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 is an element of q = Lie(Q) and consider the centraliser q(x) = {y is an element of q: [xy] = 0}. We show that the invariant algebra S(q(x))(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 Zassenhaus' 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 q(x)-modules. An application to the geometry of the Zassenhaus variety is given. When g is of type A and g = t circle plus p is a symmetric decomposition of orthogonal type we use similar methods to show that for every nilpotent e is an element of t the invariant algebra S(p(e))(te) is generated by the pth power subalgebra and S(p(e))(Ke) which is also shown to be polynomial. (C) 2013 Elsevier Inc. All rights reserved.