2.1 The map $\mathcal {U}(\mathfrak {g}) \otimes _{\mathcal {Z}(\mathfrak {g})} \mathcal {U}(\mathfrak {h}) \rightarrow \Gamma (\widetilde {X}, \mathcal {D}_{\widetilde {X}})$

Our strategy for gaining intuition about the $\mathcal {D}_{\widetilde {X}}$ -modules arising in the construction of the Jantzen filtration is to illustrate the $\mathfrak {g}$ -module structure on their global sections. This gives us an algebraic snapshot as to what is happening at each step in the construction sketched in Section 1.3. The first step is to differentiate the actions (2-6) and (2-7) to obtain a map $\mathcal {U}(\mathfrak {g}) \otimes _{\mathcal {Z}(\mathfrak {g})} \mathcal {U}(\mathfrak {h}) \rightarrow \Gamma (\widetilde {X}, \mathcal {D}_{\widetilde {X}})$ . This map provides the $\mathfrak {g}$ -module structure on the global sections of $\mathcal {D}_{\widetilde {X}}$ -modules. We dedicate this section to the computation of this map.

By differentiating the left action of G in (2-6), we obtain an algebra homomorphism

(2-9) $$ \begin{align} L:\mathcal{U}(\mathfrak{g}) \rightarrow \Gamma(\widetilde{X}, \mathcal{D}_{\widetilde{X}}),\quad g \mapsto L_g \end{align} $$

given by the formula

$$ \begin{align*} L_g f(x) = \frac{d}{dt} \bigg|_{t=0} f (\mathrm{exp}(tg)^{-1}x) \end{align*} $$

for $g \in G$ , $f \in \Gamma (\widetilde {X}, \mathcal {O}_{ \widetilde {X}})$ , $x \in \widetilde {X}$ . Computing the image of the basis (2-1) under the homomorphism (2-9) is straightforward. For example, the image of e is given by the following computation using (2-6):

$$ \begin{align*} e \cdot f(x_1, x_2) &= \frac{d}{dt} \bigg|_{t=0} f \left( \begin{pmatrix} 1 & -t \\ 0 & 1 \end{pmatrix} \cdot (x_1, x_2 ) \right) \\ &= \frac{d}{dt} \bigg|_{t=0} f(x_1 - t x_2, x_2) \\ &= - x_2 \partial_1 f(x_1, x_2). \end{align*} $$

Similar computations determine the images of f and h:

(2-10) $$ \begin{align} L_e = -x_2 \partial_1, \quad L_f = -x_1 \partial_2, \quad L_h = -x_1 \partial_1 + x_2 \partial_2. \end{align} $$

It is also useful to compute the image of the Casimir element (2-2) under the homomorphism L:

(2-11) $$ \begin{align} L_\Omega = x_1^2 \partial_1^2 + 3 x_1 \partial_1 + 3 x_2 \partial_2 + x_2^2 \partial_2^2 + 2x_1 x_2 \partial_1 \partial_2. \end{align} $$

Similarly, the right action of H determines an algebra homomorphism

(2-12) $$ \begin{align} R: \mathcal{U}(\mathfrak{h}) \rightarrow \Gamma(\widetilde{X}, \mathcal{D}_{\widetilde{X}}), \quad g \mapsto R_g. \end{align} $$

Under this homomorphism, h is sent to the Euler operator

(2-13) $$ \begin{align} R_h = x_1 \partial_1 + x_2 \partial_2. \end{align} $$

Combining the homomorphisms L in (2-9) and R in (2-12), we obtain an algebra homomorphism

(2-14) $$ \begin{align} \mathcal{U}(\mathfrak{g}) \otimes_{\mathbb{C}} \mathcal{U}(\mathfrak{h}) \rightarrow \Gamma(\widetilde{X},\mathcal{D}_{\widetilde{X}}); \quad g \otimes g' \mapsto L_g R_{g'}. \end{align} $$

Lemma 2.2. The homomorphism (2-14) factors through the quotient

(2-15) $$ \begin{align} \widetilde{\mathcal{U}}:= \mathcal{U}(\mathfrak{g}) \otimes _{\mathcal{Z}(\mathfrak{g})} \mathcal{U}(\mathfrak{h}), \end{align} $$

where $\mathcal {Z}(\mathfrak {g})$ acts on $\mathcal {U}(\mathfrak {h})$ via the Harish-Chandra projection $\gamma _{\mathrm {HC}}$ in (2-3).

Proof. Direct computation shows that the images of $\Omega \otimes 1$ and $1 \otimes \gamma _{\mathrm {HC}}(\Omega )$ agree. Indeed,

$$ \begin{align*} 1 \otimes \gamma_{\mathrm{HC}}(\Omega) = 1 \otimes (h^2 + 2h) \mapsto &R_h^2 + 2 R_h \\ &=(x_1 \partial_1 + x_2 \partial_2)^2 + 2(x_1 \partial_1 + x_2 \partial_2) \\ &=x_1^2 \partial _1^2 + 3 x_1 \partial_1 + 2 x_1 x_2 \partial_1 \partial_2 + x_2^2 \partial_x^2 + 3 x_2 \partial_x \\ &= L_\Omega.\\[-2.8pc] \end{align*} $$

We refer to the algebra $\widetilde {\mathcal {U}}$ as the extended universal enveloping algebra. By Lemma 2.2, we have an algebra homomorphism

(2-16) $$ \begin{align} \alpha: \widetilde{\mathcal{U}} \rightarrow \Gamma(\widetilde{X}, \mathcal{D}_{\widetilde{X}}); \quad g \otimes g' \mapsto L_g R_{g'}. \end{align} $$

Global sections of $\mathcal {D}_{\widetilde {X}}$ -modules have the structure of $\widetilde {\mathcal {U}}$ -modules via $\alpha .$

2.2 Monodromic $\mathcal {D}_{X}$ -modules

The $\mathcal {D}$ -modules that play a role in our story have an additional structure: they are ‘H-monodromic’. It is necessary for our purposes to work with H-monodromic $\mathcal {D}$ -modules on base affine space instead of $\mathcal {D}$ -modules on the flag variety. This is due to the fact that the $\mathfrak {g}$ -modules in the construction of the Jantzen filtration have generalized infinitesimal characters, so they do not arise as global sections of modules over twisted sheaves of differential operators on the flag variety.

The machinery of H-monodromic $\mathcal {D}$ -modules is rather technical, and the details of the construction are not strictly necessary for our computation of the Jantzen filtration below. However, we thought that it would be useful to describe this construction in a specific example to illustrate that the equivalences established in [Reference Beilinson, Bernstein and Gel’fandBB93, Section 2.5] are quite clear for $\mathfrak {sl}_2(\mathbb {C})$ . In this section, we describe the construction of H-monodromic $\mathcal {D}$ -modules for $\mathfrak {sl}_2(\mathbb {\mathbb {C}})$ and explain how it relates to representations of Lie algebras. More details on the general construction can be found in [Reference Beilinson, Bernstein and Gel’fandBB93, Reference Beilinson and GinzburgBG99].

A weakly H-equivariant $\mathcal {D}_{\widetilde {X}}$ -module is an H-equivariant sheaf $\mathcal {V}$ equipped with a $\mathcal {D}_{\widetilde {X}}$ -module structure so that the isomorphism $\mathrm {act}^*\mathcal {V} \rightarrow p^* \mathcal {V}$ given by the equivariant sheaf structure on $\mathcal {V}$ is a morphism of $\mathcal {D}_{\widetilde {X}} \boxtimes \mathcal {O}_H$ -modules. Here, $\mathrm {act}:\widetilde {X} \times H \rightarrow \widetilde {X}$ is the action map and $p: \widetilde {X} \times H \rightarrow \widetilde {X}$ is the projection map. For a reference on weakly equivariant $\mathcal {D}$ -modules, see [Reference Miličić, Pandžić, Tirao, Vogan and WolfMP98, Section 4].

There is an equivalent characterization of H-monodromic $\mathcal {D}_X$ -modules in terms of H-invariant differential operators, which is established in [Reference Beilinson, Bernstein and Gel’fandBB93, Section 2.5.2]. This perspective makes the structures of our examples more transparent, so we take this approach to monodromicity. Below we describe the construction for $\mathfrak {g}=\mathfrak {sl}_2(\mathbb {C})$ .

