be modelled by the spectrum of equivariant cohomology and big algebras, we propose a host of computational tests of conjectured mirror branes of Kapustin–Witten in Langlands dual Hitchin systems. We explore several applications, including polynomial relationships between quantum numbers in baryon multiplets; a big algebra approach to Kashiwara’s conjecture on affine crystals; compatibility with Langlands duality, endoscopy, transfer and character formulas in the relative Langlands program; geometrization of various q = −1 phenomena and cyclic sieving in algebraic combinatorics.