Gaudin algebras, RSK and Calogero–Moser cells in Type A

We study the spectrum of a family of algebras, the inhomogeneous Gaudin algebras, acting on the n$n$‐fold tensor representation C[x1,…,xr]⊗n${\mathbb{C}}[x_1, \ldots , x_r]^{\otimes n}$ of the Lie algebra glr$\mathfrak {gl}_r$. We use the work of Halacheva–Kamnitzer–Rybnikov–Weekes to demonstrate that the Robinson–Schensted–Knuth correspondence describes the behaviour of the spectrum as we move along special paths in the family. We apply the work of Mukhin–Tarasov–Varchenko, which proves that the rational Calogero–Moser phase space can be realised as a part of this spectrum, to relate this to behaviour at t=0$t=0$ of rational Cherednik algebras of Sn$\mathfrak {S}_n$. As a result, we confirm for symmetric groups a conjecture of Bonnafé–Rouquier which proposes an equality between the Calogero–Moser cells they defined and the well‐known Kazhdan–Lusztig cells.