Invariants in divided power algebras

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 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 ≤ n of the centre of the hyperalgebra of the Frobenius kernel G s .