  Let G be an infinitesimal group scheme over a field k of positive
characteristic p. We introduce the global p-nilpotent operator $\Theta_G: k[G]
\to k[V(G)]$, where V(G) is the scheme which represents 1-parameter subgroups
of G. This operator applied to M encodes the local Jordan type of M, and leads
to computational insights into the representation theory of G. For certain
G-modules (including those of constant Jordan type), we employ the global
p-nilpotent operator to associate various algebraic vector bundles on the
projective scheme $\bP(G)$, the projectivization of the scheme of one-parameter
subgroups of G. These vector bundles not only distinguish certain
representations with the same local Jordan type, but also provide a method of
constructing algebraic vector bundles on $\bP(G)$.