The right H-action in (2-7) induces a left H-action on $\mathcal {O}_{\widetilde {X}}$ and $\mathcal {D}_{\widetilde {X}}$ . The H-action on $\mathcal {D}_{\widetilde {X}}$ satisfies the following relation: for $g \in H$ , $\theta \in \mathcal {D}_{\widetilde {X}}$ , and $f \in \mathcal {O}_{\widetilde {X}}$ ,

(2-17) $$ \begin{align} (g \cdot \theta) (g \cdot f) = g \cdot (\theta(f)). \end{align} $$

The H-action on $\mathcal {D}_{\widetilde {X}}$ induces an H-action on the sheaf $\pi _*(\mathcal {D}_{\widetilde {X}})$ by algebra automorphisms, where $\pi :\widetilde {X}\rightarrow X$ is the quotient map (2-8). Here, $\pi _*$ is the $\mathcal {O}$ -module direct image. Denote the sheaf of H-invariant sections of $\pi _*\mathcal {D}_{\widetilde {X}}$ by

(2-18) $$ \begin{align} \widetilde{\mathcal{D}}: = [\pi_*\mathcal{D}_{\widetilde{X}}]^H. \end{align} $$

This is a sheaf of algebras on X. Explicitly, on an open set $U \subseteq X$ ,

(2-19) $$ \begin{align} \widetilde{\mathcal{D}}(U) = \mathcal{D}_{\widetilde{X}}(\pi^{-1}(U))^H. \end{align} $$

Note that because $\pi $ is an H-torsor, $\pi ^{-1}(U)$ is H-stable for any set U, so this construction is well defined.

Let $\mathcal {M}(\mathcal {D}_{\widetilde {X}}, H)_{\mathrm {weak}}$ be the category of weakly H-equivariant $\mathcal {D}_{\widetilde {X}}$ -modules, and $\mathcal {M}(\widetilde {\mathcal {D}})$ be the category of $\widetilde {\mathcal {D}}$ -modules. By [Reference Beilinson, Bernstein and Gel’fandBB93, Sections 1.8.9, 2.5.2], there is an equivalence of categories

(2-20) $$ \begin{align} \mathcal{M}(\mathcal{D}_{\widetilde{X}}, H)_{\mathrm{weak}} \simeq \mathcal{M}(\widetilde{\mathcal{D}}). \end{align} $$

Hence, we can study monodromic $\mathcal {D}_X$ -modules by instead considering $\widetilde {\mathcal {D}}$ -modules. For the remainder of the paper, we take this to be our definition of monodromicity.

Modules over $\widetilde {\mathcal {D}}$ are directly related to modules over the extended universal enveloping algebra (2-15) via the global sections functor. The relationship is as follows. Because the left G-action and right H-action commute, the differential operators $L_e, L_f,$ and $L_h$ in (2-10) are H-invariant. This can also be shown via direct computation using (2-10) and (2-13). Hence, the image of the homomorphism (2-16) is contained in H-invariant differential operators:

$$ \begin{align*} \alpha(\widetilde{\mathcal{U}}) \subseteq \Gamma(\widetilde{X}, \mathcal{D}_{\widetilde{X}})^H. \end{align*} $$

Composing $\alpha $ with $\Gamma (\pi _*)$ , we obtain a map

(2-21) $$ \begin{align} \widetilde{\mathcal{U}} \rightarrow \Gamma(X, \widetilde{\mathcal{D}}). \end{align} $$

Proof. The theorem holds for general $\mathfrak {g}$ . We prove the theorem for $\mathfrak {g}=\mathfrak {sl}_2(\mathbb {C})$ by direct computation.

We start by describing the sheaves $\pi _*\mathcal {D}_{\widetilde {X}}$ and $\widetilde {\mathcal {D}}$ on $X=\mathbb {C} \mathbb {P} ^1$ by describing them on the open patches

$$ \begin{align*} U_1 := \mathbb{C} \mathbb{P}^1 \backslash \{(0:1) \} \quad \text{and} \quad U_2:= \mathbb{C} \mathbb{P}^1 \backslash \{ (1:0)\} \end{align*} $$

and giving gluing conditions. Set

$$ \begin{align*} V_1:= \pi^{-1}(U_1) = \mathbb{C}^2 \backslash V(x_1) \quad \text{and} \quad V_2:= \pi^{-1}(U_2) =\mathbb{C}^2 \backslash V(x_2), \end{align*} $$

where $V(f(x_1, x_2))$ denotes the vanishing of the polynomial $f(x_1, x_2)$ . By definition,

$$ \begin{align*} \pi_*\mathcal{D}_{\widetilde{X}}(U_1) &= \mathcal{D}_{\widetilde{X}}(V_1) = \mathcal{D}(\mathbb{C}^2)[x_1^{-1}], \end{align*} $$

$$ \begin{align*} \pi_*\mathcal{D}_{\widetilde{X}}(U_2) &= \mathcal{D}_{\widetilde{X}}(V_2) = \mathcal{D}(\mathbb{C}^2)[x_2^{-1}], \end{align*} $$

with obvious gluing conditions.

Using (2-17), we conclude that the H-action on $\mathcal {D}_{\widetilde {X}}$ is given by the local formulas

$$ \begin{align*} g \cdot x_i = gx_i \quad \text{and} \quad g \cdot \partial_i = g^{-1} \partial_i, \end{align*} $$

where $g \in H$ is regarded as an element of $\mathbb {C}^\times $ under the identification

$$ \begin{align*} H \simeq \mathbb{C}^\times, \begin{pmatrix} a & 0 \\ 0 & a^{-1} \end{pmatrix} \mapsto a. \end{align*} $$

From this, we obtain a local description of $\widetilde {\mathcal {D}}$ , using (2-19):

$$ \begin{align*} \widetilde{\mathcal{D}}(U_1) &= \langle x_1^{-1}x_2, x_1 \partial_1, x_1 \partial_2, x_2 \partial_1, x_2 \partial_2 \rangle \subseteq \mathcal{D}_{\widetilde{X}}(V_1) \end{align*} $$

$$ \begin{align*} \widetilde{\mathcal{D}}(U_2)&= \langle x_1 x_2^{-1}, x_1 \partial_1, x_1 \partial_2, x_2 \partial_1, x_2 \partial_2 \rangle \subseteq \mathcal{D}_{\widetilde{X}}(V_2). \end{align*} $$

Hence, the global sections are given by

$$ \begin{align*} \Gamma(X, \widetilde{\mathcal{D}}) \simeq \langle x_1 \partial_1, x_1 \partial_2, x_2 \partial_1, x_2 \partial_2 \rangle =\Gamma(\widetilde{X}, \mathcal{D}_{\widetilde{X}})^H \subseteq \Gamma(\widetilde{X}, \mathcal{D}_{\widetilde{X}}). \end{align*} $$

Now, it is clear that

$$ \begin{align*} L_e = -x_2 \partial_1, \quad L_f = -x_1 \partial_2, \quad L_h + R_h = 2x_2 \partial_2 \quad \text{and} \quad L_h-R_h = -2x_1 \partial_1, \end{align*} $$

so the operators $L_e, L_f, L_h$ , and $R_h$ generate $\Gamma (\widetilde {X}, \mathcal {D}_{\widetilde {X}})^H$ . Since $e\otimes 1, f \otimes 1, h \otimes 1$ and $1 \otimes h$ generate $\widetilde {\mathcal {U}}$ , this shows that the map (2-21) is surjective. Direct computations establish that

$$ \begin{align*}[L_e,L_f]=x_2\partial_1x_1\partial_2-x_1\partial_2x_2\partial_1=x_2\partial_2-x_1\partial_1=L_h,\end{align*} $$

$$ \begin{align*}[L_e,L_h]=x_2\partial_1(x_1\partial_1-x_2\partial_2)-(x_1\partial_1-x_2\partial_2)x_2\partial_1=2x_2\partial_1=-2L_e,\end{align*} $$

$$ \begin{align*}[L_f,L_h]=x_1\partial_2(x_1\partial_1-x_2\partial_2)-(x_1\partial_1-x_2\partial_2)x_1\partial_2=-2x_1\partial_2=2L_f,\end{align*} $$

$$ \begin{align*}[L_e,R_h]=[L_f,R_h]=[L_h,R_h]=0.\end{align*} $$

Combining these computations with the fact that $L_e,L_f,L_h$ , and $R_h$ are linearly independent shows that the relations satisfied by $L_e, L_f, L_h,$ and $R_h$ are precisely those satisfied by $e\otimes 1,f\otimes 1,h\otimes 1$ , and $1\otimes h.$ Therefore, the map (2-21) is also injective.

The relationships described in this section can be summarized with the following commuting diagrams.

The composition of the top two arrows and the right-most arrow is the equivalence (2-20). (See [Reference Beilinson, Bernstein and Gel’fandBB93, Sections 1.8.9, 2.5.3] for more details.)

2.3 Verma modules and dual Verma modules

Using the map (2-16) constructed in Section 2.1, we can describe the $\widetilde {\mathcal {U}}$ -module structure on various classes of $\mathcal {D}_{\widetilde {X}}$ -modules. We start by examining the $\mathcal {D}_{\widetilde {X}}$ -modules $j_+\mathcal {O}_U$ and $j_!\mathcal {O}_U$ , where $j:U \hookrightarrow \widetilde {X}$ is inclusion of the open union of N-orbits

(2-22) $$ \begin{align} U := \mathbb{C}^2 \backslash V(x_2). \end{align} $$

Here the $+$ and $!$ indicate the $\mathcal {D}$ -module push-forward functors; see [Reference MiličićMil]. These are the $\mathcal {D}_{\widetilde {X}}$ -modules that are eventually endowed with geometric Jantzen filtrations in Section 2.5. In this section, we describe the $\widetilde {\mathcal {U}}$ -module structure on $\Gamma (\widetilde {X}, j_+\mathcal {O}_U)$ and $\Gamma (\widetilde {X}, j_!\mathcal {O}_U)$ .

