  The representation of the conformal group (PSU(2,2)) on the space of
solutions to Maxwell's equations on the conformal compactification of Minkowski
space is shown to break up into four irreducible unitarizable smooth Fr\'echet
representations of moderate growth. An explicit inner product is defined on
each representation. The frequency spectrum of each of these representations is
analyzed. These representations have notable properties; in particular they
have positive or negative energy, they are of type $A_{\frak q}(\lambda)$ and
are quaternionic. Physical implications of the results are explained.
