Irreducible \(Y(\mathfrak {gl}_2)\)-modules arising from free modules

We classify a class of modules for the Yangian \(Y(\mathfrak {gl}_2)\) $Y({\mathfrak{gl}}_2)$ , where the element \(e_{21}\in Y(\mathfrak {gl}_2)\) $e_{21}\in Y({\mathfrak{gl}}_2)$ acts regularly. These modules can be realized by use of the differential operators. Moreover, we compute the central characters of these modules by employing the trick of formal generating series.