Because j is an open embedding, the $\mathcal {D}_{\widetilde {X}}$ -module $j_+\mathcal {O}_U$ is just the sheaf $\mathcal {O}_U$ with $\mathcal {D}_{\widetilde {X}}$ -module structure given by the restriction of $\mathcal {D}_U$ to $\mathcal {D}_{\widetilde {X}} \subseteq \mathcal {D}_U$ . Hence, the global sections of $j_+\mathcal {O}_U$ can be identified with the ring

$$ \begin{align*} \Gamma(\widetilde{X}, j_+\mathcal{O}_U) = \mathbb{C}[x_1, x_2, x_2^{-1}]. \end{align*} $$

Figure 1 Dual Verma modules arise as global sections of $j_+\mathcal {O}_U$ .

The operators $L_e, L_f, L_h$ , and $R_h$ from (2-10) and (2-13) act on monomials $x_1^m x_2^n$ for $m \geq 0$ , $n \in \mathbb {Z}$ by the formulas

(2-23) $$ \begin{align} L_e\cdot x_1^mx_2^n&=-mx_1^{m-1}x_2^{n+1}, \end{align} $$

(2-24) $$\begin{align*}L_f\cdot x_1^mx_2^n&=-nx_1^{m+1}x_2^{n-1}, \end{align*}$$

(2-25) $$ \begin{align} L_h\cdot x_1^mx_2^n&=(n-m)x_1^mx_2^n, \end{align} $$

(2-26) $$ \begin{align} R_h\cdot x_1^mx_2^n&=(m+n)x_1^mx_2^n. \end{align} $$

Using (2-23)–(2-26), we can illustrate the $\widetilde {\mathcal {U}}$ -module structure on $\Gamma (\widetilde {X}, j_+\mathcal {O}_U)$ using nodes and colored arrows. We do this in Figure 1. The monomials $x_1^m x_2^n$ for $m \in \mathbb {Z}_{\geq 0}$ and $n \in \mathbb {Z}$ form a basis for $\Gamma (\widetilde {X}, j_+\mathcal {O}_U)$ . The green (dashed) arrows illustrate the action of the operator $ L_e$ on basis elements, the red (dotted) arrows the action of $L_f$ , and the blue (solid) arrows the action of $L_h$ . If an operator acts by zero, no arrow is included. The $R_h$ -eigenspaces are highlighted in gray, with corresponding eigenvalues listed below.

Next we describe $\Gamma (\widetilde {X}, j_!\mathcal {O}_{\widetilde {X}})$ . This is slightly more involved. By definition,

(2-27) $$ \begin{align} j_!= \mathbb{D}_{\widetilde{X}} \circ j_+ \circ \mathbb{D}_U, \end{align} $$

where $\mathbb {D}$ denotes the holonomic duality functor. Explicitly, for a smooth algebraic variety Y and a holonomic $\mathcal {D}_Y$ -module $\mathcal {V}$ ,

(2-28) $$ \begin{align} \mathbb{D}_Y(\mathcal{V}):= \operatorname{\mathrm{Ext}}_{\mathcal{D}_Y}^{\dim Y}(\mathcal{V}, \mathcal{D}_X). \end{align} $$

This is a well-defined functor from the category of holonomic $\mathcal {D}_Y$ -modules to itself [Reference Hotta, Takeuchi and TanisakiHTT08, Corollary 2.6.8].

The first two steps of the composition in (2-27) are straightforward to compute. The right $\mathcal {D}_U$ -module $\mathbb {D}_U \mathcal {O}_U$ is just the sheaf $\mathcal {O}_U$ , viewed as a right $\mathcal {D}_U$ -module via the natural right action. Then, since j is an open immersion, $j_+ \mathbb {D}_U \mathcal {O}_U$ is the sheaf $\mathcal {O}_U$ with right $\mathcal {D}_{\widetilde {X}}$ -module structure given by restriction to $\mathcal {D}_{\widetilde {X}} \subset \mathcal {D}_U$ .

To apply $\mathbb {D}_{\widetilde {X}}$ to $j_+ \mathbb {D}_U \mathcal {O}_U$ , we must take a projective resolution of $j_+\mathbb {D}_U \mathcal {O}_U$ . First, we make the identification

$$ \begin{align*} j_+ \mathbb{D}_U \mathcal{O}_U \simeq \langle \partial_1, \partial_2 \rangle \mathcal{D}_U \backslash \mathcal{D}_U. \end{align*} $$

We take the following free (hence, projective) resolution of $\langle \partial _1, \partial _2 \rangle \mathcal {D}_U \backslash \mathcal {D}_U$ :

$$ \begin{align*} 0 \leftarrow \langle \partial_1, \partial_2 \rangle \mathcal{D}_U \backslash \mathcal{D}_U \xleftarrow{\epsilon} \mathcal{D}_{\widetilde{X}} \xleftarrow{d_0} \mathcal{D}_{\widetilde{X}} \oplus \mathcal{D}_{\widetilde{X}} \xleftarrow{d_1} \mathcal{D}_{\widetilde{X}} \xleftarrow{d_2} 0, \end{align*} $$

where the maps are defined by

$$ \begin{align*} \epsilon: 1 &\mapsto x_2^{-1}, \\ d_0: (\theta_1, \theta_2) &\mapsto \partial_1 \theta_1 - x_2 \partial_2 \theta_2, \\ d_1: 1 &\mapsto (x_2 \partial_2, \partial_1). \end{align*} $$

Applying $\mathrm {Hom}_{\mathcal {D}_{\widetilde {X}}, r}( - , \mathcal {D}_{\widetilde {X}})$ to this complex, we obtain the complex

(2-29) $$ \begin{align} 0 \rightarrow \mathrm{Hom}_{\mathcal{D}_{\widetilde{X}},r}(\mathcal{D}_{\widetilde{X}}, \mathcal{D}_{\widetilde{X}}) \xrightarrow{d_0^*} \mathrm{Hom}_{\mathcal{D}_{\widetilde{X}},r}(\mathcal{D}_{\widetilde{X}}\oplus \mathcal{D}_{\widetilde{X}}, \mathcal{D}_{\widetilde{X}}) \xrightarrow{d_1^*} \mathrm{Hom}_{\mathcal{D}_{\widetilde{X}},r}(\mathcal{D}_{\widetilde{X}}, \mathcal{D}_{\widetilde{X}}) \xrightarrow{d_2^*} \ 0, \end{align} $$

where $d_i^*$ sends a morphism f to $f \circ d_i$ .

Because the module $j_+\mathbb {D}_U \mathcal {O}_U$ is holonomic, the complex (2-29) only has nonzero cohomology in degree $2$ . This can also be seen by direct computation. By identifying $\mathrm {Hom}_{\mathcal {D}_{\widetilde {X}},r}(\mathcal {D}_{\widetilde {X}}, \mathcal {D}_{\widetilde {X}}) \simeq \mathcal {D}_{\widetilde {X}}$ via $f \mapsto f(1)$ and $\mathrm {Hom}_{\mathcal {D}_{\widetilde {X}},r}(\mathcal {D}_{\widetilde {X}}\oplus \mathcal {D}_{\widetilde {X}}, \mathcal {D}_{\widetilde {X}}) \simeq \mathcal {D}_{\widetilde {X}} \oplus \mathcal {D}_{\widetilde {X}}$ via $ f \mapsto (f(1,0), f(0,1))$ , we see that

$$ \begin{align*} \ker d_2^* \simeq \mathcal{D}_{\widetilde{X}}\quad\text{and}\quad \operatorname{\mbox{im }} d_1^* \simeq \mathcal{D}_{\widetilde{X}} \langle \partial_1, x_2 \partial_2 \rangle. \end{align*} $$

Hence,

$$ \begin{align*} j_!\mathcal{O}_U \simeq \mathcal{D}_{\widetilde{X}} / \mathcal{D}_{\widetilde{X}} \langle \partial_1, x_2 \partial_2 \rangle. \end{align*} $$

Now we can describe the global sections of $j_! \mathcal {O}_U$ and illustrate their $\widetilde {\mathcal {U}}$ -module structure, as we did for $\mathcal {O}_{\widetilde {X}}$ and $j_+\mathcal {O}_U$ . The monomials $x_1^m x_2^n$ and $x_1^m \partial _2^n$ for $m, n \geq 0$ form a basis for $\Gamma (\widetilde {X}, j_! \mathcal {O}_U)$ . The action of $L_e, L_f, L_h$ , and $R_h$ on $x^m_1 x_2^n$ for $m \geq 0$ and $n> 0$ is given by (2-23)–(2-26). The action of $L_e, L_f, L_h$ , and $R_h$ on $x^m_1 \partial _2^n$ for $m \geq 0$ and $n> 0$ is given by

$$ \begin{align*}\quad L_e \cdot x_1^m \partial_2^n &= m(n-1) x_1 ^{m-1} \partial_2^{n-1}, \end{align*} $$

(2-30) $$ \begin{align} L_f \cdot x_1^m \partial_2^n &= -x_1^{m+1} \partial_2^{n+1},\qquad \end{align} $$

(2-31) $$ \begin{align} L_h \cdot x_1^m \partial_2^n &= -(m+n) x_1^m \partial_2^n,\ \ \end{align} $$

(2-32) $$ \begin{align} R_h \cdot x_1^m \partial_2^n &= (m-n) x_1^m \partial_2^n.\quad\kern0.3pt \end{align} $$

The action of $L_e$ on $x_1^m$ is given by (2-23), the action of $L_f$ on $x_1^m$ is given by (2-30), and the actions of $L_h$ and $R_h$ on $x^m$ are given by either (2-25)–(2-26) or (2-31)–(2-32).

