Poles of finite-dimensional representations of Yangians in type A

Sachin Gautam, Curtis Wendlandt

Published 2020-09-14Version 1

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $Y_{\hbar}(\mathfrak{g})$ be the Yangian of $\mathfrak{g}$. In this paper, we initiate the study of the set of poles of the rational currents defining the action of $Y_{\hbar}(\mathfrak{g})$ on an arbitrary finite-dimensional vector space $V$. We prove that this set is completely determined by the eigenvalues of the commuting Cartan currents of $Y_{\hbar}(\mathfrak{g})$, and therefore encodes the singularities of the components of the $q$-character of $V$. In type $\mathsf{A}$, we explicitly determine the set of poles of every irreducible $V$ in terms of the roots of the underlying Drinfeld polynomials. In particular, our results yield a complete classification of the finite-dimensional irreducible representations of the Yangian double in type $\mathsf{A}$.