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The $sp(8, R)$ invariant formulation of free field equations of masslessfields of all spins in $AdS_4$ available previously in terms of gauge invariantfield strengths is extended to gauge potentials. As a by-product, free fieldequations for a massless gauge field are shown to possess both $su(2,2)\simo(4,2)$ and $sl(4,R)\sim o(3,3)$ symmetry. The proposed formulation iswell-defined in the $AdS_4$ background but experiences certain degeneracy inthe flat limit that does not allow conformal invariant field equations for spin$s>1$ gauge fields in Minkowski space. The basis model involves the doubled setof fields of all spins. It is manifestly invariant under U(1) electric-magneticduality extended to higher spins. Reduction to a single massless field containsthe equations that relate its electric and magnetic potentials which are mixedby the conformal transformations for s>1. We use the unfolded formulationapproach recalled in the paper with some emphasis on the role ofChevalley-Eilenberg cohomology of a Lie algebra $g$ in $g$-invariant fieldequations. This method makes it easy to guess a form of the 4d $sp(8, R)$invariant massless field equations and then to extend them to the tendimensional $sp(8,R)$ invariant space-time. Dynamical content of the fieldequations is analyzed in terms of $\sigma_-$ cohomology.

On Conformal, SL(4,R) and Sp(8,R) Symmetries of 4d Massless Fields