Dynkin automorphism actions on Gaudin algebras

We study the action of Dynkin diagram automorphisms $\sigma$ on generalized Gaudin algebras, focusing in particular on the big Gaudin algebra $\mathcal{B}(\mathfrak{g})\subset (U(\mathfrak{g})\otimes S(\mathfrak{g}){)}^{\mathfrak{g}}$ and its evaluated versions ${\mathcal{B}}^{\lambda }(\mathfrak{g})$ and ${\mathcal{B}}_{\chi }(\mathfrak{g})$. We show isomorphisms between the coinvariants of the generalized Gaudin algebras associated with ${\mathfrak{g}}^{\vee }$ and the generalized Gaudin algebras associated with ${\mathfrak{g}}_{\sigma }^{\vee }$, where ${\mathfrak{g}}_{\sigma }$ is the fixed point subalgebra. In particular, we get an isomorphism ${\mathcal{B}}^{\lambda }({\mathfrak{g}}^{\vee }{)}_{\sigma }\simeq {\mathcal{B}}^{\lambda }({\mathfrak{g}}_{\sigma }^{\vee })$ for any $\sigma$-invariant dominant weight $\lambda$, which allows us to reprove Jantzen’s twining formula. Our approach relies on interpreting generalized Gaudin algebras via spaces of opers, which explains the appearance of the Langlands duals in our results and in Jantzen’s twining formula.