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Gelfand–Tsetlin Degeneration of Shift of Argument Subalgebras in Types B, C and D

Arnold Mathematical Journal. 2019. Vol. 5. No. 2-3. P. 285-313.
Leonid Rybnikov, Mikhail Zavalin.

The universal enveloping algebra of any semisimple Lie algebra gg contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of gg. For g=glng=gln the Gelfand–Tsetlin commutative subalgebra in U(g)U(g) arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for classical Lie algebras (g=sp2ng=sp2n or sonson). The limit subalgebra is described explicitly in terms of Bethe subalgebras in twisted Yangians Y−(2)Y−(2) and Y+(2)Y+(2), respectively. We index the eigenbasis of such limit subalgebra in any irreducible finite-dimensional representation of gg by Gelfand–Tsetlin patterns of the corresponding type, and conjecture that this indexing is, in appropriate sense, natural. According to Halacheva et al. (Crystals and monodromy of Bethe vectors. arXiv:1708.05105, 2017) such eigenbasis has a natural gg-crystal structure. We conjecture that this crystal structure coincides with that on Gelfand–Tsetlin patterns defined by Littelmann in Cones, crystals, and patterns (Transform Groups 3(2):145–179, 1998).