Invariants of Centralisers in Positive Characteristic

TL;DR

This paper investigates the structure of invariant algebras associated with centralisers in simple algebraic Lie algebras over fields of positive characteristic, establishing their generation by p-th powers and polynomiality, with applications to module dimensions.

Contribution

It proves that invariant algebras of centralisers are generated by p-th powers and known invariants, extending polynomiality results to positive characteristic and analyzing module dimensions.

Findings

Invariant algebra S(q_x)^{q_x} is generated by p-th powers and characteristic zero invariants.

The invariant algebra remains polynomial after reduction in positive characteristic.

Provides bounds on dimensions of simple modules associated with centralisers.

Abstract

Let \q be a simple algebraic group of type A or C over a field of good positive characteristic. We show for any x \in \q =\Lie(Q) that the invariant algebra S(\q_x)^{\q_x} is generated by the p^{th} power subalgebra and the mod p reduction of the characteristic zero invariant algebra. The latter algebra is known to be polynomial \cite{PPY} 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 p^{th} power subalgebra. In Zassenhaus' foundational work \cite{Zas}, 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. When \g is of type A and \g = \k \oplus \p is a symmetric decomposition of orthogonal type we use similar methods to show that for every nilpotent e \in \k the invariant algebra S(\p_e)^{\k_e} is generated by the p^{th} power subalgebra and S(\p_e)^{K_e} which is also shown to be polynomial.