Action of the conformal group on steady state solutions to Maxwells equations and background radiation

The representation of the conformal group (PSU(2,2)) on the space of solutions to Maxwells equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Frechet 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.

On the algebraic set of singular elements in a complex simple Lie algebra

Let $G$ be a complex simple Lie group and let $g = hbox{rm Lie},G$. Let $S(g)$ be the $G$-module of polynomial functions on $g$ and let $hbox{rm Sing},g$ be the closed algebraic cone of singular elements in $g$. Let ${cal L}s S(g)$ be the (graded) ideal defining $hbox{rm Sing},g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $g$. Then ${cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $Ms {cal L}^r$ which already defines $hbox{rm Sing},g$. The main results of this paper are a determination of the structure of $M$.