Harish-Chandra invariants and the centre of the reduced enveloping algebra

TL;DR

This paper characterizes when the Harish-Chandra map is surjective onto the centre of reduced enveloping algebras for reductive Lie algebras in good characteristic, linking regularity of p-characters to surjectivity.

Contribution

It provides a precise criterion for the surjectivity of the Harish-Chandra map based on the regularity of p-characters, completing a converse to a known theorem.

Findings

The Harish-Chandra map is surjective if and only if the p-character is regular.

The result applies to reductive algebraic groups in very good characteristic p > 2.

It extends the understanding of the centre of reduced enveloping algebras in modular representation theory.

Abstract

In this article we consider the centre of the reduced enveloping algebra of the Lie algebra of a reductive algebraic group in very good characteristic p > 2. The Harish-Chandra centre maps to the centre of each reduced enveloping algebra and, using a combination of induction and deformation arguments, we describe precisely for which p-characters this map is surjective: it is if and only if the chosen character is regular. This provides the converse to a theorem of Mirkovi\'{c} and Rumynin.