We illustrate the $\widetilde {\mathcal {U}}$ -module structure of $\Gamma (\widetilde {X}, j_! \mathcal {O}_U)$ in Figure 2. The colors (line-styles) indicate the same operators as in the earlier example: green (dashed) is $L_e$ , red (dotted) is $L_f$ , blue (solid) is $L_h$ , and $R_h$ -eigenspaces are highlighted in gray, with corresponding eigenvalues listed below.

Figure 2 Verma modules arise as global sections of $j_!\mathcal {O}_U$ .

2.4 The maximal extension $\Xi _\rho \mathcal {O}_U$

To describe the geometric Jantzen filtrations on the $\mathcal {D}_{\widetilde {X}}$ -modules $j_!\mathcal {O}_U$ and $j_+\mathcal {O}_U$ , it is necessary to introduce the maximal extension functor

$$ \begin{align*} \Xi_\rho: \mathcal{M}_{\mathrm{hol}}(\mathcal{D}_U) \rightarrow \mathcal{M}_{\mathrm{hol}}(\mathcal{D}_{\widetilde{X}}). \end{align*} $$

This functor (defined in (2-36) below) extends $j_+$ and $j_!$ (see (2-37)–(2-38)), so it is a natural way to study both modules $j_!\mathcal {O}_U$ and $j_+\mathcal {O}_U$ at once. In this section, we give the construction of $\Xi _\rho $ , then describe the $\widetilde {\mathcal {U}}$ -module structure on $\Gamma (\widetilde {X}, \Xi _\rho \mathcal {O}_U)$ .

To start, we recall the construction of maximal extension for $\mathcal {D}$ -modules, which is a special case of the construction in [Reference BeilinsonBei87], which produces the maximal extension and nearby cycle functors. Let Y be a smooth variety, $f:Y \rightarrow \mathbb {A}^1$ a regular function, and

(2-33)

the corresponding open-closed decomposition of Y. For $n \in \mathbb {N}$ , denote by

(2-34) $$ \begin{align} I^{(n)}:= ( \mathcal{O}_{\mathbb{A}^1 - \{0\}} \otimes \mathbb{C}[s]/s^n ) t^s \end{align} $$

the free rank-1 $\mathcal {O}_{\mathbb {A}^1 - \{0\}} \otimes \mathbb {C}[s]/s^n$ -module generated by the symbol $t^s$ . The action $\partial _t \cdot t^s = st^{-1}t^s$ gives $I^{(n)}$ the structure of a $\mathcal {D}_{\mathbb {A}^1-\{0\}}$ -module. Any $\mathcal {D}_U$ -module $\mathcal {M}_U$ can be deformed using $I^{(n)}$ : set

$$ \begin{align*} f^s\mathcal{M}_U^{(n)}:= f^+I^{(n)} \otimes_{\mathcal{O}_U} \mathcal{M}_U \end{align*} $$

to be the $\mathcal {D}_U$ -module obtained by twisting $\mathcal {M}_U$ by $I^{(n)}$ . Note that $f^s\mathcal {M}_U^{(1)} = \mathcal {M}_U$ , and that both $I^{(n)}$ and $f^s\mathcal {M}_U^{(n)}$ have a natural action by $s \in \mathbb {C}[s]/s^n$ .

Assume that $\mathcal {M}_U$ is holonomic. Denote by

(2-35) $$ \begin{align} \mathrm{can}: j_! f^s \mathcal{M}_U^{(n)} \rightarrow j_+f^s \mathcal{M}_{U}^{(n)} \end{align} $$

the canonical map between the $!$ and $+$ pushforwards, and

$$ \begin{align*} s^1(n): j_! f^s \mathcal{M}_U^{(n)} \rightarrow j_+f^s \mathcal{M}_{U}^{(n)} \end{align*} $$

the composition of $\mathrm {can}$ with multiplication by s. For large enough n, the cokernel of $s^1(n)$ stabilizes; that is, $\operatorname {\mathrm {coker}} s^1(n) = \operatorname {\mathrm {coker}} s^1(n+k)$ for all $k>0$ . For $n\gg 0$ , define the $\mathcal {D}_Y$ -module

(2-36) $$ \begin{align} \Xi_f \mathcal{M}_U := \operatorname{\mathrm{coker}} s^1(n), \end{align} $$

called the maximal extension of $\mathcal {M}_U$ . By construction, this module comes equipped with the nilpotent endomorphism s. The corresponding functor

$$ \begin{align*} \Xi_f: \mathcal{M}_{\mathrm{hol}}(\mathcal{D}_U) \rightarrow \mathcal{M}_{\mathrm{hol}}(\mathcal{D}_Y) \end{align*} $$

is exact [Reference Beilinson, Bernstein and Gel’fandBB93, Lemma 4.2.1(i)]. Moreover, there are canonical short exact sequences [Reference Beilinson, Bernstein and Gel’fandBB93, Lemma 4.2.1(ii)’]

(2-37) $$ \begin{align} &0 \rightarrow j_!\mathcal{M}_U \rightarrow \Xi_f\mathcal{M}_U \rightarrow \operatorname{\mathrm{coker}}(\mathrm{can}) \rightarrow 0 \end{align} $$

(2-38) $$ \begin{align} &0 \rightarrow \operatorname{\mathrm{coker}}(\mathrm{can}) \rightarrow \Xi_f \mathcal{M}_U \rightarrow j_+ \mathcal{M}_U \rightarrow 0 \end{align} $$

with $j_! = \ker (s: \Xi _f \rightarrow \Xi _f)$ and $j_+ = \operatorname {\mathrm {coker}}(s: \Xi _f \rightarrow \Xi _f)$ .

Now, we apply this general construction in the setting of our example. Let $\widetilde {X}$ and U be as above (see (2-5) and (2-22)), and let $f_\rho $ be the function

$$ \begin{align*} f_\rho: \widetilde{X} \rightarrow \mathbb{A}^1; (x_1, x_2) \mapsto x_2. \end{align*} $$

This choice of function corresponds to the deformation direction $\rho \in \mathfrak {h}^*$ ; see Remarks 1.3 and 2.10.

For a variety Y, set

$$ \begin{align*} \mathcal{A}_Y:= \mathcal{D}_Y \otimes \mathbb{C}[s]/s^n. \end{align*} $$

We compute the maximal extension $\Xi _{\rho }\mathcal {O}_U:= \Xi _{f_\rho } \mathcal {O}_U$ of the structure sheaf $\mathcal {O}_U$ using the construction above, then describe the $\widetilde {\mathcal {U}}$ -module structure on its global sections. To clarify the exposition, we list each step as a subsection.

2.4.1 Step 1: Deformation

Let $I^{(n)}$ be as in (2-34). The deformed version of $\mathcal {O}_U$ is

$$ \begin{align*} f^s \mathcal{O}_U^{(n)} = f^+I^{(n)} = \mathcal{O}_U \otimes_{f^{-1}(\mathcal{O}_{\mathbb{A}^1 - \{0\}})} f^{-1}(I^{(n)}). \end{align*} $$

The global sections of $f^s \mathcal {O}_U^{(n)}$ are

(2-39) $$ \begin{align} (\mathbb{C}[x_1, x_2, x_2^{-1}] \otimes \mathbb{C}[s]/s^n )t^s, \end{align} $$

where the differentials $\partial _1,\partial _2 \in \Gamma (\widetilde {X}, \mathcal {D}_{\widetilde {X}})$ act on the generator $t^s$ by

$$ \begin{align*} \partial_1 \cdot t^s = 0 \quad\text{and}\quad \partial_2 \cdot t^s = sx_2^{-1} t^s. \end{align*} $$

Alternatively, we can identify $f^s \mathcal {O}_U^{(n)}$ with a quotient of $\mathcal {A}_U$ :

(2-40) $$ \begin{align} f^s \mathcal{O}_U ^{(n)} = \mathcal{A}_U / \mathcal{A}_U \langle \partial_1, x_2 \partial_2 - s\rangle. \end{align} $$

Both descriptions are useful below.

