Primitive Ideals and Duflo Theorem - Examples

Given: A finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a dominant integral weight $\lambda$, and the finite-dimensional simple module $L(\lambda )$ with $d:={\dim }_{\mathbb{C}}L(\lambda )\ge 1$.

$I(\lambda )$ is the kernel of the unital algebra homomorphism $U(\mathfrak{g})\to {\mathrm{End}}_{\mathbb{C}}(L(\lambda ))$ given by the action, so it is a proper two-sided ideal and $U(\mathfrak{g})\mathrm{/}I(\lambda )$ embeds into the finite-dimensional algebra ${\mathrm{End}}_{\mathbb{C}}(L(\lambda ))$ (The annihilator of a module over an enveloping algebra, Finite-dimensional simple modules are classified by dominant highest weights).

Cayley-Hamilton: for every endomorphism $T$ of the finite-dimensional space $L(\lambda )$, the characteristic polynomial ${\chi }_T$ satisfies ${\chi }_T(T)=0$; it is monic of degree $d$ (Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, ${\chi }_T(T)=0$, Polynomial evaluation at an endomorphism: $p(T)={\sum }_k{a}_k{T}^k$).

The symbol of an element $p(y)=y^d+\text{(terms of degree }<d)$ of the PBW filtration is $y^d\in S^d(\mathfrak{g})$; symbols multiply, and $\mathcal{V}(I)$ is the zero locus of $\mathrm{gr}I$ in ${\mathfrak{g}}^{\ast }$ with ${\mathfrak{g}}^{\ast }$ in duality with the degree-one symbols (The associated graded variety of a two-sided ideal, The PBW filtration by tensor degree on the enveloping algebra, PBW gives an ordered monomial basis for the enveloping algebra, Classical affine zero loci form the Zariski closed sets).

Given: $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with Casimir $\mathrm{\Omega }$, the finite-dimensional simple modules $L(0)=\mathbb{C}$ and $L(1)$ (the standard two-dimensional module), and the ideals $I_0={\mathrm{Ann}}_{U(\mathfrak{g})}L(0)$, $I_1={\mathrm{Ann}}_{U(\mathfrak{g})}L(1)$.

Annihilators of the simple modules $L(0)$ and $L(1)$ are 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).

A primitive ideal $I$ satisfies $I\cap Z(U(\mathfrak{g}))=\ker {\chi }_I$, and this intersection is a maximal ideal of $Z(U(\mathfrak{g}))$ (A primitive ideal determines a central character, Prime ideals and maximal ideals in a commutative ring).

Put $\mathrm{\Omega }:=ef+fe+\frac{1}{2}{h}^2\in Z(U(\mathfrak{g}))$; it is central because the relations $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$ give $[h,\mathrm{\Omega }]=[e,\mathrm{\Omega }]=[f,\mathrm{\Omega }]=0$ (The special linear Lie algebra sl_2). The finite-dimensional simple modules are the $L(n)$, $n\ge 0$ (Finite-dimensional simple modules are classified by dominant highest weights), and on the highest-weight vector of $L(n)$ one has $ev=0$, $efv=[e,f]v+f(ev)=hv=nv$, $fev=0$, so $\mathrm{\Omega }v=(n+0+\frac{1}{2}{n}^2)v=\frac{n(n+2)}{2}v$; in particular ${\chi }_0(\mathrm{\Omega })=0$ and ${\chi }_1(\mathrm{\Omega })=\frac{3}{2}$ (Central character of a Lie algebra module).

In a commutative ring, two distinct maximal ideals have non-maximal intersection: if $M=\mathfrak{m}\cap \mathfrak{n}$ with $\mathfrak{m}\mathrm{\ne }\mathfrak{n}$ maximal were maximal, then $\mathfrak{m}\supseteq M$ and maximality of $M$ would force $M=\mathfrak{m}$, so $\mathfrak{m}\subseteq \mathfrak{n}$ and maximality of $\mathfrak{m}$ would force $\mathfrak{m}=\mathfrak{n}$, a contradiction (Prime ideals and maximal ideals in a commutative ring).

