Invariants of centralisers in positive characteristic

Let Q be a simple algebraic group of type A or C over a field of good positive characteristic. Let x∈q=Lie(Q) and consider the centraliser qx={y∈q:[xy]=0}. We show that the invariant algebra S(qx)qx is generated by the pth power subalgebra and the mod p reduction of the characteristic zero invariant algebra. The latter algebra is known to be polynomial [17] and we show that it remains so after reduction. Using a theory of symmetrisation in positive characteristic we prove the analogue of this result in the enveloping algebra, where the p-centre plays the role of the pth power subalgebra. In Zassenhausʼ foundational work [30], the invariant theory and representation theory of modular Lie algebras were shown to be explicitly intertwined. We exploit his theory to give a precise upper bound for the dimensions of simple qx-modules. An application to the geometry of the Zassenhaus variety is given.When g is of type A and g=k⊕p is a symmetric decomposition of orthogonal type we use similar methods to show that for every nilpotent e∈k the invariant algebra S(pe)ke is generated by the pth power subalgebra and S(pe)Ke which is also shown to be polynomial.