Beilinson–Bernstein Localization and Category O: Examples and Counterexamples

For $\alpha$ the unique positive root, $⟨\kappa ,{\alpha }^{\vee }⟩=c>0$, so $\kappa$ is regular and non-antidominant. By [F1], $M=\mathcal{O}(-c-1)$ is a left ${\mathcal{D}}_{\kappa }$-module, and it is nonzero because it is locally free of rank one.

With $z=y\mathrm{/}x$, $w=x\mathrm{/}y$ and transition $e_1=z^d{e}_0$ for $d=-c-1$, the Cech differential has image spanned by nonnegative powers $z^n$ and powers $z^{d-n}$ for $n\ge 0$. The first summand gives every exponent $n\ge 0$, the second every exponent $n\le -c-1$; the quotient has basis $z^{-c},z^{-c+1},\dots ,z^{-1}$, of size $c$. No exponent belongs to the intersection of those two ranges, so $H^0=0$. Thus $H^1(X,M)\cong {\mathbb{C}}^c$ and $\mathrm{\Gamma }(X,M)=0$.

Since $M$ is nonzero and quasi-coherent but $\mathrm{\Gamma }(M)=0$, the map $0\to M$ proves failure of conservativity. For the standard chart inclusions $j_0,j_1,j_{01}$, put $E=j_{0\ast }(M{\mathrm{\mid }}_{U_0})\oplus j_{1\ast }(M{\mathrm{\mid }}_{U_1})$ and $F=j_{01\ast }(M{\mathrm{\mid }}_{U_0\cap U_1})$. Restriction and difference give a ${\mathcal{D}}_{\kappa }$-linear exact sequence $0\to M\to E\to F\to 0$: differential operators act on direct images by restriction, the kernel consists of sections that glue, and on $U_i$ the other direct-image summand maps isomorphically onto $F{\mathrm{\mid }}_{U_i}$. These sheaves are quasi-coherent by [F4]: on $U_0=\mathrm{Spec}\mathbb{C}[z]$ their distinguished-open sections are the localizations of $\mathbb{C}[z]e_0$, $(\mathbb{C}[z]\oplus \mathbb{C}[z,z^{-1}])e_0$ and $\mathbb{C}[z,z^{-1}]e_0$, respectively; the same calculation holds with $w,e_1$ on $U_1$. The map $\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(F)$ is exactly the computed Cech differential with nonzero cokernel. Thus global sections is not exact on quasi-coherent ${\mathcal{D}}_{\kappa }$-modules.

The weight of the unique positive root is $2\rho$, so $\rho (h)=1$. With $d=-c-1$, the associated line is ${\mathcal{O}}_X((c+1)\rho )={\mathcal{O}}_{{\mathbb{P}}^1}(d)$ by [F2]. On $U_0$, the inverse upper-unipotent matrix sends $(1,z)$ to $(1-tz,z)$; therefore its action on a local coefficient $g(z)e_0$ is $(1-tz{)}^{d}g(z\mathrm{/}(1-tz))e_0$. Differentiating at $t=0$ gives $z^2{\mathrm{\partial }}_z-dz=z^2{\mathrm{\partial }}_z+(c+1)z$ for $e$. The inverse lower unipotent sends $(1,z)$ to $(1,z-t)$, giving $-{\mathrm{\partial }}_z$ for $f$. The inverse diagonal matrix sends $(1,z)$ to $(e^{-t},e^{t}z)$; the homogeneous frame contributes $e^{-dt}$, so differentiation gives $2z{\mathrm{\partial }}_z-d=2z{\mathrm{\partial }}_z+(c+1)$ for $h$.

On $U_0\cap U_1$, the frames satisfy $e_1=z^d{e}_0$, so a coefficient $g_1(w)$ on $U_1$ corresponds to $g_0(z)=z^d{g}_1(1\mathrm{/}z)$ on $U_0$. If $T{g}_1=z^d{g}_1(1\mathrm{/}z)$, direct differentiation gives $T^{-1}{\mathrm{\partial }}_{z}T=dw-w^2{\mathrm{\partial }}_w\mathrm{.}$ Conjugating the three operators from step 1.1 by $T$ and using $d=-(c+1)$ gives, respectively, $-{\mathrm{\partial }}_w$, $w^2{\mathrm{\partial }}_w+(c+1)w$ and $-2w{\mathrm{\partial }}_w-(c+1)$. Thus the two formulas glue. Since they are obtained by differentiating the algebraic $G$-action, their commutators are the brackets in [F1]; direct substitution gives the same relations.

