Gelfand-Tsetlin degeneration of shift of argument subalgebras in types B and C

TL;DR

This paper explores the limits of shift of argument subalgebras in types B and C Lie algebras, describing their structure via Bethe subalgebras in twisted Yangians and linking eigenbasis indexing to Gelfand-Tsetlin patterns and crystal structures.

Contribution

It explicitly describes the limit subalgebras for types B and C and connects their eigenbasis to Gelfand-Tsetlin patterns and crystal structures, extending known results from type A.

Findings

Limit subalgebras described in terms of Bethe subalgebras in twisted Yangians.

Eigenbasis indexed by Gelfand-Tsetlin patterns of the corresponding type.

Conjecture that the crystal structure matches Littelmann's on Gelfand-Tsetlin patterns.

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 the Lie algebras $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $\mathfrak{g}=\mathfrak{so}_{2n+1}$. 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 arXiv:1708.05105 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 https://doi.org/10.1007/BF01236431 .