2.1 $\mathcal{D}$ -modules, mixed Hodge modules, and local cohomology

Let X be a smooth complex variety of dimension $d_{X}$ , with sheaf of algebraic differential operators $\mathcal{D}_{X}$ . A $\mathcal{D}_X$ -module is a sheaf of left modules over $\mathcal{D}_X$ . We write ${\rm MHM}(X)$ for the category of algebraic mixed Hodge modules on X (see [Reference Hotta, Takeuchi and TanisakiHTT08, § 8.3.3] or [Reference SaitoSai88, Reference SaitoSai90]), with bounded derived category ${\rm D}^b{\rm MHM}(X)$ . These categories are also defined on singular varieties.

Given an irreducible closed subvariety $Z\subseteq X$ , we write $\mathcal{L}(Z,X)$ for the intersection cohomology $\mathcal{D}_X$ -module associated to the trivial local system on the regular locus $Z_{{\rm reg}}\subseteq Z$ (see [Reference Hotta, Takeuchi and TanisakiHTT08, Definition 3.4.1]), and we write $IC^H_Z$ for the pure Hodge module associated to the trivial variation of Hodge structure on $Z_{{\rm reg}}$ (see [Reference Hotta, Takeuchi and TanisakiHTT08, § 8.3.3(m13)]), which has weight $d_Z$ . For instance, we have $\mathcal{L}(X,X)=\mathcal{O}_X$ and $IC^H_X=\mathcal{O}_X^H$ . For $k\in \mathbb{Z}$ , we write $IC^H_Z(k)$ for the kth Tate twist of $IC^H_Z$ (see [Reference Hotta, Takeuchi and TanisakiHTT08, § 8.3.3(m5)]), which is pure of weight $d_Z-2k$ . The modules $IC^H_Z(k)$ provide a complete list of polarizable pure Hodge modules that may overlie the $\mathcal{D}_X$ -module $\mathcal{L}(Z,X)$ (see [Reference Hotta, Takeuchi and TanisakiHTT08, § 8.3.3(m13)]).

Let $f:X\to Y$ be a morphism between complex varieties. We write the following for the corresponding direct and inverse image functors for mixed Hodge modules [Reference SaitoSai90, § 4]:

$$f_{\ast}:{\rm D}^b{\rm MHM}(X)\to {\rm D}^b{\rm MHM}(Y) \quad{\rm and} \quad f^{!}:{\rm D}^b{\rm MHM}(Y)\to {\rm D}^b{\rm MHM}(X).$$

If X and Y are smooth, then these functors lift the $\mathcal{D}$ -module functors $\int_f$ and $f^{\dagger}$ from [Reference Hotta, Takeuchi and TanisakiHTT08, Chapter 1.5], and if f is a closed immersion of codimension c, then $f^{!}\mathcal{O}_Y^H=\mathcal{O}_X^H(-c)[-c]$ .

Let X be smooth and let $i:Z\hookrightarrow X$ be a locally closed subvariety of X. Given $\mathcal{M}\in {\rm D}^b{\rm MHM}(X)$ , we define the local cohomology modules of $\mathcal{M}$ with support in Z via

$$\mathscr{H}_{Z}^q(\mathcal{M}):=\mathcal{H}^q(i_{\ast}i^{!}\mathcal{M}),\quad q\in \mathbb{Z}.$$

In particular, for $\mathcal{M}=\mathcal{O}^H_X$ , this endows each local cohomology module $\mathscr{H}_{Z}^q(\mathcal{O}_X)$ with a mixed Hodge module structure, which we denote by $\mathscr{H}_{Z}^q(\mathcal{O}^H_X)$ . For more information, see [Reference Mustaţă and PopaMP22, § B].

Now assume that $Z\subseteq X$ is a closed subvariety of X, with complement $U=X\setminus Z$ , and open immersion $j:U\to X$ . Given $\mathcal{M}\in {\rm D}^b{\rm MHM}(X)$ , there is an exact triangle [Reference SaitoSai90, (4.4.1)]

(2.1) \begin{equation}i_{\ast}i^{!}(\mathcal{M})\longrightarrow \mathcal{M}\longrightarrow j_{\ast}j^{!}(\mathcal{M})\overset{+1}\longrightarrow.\end{equation}

In particular, if $\mathcal{M}$ is a mixed Hodge module, we obtain an exact sequence

$$0\longrightarrow \mathscr{H}_{Z}^0(\mathcal{M})\longrightarrow \mathcal{M}\longrightarrow \mathcal{H}^0(j_{\ast}(\mathcal{M}|_U))\longrightarrow \mathscr{H}_{Z}^1(\mathcal{M})\longrightarrow 0,$$

and isomorphisms

$$\mathscr{H}_{Z}^q(\mathcal{M})\cong \mathcal{H}^{q-1}(j_{\ast}(\mathcal{M}|_U))\quad \mbox{for $q\geqslant 2$.}$$

We write $F_{\bullet}$ and $W_{\bullet}$ for the Hodge and weight filtrations of $\mathscr{H}_{Z}^q(\mathcal{M})$ .

2.2 Construction of the Grothendieck–Cousin complex

Lemma 2.1. Let $\mathcal{M}$ be a mixed Hodge module on a smooth complex variety X, and let $Z'\subsetneq Z$ be closed subvarieties of X. We have the following exact sequence of mixed Hodge modules:

$$0 \to \mathscr{H}^0_{Z'}(\mathcal{M})\to \mathscr{H}^0_Z(\mathcal{M}) \to \mathscr{H}^0_{Z\setminus Z'}(\mathcal{M})\to \mathscr{H}^{1}_{Z'}(\mathcal{M})\to \mathscr{H}^{1}_Z(\mathcal{M}) \to \mathscr{H}^{1}_{Z\setminus Z'}(\mathcal{M})\to \cdots$$

Proof. We set $U=X\setminus Z'$ with open immersion j, and we write $i':Z'\to X$ , $i:Z\to X$ for the natural inclusions. By (2.1) applied to $i'=i$ and $i_{\ast}i^!\mathcal{M}$ , we obtain an exact triangle

$$i'_{\ast}i'^!i_{\ast}i^!\mathcal{M}\longrightarrow i_{\ast}i^!\mathcal{M}\longrightarrow j_{\ast}j^!i_{\ast}i^!\mathcal{M}\overset{+1}\longrightarrow.$$

Let f be the inclusion of Z’ into Z. Since $i'=i\circ f$ , we have $i'_{\ast}i'^!i_{\ast}i^!= i'_{\ast}f^!i^!i_{\ast}i^!$ , which by adjunction of $i^!$ and $i_{\ast}$ is equal to $i_{\ast}'i'^!$ . Let $\tilde{j}$ be the inclusion of $Z\setminus Z'$ into Z, and let $\tilde{i}$ be the inclusion of $Z\setminus Z'$ into U. By base change [Reference SaitoSai90, (4.4.3)], we have that $j_{\ast}j^!i_{\ast}i^!=k_{\ast}k^!$ . Therefore, the above triangle becomes

(2.2) \begin{equation}i'_{\ast}i'^!\mathcal{M}\longrightarrow i_{\ast}i^!\mathcal{M}\longrightarrow k_{\ast}k^!\mathcal{M}\overset{+1}\longrightarrow.\end{equation}

The desired exact sequence is obtained as the long exact sequence of cohomology of (2.2).

