Gauge-natural field theories and Noether Theorems: canonical covariant conserved currents

Recently we found that canonical gauge-natural superpotentials are obtained as global sections of the {\em reduced} $(n-2)$-degree and $(2s-1)$-order quotient sheaf on the fibered manifold $\bY_{\zet} \times_{\bX} \mathfrak{K}$, where $\mathfrak{K}$ is an appropriate subbundle of the vector bundle of (prolongations of) infinitesimal right-invariant automorphisms $\bar{\Xi}$. In this paper, we provide an alternative proof of the fact that the naturality property $\cL_{j_{s}\bar{\Xi}_{H}}\omega (\lambda, \mathfrak{K})=0$ holds true for the {\em new} Lagrangian $\omega (\lambda, \mathfrak{K})$ obtained contracting the Euler--Lagrange form of the original Lagrangian with $\bar{\Xi}_{V}\in \mathfrak{K}$. We use as fundamental tools an invariant decomposition formula of vertical morphisms due to Kol\'a\v{r} and the theory of iterated Lie derivatives of sections of fibered bundles. As a consequence, we recover the existence of a canonical generalized energy--momentum conserved tensor density associated with $\omega (\lambda, \mathfrak{K})$.