Primitive ideals in the affinoid enveloping algebras of semisimple Lie algebras

TL;DR

This paper proves that in the affinoid enveloping algebra of a semisimple Lie algebra over a discrete valuation ring, primitive ideals with rational central characters are exactly the annihilators of affinoid highest weight modules.

Contribution

It establishes a correspondence between primitive ideals with rational central characters and affinoid highest weight modules in the affinoid enveloping algebra setting.

Findings

Primitive ideals with rational central characters are annihilators of affinoid highest weight modules.

The result extends classical representation theory to the affinoid algebra context.

Provides a structural understanding of primitive ideals in affinoid enveloping algebras.

Abstract

For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions $K$, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, $\widehat{U(\mathfrak{g})_{K}},$ is the annihilator of an affinoid highest weight module.