At $c=-1$ one has $d=0$, so the line bundle is the trivial bundle and [F2] gives ${\mathcal{D}}_{-\rho }=\mathrm{Diff}({\mathcal{O}}_X)$, as claimed.

In type $A_1$, $\rho$ is the fundamental weight and ${\mathcal{O}}_X(\rho )=\mathcal{O}(-1)$. Applying [F1] with $\kappa =0$ and $\eta =\rho$ sends $M=\mathcal{O}(-1)$ to $M\otimes {\mathcal{O}}_X(\rho )=\mathcal{O}(-2)$ at the new twist $\kappa +\eta =\rho$.

The Harish–Chandra polynomial from [F2] is $p(t)=t+\frac{1}{2}{t}^2$ in the $h$-coordinate. At $\kappa =0$, the evaluation point is $\kappa -\rho =-1$, so $p(-1)=-\frac{1}{2}$. At ${\kappa }^{\mathrm{\prime }}=\rho$, the evaluation point is $0$, so $p(0)=0$. Thus the central characters differ; also $⟨\rho ,{\alpha }^{\vee }⟩=1$, so the target parameter is non-antidominant.

Put $z=y\mathrm{/}x$ on $U_0$ and $w=x\mathrm{/}y$ on $U_1$. For $d=-2$, the Cech complex has $C^0=\mathbb{C}[z]e_0\oplus \mathbb{C}[w]e_1$ and $C^1=\mathbb{C}[z,z^{-1}]e_0$, with transition $e_1=z^{-2}{e}_0$. The image contains powers $z^n$ for $n\ge 0$ and $z^{-2-n}$ for $n\ge 0$, leaving the single quotient basis $z^{-1}$. Hence $H^0(\mathcal{O}(-2))=0$ and $\dim H^1(\mathcal{O}(-2))=1$ by [F3]. The nonzero identity of $M=\mathcal{O}(-2)$ is killed by global sections, so that functor is not faithful. For the standard chart inclusions $j_0,j_1,j_{01}$, put $E=j_{0\ast }(M{\mathrm{\mid }}_{U_0})\oplus j_{1\ast }(M{\mathrm{\mid }}_{U_1})$ and $F=j_{01\ast }(M{\mathrm{\mid }}_{U_0\cap U_1})$. Restriction and difference give a ${\mathcal{D}}_{\rho }$-linear exact sequence $0\to M\to E\to F\to 0$: differential operators act on direct images by restriction, the kernel consists of sections that glue, and on $U_i$ the other direct-image summand maps isomorphically onto $F{\mathrm{\mid }}_{U_i}$. These sheaves are quasi-coherent by [F5]: on $U_0=\mathrm{Spec}\mathbb{C}[z]$ their distinguished-open sections are the localizations of $\mathbb{C}[z]e_0$, $(\mathbb{C}[z]\oplus \mathbb{C}[z,z^{-1}])e_0$ and $\mathbb{C}[z,z^{-1}]e_0$, respectively; the same calculation holds with $w,e_1$ on $U_1$. The map $\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(F)$ is exactly the computed Cech differential with nonzero cokernel. Thus global sections is not exact on quasi-coherent ${\mathcal{D}}_{\rho }$-modules. This computation makes no claim that line-bundle translation preserves global-section faithfulness.