Proposition 2.2. Let X be a smooth complex variety and let

$$X=Z_0\supsetneq Z_1 \supsetneq \cdots \supsetneq Z_{\ell} \supsetneq Z_{\ell+1}=\emptyset$$

be a decreasing sequence of closed subsets. If $\mathcal{M}$ is a mixed Hodge module on X satisfying $\mathscr{H}^{j-1}_{Z_j}(\mathcal{M})=\mathscr{H}^{j+1}_{Z_j}(\mathcal{M})=0$ for all $0\leqslant j\leqslant \ell$ , then there exists a complex

$$\mathcal{GC}^{\bullet}_{\{Z\}}(\mathcal{M}):\quad 0 \to \mathscr{H}^0_{Z_0\setminus Z_1}(\mathcal{M})\to \mathscr{H}^1_{Z_1\setminus Z_2}(\mathcal{M}) \to \cdots \to \mathscr{H}^{\ell}_{Z_{\ell}\setminus Z_{\ell+1}}(\mathcal{M})\to 0,$$

which is a right resolution of $\mathcal{M}$ in the category of mixed Hodge modules. This complex is known as the Grothendieck–Cousin of $\mathcal{M}$ complex associated to the filtration $\{Z\}$ .

Proof. By Lemma 2.1 and the hypothesis on $\mathcal{M}$ , we obtain exact sequences

$$0 \to \mathscr{H}^j_{Z_j}(\mathcal{M}) \overset{\alpha^j}\longrightarrow \mathscr{H}^j_{Z_j\setminus Z_{j+1}}(\mathcal{M})\overset{\beta^j}\longrightarrow \mathscr{H}^{j+1}_{Z_{j+1}}(\mathcal{M})\to 0.$$

For $0\leqslant j\leqslant \ell$ , we set $\mathcal{GC}^{j}_{\{Z\}}(\mathcal{M})=\mathscr{H}^j_{Z_j\setminus Z_{j+1}}(\mathcal{M})$ , and we define $d^j:\mathcal{GC}^{j}_{\{Z\}}(\mathcal{M})\to \mathcal{GC}^{j+1}_{\{Z\}}(\mathcal{M})$ to be the composition of maps

$$d^j=\alpha^{j+1}\circ \beta^j.$$

By construction, the image of $d^j$ is $\mathscr{H}^{j+1}_{Z_{j+1}}(\mathcal{M})$ , which is the kernel of $d^{j+1}$ . Thus, $\mathcal{GC}^{\bullet}_{\{Z\}}(\mathcal{M})$ is exact in cohomological degrees $j\geqslant 1$ . Since the kernel of $\beta^0$ is $\mathscr{H}^0_{X}(\mathcal{M})=\mathcal{M}$ , we conclude that $\mathcal{GC}^{\bullet}_{\{Z\}}(\mathcal{M})$ is a resolution of $\mathcal{M}$ .

The Grothendieck–Cousin complex was introduced and studied in greater generality in [Reference KempfKem78]. In particular, Kempf considers more general input sheaves. We are interested in the following case.

Corollary 2.3. Let X be an irreducible smooth complex variety of dimension d, and let

$$X=Z_0\supsetneq Z_1 \supsetneq \cdots \supsetneq Z_{d} \supsetneq Z_{d+1}=\emptyset$$

be a decreasing sequence of equidimensional closed subsets with $\operatorname{{\rm codim}}(Z_j,X)=j$ . If $Z_j\setminus Z_{j+1}$ is smooth and the inclusion $Z_j\setminus Z_{j+1}\subseteq X$ is affine for all $0\leqslant j \leqslant d$ , then $\mathcal{GC}^{\bullet}_{\{Z\}}:=\mathcal{GC}^{\bullet}_{\{Z\}}(\mathcal{O}_X^H)$ is a right resolution of $\mathcal{O}_X^H$ in the category of mixed Hodge modules.

Let k be the inclusion of $Z_j\setminus Z_{j+1}$ into X. Since $Z_j\setminus Z_{j+1}$ is smooth, we have $k^{!}\mathcal{O}_X^H=\mathcal{O}_{Z_j\setminus Z_{j+1}}^H(-j)[-j]$ , and since k is affine, we have that $k_{\ast}k^{!}\mathcal{O}_X^H$ only has cohomology in degree j, which is $\mathscr{H}^j_{Z_j\setminus Z_{j+1}}(\mathcal{O}_X^H)$ . By Lemma 2.1, we obtain that $\mathscr{H}^i_{Z_j}(\mathcal{O}_X^H)\cong \mathscr{H}^i_{Z_{j+1}}(\mathcal{O}_X^H)$ for all $i \gt j+1$ . Since $\mathscr{H}^i_X(\mathcal{O}_X^H)=0$ for all $i \gt 0$ , we conclude that, for all $0\leqslant j\leqslant d$ , we have $\mathscr{H}^i_{Z_j}(\mathcal{O}_X^H)=0$ for $i \gt j$ . In particular, $\mathscr{H}^{j+1}_{Z_j}(\mathcal{O}_X^H)=0$ .

3.1 Lie algebras and Weyl groups

We let $\mathfrak{g}$ be a complex simple Lie algebra of rank n. We fix a Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$ , and we write $\Phi\subset \mathfrak{h}^{\ast}$ for the root system of $\mathfrak{g}$ with respect to $\mathfrak{h}$ . We have the root space decomposition of $\mathfrak{g}$ :

$$\mathfrak{g}=\mathfrak{h}\oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}.$$

We denote by $\Delta=\{\alpha_1,\ldots, \alpha_n\}\subset \Phi$ a choice of simple roots, and we write $\Phi^{+}\subset \Phi$ for the set of positive roots with respect to $\Delta$ . The standard Borel subalgebra of $\mathfrak{g}$ associated to this data is

$$\mathfrak{b}=\mathfrak{h}\oplus \bigoplus_{\alpha \in \Phi^+} \mathfrak{g}_{\alpha}.$$

A subset $I\subset \Delta$ determines a root system $\Phi_I$ via

$$\Phi_I:=\Phi\cap \sum_{\alpha\in I}\mathbb{Z}\alpha,$$

and thus determines the following three subalgebras of $\mathfrak{g}$ :

(3.1) \begin{equation}\mathfrak{l}_I=\mathfrak{h}\oplus \bigoplus_{\alpha \in \Phi_I} \mathfrak{g}_{\alpha},\quad \mathfrak{u}_I= \bigoplus_{\alpha \in \Phi^+\setminus \Phi_I}\mathfrak{g}_{\alpha},\quad \mathfrak{p}_I=\mathfrak{l}_I\oplus \mathfrak{u}_I.\end{equation}

We say that $\mathfrak{p}_I$ is a parabolic subalgebra of $\mathfrak{g}$ , with Levi factor $\mathfrak{l}_I$ and nilradical $\mathfrak{u}_I$ . When the set I is understood, we will sometimes omit the subscript I and simply write $\mathfrak{p}=\mathfrak{l}\oplus \mathfrak{u}$ .