2.4.2 Step 2: $+$ -pushforward

Because $j: U \hookrightarrow \widetilde {X}$ is an open embedding, the $\mathcal {D}_{\widetilde {X}}$ -module $j_+ f^s \mathcal {O}_U^{(n)}$ is the sheaf $f^s \mathcal {O}_U^{(n)}$ with $\mathcal {D}_{\widetilde {X}}$ -module structure given by restriction to $\mathcal {D}_{\widetilde {X}} \subset \mathcal {D}_U$ . Under the identification (2-40),

$$ \begin{align*} j_+ f^s \mathcal{O}_U ^{(n)} = \mathcal{A}_U / \mathcal{A}_U \langle \partial_1, x_2 \partial_2 - s \rangle, \end{align*} $$

with $\mathcal {D}_{\widetilde {X}}$ -action given by left multiplication.

It is interesting to examine the $\widetilde {\mathcal {U}}$ -module structure on the global sections of this module. The operators $L_e, L_f, L_h$ , and $R_h$ in (2-10), (2-13) act on the monomial basis elements of (2-39) by the following formulas:

(2-41) $$ \begin{align} L_e \cdot x_1^k x_2^\ell s^m t^s &= -k x_1^{k-1} x_2^{\ell+1} s^m t^s;\qquad \end{align} $$

(2-42) $$\begin{align}\kern0.2pt L_f \cdot x_1^k x_2^\ell s^m t^s &= (-s -\ell) x_1^{k+1} x_2^{\ell-1} s^m t^s; \end{align}$$

(2-43) $$\begin{align} L_h \cdot x_1^k x_2^\ell s^m t^s &= (s -k+\ell) x_1^k x_2^\ell s^m t^s;\ \ \kern0.1pt \end{align}$$

(2-44) $$ \begin{align} R_h \cdot x_1^k x_2^\ell s^m t^s &= (s +k + \ell) x_2^k x_2^\ell s^m t^s.\ \ \kern0.2pt\end{align} $$

The resulting $\widetilde {\mathcal {U}}$ -module has a natural filtration given by powers of s, and it decomposes into a direct sum of submodules spanned by monomials $\{x_1^k x_2^\ell s^m t^s\}$ for fixed integers $k + \ell $ . Each of these submodules has the structure of a deformed dual Verma module, as illustrated in Figure 3 for $k + \ell = 0$ . Note that in Figure 3, we omit the generator $t^s$ and the arrows corresponding to the $R_h$ -action for clarity.

Figure 3 Deformed dual Verma modules arise as global sections of $j_+f^s\mathcal {O}_U^{(n)}$ .

Moreover, one can compute that the Casimir element $L_\Omega $ in (2-11) acts by

(2-45) $$ \begin{align} L_\Omega \cdot x_1^k x_2^\ell s^m t^s = ( (k + \ell)^2 + 2(k + \ell) + 2s(1 + k + \ell) + s^2 ) x_1^k x_2^\ell s^m t^s. \end{align} $$

Since s is nilpotent, we can see from this computation that a high enough power of the operator

(2-46) $$ \begin{align} L_\Omega - \gamma_{\mathrm{HC}}(k + \ell) = 2s(1 + k + \ell) + s^2 \end{align} $$

annihilates any monomial basis element. (Here, $\gamma _{\mathrm {HC}}$ is the Harish-Chandra projection in (2-3).) Hence, the global sections of the submodules of $j_+ f^s \mathcal {O}_U ^{(n)}$ spanned by monomials $\{x_1^k x_2^\ell s^m t^s\}$ for fixed integers $k + \ell $ have generalized, but not strict, infinitesimal character.

2.4.3 Step 3: $!$ -pushforward

Recall that $j_! = \mathbb {D}_{\widetilde {X}} \circ j_+ \circ \mathbb {D}_U$ , where $\mathbb {D}$ denotes holonomic duality, as in (2-28). We begin by computing the right $\mathcal {D}_U$ -module $\mathbb {D}_U f^s \mathcal {O}_U^{(n)}$ by taking a projective resolution of $f^s \mathcal {O}_U^{(n)}$ as a left $\mathcal {A}_U$ -module. This is straightforward using the description (2-39). The complex

$$ \begin{align*} 0 \xrightarrow{d_2} \mathcal{A}_U \xrightarrow{d_1} \mathcal{A}_U \oplus \mathcal{A}_U \xrightarrow{d_0} \mathcal{A}_U \xrightarrow{\epsilon} \mathcal{A}_U / \mathcal{A}_U \langle \partial_1, x_2 \partial_2 - s \rangle \rightarrow 0, \end{align*} $$

where $\epsilon $ is the canonical quotient map, $d_0$ sends $(\theta _1, \theta _2) \in \mathcal {A}_U \oplus \mathcal {A}_U$ to $\theta _1 \partial _1 - \theta _2 (x_2 \partial _2 -s)$ , and $d_1$ sends $1 \mapsto (x_2 \partial _2 - s, \partial _1)$ , is a free resolution of the left $\mathcal {A}_U$ -module $f^s \mathcal {O}_U^{(n)}$ . Applying the functor $\mathrm {Hom}_{\mathcal {A}_U}( - , \mathcal {A}_U)$ and making the natural identification

$$ \begin{align*} \mathrm{Hom}_{\mathcal{A}_U}(\mathcal{A}_U, \mathcal{A}_U) \simeq \mathcal{A}_U; \varphi \mapsto \varphi(1) \end{align*} $$

of right $\mathcal {A}_U$ -modules, we see that

$$ \begin{align*} \mathbb{D}_U f^s \mathcal{O}_U^{(n)} = \mathrm{Ext}_{\mathcal{A}_U}^2(f^s \mathcal{O}_U^{(n)}, \mathcal{A}_U) = \operatorname{\mbox{im }} d_1^* \backslash \ker d_2^* = \langle \partial_1, x_2 \partial_2 - s \rangle \mathcal{A}_U \backslash \mathcal{A}_U. \end{align*} $$

Here, $d_i^* (\varphi ) = \varphi \circ d_i$ for an appropriate homomorphism $\varphi $ , and the right $\mathcal {A}_U$ -module structure is given by right multiplication.

To finish the computation of $j_! f^s \mathcal {O}_U^{(n)}$ , we must take a projective resolution of this module. We do so following a similar process to the $!$ -pushforward computation in Section 2.3. Denote by I the right ideal $\langle \partial _1, x_2 \partial _2 - s \rangle \mathcal {A}_U$ in $\mathcal {A}_U$ . The complex

$$ \begin{align*} 0 \leftarrow I \backslash \mathcal{A}_U \xleftarrow{\epsilon} \mathcal{A}_{\widetilde{X}} \xleftarrow{d_0} \mathcal{A}_{\widetilde{X}} \oplus \mathcal{A}_{\widetilde{X}} \xleftarrow{d_1} \mathcal{A}_{\widetilde{X}} \xleftarrow{d_2} 0 \end{align*} $$

with maps given by

$$ \begin{align*} \,\epsilon: 1 &\mapsto Ix_2^{-1};\\d_0: (\theta_1, \theta_2) &\mapsto x_2 \partial_1 \theta_1 - (x_2^2 \partial_2 - x_2 s) \theta_2;\\d_1: 1 &\mapsto (x_2 \partial_2 - s, \partial_1) \end{align*} $$

is a free resolution of $\mathbb {D}_U f^s \mathcal {O}_U^{(n)}$ by right $\mathcal {A}_{\widetilde {X}}$ -modules. Applying $\mathrm {Hom}_{\mathcal {A}_{\widetilde {X}}, r}( - , \mathcal {A}_{\widetilde {X}})$ and making the natural identifications as above,

$$ \begin{align*} j_! f^s \mathcal{O}_U^{(n)} = \ker d_2^* / \operatorname{\mbox{im }} d_1^* = \mathcal{A}_{\widetilde{X}} / \mathcal{A}_{\widetilde{X}} \langle \partial_1, x_2 \partial_2 - s \rangle. \end{align*} $$

The left $\mathcal {A}_{\widetilde {X}}$ -module structure is given by left multiplication.