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with Casimir $\mathrm{\Omega }$, an integer $n\ge 0$, the finite-dimensional simple module $L(n)$, and the Verma module $M(-n-2)$.

$L(n)$ is a finite-dimensional simple module, and $I(n)={\mathrm{Ann}}_{U(\mathfrak{g})}L(n)$ is primitive; here $\mathrm{\Omega }:=ef+fe+\frac{1}{2}{h}^2$ is the normalized central Casimir, and the central-reduction lemma also gives $Z(U(\mathfrak{g}))=\mathbb{C}[\mathrm{\Omega }]$ and the eigenvalue $\mathrm{\Omega }\mapsto n(n+2)\mathrm{/}2$ on $L(n)$ (Finite-dimensional simple modules are classified by dominant highest weights, Annihilators of simple highest-weight modules are primitive, The central reduction of U(sl2) is simple away from the finite-dimensional central characters).

$M(-n-2)$ is simple — its highest weight is antidominant, $⟨\lambda +\rho ,{\alpha }^{\vee }⟩<0$ for the positive root — so $J(n)={\mathrm{Ann}}_{U(\mathfrak{g})}M(-n-2)$ is primitive; $\mathrm{\Omega }$ acts on $M(-n-2)$ by $(-n-2)(-n)\mathrm{/}2=n(n+2)\mathrm{/}2$, and every weight space of $M(-n-2)$ is finite-dimensional with weights $(-n-2)-2k$, $k\in {\mathbb{Z}}_{\ge 0}$, so $M(-n-2)$ is infinite-dimensional (Antidominant regular Verma modules are simple, Verma modules, The central reduction of U(sl2) is simple away from the finite-dimensional central characters, Weights of a Verma module lie below lambda, The Axiom of Choice).

For ${\mathfrak{s}\mathfrak{l}}_2$, equal values of the character on $\mathrm{\Omega }$ give equal central characters: ${\chi }_n={\chi }_{-n-2}$ by [F1], [F2] and Central characters are dot-Weyl orbits, because $0$ and $-2$ (and generally $n$ and $-n-2$) lie in the same dot orbit. [F1, F2]

If $K$ is the annihilator of a module $M$, then $U(\mathfrak{g})\mathrm{/}K$ embeds into ${\mathrm{End}}_{\mathbb{C}}(M)$ and acts faithfully on $M$; the image of a finite-dimensional vector space under a linear map is finite-dimensional, since the images of a finite basis span the image (The annihilator of a module over an enveloping algebra, Primitive ideals of an enveloping algebra).

If the normalized Casimir value $c$ is not $m(m+2)\mathrm{/}2$ for any integer $m\ge 0$, every Verma module with that value is simple and its annihilator is the central ideal (The central reduction of U(sl2) is simple away from the finite-dimensional central characters). A simple Verma module equals its unique simple quotient $L(\lambda )$ (Annihilators of simple highest-weight modules are primitive).

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with basis $e,f,h$, the trivial module $\mathbb{C}$, and the ideal ${\mathrm{Ann}}_{U(\mathfrak{g})}\mathbb{C}$ of operators acting by $0$ on $\mathbb{C}$.

The trivial module is the unital module on which $e,f,h$ act by $0$; the corresponding algebra homomorphism is the augmentation $\epsilon \text{\hspace{0.17em}⁣}:U(\mathfrak{g})\to \mathbb{C}$ with $\epsilon (\mathfrak{g})=0$, so its kernel is the two-sided ideal generated by $\mathfrak{g}$: after killing $e,f,h$, every nonempty tensor word is zero and the quotient is spanned by $1$; $\epsilon (1)=1$ makes the remaining scalar quotient exactly $\mathbb{C}$ (The annihilator of a module over an enveloping algebra, The special linear Lie algebra sl_2).