Let W denote the Weyl group of $\mathfrak{g}$ , generated by the simple reflections $s_{\alpha}$ , where $\alpha\in \Delta$ . Given $w\in W$ , we write $\ell(w)$ for the length of w, which is the length of the shortest word for w in the simple reflections. The group W has a unique longest element, which we will denote by $w_{\circ}$ , with length $\ell(w_{\circ})=|\Phi^+|$ . Given a parabolic subalgebra $\mathfrak{p}_I=\mathfrak{l}\oplus \mathfrak{u}$ , we write $W_I=\langle s_{\alpha}\mid \alpha \in I\rangle$ for the Weyl group of the Levi factor $\mathfrak{l}$ , with longest element $w_I$ . In each left coset $wW_I\in W/W_I$ , there exists a unique element of minimal length. We consider the set of minimal-length coset representatives:

(3.2) \begin{equation}W^I=\{ w\in W\mid \ell(ww)=\ell(w)+\ell(w')\ \mbox{for all $w'\in W_I$}\}.\end{equation}

Each $x\in W$ may be expressed uniquely as $x=ww'$ , where $w\in W^I$ and $w'\in W_I$ . We write $w_{\circ}(I)$ for the longest element in $W^I$ , which satisfies $w_{\circ}(I)w_I=w_{\circ}$ and $\ell(w_{\circ}(I))=|\Phi^+\setminus \Phi_I|$ .

3.2 Generalized flag varieties and Schubert varieties

Let G be the simply connected complex simple linear algebraic group with Lie algebra $\mathfrak{g}$ , and with Borel subgroup B corresponding to $\mathfrak{b}\subset \mathfrak{g}$ . For $I\subset \Phi$ , we write $P=P_I$ for the parabolic subgroup of G corresponding to $\mathfrak{p}=\mathfrak{p}_I$ . We refer to $G/B$ as the complete flag variety of G, and we call $G/P$ the generalized flag variety associated to P. The dimension of $G/P$ is $\dim(G/P)=\ell(w_{\circ}(I))$ .

Let T be the maximal torus of G associated to the fixed Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ . With this notation, the Weyl group of G is $W=N_G(T)/T$ . The group G has the Bruhat decomposition

$$G=\bigcup_{w\in W} BwB,$$

where the w in BwB is any representative in G of the class of w. Furthermore, $G/P$ has Bruhat decomposition

$$G/P=\bigcup_{w\in W^I}BwP/P.$$

The sets $BwP/P$ are the B-orbits on $G/P$ , known as the Schubert cells. For $w\in W^I$ , we set

(3.3) \begin{equation}O_{w}=BwP/P\subseteq G/P.\end{equation}

The Zariski closure $Z_{w}$ of $O_{w}$ is the Schubert variety associated to $w\in W^I$ in $G/P$ . It has dimension $\ell(w)$ and codimension $c(w):=\operatorname{{\rm codim}}(Z_{w},X)$ . We say that $v\leqslant w$ if $O_v\subseteq Z_w$ .

3.3 Structure of the Grothendieck–Cousin complex

We let $X=G/P_I$ and $d=\dim X$ . For $0\leqslant j\leqslant d$ , we consider the closed sets

$$Z_j=\bigcup_{\substack{v \in W^I\\ c(v)=j}} Z_{v}.$$

The sets $Z_j$ fit into a descending chain

$$X=Z_0\supset Z_1 \supset \cdots \supset Z_d \supset Z_{d+1}=\emptyset,$$

where the inclusion $Z_j\setminus Z_{j+1}\subseteq X$ is affine. Each $Z_j\setminus Z_{j+1}$ is the disjoint union of closed smooth subvarieties $O_v$ over $v\in W^I$ with $c(v)=j$ , so $Z_j\setminus Z_{j+1}$ is smooth and $Z_j$ satisfies $\operatorname{{\rm codim}}(Z_j,X)=j$ .

By Corollary 2.3, the mixed Hodge module $\mathcal{O}^H_X$ admits a right resolution

$$\mathcal{GC}^{\bullet}_{\{Z\}}:\quad 0 \longrightarrow \mathscr{H}^0_{Z_0\setminus Z_1}(\mathcal{O}_X^H)\longrightarrow \mathscr{H}^1_{Z_1\setminus Z_2}(\mathcal{O}_X^H) \longrightarrow \cdots \longrightarrow \mathscr{H}^d_{Z_d\setminus Z_{d+1}}(\mathcal{O}_X^H)\longrightarrow 0,$$

where we view $\mathscr{H}^j_{Z_j\setminus Z_{j+1}}(\mathcal{O}_X^H)$ as being in cohomological degree j. For $w\in W^I$ , we define

(3.4) \begin{equation}\mathcal{GC}^{\bullet}_{w}=\mathscr{H}^0_{Z_{w}}(\mathcal{GC}^{\bullet}_{\{Z\}}).\end{equation}

The main result of this subsection is the following.

1. (i) For all $0\leqslant j\leqslant d$ , we have

   $$\mathcal{H}^j(\mathcal{GC}^{\bullet}_{w})=\mathscr{H}^j_{Z_{w}}(\mathcal{O}_X^H).$$

2. (ii) $\mathcal{GC}^j_{w}=0$ for $j<c(w)$ , and for $j\geqslant c(w)$ we have an isomorphism

   $$\mathcal{GC}^j_{w}=\bigoplus_{\substack{v \leqslant w \\ c(v)=j}} \mathscr{H}^j_{O_{v}}(\mathcal{O}_X^H).$$

3. (iii) For $u<v \leqslant w$ with $j=c(v)=c(u)-1$ , the differential in $\mathcal{GC}^{\bullet}_{w}$ induces a nonzero morphism:

   $$\mathscr{H}^j_{O_{v}}(\mathcal{O}_X^H)\longrightarrow \mathscr{H}^{j+1}_{O_{u}}(\mathcal{O}_X^H),$$
   with $\mathcal{L}(Z_u,X)$ in its image.