Again, it is interesting to examine the $\widetilde {\mathcal {U}}$ -module structure on the global sections of this module. The global sections of $j_! f^s \mathcal {O}_U^{(n)}$ are spanned by monomials $x_1^k x_2^{\ell } s^m$ for $k, \ell \geq 0$ and $0 \leq m < n$ , and $x_1^a \partial _2^ b s^m$ for $a, b \geq 0$ and $0 \leq m < n$ . For $\ell>0$ , the $L_e, L_f, L_h$ , and $R_h$ -actions on the monomials $x_1^k x_2^{\ell } s^m$ are as in (2-41)–(2-44) (where we identify the generator $t^s$ of $j_+ f^s \mathcal {O}_U^{(n)}$ with the coset containing $1$ in $j_!f^s \mathcal {O}_U^{(n)}$ ), and the actions on the monomials $x_1^a \partial _2^ b s^m$ are given by the following formulas:

$$ \begin{align*} L_e \cdot x_1^a \partial_2^ b s^m &= a(b - 1 - s) x_1^{a-1} \partial_2 ^{b-1} s^m;\\L_f \cdot x_1^a \partial_2^ b s^m &= -x_1^{a+1} \partial_2^{b+1} s^m; \end{align*} $$

(2-47) $$ \begin{align} L_h \cdot x_1^a \partial_2^ b s^m &= (s - a - b) x_1^a \partial_2^b s^m;\qquad \end{align} $$

(2-48) $$ \begin{align}R_h \cdot x_1^a \partial_2^ b s^m &= (s+ a - b) x_1^a \partial_2^b s^m.\qquad \end{align} $$

For $\ell =b= 0$ , the actions of $L_h$ and $R_h$ are as in (2-47)–(2-48), and the actions of $L_e$ and $L_f$ are given by

$$ \begin{align*} L_e \cdot x_1^k s^m &= -k x_1^{k-1} x_2 s^m; \end{align*} $$

$$ \begin{align*} L_f \cdot x_1^k s^m &= -x_1^{k+1} \partial_2 s^m.\ \ \end{align*} $$

As in Section 2.4.2, this $\widetilde {\mathcal {U}}$ -module has an n-step filtration by powers of s, and decomposes into a direct sum of $\widetilde {\mathcal {U}}$ -submodules, each spanned by the set of monomials $\{x_1^k x_2^{\ell } s^m\}$ and $\{x_1^a \partial _2^ b s^m\}$ such that $k + \ell = a-b$ is a fixed integer. For $k + \ell = a-b=\lambda $ , this submodule is isomorphic to a deformed Verma module of highest weight $\lambda $ . We illustrate the module corresponding to $\lambda = 0$ in Figure 4.

Figure 4 Deformed Verma modules arise as global sections of $j_!f^s\mathcal {O}_U^{(n)}$ .

2.4.4 Step 4: Image of the canonical map

Set $I_U = \mathcal {A}_U \langle \partial _1, x_2 \partial _2 - s \rangle $ and $I_{\widetilde {X}} = \mathcal {A}_{\widetilde {X}} \langle \partial _1, x_2 \partial _2 - s \rangle $ to be the left ideals generated by the operators $\partial _1$ and $x_2 \partial _2 - s$ in $\mathcal {A}_U$ and $\mathcal {A}_{\widetilde {X}}$ , respectively. The canonical map between the $!$ - and $+$ -pushforwards is given by

$$ \begin{align*} j_! f^s \mathcal{O}_U^{(n)} = \mathcal{A}_{\widetilde{X}} / I_{\widetilde{X}} &\xrightarrow{\mathrm{can}} \mathcal{A}_U / I_U =j_+ f^s \mathcal{O}_U^{(n)} \\ 1I_{\widetilde{X}} &\longmapsto 1 I_U. \end{align*} $$

Since $1\mathcal {A}_{\widetilde {X}}$ generates $j_! f^s \mathcal {O}_U^{(n)}$ as an $\mathcal {A}_{\widetilde {X}}$ -module, its image completely determines the morphism $\mathrm {can}$ . On the monomial basis elements $x_1^k x_2^\ell s^m$ and $x_1^k \partial _2^\ell s^m$ of $j_! f^s \mathcal {O}_U^{(n)}$ , the canonical map acts by

(2-49)

for $b>1$ . For $b=1$ , .

The image of the morphism $\mathrm {can}$ is the $\mathcal {A}_{\widetilde {X}}$ -submodule

$$ \begin{align*} \operatorname{\mbox{im }} (\mathrm{can}) = \mathcal{A}_{\widetilde{X}}/I_U \subset \mathcal{A}_U / I_U. \end{align*} $$

In the description of the global sections of $j_+ f^s \mathcal {O}_U^{(n)}$ in (2-39), the global sections of $\operatorname {\mbox {im }} (\mathrm {can})$ can be identified with

$$ \begin{align*} (\mathbb{C}[x_1, x_2] \otimes \mathbb{C}[s]/s^n + \mathbb{C}[x_1, x_2, x_2^{-1}] \otimes s \mathbb{C}[s]/s^n)t^s. \end{align*} $$

2.4.5 Step 5: The maximal extension

Composing the canonical map $\mathrm {can}$ with s gives

(2-50) $$ \begin{align} s^1(n): \mathcal{A}_{\widetilde{X}}/I_{\widetilde{X}} \xrightarrow{\mathrm{can}} \mathcal{A}_U/I_U \xrightarrow{s} \mathcal{A}_U/I_U. \end{align} $$

The global sections of the image of $s^1(n)$ (as a submodule of (2-39)) are

$$ \begin{align*} \Gamma(\widetilde{X}, \operatorname{\mbox{im }} s^1(n)) \simeq ( \mathbb{C}[x_1, x_2] \otimes s\mathbb{C}[s]/s^n + \mathbb{C}[x_1, x_2, x_2^{-1}] \otimes s^2 \mathbb{C}[s]/s^n )t^s. \end{align*} $$

This gives us an explicit description of $\Xi _\rho \mathcal {O}_U = \operatorname {\mathrm {coker}} s^1(n)$ :

(2-51) $$ \begin{align} \Gamma(\widetilde{X}, \Xi_\rho \mathcal{O}_U) &= (\mathbb{C}[x_1, x_2, x_2^{-1}] \otimes \mathbb{C}[s]/s^n ) t^s / \Gamma(\widetilde{X}, \operatorname{\mbox{im }} s^1(n))\nonumber\\ &= ( \mathbb{C}[x_1, x_2, x_2^{-1}] \otimes \mathbb{C}[s]/s^2 )t^s / ( \mathbb{C}[x_1, x_2]\otimes s \mathbb{C}[s]/s^2 )t^s. \end{align} $$

Figure 5 Caricature of the maximal extension $\Xi _\rho \mathcal {O}_U$ .

A caricature of the $\Gamma (\widetilde {X}, \mathcal {A}_{\widetilde {X}})$ -module (2-51) is illustrated as in Figure 5. It has two layers, corresponding to the two nonzero powers of s, and action by s moves layers up. As vector spaces, the bottom layer is isomorphic to $\mathbb {C}[x_1, x_2, x_2^{-1}]$ and the top layer to $sx_2^{-1} \mathbb {C}[x_1, x_2^{-1}]$ .

Our final step is to examine the $\widetilde {\mathcal {U}}$ -module structure on $\Gamma (\widetilde {X}, \Xi _\rho \mathcal {O}_U)$ . The module (2-51) has a basis given by monomials $x_1^k x_2^\ell t^s$ for $k \in \mathbb {Z}_{\geq 0}$ and $\ell \in \mathbb {Z}$ , and $x_1^k x_2^\ell s t^s$ for $k \in \mathbb {Z}_{\geq 0}$ and $\ell \in \mathbb {Z}_{< 0}$ . The actions of the operators $L_e, L_f, L_h$ , and $R_h$ in (2-10) on these monomials are given by applying the formulas (2-41)–(2-44) and taking the image of the resulting monomials in the quotient (2-51).

The $\widetilde {\mathcal {U}}$ -module $\Gamma (\widetilde {X}, \Xi _\rho \mathcal {O}_U)$ splits into a direct sum of submodules spanned by monomials $x_1^k x_2^\ell t^s$ and $x_1^k x_2^\ell s t^s$ such that $k + \ell $ is a fixed integer. We illustrate the submodule for $k + \ell = 0$ in Figure 6. For clarity, we drop the generator $t^s$ from our notation in Figure 6. If $\lambda \geq 0$ , the submodule corresponding to the integer $\lambda = k + \ell $ has the Verma module of highest weight $\lambda $ as a submodule, and the dual Verma module corresponding to $\lambda $ as a quotient. As a $\mathcal {U}(\mathfrak {g})$ -module, it is isomorphic to the big projective module $P(w_0\lambda )$ in the corresponding block of category $\mathcal {O}$ . (The big projective module is the projective cover of the irreducible highest-weight module $L(w_0 \lambda )$ , where $w_0$ is the longest element of the Weyl group. It is the longest indecomposable projective object in the block $\mathcal {O}_\lambda $ of category $\mathcal {O}$ [Reference HumphreysHum08, Section 3.12].)

Figure 6 Big projective modules arise as global sections of slices of $\Xi _\rho \mathcal {O}_U$ .

