SPACES OF QUASI-EXPONENTIALS AND REPRESENTATIONS OF THE YANGIAN $Y\left( {\mathfrak{g}{{\mathfrak{l}}_N}} \right)$

Abstract

We consider a tensor product $V(b)=\otimes_{i=1}^n{{\mathbb{C}}^N}\left( {{b_i}} \right)$ of the Yangian $Y\left( {\mathfrak{g}{{\mathfrak{l}}_N}} \right)$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra ${{\mathcal{B}}^q}\subset Y\left( {\mathfrak{g}{{\mathfrak{l}}_N}} \right)$ on a $\mathfrak{g}{{\mathfrak{l}}_N}$-weight subspace $V{(b)_{\uplambda}}\subset V(b)$ of weight λ. Here the Bethe algebra depends on the parameters q = (q ₁, . . . , q _N ). We identify the ${{\mathcal{B}}^q}$ -module V (b) _λ with the regular representation of the algebra of functions on a fiber of a suitable discrete Wronski map. For q = (1, . . . , 1), we study the action of ${{\mathcal{B}}^q}$ on the space $V(b)_{\lambda}^{\mathrm{sing}}$ of singular vectors of a certain weight and identify the ${{\mathcal{B}}^q}$ -module $V(b)_{\lambda}^{\mathrm{sing}}$ with the regular representation of the algebra of functions on a fiber of another suitable discrete Wronski map.

These results we announced earlier in relation with a description of the quantum equivariant cohomology of the cotangent bundle of a partial flag variety and a description of commutative subalgebras of the group algebra of a symmetric group.