Gelfand-Tsetlin degeneration of shift of argument subalgebras in type D

TL;DR

This paper extends the understanding of limit shift of argument subalgebras in type D Lie algebras, describing their structure via Bethe subalgebras of twisted Yangians and parametrizing eigenbases with Gelfand-Tsetlin patterns.

Contribution

It provides a new description of the limit subalgebras for type D Lie algebras and relates them to Bethe subalgebras in twisted Yangians, completing the analysis for all classical types.

Findings

Describes the limit shift of argument subalgebra for type D Lie algebras.

Parametrizes eigenbases using type D Gelfand-Tsetlin patterns.

Connects the subalgebra structure to Bethe subalgebras in twisted Yangians.

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. In our previous work (arXiv:1807.11126) we studied the analogous limit of shift of argument subalgebras for the Lie algebras $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $\mathfrak{g}=\mathfrak{o}_{2n+1}$. We described the limit subalgebras in terms of Bethe subalgebras of twisted Yangians $Y^-(2)$ and $Y^+(2)$, respectively, and parametrized the eigenbases of these limit subalgebras in the finite dimensional irreducible highest weight representations by Gelfand-Tsetlin patterns of types C and B. In this note we state and prove similar results for the last case of classical Lie algebras, $\mathfrak{g}=\mathfrak{o}_{2n}$. We describe the limit shift of argument subalgebra in terms of the Bethe subalgebra in the twisted Yangian $Y^+(2)$ and give a natural indexing of its eigenbasis in any finite dimensional irreducible highest weight $\mathfrak{g}$-module by type D Gelfand-Tsetlin patterns.