We note that part (iii) is not formal: its proof uses the fact that Schubert varieties are normal. In order to prove Theorem 3.1, we first need the following lemma.

Lemma 3.2. For $0\leqslant j\leqslant d$ , we have

$$\mathscr{H}^j_{Z_j\setminus Z_{j+1}}(\mathcal{O}_X^H)=\bigoplus_{\substack{v \in W^I\\ c(v)=j}} \mathscr{H}^j_{O_{v}}(\mathcal{O}_X^H).$$

Proof. Let $k^j:Z_j\setminus Z_{j+1}\hookrightarrow X$ , $i_v:O_v\hookrightarrow X$ , and $i^j_v: O_v\hookrightarrow Z_j\setminus Z_{j+1}$ . The variety $Z_j\setminus Z_{j+1}$ is the disjoint union of closed smooth subvarieties $O_v$ over $v\in W^I$ with $c(v)=j$ . Thus,

$$\mathcal{O}^H_{Z_j\setminus Z_{j+1}}=\bigoplus_{\substack{v \in W^I\\ c(v)=j}} i^j_{v\ast}\mathcal{O}^H_{O_v}.$$

Since $\mathscr{H}^j_{Z_j\setminus Z_{j+1}}(\mathcal{O}_X^H)=k^j_{\ast} \mathcal{O}^H_{Z_j\setminus Z_{j+1}}(-j)[-j]$ and $\mathscr{H}^j_{O_{v}}(\mathcal{O}_X^H)=i_{v\ast}\mathcal{O}^H_{O_v}(-j)[-j]$ , we obtain the desired result using $k^j_{\ast}\circ i^j_{v\ast}=i_{v\ast}$ .

Proof of Theorem 3.1. Since $\mathcal{GC}^{\bullet}_{\{Z\}}$ is a resolution of $\mathcal{O}_X^H$ , to prove parts (i) and (ii) it suffices to show that

$$\mathscr{H}^0_{Z_{w}}( \mathscr{H}^j_{Z_j\setminus Z_{j+1}}(\mathcal{O}_X^H))=\bigoplus_{\substack{v \leqslant w \\ c(v)=j}}\mathscr{H}^j_{O_{v}}(\mathcal{O}_X^H),$$

and $\mathscr{H}^i_{Z_{w}}(\mathscr{H}^{j}_{Z_j\setminus Z_{j+1}}(\mathcal{O}_X^H))=0$ for $i \gt 0$ . By Lemma 3.2, we have that

$$\mathscr{H}^j_{Z_j\setminus Z_{j+1}}(\mathcal{O}_X^H)=\bigoplus_{\substack{v \in W^I\\ c(v)=j}} \mathscr{H}^j_{O_{v}}(\mathcal{O}_X^H).$$

Let $v\in W^I$ with $c(v)=j$ . If $v\leqslant w$ , then $O_{v}\cap Z_{w}=O_{v}$ . Let $\beta:O_{v}\hookrightarrow Z_{w}$ , $\alpha:O_{v}\hookrightarrow X$ , $\gamma:Z_{w}\hookrightarrow X$ , so that $\alpha=\gamma\circ \beta$ . By base change [Reference SaitoSai90, (4.4.3)], we have $\gamma_{\ast}\gamma^!\alpha_{\ast}\alpha^!=\gamma_{\ast}\beta_{\ast}\alpha^!=\alpha_{\ast}\alpha^!$ . Thus, there is a Grothendieck spectral sequence

$$E^{p,q}_2=\mathscr{H}^p_{Z_{w}}(\mathscr{H}^q_{O_{v}}(\mathcal{O}_X^H)) \implies \mathscr{H}^{p+q}_{O_{v}\cap Z_{w}}(\mathcal{O}_X^H)=\mathscr{H}^{p+q}_{O_{v}}(\mathcal{O}_X^H).$$

Since $\mathscr{H}^q_{O_{v}}(\mathcal{O}_X^H)\neq 0$ only if $q=j$ , this spectral sequence has only one nonzero column on the $E_2$ -page, and has only one nonzero row on the $E_{\infty}$ -page. Thus, the spectral sequence degenerates at $E_2$ , so we have $\mathscr{H}^0_{Z_{w}}(\mathscr{H}^j_{O_{v}}(\mathcal{O}_X^H))=\mathscr{H}^{j}_{O_{v}}(\mathcal{O}_X^H)$ , and $\mathscr{H}^i_{Z_{w}}(\mathscr{H}^{j}_{O_{v}}(\mathcal{O}_X^H))=0$ for $i \gt 0$ . On the other hand, if $v\nleq w$ , then $O_v\cap Z_w=\emptyset$ , so by base change we obtain a spectral sequence

$$E^{p,q}_2=\mathscr{H}^p_{Z_{w}}(\mathscr{H}^q_{O_{v}}(\mathcal{O}_X^H)) \implies 0.$$

Since $\mathscr{H}^q_{O_{v}}(\mathcal{O}_X^H)\neq 0$ only if $q=j$ , this spectral sequence is degenerate, yielding $\mathscr{H}^i_{Z_{w}}(\mathscr{H}^{j}_{O_{v}}(\mathcal{O}_X^H))=0$ for all i and j, which completes the proof of parts (i) and (ii).

It suffices to prove part (iii) in the case $w=w_{\circ}(I)$ is the longest element of $W^I$ so that $\mathcal{GC}^{\bullet}_{w}=\mathcal{GC}^{\bullet}_{\{Z\}}$ , and after restricting to the following open set:

$$U=X \setminus \bigcup_{x \not\geqslant u} Z_{x}.$$

We note that $O_u$ is closed in U. By construction of $\mathcal{GC}^{\bullet}_{\{Z\}}$ , the map from $\mathscr{H}^j_{O_{v}}(\mathcal{O}_U^H)$ to $\mathscr{H}^{j+1}_{O_{u}}(\mathcal{O}_U^H)$ in $\mathcal{GC}^{\bullet}_{\{Z\}}|_U$ fits into the following exact sequence (see Lemma 2.1):

$$ 0\longrightarrow \mathscr{H}^j_{\overline{O}_v}(\mathcal{O}_U^H)\longrightarrow \mathscr{H}^j_{O_{v}}(\mathcal{O}_U^H)\longrightarrow \mathscr{H}^{j+1}_{O_{u}}(\mathcal{O}_U^H)\longrightarrow \mathscr{H}^{j+1}_{\overline{O}_v}(\mathcal{O}_U^H)\longrightarrow 0. $$

Since $Z_{v}$ is normal [Reference Ramanan and RamanathanRR85, Theorem 3], its singular locus must have dimension $\leqslant \ell(v)-2$ , and by equivariance it must be a union of Schubert varieties. Since $\ell(v)=\ell(u)+1$ , it follows that $\overline{O}_v$ is smooth in U, so we have $\mathscr{H}^{j+1}_{\overline{O}_v}(\mathcal{O}_U^H)=0$ . As $\mathscr{H}^{j+1}_{O_{u}}(\mathcal{O}_U^H)\cong \mathcal{L}(Z_u,X)|_U\neq 0$ , we conclude that the map $\mathscr{H}^j_{O_{v}}(\mathcal{O}_U^H)\to \mathscr{H}^{j+1}_{O_{u}}(\mathcal{O}_U^H)$ is surjective, and therefore the desired map is nonzero, with $\mathcal{L}(Z_u,X)$ in its image.