2.5 The monodromy filtration and the geometric Jantzen filtration

The maximal extension $\Xi _\rho \mathcal {O}_U$ naturally comes equipped with a nilpotent endomorphism s, giving it a corresponding monodromy filtration. This is the source of the geometric Jantzen filtrations on $j_!\mathcal {O}_U$ and $j_+\mathcal {O}_U$ . In this section, we use the monodromy filtration on $\Xi _\rho \mathcal {O}_U$ to compute the geometric Jantzen filtration on $j_!\mathcal {O}_U$ . Using the computations of Section 2.4, we then describe the corresponding $\widetilde {\mathcal {U}}$ -module filtration on global sections.

We begin by recalling monodromy filtrations in abelian categories, following [Reference DeligneDel80, Section 1.6]. Given an object A in an abelian category $\mathcal {A}$ and a nilpotent endomorphism $s: A \rightarrow A$ , it follows from the Jacobson–Morosov theorem [Reference DeligneDel80, Proposition 1.6.1] that there exists a unique increasing exhaustive filtration $\mu ^\bullet $ on A such that $s\mu ^n \subset \mu ^{n-2}$ , and for $k \in \mathbb {N}$ , $s^k$ induces an isomorphism $\mathrm {gr}_\mu ^k A \simeq \mathrm {gr}_\mu ^{-k} A.$ This unique filtration is called the monodromy filtration of A.

Following Deligne’s proof in [Reference DeligneDel80, Section 1.6], the monodromy filtration can be described explicitly in terms of powers of $s.$ Namely, if we set

$$ \begin{align*} \mathscr{K}^p A: = \begin{cases} \ker s^{p+1} & \text{for } p \geq 0; \\ 0 & \text{for } p<0 \end{cases} \end{align*} $$

to be the kernel filtration of A and

$$ \begin{align*} \mathscr{I}^q A:= \begin{cases} \operatorname{\mbox{im }} s^q & \text{for } q>0; \\ A & \text{for } q \leq 0, \end{cases} \end{align*} $$

to be the image filtration of A, then $\mu ^\bullet $ is the convolution of the kernel and image filtrations; that is,

(2-52) $$ \begin{align} \mu^r = \sum_{p-q = r} \mathscr{K}^p \cap \mathscr{I}^q. \end{align} $$

The monodromy filtration $\mu ^\bullet $ induces filtrations $J_!^\bullet $ on $\ker s$ and $J_+^\bullet $ on $\operatorname {\mathrm {coker}} s$ . By (2-52), these can be seen to be

(2-53) $$ \begin{align} J_!^i = \ker s \cap \mathscr{I}^{-i} \quad\text{and}\quad J_+^i = (\mathscr{K}^i + \operatorname{\mbox{im }} s) / \operatorname{\mbox{im }} s. \end{align} $$

In the setting of holonomic $\mathcal {D}$ -modules, the filtrations $J_!^\bullet $ and $J_+^\bullet $ define the geometric Jantzen filtrations.

Now we return to our running example. The monodromy filtration $\mu ^\bullet $ on $\Xi _\rho \mathcal {O}_U$ is

$$ \begin{align*} \mu^{-2} = 0 \subset \mu^{-1} = \operatorname{\mbox{im }} s \subset \mu^0 = j_! \mathcal{O}_U \subset \mu^1 =\Xi_\rho \mathcal{O}_U. \end{align*} $$

Restricting this to $\ker s = j_! \mathcal {O}_U$ , we obtain the geometric Jantzen filtration of $j_! \mathcal {O}_U$ :

$$ \begin{align*} 0 \subset \operatorname{\mbox{im }} s \subset j_! \mathcal{O}_U. \end{align*} $$

The induced filtration on $\operatorname {\mathrm {coker}} s = j_+ \mathcal {O}_U$ gives the geometric Jantzen filtration on $j_+ \mathcal {O}_U$ :

$$ \begin{align*} 0 \subset \ker s / \operatorname{\mbox{im }} s \subset j_+ \mathcal{O}_U. \end{align*} $$

Using the computations in Section 2.4, we can examine the $\widetilde {\mathcal {U}}$ -module filtrations that we obtain on global sections. Recall that $\Gamma (\widetilde {X}, \Xi _\rho \mathcal {O}_U)$ decomposes into a direct sum of submodules spanned by monomials $x_1^k x_2^\ell t^s$ and $x_1^k x_2^\ell s t^s$ such that $k + \ell $ is a fixed nonnegative integer. Figure 6 illustrates the submodule corresponding to $k + \ell = 0$ . Looking at this figure, it is clear that $\ker s = \mathrm {span}\{ x_1^k x_2^\ell s t^s, t^s \}$ is isomorphic to the Verma module of highest weight $0$ , and $\operatorname {\mathrm {coker}} s = \mathrm {span}\{x_1^k x_2^\ell t^s\}$ is isomorphic to the corresponding dual Verma module. Moreover, the global sections of the monodromy filtration on $\Xi _\rho \mathcal {O}_U$ restricted to the submodule corresponding to $k + \ell = \lambda $ is the composition series of the corresponding big projective module $P( w_0\lambda )$ when $\lambda \geq 0$ . This is illustrated in Figure 7 for $\lambda = 0$ . We conclude that the filtrations on the Verma module $M(\lambda )$ and dual Verma module $I(\lambda )$ obtained by taking global sections of the geometric Jantzen filtrations are the composition series. Note that this is an $\mathfrak {sl}_2(\mathbb {C})$ phenomenon. For larger groups, this procedure yields a filtration different from the composition series.

Figure 7 Global sections of the monodromy filtration on $\Xi _\rho \mathcal {O}_U$ are the composition series of the big projective module.

2.6 Relation to the algebraic Jantzen filtration

The geometric Jantzen filtrations described above have an algebraic analogue, due to Jantzen [Reference JantzenJan79]. In this section, we recall the construction of the algebraic Jantzen filtration of a Verma module, then explain its relation with the geometric construction in Section 2.5.

2.6.1 The algebraic Jantzen filtration

We follow [Reference SoergelSoe08]. Another nice reference for Jantzen filtrations is [Reference Iohara and KogaIK11].

Let $\mathfrak {g}$ be a complex semisimple Lie algebra, $\mathfrak {b}$ a fixed Borel subalgebra, $\mathfrak {n} = [\mathfrak {b}, \mathfrak {b}]$ the nilpotent radical of $\mathfrak {b}$ , and $\mathfrak {h}$ a Cartan subalgebra so that $\mathfrak {b}= \mathfrak {h} \oplus \mathfrak {n}$ . Denote by $\bar {\mathfrak {b}}$ the opposite Borel subalgebra to $\mathfrak {b}$ . For $\lambda \in \mathfrak {h}^*$ , denote the Verma module of highest weight $\lambda $ by

$$ \begin{align*} M(\lambda) := \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{b})} \mathbb{C}_\lambda. \end{align*} $$

Denote by $I(\lambda )$ the corresponding dual Verma module, defined to be the direct sum of weight spaces in

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\mathcal{U}(\overline{\mathfrak{b}})} (\mathcal{U}(\mathfrak{g}), \mathbb{C}_\lambda). \end{align*} $$

Set $T = \mathcal {O}(\mathbb {C}\rho )$ to be the ring of regular functions on the line $\mathbb {C} \rho \subset \mathfrak {h}^*$ , where $\rho $ is the half-sum of positive roots in the root system determined by $\mathfrak {b}$ . A choice of linear functional $s: \mathbb {C} \rho \rightarrow \mathbb {C}$ gives an isomorphism $T \simeq \mathbb {C}[s]$ . Fix such an identification. Set $A:=T_{(s)}$ to be the local $\mathbb {C}$ -algebra obtained from T by inverting all polynomials not divisible by s, and

(2-54) $$ \begin{align} \varphi: \mathcal{O}(\mathfrak{h}^*) \rightarrow A \end{align} $$

to be the composition of the restriction map $\mathcal {O}(\mathfrak {h}^*) \rightarrow T$ with the inclusion $T \hookrightarrow A$ . Note that under the identification $\mathcal {U}(\mathfrak {h}) \simeq \mathcal {O}(\mathfrak {h}^*)$ , $\varphi (\mathfrak {h}) \subseteq (s)$ , the unique maximal ideal of A.

Let V be a $(\mathfrak {g},A)$ -bimodule on which the right and left actions of $\mathbb {C}$ agree. The deformed weight space $V^\mu $ of V corresponding to a weight $\mu $ is the subspace

(2-55) $$ \begin{align} V^\mu:= \{ v \in V \mid (h - \mu(h))v=v \varphi(h) \text{ for all } h \in \mathfrak{h} \}. \end{align} $$

The direct sum of all deformed weight spaces of V is a $(\mathfrak {g},A)$ -submodule of V [Reference SoergelSoe08, Section 2.3].

