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

Abstract

The universal enveloping algebra of any semisimple Lie algebra $$\mathfrak {g}$$ contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of $$\mathfrak {g}$$. For $$\mathfrak {g}=\mathfrak {gl}_n$$ the Gelfand–Tsetlin commutative subalgebra in $$U(\mathfrak {g})$$ arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for classical Lie algebras ($$\mathfrak {g}=\mathfrak {sp}_{2n}$$ or $$\mathfrak {so}_{n}$$). The limit subalgebra is described explicitly in terms of Bethe subalgebras in twisted Yangians $$Y^-(2)$$ and $$Y^+(2)$$, respectively. We index the eigenbasis of such limit subalgebra in any irreducible finite-dimensional representation of $$\mathfrak {g}$$ 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 $$\mathfrak {g}$$-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).