Since $\mathcal{GC}_w^{\bullet}$ is a complex of mixed Hodge modules, it is endowed with a Hodge filtration $F_{\bullet}$ and a weight filtration $W_{\bullet}$ . The next result gives our main tool to get information about the mixed Hodge module structure on local cohomology. The following result implies Theorem A.

Theorem 3.3. Let $w\in W^I$ , let $p\in \mathbb{Z}$ , and let $q\geqslant 0$ . The following is true about the Hodge and weight filtrations on local cohomology with support in $Z_{w}$ :

$$F_p(\mathscr{H}_{Z_w}^{q}(\mathcal{O}_X^H))\cong \mathcal{H}^q(F_p(\mathcal{GC}_w^{\bullet})) \quad{and}\quad W_p(\mathscr{H}_{Z_w}^{q}(\mathcal{O}_X^H))\cong \mathcal{H}^q(W_p(\mathcal{GC}_w^{\bullet})).$$

Furthermore, the following is true for $i\in \mathbb{Z}$ :

$$F_pW_i(\mathscr{H}_{Z_w}^{q}(\mathcal{O}_X^H))\cong \mathcal{H}^q(F_pW_i(\mathcal{GC}_w^{\bullet})),$$

where the left-hand side is the intersection of the two filtrations.

3.4 Parabolic Verma modules, P-orbits, and localization

Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$ , and let $\mathfrak{p}=\mathfrak{p}_I$ be a parabolic subalgebra of $\mathfrak{g}$ with Levi decomposition $\mathfrak{p}=\mathfrak{l}\oplus \mathfrak{u}$ and adjoint group $P=P_I$ . Let ${\rm mod}_P(\mathfrak{g},\chi_{0})$ denote the category of P-equivariant finitely generated $U(\mathfrak{g})$ -modules with trivial central character (see [Reference Hotta, Takeuchi and TanisakiHTT08, § 11.5]). We write $\rho=({1}/{2})\sum_{\alpha\in \Phi^+}\alpha$ .

For $w\in W^I$ , we let $F_{w}$ denote the finite-dimensional irreducible $\mathfrak{l}$ -representation with highest weight $ww_I\rho-\rho$ , where $w_I$ is the longest element in $W_I$ . We view $F_{w}$ as a representation of $\mathfrak{p}$ , with $\mathfrak{u}$ acting trivially. The parabolic Verma module (sometimes called the generalized Verma module) with highest weight $ww_I\rho-\rho$ is the left $U(\mathfrak{g})$ -module

(3.5) \begin{equation}M_I(w)=U(\mathfrak{g})\otimes _{U(\mathfrak{p})}F_{w}.\end{equation}

The module $M_I(w)$ is a quotient of the Verma module $M_{\emptyset}(w)$ , and thus has unique simple quotient L(w), the irreducible representation of $\mathfrak{g}$ with highest weight $ww_I\rho-\rho$ . We define the BGG dual of $M_I(w)$ as $N_I(w)=M_I(w)^{\vee}$ (see [Reference HumphreysHum08, § 3.2]). The module $N_I(w)$ is dual to $M_I(w)$ in the sense that they have the same composition factors and multiplicities of factors, and the submodule lattice of $N_I(w)$ is dual to that of $M_I(w)$ (see [Reference HumphreysHum08, § 3.2 Theorem]).

We now state the Beilinson–Bernstein theorem [Reference Beilinson and BernsteinBB81] for trivial central characters. Consider the natural left action of P on the complete flag variety $G/B$ , with orbits

(3.6) \begin{equation}G/B=\bigcup_{w\in W^I}\mathbb{O}_w\quad{\rm where}\ \mathbb{O}_w=\bigcup_{z\in W_I} O_{wz},\end{equation}

in which $O_{y}$ denotes the B-orbit of yB in $G/B$ . Here, $O_{ww_I}$ is dense in $\mathbb{O}_w$ , so that $\overline{O}_{ww_I}=\overline{\mathbb{O}}_w=Z_{w}$ in $G/B$ . For $w\in W^I$ , we define the $\mathcal{D}_{G/B}$ -modules:

(3.7) \begin{equation}\mathcal{N}_w=\mathscr{H}_{\mathbb{O}_w}^{c(w)}(\mathcal{O}_{G/B}),\quad \mathcal{M}_w=\mathbb{D}\mathcal{N}_w,\quad \mathcal{L}_w=\mathcal{L}(\mathbb{O}_w,G/B).\end{equation}

For the following statements, we write ${\rm mod}_P(\mathcal{D}_{G/B})$ for the category of P-equivariant coherent $\mathcal{D}_{G/B}$ -modules (see [Reference Lõrincz and WaltherLW19, § 2.1] or [Reference Hotta, Takeuchi and TanisakiHTT08, § 11.5]), and we write ${\rm mod}_B(\mathcal{D}_{G/P})$ for the category of B-equivariant coherent $\mathcal{D}_{G/P}$ -modules.

1. (i) The global sections functor $\Gamma(G/B,-)$ induces an equivalence of categories

   $${\rm mod}_P(\mathcal{D}_{G/B})\cong {\rm mod}_P(\mathfrak{g},\chi_0).$$

2. (ii) Furthermore, given $w\in W^I$ , we have

   $$\Gamma(G/B,\mathcal{N}_w)\cong N_I(w),\quad \Gamma(G/B,\mathcal{M}_w)\cong M_I(w),\quad \Gamma(G/B,\mathcal{L}_w)\cong L(w).$$

For the remainder of the subsection, we write $\pi^{\ast}$ for the (unshifted) pullback functor of $\mathcal{D}$ -modules [Reference Hotta, Takeuchi and TanisakiHTT08, § 1.3] and we write $\pi_{\ast}$ for the direct image functor of quasi-coherent $\mathcal{O}$ -modules.

