Spaces of quasi-exponentials and representations of the Yangian Y(gl_N)

Abstract

We consider a tensor product $V(b)= \otimes_{i=1}^n\C^N(b_i)$ of the Yangian $Y(g{l}_N)$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra $B^q\subset Y(g{l}_N)$ on a $g{l}_N$-weight subspace $V(b{)}_{\hat{I}}»\subset V(b)$ of weight $\hat{I}»$. Here the Bethe algebra depends on the parameters $q=(q_1,...,q_N)$. We identify the $B^q$-module $V(b{)}_{\hat{I}}»$ with the regular representation of the algebra of functions on a fiber of a suitable discrete Wronski map. If $q=(1,...,1)$, we study the action of $B^{q=1}$ on a space $V(b{)}_{\hat{I}}^{sing}»$ of singular vectors of a certain weight. Again, we identify the $B^{q=1}$-module $V(b{)}_{\hat{I}}^{sing}»$ 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.

Latex, 23 pages, misprints corrected