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

TL;DR

This paper explores the connection between the KdV hierarchy and the Drinfeld-Sokolov reduction with spectral parameter, revealing a natural mapping to loop groups and analyzing the Feigin-Frenkel action on conserved densities.

Contribution

It introduces a spectral parameter version of the Drinfeld-Sokolov reduction and computes the Feigin-Frenkel action on conserved densities in the $sl_2$ case.

Findings

Mapping from KdV phase space to loop groups via spectral parameter

Explicit computation of Feigin-Frenkel action on conserved densities

Connection between conserved densities and affine nilpotent group actions

Abstract

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.