Theorem 3.5. Consider the natural projection $\pi: G/B \to G/P$ . Then, $\pi^{\ast}$ is exact and induces an equivalence of categories

$${\rm mod}_B(\mathcal{D}_{G/P})\cong {\rm mod}_P(\mathcal{D}_{G/B}).$$

Furthermore, given $w\in W^I$ , we have

$$\pi^{\ast}\mathscr{H}_{O_{w}}^{c(w)}(\mathcal{O}_{G/P})\cong \mathcal{N}_w,\quad \pi^{\ast}(\mathbb{D}(\mathscr{H}_{O_{w}}^{c(w)}(\mathcal{O}_{G/P})))\cong \mathcal{M}_w,\quad \pi^{\ast}\mathcal{L}(O_{w},G/P)\cong \mathcal{L}_w.$$

It follows from (3.6) that $\pi^{-1}(O_w)=\mathbb{O}_w$ , so base change [Reference Hotta, Takeuchi and TanisakiHTT08, Theorem 1.7.3] yields the first two isomorphisms. The third isomorphism follows from the first and the equivalence of categories.

Combining Theorems 3.4 and 3.5, we obtain the following.

1. (i) The global sections functor $\Gamma(G/P,-)$ induces an equivalence of categories

   $${\rm mod}_B(\mathcal{D}_{G/P})\cong {\rm mod}_P(\mathfrak{g},\chi_0).$$

2. (ii) Furthermore, given $w\in W^I$ , we have

   \begin{align*}&\Gamma(G/P,\mathscr{H}_{O_{w}}^{c(w)}(\mathcal{O}_{G/P}))\cong N_I(w),\quad \Gamma(G/P,\mathbb{D}(\mathscr{H}_{O_{w}}^{c(w)}(\mathcal{O}_{G/P})))\cong M_I(w),\\& \Gamma(G/P,\mathcal{L}_w)\cong L(w).\end{align*}

Proof. We follow the strategy described in [Reference Borho and BrylinskiBB85, footnote p. 14] (see also [Reference Holland and PoloHP96, Proposition 2.14]). Let $\pi:G/B\to G/P$ be the projection. For $\mathcal{M}\in {\rm mod}_B(\mathcal{D}_{G/P})$ , we have that $\mathbb{R}^j\pi_{\ast}(\pi^{\ast}\mathcal{M})=0$ for $j \gt 0$ (see [Reference Holland and PoloHP96, Proposition 2.14(a)]). By Theorem 3.4(1), we have that the Leray–Serre spectral sequence

$$H^i(G/P,\mathbb{R}^j\pi_{\ast}(\pi^{\ast}\mathcal{M}))\quad \implies \quad H^{i+j}(G/B,\pi^{\ast}\mathcal{M})$$

degenerates to yield an isomorphism $\Gamma(G/P,\mathcal{M})\cong \Gamma(G/B, \pi^{\ast}\mathcal{M})$ as $U(\mathfrak{g})$ -modules. By Theorems 3.4 and 3.5, this proves both part (i) and part (ii).

For $w\in W^I$ , we consider the global Grothendieck–Cousin complex $GC_w^{\bullet}=\Gamma(X,\mathcal{GC}^{\bullet}_w)$ :

(3.8) \begin{equation}GC^{\bullet}_w:\quad 0 \longrightarrow N_I(w) \longrightarrow \bigoplus_{\substack{v\leqslant w\\ c(v)=c(w)+1}} N_I(v)\longrightarrow \cdots \longrightarrow \bigoplus_{\substack{v\leqslant w\\ c(w)=\ell(w_I)-1}} N_I(v) \longrightarrow N_I(e)\longrightarrow 0.\end{equation}

We define the Hodge filtration $F_{\bullet}$ and weight filtration $W_{\bullet}$ on $N_I(w)$ via (see [Reference Davis and VilonenDV23, Theorem 1.1])

(3.9) \begin{equation}F_p(N_I(w))=\Gamma(F_p(\mathscr{H}_{O_{w}}^{c(w)}(\mathcal{O}_{X}))) \quad {\rm and}\quad W_p(N_I(w))=\Gamma(W_p(\mathscr{H}_{O_{w}}^{c(w)}(\mathcal{O}_{X}))).\end{equation}

We endow $M_I(w)$ with filtrations in similar manner. The weight filtration on $M_I(w)$ is called the Jantzen filtration (see [Reference Beilinson and BernsteinBB93, Reference HumphreysHum08]). Similarly, we define Hodge and weight filtrations on the global local cohomology modules for $q\geqslant 0$ :

(3.10) \begin{equation}F_p(H_{Z_w}^{q}(X,\mathcal{O}_X))=\Gamma(F_p(\mathscr{H}^q_{Z_w}(\mathcal{O}_X))) \quad{\rm and}\quad W_p(H_{Z_w}^{q}(X,\mathcal{O}_X))=\Gamma(W_p(\mathscr{H}^q_{Z_w}(\mathcal{O}_X))).\end{equation}

By Theorem 3.1, Corollary 3.6, and [Reference Davis and VilonenDV23, Theorem 1.1], we obtain the following.

We do not pursue the Hodge filtration further in this work.

4.1 The Grassmannian and minimal-length coset representatives

Let $\mathfrak{g}=\mathfrak{sl}_n(\mathbb{C})$ , with simple roots $\Delta=\{\alpha_1,\ldots, \alpha_{n-1}\}$ , where $\alpha_i=\epsilon_i-\epsilon_{i+1}$ . Let $I=\Delta\setminus \{\alpha_k\}$ and let $\mathfrak{p}=\mathfrak{p}_I$ . We write $G=\operatorname{{\rm SL}}_n(\mathbb{C})$ and P for the groups corresponding to $\mathfrak{g}$ and $\mathfrak{p}$ , so that $X\cong G/P$ .

The set of minimal-length coset representatives of $W/W_I$ is

(4.1) \begin{equation}W^I=\{(i_1,\ldots, i_n)\in \mathfrak{S}_n\mid i_1<\cdots <i_k, i_{k+1}<\cdots<i_n\},\end{equation}

where $\mathfrak{S}_n$ is the symmetric group on n elements. In our notation, the identity e is represented by $(1,2,\ldots, n)$ , and the longest element $w_{\circ}(I)$ is represented by $(n-k+1,\ldots, n,1,\ldots, n-k)$ . To each $w=(i_1,\ldots, i_n)\in W^I$ , we associate a Young diagram $\underline{a}=\underline{a}(w)$ :

$$a_1=i_k-k,\quad a_2=i_{k-1}-(k-1),\quad \ldots, \quad a_k=i_1-1,$$

i.e. $a_j=i_{k+1-j}-(k+1-j)$ for $1\leqslant j\leqslant k$ and $a_j=0$ for $j \gt k$ . For instance, we have

$$e \;\longleftrightarrow \;(0,\ldots, 0) \quad {\rm and}\quad w_{\circ}(I)\; \longleftrightarrow \; (n-k,\ldots, n-k).$$

We obtain a bijection between $W^I$ and the set

(4.2) \begin{equation}\mathcal{R}_{k,n}=\{\underline{a} \mid n-k\geqslant a_1\geqslant a_2\geqslant \cdots \geqslant a_k\geqslant 0\}.\end{equation}

We think of $\mathcal{R}_{k,n}$ as the set of Young diagrams that fit into a $k\times (n-k)$ rectangle.

