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

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