For $\lambda \in \mathfrak {h}^*$ , the deformed Verma module corresponding to $\lambda $ is the $(\mathfrak {g},A)$ -bimodule

$$ \begin{align*} M_A(\lambda):= \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{b})} A_\lambda, \end{align*} $$

where the $\mathcal {U}(\mathfrak {b})$ -module structure on $A_\lambda $ is given by extending the $\mathfrak {h}$ -action

$$ \begin{align*} h \cdot a = (\lambda + \varphi)(h) a \end{align*} $$

trivially to $\mathfrak {b}$ . Here, $h \in \mathfrak {h}$ , $a \in A$ , and $\varphi $ is as in (2-54). The deformed Verma module $M_A(\lambda )$ is equal to the direct sum of its deformed weight spaces.

The deformed dual Verma module $I_A(\lambda )$ corresponding to $\lambda $ is the direct sum of deformed weight spaces in the $(\mathfrak {g},A)$ -bimodule

(2-56) $$ \begin{align} \operatorname{\mathrm{Hom}}_{\mathcal{U}(\bar{\mathfrak{b}})}(\mathcal{U}(\mathfrak{g}), A_\lambda). \end{align} $$

There is a canonical isomorphism [Reference SoergelSoe08, Proposition 2.12]

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{(\mathfrak{g},A) \mathrm{-mod}}(M_A(\lambda), I_A(\lambda)) \simeq A. \end{align*} $$

Under this isomorphism, $1 \in A$ distinguishes a canonical $(\mathfrak {g},A)$ -bimodule homomorphism

(2-57) $$ \begin{align} \psi_{A, \lambda}: M_{A}(\lambda) \rightarrow I_A(\lambda). \end{align} $$

For any A-module M, there is a descending A-module filtration $M^i:=s^iM$ with associated grading $gr^iM = M^i/M^{i+1}$ . Hence, there is a reduction map

$$ \begin{align*} \mathrm{red}: M \rightarrow gr^0M = M/sM. \end{align*} $$

For $M_A(\lambda )$ and $I_A(\lambda )$ , the layers of this filtration are $\mathfrak {g}$ -stable, so we obtain surjective $\mathfrak {g}$ -module homomorphisms

(2-58) $$ \begin{align} \mathrm{red}: M_A(\lambda) \rightarrow M(\lambda) = gr^0 M_A(\lambda) \text{ and } \mathrm{red}: I_A(\lambda) \rightarrow I(\lambda) = gr^0 I_A(\lambda). \end{align} $$

Pulling back the filtration above along the canonical map $\psi _{A, \lambda }$ in (2-57) gives a $(\mathfrak {g}, A)$ -bimodule filtration of $M_A(\lambda )$ .

Definition 2.11. The algebraic Jantzen filtration of $M_A(\lambda )$ is the $(\mathfrak {g}, A)$ -bimodule filtration

$$ \begin{align*} M_A(\lambda)^i:= \{m \in M_A(\lambda) \mid \psi_{A, \lambda} (m) \in s^i I_A(\lambda)\}, \end{align*} $$

where $\psi _{A, \lambda }$ is the canonical map (2-54). By applying the reduction map in (2-58) to the filtration layers, we obtain a filtration $M(\lambda )^\bullet $ of $M(\lambda )$ .

2.6.2 Relationship between algebraic and geometric Jantzen filtrations

Though the constructions seem quite different at first glance, the geometric Jantzen filtration in Section 2.5 aligns with the algebraic Jantzen filtration described in Section 2.6.1 under the global sections functor. In this final section, we illustrate this relationship through our running example.

Recall the canonical map (2-35):

$$ \begin{align*} \mathrm{can}:j_! f^s \mathcal{O}_U^{(n)} \rightarrow j_+ f^s \mathcal{O}_U^{(n)}. \end{align*} $$

As illustrated in Figures 3 and 4, the global sections of $j_!f^s \mathcal {O}_U^{(n)}$ and $j_+ f^s \mathcal {O}_U^{(n)}$ decompose into direct sums of deformed dual Verma and Verma modules, respectively. The global sections of $\mathrm {can}$ are the direct sum of $\psi _{A,\lambda }$ in (2-57) for all integral $\lambda $ .

Remark 2.12. To be more precise, the submodules of $\Gamma (\widetilde {X}, j_!f^s \mathcal {O}_U^{(n)})$ and $\Gamma (\widetilde {X}, j_+ f^s \mathcal {O}_U^{(n)})$ corresponding to an integer $\lambda $ are truncated versions of $M_A(\lambda )$ in (2-56) and $I_A(\lambda )$ in (2-56) obtained by taking a quotient so that $s^n = 0$ .

There are two natural filtrations of $j_! \mathcal {O}_U$ that we have described using this set-up.

Filtration 1: (algebraic Jantzen filtration)

We obtain a filtration of $j_! f^s \mathcal {O}_U^{(n)}$ by pulling back the ‘powers of s’ filtration along $\mathrm {can}$ . This induces a filtration on the quotient

(2-59) $$ \begin{align} j_!(\mathcal{O}_U) \simeq j_! f^s \mathcal{O}_U^{(n)} / s j_! f^s \mathcal{O}_U^{(n)}. \end{align} $$

This is exactly the $\mathcal {D}$ -module analogue of the algebraic Jantzen filtration described in Section 2.6.1. On global sections, it is the filtration

(2-60) $$ \begin{align} F^i \Gamma(\widetilde{X}, j_! \mathcal{O}_U) = \{ v \in \Gamma(\widetilde{X}, j_! \mathcal{O}_U) \mid \Gamma(\mathrm{can})(v) \in s^i \Gamma(\widetilde{X}, j_+ f^s \mathcal{O}_U^{(n)}) \}. \end{align} $$

Filtration 2: (geometric Jantzen filtration)

There is a unique monodromy filtration on $\Xi _\rho \mathcal {O}_U = \operatorname {\mathrm {coker}} (s \circ \mathrm {can})$ . Restricting this to the kernel of s, we obtain a filtration on

(2-61) $$ \begin{align} j_! \mathcal{O}_U \simeq \ker (s: \Xi_\rho \mathcal{O}_U \rightarrow \Xi_\rho \mathcal{O}_U). \end{align} $$

This is the geometric Jantzen filtration. It can be realized explicitly in terms of the image of powers of s using (2-53). On global sections, this gives

(2-62) $$ \begin{align} G^i \Gamma(\widetilde{X}, j_! \mathcal{O}_U) = \{ w \in \ker (s \circlearrowright \Gamma( \widetilde{X}, \Xi_\rho \mathcal{O}_U)) \mid w \in s^i \Gamma( \widetilde{X}, \Xi_\rho \mathcal{O}_U) \}. \end{align} $$

Figure 8 Relationship between the algebraic and geometric Jantzen filtrations.

It is helpful to see these filtrations in a picture. Figure 8 illustrates the set-up when restricted to the submodule corresponding to $\lambda = 0$ .

The map $\mathrm {can}$ is described on basis elements in (2-49). Computing these actions for $\lambda = 0$ , we see in Figure 8 that $\mathrm {can}$ fixes the right-most column and sends any other monomial on the left to a linear combination of monomials directly above the corresponding monomial on the right. The image of $s_1(n)=s \circ \mathrm {can}$ in (2-50) is highlighted in gray. The quotient by this image is the maximal extension, which is outlined in the black box. The quotient in (2-59) is highlighted in blue (darker shading) in the left hand module, and the submodule in (2-61) is highlighted in blue (darker shading) in the right-hand module.

We see that there are two copies of $j_! \mathcal {O}_U$ (each highlighted in blue (darker shading) in Figure 8) in this set-up: one as a quotient of the left-hand module $j_! f^s \mathcal {O}_U^{(n)}$ , and one as a submodule of a quotient of the right-hand module $j_+ f^s \mathcal {O}_U^{(n)}$ . These two copies can be naturally identified as follows.

Because the submodule $s j_! f^s \mathcal {O}_U^{(n)}$ is in the kernel of the composition of $\mathrm {can}$ with the quotient $j_! f^s \mathcal {O}_U^{(n)} \rightarrow \operatorname {\mathrm {coker}}( s \circ \mathrm {can}) = \Xi _\rho \mathcal {O}_U$ , the map $\mathrm {can}$ descends to a map on the quotient:

$$ \begin{align*} \overline{\mathrm{can}}: j_! \mathcal{O}_U \simeq j_! f^s \mathcal{O}_U ^{(n)} / s j_! f^s \mathcal{O}_U^{(n)} \rightarrow \Xi_\rho \mathcal{O}_U. \end{align*} $$

By construction, the map $\overline {\mathrm {can}}$ is injective. Its image is exactly $\ker (s: \Xi _\rho \mathcal {O}_U \rightarrow \Xi _\rho \mathcal {O}_U)$ . This is immediately apparent in Figure 8. Hence, $\overline {\mathrm {can}}$ provides an explicit isomorphism that can be used to identify the two copies of $j_! \mathcal {O}_U$ . Under this identification, the algebraic Jantzen filtration in (2-60) and the geometric Jantzen filtration in (2-62) clearly align.