Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebras

TL;DR

This paper links the Goldie rank of primitive ideals in semisimple Lie algebras to the dimensions of associated W-algebra representations and introduces a new relation involving equivariant Azumaya algebra indexes.

Contribution

It establishes a novel relation between Goldie ranks and representation dimensions via equivariant Azumaya algebra indexes, and computes these indexes in representation-theoretic terms.

Findings

Proved an inequality relating Goldie rank and W-algebra representation dimension.

Computed the index of equivariant Azumaya algebras in representation-theoretic terms.

Established a new connection between primitive ideals and algebraic indexes.

Abstract

Let $\mathfrak{g}$ be a semisimple Lie algebra. We establish a new relation between the Goldie rank of a primitive ideal $\mathcal{J}\subset U(\mathfrak{g})$ and the dimension of the corresponding irreducible representation $V$ of an appropriate finite W-algebra. Namely, we show that $\operatorname{Grk}(\mathcal{J}) \leqslant \dim V/d_V$, where $d_V$ is the index of a suitable equivariant Azumaya algebra on a homogeneous space. We also compute $d_V$ in representation theoretic terms.