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

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 = (q1, . . . , qN ). 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.