In a new paper on arxiv: “Dynkin automorphisms actions on Gaudin algebras” 2022 ISTern and 2023 ISTA summer intern Vladyslav Zveryk gives a proof of the twining conjecture on big algebras.
“In this paper, we study the action of a Dynkin automorphism σ of a simple Lie algebra 𝔤 on the generalized Gaudin algebras. In particular, we show isomorphisms between the coinvariants of the generalized Gaudin algebras associated with Lie(𝔤∨) and the generalized Gaudin algebras associated with Lie(𝔤∨σ), where 𝔤σ is the fixed point subalgebra.”
Here is a list of Dynkin automorphisms of root systems, their twining root systems and endoscopy algebras (originally from slide 8 of talk “Higgs bundle tournaments“)