On the affine chart $U_0$, the module supported at $z=0$ is the Weyl algebra quotient $\mathbb{C}[z]⟨{\mathrm{\partial }}_z⟩\mathrm{/}\mathbb{C}[z]⟨{\mathrm{\partial }}_z⟩z$. Ordered monomials ${\mathrm{\partial }}_z^n{z}^m$ show that its global sections have basis $\delta ,{\mathrm{\partial }}_z\delta ,{\mathrm{\partial }}_z^2\delta ,\dots$; sections vanish off the support point. The relation $z\delta =0$ gives $z{\mathrm{\partial }}_z\delta =-\delta$. Therefore [F1] gives $e\delta =0,h\delta =(c+1-2)\delta =(c-1)\delta ,f\delta =-{\mathrm{\partial }}_z\delta \mathrm{.}$

The universal property in [F2] gives a surjection from the Verma module of highest weight $(c-1)\rho$ to $\mathrm{\Gamma }(X,{\delta }_{x_0})$, sending its highest vector to $\delta$. Its PBW basis is $f^{n}v$ for $n\ge 0$, and $f^n\delta =(-1{)}^n{\mathrm{\partial }}_z^n\delta$ is the basis in step 1.1. The map is therefore an isomorphism.

Since $e\delta =0$ and $h\delta =(c-1)\delta$, the Casimir acts by $ef\delta +fe\delta +\frac{1}{2}{h}^2\delta =((c-1)+\frac{1}{2}(c-1{)}^2)\delta =\frac{1}{2}(c^2-1)\delta \mathrm{.}$ In the chosen PBW order, $ef=fe+h$ and the Harish–Chandra projection of $C$ is $h+\frac{1}{2}{h}^2$. Evaluating it at $\kappa -\rho =(c-1)\rho$ gives the same scalar by [F4]. Thus the module has the exact central character stated.

The twist parameter is $\kappa =0$, so $⟨\kappa ,{\alpha }^{\vee }⟩=0$: it is antidominant and singular under the definitions in Flag-variety twisted differential operators and their modules. The line bundle $\mathcal{O}(-1)$ is nonzero and is a ${\mathcal{D}}_0$-module by [F1].

Let $z=y\mathrm{/}x$ on $U_0$ and $w=x\mathrm{/}y$ on $U_1$, with frames related by $e_1=z^{-1}{e}_0$. The Cech complex has $C^0=\mathbb{C}[z]e_0\oplus \mathbb{C}[w]e_1,C^1=\mathbb{C}[z,z^{-1}]e_0,$ and differential $(f_0,f_1)\mapsto z^{-1}{f}_1(1\mathrm{/}z)-f_0(z)$. A global section would identify a polynomial in nonnegative powers of $z$ with a sum of strictly negative powers, so $H^0=0$. In degree one the image contains all nonnegative powers from $f_0$ and all negative powers from $z^{-1}{f}_1(1\mathrm{/}z)$; these span the Laurent polynomials, so $H^1=0$.

By [F3], $p_{\alpha }:X\to G\mathrm{/}{P}_{\alpha }$ is the map to a point. Hence $p_{\alpha \ast }\mathcal{O}(-1)=H^0(X,\mathcal{O}(-1))=0$, while $\mathcal{O}(-1)\mathrm{\ne }0$. The global-sections kernel and the parabolic-pushforward kernel are therefore nonzero, contradicting the refuted assertion. This is the singular Serre-kernel branch, not a counterexample to exactness: the antidominant global-sections functor remains exact by Antidominant twists are D-affine and regular ones are conservative.

By [F1], $V(n\rho )$ is finite-dimensional and irreducible. Its Cartan action is diagonalizable, so [F2] places it in $\mathcal{O}$. Schur's lemma makes every central element act by a scalar; evaluating on a highest vector and using [F3], its central character is ${\chi }_{n\rho +\rho }={\chi }_{(n+1)\rho }$.

In type $A_1$, $s\rho =-\rho$, so $s((n+1)\rho )=-(n+1)\rho =\kappa$. W-invariance in [F3] gives ${\chi }_{\kappa }={\chi }_{(n+1)\rho }$. The pairing $⟨\kappa ,{\alpha }^{\vee }⟩=-(n+1)$ is nonzero and nonpositive, so $\kappa$ is regular antidominant.

The central ideal $\ker {\chi }_{\kappa }$ annihilates $V(n\rho )$ exactly, not merely by a power, because its central character is ${\chi }_{\kappa }$ by step 2.1. Thus it belongs to the exact subcategory in [F4], and that corollary gives the asserted coherent strongly B-equivariant localization. The generalized-character condition permits nilpotent action of the central ideal and is not used in this example.