Under AC the center is $\mathbb{C}[\mathrm{\Omega }]$, so $\ker {\chi }_0=(\mathrm{\Omega })$ and $U(\mathfrak{g})\ker {\chi }_0=(\mathrm{\Omega })U(\mathfrak{g})$; the trivial module is simple and nonzero, so its annihilator is primitive and the center acts on it through the central character ${\chi }_0$ with ${\chi }_0(\mathrm{\Omega })=0$, the Casimir eigenvalue of the trivial module (The central reduction of U(sl2) is simple away from the finite-dimensional central characters, The center of the enveloping algebra is polynomial on rank-many generators, The Axiom of Choice, Primitive ideals of an enveloping algebra, A primitive ideal determines a central character, Central character of a Lie algebra module, The quadratic Casimir eigenvalue on a highest-weight module is $(\lambda ,\lambda +2\rho )$).

$M(-2)$ is a simple Verma module by the Verma irreducibility criterion, since $-2+1=-1\mathrm{\notin }{\mathbb{Z}}_{>0}$; its highest vector $v_{-2}$ satisfies $h{v}_{-2}=-2v_{-2}$, and its central character is ${\chi }_{-2}$, which equals ${\chi }_0$ because the normalized Casimir $\mathrm{\Omega }=ef+fe+\frac{1}{2}{h}^2$ acts on both by $0$, by the eigenvalue formula $\mathrm{\Omega }\mapsto \lambda (\lambda +2)\mathrm{/}2$ and $-2\cdot 0\mathrm{/}2=0$ (Verma modules, The Verma irreducibility criterion from Shapovalov determinants, Highest-weight vectors and cyclic highest-weight modules, The central reduction of U(sl2) is simple away from the finite-dimensional central characters).

$U(\mathfrak{g})\ker {\chi }_{\lambda }\subseteq {\mathrm{Ann}}_{U(\mathfrak{g})}M(\lambda )$ for every $\lambda$ (The Verma annihilator contains the central-character ideal).

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$, a central character $\chi$ that is not the central character of any nonzero finite-dimensional $U(\mathfrak{g})$-module, and the central reduction $U_{\chi }=U(\mathfrak{g})\mathrm{/}U(\mathfrak{g})\ker \chi$.

Under AC, $U_{\chi }$ is simple if and only if $\chi$ is not a finite-dimensional central character; when $U_{\chi }$ is simple, every nonzero module with central character $\chi$ is faithful over $U_{\chi }$, every Verma module $M(\lambda )$ with ${\chi }_{\lambda }=\chi$ is simple, and ${\mathrm{Ann}}_{U(\mathfrak{g})}M(\lambda )=U(\mathfrak{g})\ker \chi$ (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).

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

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

Put $\mathrm{\Omega }=ef+fe+\frac{1}{2}{h}^2$. Under AC the center of $U(\mathfrak{g})$ is $\mathbb{C}[\mathrm{\Omega }]$, so $\ker \chi =(\mathrm{\Omega }-\chi (\mathrm{\Omega }))$ and $U(\mathfrak{g})\ker \chi =(\mathrm{\Omega }-\chi (\mathrm{\Omega }))U(\mathfrak{g})$; moreover $\mathrm{\Omega }$ acts on $M(\lambda )$ by $\lambda (\lambda +2)\mathrm{/}2$, so every $\lambda$ with ${\chi }_{\lambda }=\chi$ satisfies $\lambda (\lambda +2)=2\chi (\mathrm{\Omega })$, where $\lambda$ denotes $\lambda (h)$. Conversely, algebraic closure supplies a root $\lambda$ of this quadratic; the Casimir eigenvalue then equals $\chi (\mathrm{\Omega })$, and since $\mathrm{\Omega }$ generates the center, ${\chi }_{\lambda }=\chi$ (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 $(\lambda ,\lambda +2\rho )$, Verma modules, The special linear Lie algebra sl_2).