Nilpotent action on the KdV variables and 2-dimensional Drinfeld–Sokolov reduction

We note that a version “with spectral parameter” of the Drinfeld–Sokolov reduction gives a natural mapping from the KdV phase space to the group of loops with values in $\widehat N_{+}/A, \widehat N_{+}$: affine nilpotent and $A$ principal commutative (or anisotropic Cartan) subgroup; this mapping is connected to the conserved densities of the hierarchy. We compute the Feigin–Frenkel action of $\widehat n_{+}$ (defined in terms of screening operators) on the conserved densities, in the $sl_2$ case.