Given $w\in W^I$ with $\underline{a}=\underline{a}(w)$ , we write $O_{\underline{a}}=O_{w}$ and $Z_{\underline{a}}=Z_{w}$ for the corresponding Schubert cell and Schubert variety. We have $\dim Z_{\underline{a}}=|\underline{a}|=\ell(w)$ . We write $M(\underline{a})=M_I(w)$ , $N(\underline{a})=N_I(w)$ , $L(\underline{a})=L_I(w)$ , so that $M(\underline{a})$ and $L(\underline{a})$ have highest weight

(4.3) \begin{equation}\lambda_{\underline{a},n}:=ww_I\rho-\rho=(-a_k^c,-a_{k-1}^c,\ldots, -a_1^c\mid (a^c)_1',(a^c)_2',\ldots, (a^c)_{n-k}'),\end{equation}

where $\underline{a}^c=(n-k-a_k,\ldots, n-k-a_1)$ is the complementary partition of $\underline{a}$ in $\mathcal{R}_{k,n}$ and $(\underline{a}^c)'$ is the conjugate partition to $\underline{a}^c$ . We use the notation $\lambda_{\underline{a},n}$ to emphasize the dependence on n.

4.2 Properties of the parabolic Verma modules

Let $\underline{a}\in \mathcal{R}_{k,n}$ . We freely identify the modules $N(\underline{a})$ , $M(\underline{a})$ , and $L(\underline{a})$ with their $\mathcal{D}_X$ -module-theoretic counterparts via Corollary 3.6, i.e.

$$ N(\underline{a})\;\longleftrightarrow \;\mathscr{H}^{c(\underline{a})}_{O_{\underline{a}}}(\mathcal{O}_X),\quad M(\underline{a})\;\longleftrightarrow \; \mathbb{D}(\mathscr{H}^{c(\underline{a})}_{O_{\underline{a}}}(\mathcal{O}_X)),\quad L(\underline{a})\;\longleftrightarrow \; \mathcal{L}(O_{\underline{a}},X),$$

and we recall that $M(\underline{a})$ and $N(\underline{a})$ are endowed with weight filtrations $W_{\bullet}$ via this equivalence.

A Loewy filtration on a finite length $U(\mathfrak{g})$ -module M is a filtration of minimal length such that the subquotients are semi-simple. Two examples of Loewy filtrations are the socle and radical filtrations (see [Reference HumphreysHum08, § 8.14]). We say that M is rigid if the socle and radical filtrations coincide, in which case there is a unique Loewy filtration on M. For $p\in \mathbb{Z}$ , we write

(4.4) \begin{equation}\mathcal{Z}_p(\underline{a})=\{\mathbb{D}\in {\rm Dyck}(\underline{a}) \mid |\mathbb{D}|=p+|\underline{a}|-2d_{X} \}.\end{equation}

We emphasize that elements of $\mathcal{Z}_p(\underline{a})$ may have Dyck paths of length 1, and they have no bullets. Recall that $\underline{a}^{\mathbb{D}}$ is the partition obtained from $\underline{a}$ by removing the boxes in the support of $\mathbb{D}$ (see (1.6)).

The following result is a collection of facts from the literature.

Theorem 4.1. The modules $M(\underline{a})$ and $N(\underline{a})$ are rigid, and the weight filtration $W_{\bullet}$ is the unique Loewy filtration. For $p\in \mathbb{Z}$ , the graded pieces of the weight filtration are given by

$$ {\rm gr}_p^W(M(\underline{a}))=\bigoplus_{\mathbb{D}\in \mathcal{Z}_{-p}(\underline{a})} L(\underline{a}^{\mathbb{D}}) \quad{and}\quad {\rm gr}_p^W(N(\underline{a}))=\bigoplus_{\mathbb{D}\in \mathcal{Z}_p(\underline{a})} L(\underline{a}^{\mathbb{D}}).$$

In particular, each composition factor of $M(\underline{a})$ and $N(\underline{a})$ appears with multiplicity one.

We remark that the original combinatorial description of the relevant Kazhdan–Lusztig polynomials may be found in [Reference Lascoux and SchützenbergerLS81].

It follows from Theorem 4.1 that the weight filtration on $N(\underline{a})$ is given by

\begin{align*}\operatorname{{\rm gr}}^W_{11}(N(\underline{a})) & =L(3,2,2),\quad \operatorname{{\rm gr}}^W_{12}(N(\underline{a}))=L(2,2,2)\oplus L(3,2,1),\\\operatorname{{\rm gr}}^W_{13}(N(\underline{a})) & =L(3,0,0)\oplus L(2,2,1)\oplus L(1,0,0),\quad \operatorname{{\rm gr}}^W_{14}(N(\underline{a}))=L(2,0,0).\end{align*}

The composition factors of $M(\underline{a})$ are the same and $\operatorname{{\rm gr}}_p^W(M(\underline{a}))=\operatorname{{\rm gr}}^W_{-p}(N(\underline{a}))$ for all $p\in \mathbb{Z}$ .

For a larger example, see [Reference HuangHua20, Example 2.19] for composition factors of N(5,4,2,1) when $k=4$ , $n= 9$ , noting that the author uses the complementary convention for partitions. In particular, the simple L(2,2) (shown in the bottom right) has weight $p=32$ , which is maximal in N(5,4,2,1).

Given $\mathbb{D}\in {\rm Dyck}(\underline{a})$ , we write $v_{\underline{a}}^{\mathbb{D}}$ for any vector v in $M(\underline{a})$ such that the class of v in a quotient of $M(\underline{a})$ is a highest-weight vector for $L(\underline{a}^{\mathbb{D}})$ . Such vectors are called primitive vectors. For two Dyck patterns with no bullets $\mathbb{D},\mathbb{D}'\in {\rm Dyck}(\underline{a})$ , we write $\mathbb{D}\subseteq \mathbb{D}'$ if all paths in $\mathbb{D}$ are also paths in $\mathbb{D}'$ .

