Highest weights can have the same primitive ideal

Assume the Axiom of Choice (The Axiom of Choice) and let $\mathfrak{g}$ be finite-dimensional complex semisimple. The assignment $\lambda \mapsto I(\lambda )={\mathrm{Ann}}_{U(\mathfrak{g})}L(\lambda )$ from weights to primitive ideals (Annihilators of simple highest-weight modules are primitive, Primitive ideals of an enveloping algebra, The annihilator of a module over an enveloping algebra) is not injective in general, and a fixed central character can carry more than one primitive ideal; describing the fibres of this map is the content of Joseph's theory of Goldie rank polynomials and of Kazhdan-Lusztig cell theory, which is not part of the algebraic prefix on this page. For noninjectivity, already in ${\mathfrak{s}\mathfrak{l}}_2$ the distinct weights $1\mathrm{/}2$ and $-5\mathrm{/}2$ have normalized Casimir value $5\mathrm{/}8$, which is not $n(n+2)\mathrm{/}2$ for any integer $n\ge 0$: those values are $0$ for $n=0$ and at least $3\mathrm{/}2$ for $n\ge 1$. By The central reduction of U(sl2) is simple away from the finite-dimensional central characters, both Verma modules are simple and have the same annihilator, namely the central ideal. In contrast, the trivial central character carries the distinct primitive ideals $\mathrm{Ann}L(0)$ and $\mathrm{Ann}L(-2)$ — the annihilators of the trivial module and of the simple Verma module $M(-2)=L(-2)$ — so Duflo's surjectivity is not a bijection between weights and primitive ideals.

• What is recorded here and what is not. This item records a boundary: the map from weights to primitive ideals has fibres of size greater than one, and their description requires the character-polynomial machinery of Joseph, Barbasch and Vogan. No fibrewise classification is asserted, and the item is not used as a supplier by any proof on this page.
• The ${\mathfrak{s}\mathfrak{l}}_2$ witness. The two annihilators named above are distinct: the trivial module is finite-dimensional with $h$ acting by $0$, while $h$ acts with nonzero eigenvalue $-2$ on the highest vector of $M(-2)$, so the annihilators differ. Simplicity of $M(-2)$ follows from the ${\mathfrak{s}\mathfrak{l}}_2$ irreducibility criterion recorded in The central reduction of U(sl2) is simple away from the finite-dimensional central characters: $-2\mathrm{\notin }{\mathbb{Z}}_{\ge 0}$. This is the same witness that the companion page records in full; the comparison is by central character, since ${\chi }_0={\chi }_{-2}$ on ${\mathfrak{s}\mathfrak{l}}_2$.