Spectral complexes and currents on the Engel group

We study the spectral complexes arising from the de Rham complex on the Engel group $G$ viewed as a truncated multicomplex. We describe the two resulting complexes, analyse their compactly supported counterparts, and define the corresponding spectral currents by duality. We prove that the smooth spectral complexes embed naturally into the current complexes. Finally, we show that the Pansu pullback of a $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,q}$ map $\phi \text{\hspace{0.17em}⁣}:G\to G$ with $q>7$ induces a morphism into the corresponding complexes of spectral currents.