The center of the enveloping algebra of the p-Lie algebras n, n, n, when p divides n

Let , be a reductive Lie algebra over an algebraically closed field F with charF=p>0. Suppose G satisfies Jantzen's standard assumptions. Then the structure of Z, the center of the enveloping algebra , is described by (the extended) Veldkamp's theorem. We examine here the deviation of Z from this theorem, in case , or and p|n. It is shown that Veldkamp's description is valid for . This implies that Friedlander-Parshall-Donkin decomposition theorem for holds in case p is good for a semi-simple simply connected G (excluding, if p=2, A1-factors of G). In case or we prove a fiber product theorem for a polynomial extension of Z. However Veldkamp's description mostly fails for and . In particular Z is not Cohen-Macaulay if n>4, in both cases. Contrary to a result of Kac-Weisfeiler, we show for an odd prime p that and do not generate . We also show for that the codimension of the non-Azumaya locus of Z is at least 2 (if n≥3), and exceeds 2 if n>4. This refutes a conjecture of Brown-Goodearl. We then show that Z is factorial (excluding ), thus confirming a conjecture of Premet-Tange. We also verify Humphreys conjecture on the parametrization of blocks, in case p is good for G.