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-fold tensor representation C[x_1,...,x_r]
n of the Lie algebra 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 of rational Cherednik algebras of 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.