  Let $k$ be an algebraically closed field of characteristic $p>0$, let G=GL_n
be the general linear group over $k$, let g=gl_n be its Lie algebra and let
$D_s$ be subalgebra of the divided power algebra of g^* spanned by the divided
power monomials with exponents $<p^s$. We give a basis for the $G$-invariants
in $D_s$ up to degree $n$ and show that these are also the g-invariants.
  We define a certain natural \emph{restriction property} and show that it
doesn't hold when $s>1$. If $s=1$, then $D_1$ is isomorphic to the truncated
coordinate ring of g of dimension p^{dim(g)} and we conjecture that the
restriction property holds and show that this leads to a conjectural spanning
set for the invariants (in all degrees).
  We give similar results for the divided power algebras of several matrices
and of vectors and covectors, and show that in the second case the restriction
property doesn't hold.
  We also give the dimensions of the filtration subspaces of degree $\le n$ of
the centre of the hyper algebra of Dist(G_s).
