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Primitive ideals of U(sl2) at a generic central character

Example

Assume the Axiom of Choice. Let $g = {s l}_{2} (C)$ and let $χ$ be a central character which is not the central character of any nonzero finite-dimensional $U (g)$ -module. Then the central reduction $U_{χ}$ is a simple ring, and the unique primitive ideal of $U (g)$ with central character $χ$ is $U (g) \ker χ$ : for every $λ \in h^{*}$ with $χ_{λ} = χ$ the Verma module $M (λ)$ is simple and ${Ann}_{U (g)} M (λ) = U (g) \ker χ,$ so the annihilator is generated by the Casimir relation $Ω - χ (Ω)$ alone. This is the smallest instance of a Duflo annihilator: over a generic central character the primitive ideal is exactly the unavoidable central ideal.

Facts & Assumptions

Given: The Axiom of Choice, $g = {s l}_{2} (C)$ , a central character $χ$ that is not the central character of any nonzero finite-dimensional $U (g)$ -module, and the central reduction $U_{χ} = U (g) / U (g) \ker χ$ .

[F1]

Under AC, $U_{χ}$ is simple if and only if $χ$ is not a finite-dimensional central character; when $U_{χ}$ is simple, every nonzero module with central character $χ$ is faithful over $U_{χ}$ , every Verma module $M (λ)$ with $χ_{λ} = χ$ is simple, and ${Ann}_{U (g)} M (λ) = U (g) \ker χ$ (The central reduction of U(sl2) is simple away from the finite-dimensional central characters, The central reduction of the enveloping algebra at a central character, The Axiom of Choice).

[F2]

For every primitive ideal $I$ with central character $χ$ one has $I \cap Z (U (g)) = \ker χ$ and $U (g) \ker χ \subseteq I$ (A primitive ideal determines a central character, Primitive ideals of an enveloping algebra, Central character of a Lie algebra module).

[F3]

The two-sided ideals of $U_{χ}$ correspond to the two-sided ideals of $U (g)$ containing $U (g) \ker χ$ (The central reduction of U(sl2) is simple away from the finite-dimensional central characters, Left, right and two-sided ideals).

[F4]

Put $Ω = e f + f e + \frac{1}{2} h^{2}$ . Under AC the center of $U (g)$ is $C [Ω]$ , so $\ker χ = (Ω - χ (Ω))$ and $U (g) \ker χ = (Ω - χ (Ω)) U (g)$ ; moreover $Ω$ acts on $M (λ)$ by $λ (λ + 2) / 2$ , so every $λ$ with $χ_{λ} = χ$ satisfies $λ (λ + 2) = 2 χ (Ω)$ , where $λ$ denotes $λ (h)$ . Conversely, algebraic closure supplies a root $λ$ of this quadratic; the Casimir eigenvalue then equals $χ (Ω)$ , and since $Ω$ generates the center, $χ_{λ} = χ$ (The complex numbers are algebraically closed) (The central reduction of U(sl2) is simple away from the finite-dimensional central characters, The quadratic Casimir eigenvalue on a highest-weight module is $(λ, λ + 2 ρ)$ , Verma modules, The special linear Lie algebra sl_2).

Verification

technique · direct

1.1F1given

Since $χ$ is not a finite-dimensional central character, [F1] gives that $U_{χ}$ is a simple ring with no nonzero proper two-sided ideal.

2.1step 1.1F2F3algebra

Let $I$ be a primitive ideal of $U (g)$ with central character $χ$ . By [F2] one has $U (g) \ker χ \subseteq I$ ; the image $\overset{ˉ}{I}$ of $I$ in $U_{χ}$ is therefore a two-sided ideal, hence is $0$ or $U_{χ}$ by step 1.1. If $\overset{ˉ}{I} = U_{χ}$ then $I + U (g) \ker χ = U (g)$ , so $I = U (g)$ by [F2], contradicting that a primitive ideal is proper; therefore $\overset{ˉ}{I} = 0$ , i.e. $I \subseteq U (g) \ker χ$ . With the reverse inclusion from [F2], $I = U (g) \ker χ$ : the central ideal is the only primitive ideal with central character $χ$ .

3.1step 2.1F1F4algebra∎

By [F4] there exists a weight $λ$ with $χ_{λ} = χ$ , so the primitive fibre is nonempty. For any such $λ$ , by [F1] the Verma module $M (λ)$ is simple with ${Ann}_{U (g)} M (λ) = U (g) \ker χ$ , so $U (g) \ker χ$ is primitive (indeed it is the annihilator of a simple module) and is the unique primitive ideal over $χ$ by step 2.1. Since $\ker χ = (Ω - χ (Ω))$ in $Z (U (g)) = C [Ω]$ , the annihilator is the two-sided ideal generated by the single Casimir relation $Ω - χ (Ω)$ , so over a generic central character the primitive ideal is exactly the unavoidable central ideal.

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Sources

P. Etingof, Representations of Lie Groups (18.757, MIT OCW 2023 full notes) (standard reference, not scraped)
J. Gaddis, The Weyl algebra and its friends: a survey (arXiv:2305.01609) (standard reference, not scraped)
R. E. Block, The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra, Adv. Math. 39 (1981) 69-110 (standard reference, not scraped)