1. (i) If $\mathbb{D}\subseteq \mathbb{D}'$ , then $v^{\mathbb{D}'}_{\underline{a}}\in U(\mathfrak{g}) v_{\underline{a}}^{\mathbb{D}}$ .

2. (ii) If $\mathbb{D}=\{(x,y)\}$ and $v^{\mathbb{D}'}_{\underline{a}}\in U(\mathfrak{g}) v_{\underline{a}}^{\mathbb{D}}$ , then $\mathbb{D}\subseteq \mathbb{D}'$ .

We emphasize that the converse to part (i) is false in general (see [Reference Su and ZhangSZ12, Theorem 5.18] or [Reference Boe and EnrightBE86]). For instance, in Example 4.2, a primitive vector for L(2,0,0) generates a primitive vector for L(1,0,0). In the terminology of [Reference Su and ZhangSZ12], the Dyck pattern corresponding to L(1,0,0) is a ‘path with bridges’.

For each $r\geqslant n-k$ , we consider the general linear Lie algebra $\mathfrak{gl}_{k+r}$ with the parabolic subalgebra $\mathfrak{p}_{k+r}$ with Levi subalgebra $\mathfrak{gl}_{k}\times \mathfrak{gl}_{r}$ . Let $\underline{a}$ be as above, and consider the parabolic Verma module $M^{k+r}(\lambda_{\underline{a}})$ over $\mathfrak{gl}_{k+r}$ with highest weight (see §4.1)

$$\lambda_{\underline{a},n}=(-a_k^c,-a_{k-1}^c,\ldots, -a_1^c\mid (a^c)_1',(a^c)_2',\ldots, (a^c)_r'),$$

so that $M^{n}(\lambda_{\underline{a},n})=M(\underline{a})$ . We consider the irreducible $\mathfrak{gl}_{k+r}$ -module $L^{k+r}(\lambda_{\underline{a},n})$ of highest weight $\lambda_{\underline{a},n}$ , so that $L^{n}(\lambda_{\underline{a},n})=L(\underline{a})$ . Taking the limit as $r\to \infty$ , we obtain $\mathfrak{gl}_{k+\infty}$ and the modules $M^{k+\infty}(\lambda_{\underline{a},n})$ , $L^{k+\infty}(\lambda_{\underline{a},n })$ . We write $\mathcal{O}^{\mathfrak{p}_{k+r}}_{{\rm int}}$ and $\mathcal{O}^{\mathfrak{p}_{k+\infty}}_{{\rm int}}$ for the corresponding parabolic categories $\mathcal{O}$ with integral weights. Similar to Theorem 4.1, $M^{k+\infty}(\lambda_{\underline{a},n})$ has composition factors $L^{k+\infty}(-b_k,-b_{k-1},\ldots, -b_1\,|\,b_1',b_2',\ldots)$ , where $\underline{b}=(b_1,\ldots, b_k)$ is obtained from $\underline{a}^c$ by adding an admissible Dyck pattern with no bullets (over an infinite-dimensional space, it is more convenient to work with $\underline{a}^c$ rather than $\underline{a}$ ). We have an exact truncation functor $\operatorname{{\rm tr}}_{n-k}:\mathcal{O}^{\mathfrak{p}_{k+\infty}}_{{\rm int}}\to \mathcal{O}^{\mathfrak{p}_{n}}_{{\rm int}}$ sending $M^{k+\infty}(\lambda_{\underline{a},n})$ to $M(\underline{a})$ and $L^{k+\infty}(\lambda_{\underline{a},n})$ to $L(\underline{a})$ . Thus, $\operatorname{{\rm tr}}_{n-k}L^{k+\infty}(-b_k,\ldots, -b_1\,|\,b_1',b_2',\ldots)$ is nonzero if and only if $\underline{b}\in \mathcal{R}_{k,n}$ , in which case $\operatorname{{\rm tr}}_{n-k}L^{k+\infty}(-b_k,-b_{k-1},\ldots, -b_1\,|\,b_1',b_2',\ldots)=L(n-k-b_k,\ldots, n-k-b_1)=L(\underline{b}^c).$

For $s\geqslant r$ , we consider the general linear Lie superalgebra $\mathfrak{gl}_{k|s}$ with parabolic subalgebra $\mathfrak{p}_{k|s}$ and with Kac module $K^{k|s}(\lambda_{\underline{a},n}^{\sharp})$ and irreducible module $L^{k|r}(\lambda_{\underline{a},n}^{\sharp})$ , where

$$\lambda^{\sharp}_{\underline{a},n}=(-a_k^c,-a_{k-1}^c,\ldots, -a_1^c\mid a_1^c, a_2^c,\ldots, a_k^c,0^{s-k}).$$

We write $\mathfrak{gl}_{k|\infty}$ , $\mathfrak{p}_{k|\infty}$ , $K^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ , and $L^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ for their limits, and we write $\mathcal{O}^{\mathfrak{p}_{k|s}}_{{\rm int}}$ and $\mathcal{O}^{\mathfrak{p}_{k|\infty}}_{{\rm int}}$ for the corresponding parabolic categories $\mathcal{O}$ . There is an exact truncation functor $\operatorname{{\rm tr}}_s:\mathcal{O}^{\mathfrak{p}_{k|\infty}}_{{\rm int}}\to \mathcal{O}^{\mathfrak{p}_{k|s}}_{{\rm int}}$ sending $M^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ to $M^{k|s}(\lambda_{\underline{a},n}^{\sharp})$ and $L^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ to $L^{k|s}(\lambda_{\underline{a},n}^{\sharp})$ .

As a consequence of super duality (see [Reference Su and ZhangSZ12, Theorem 3.13]), there is an equivalence of categories $\mathcal{O}^{\mathfrak{p}_{k+\infty}}_{{\rm int}}\overset{\sim}\to \mathcal{O}^{\mathfrak{p}_{k|\infty}}_{{\rm int}}$ identifying the pairs

$$M^{k+\infty}(\lambda_{\underline{a},n})\;\longleftrightarrow \;K^{k|\infty}(\lambda_{\underline{a},n}^{\sharp}),\quad L^{k+\infty}(\lambda_{\underline{a},n})\;\longleftrightarrow \;L^{k|\infty}(\lambda_{\underline{a},n}^{\sharp}).$$

In particular, the composition factors of $K^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ are given by $L^{k|\infty}(-b_k,\ldots, -b_1\,|\,b_1,\ldots, b_k)$ , where $\underline{b}=(b_1,\ldots, b_k)$ is obtained from $\underline{a}^c$ by adding an admissible Dyck pattern with no bullets. Given $\mathbb{D}\in {\rm Dyck}(\underline{a})$ , we write $v_{\underline{a}}^{\mathbb{D}}$ for any vector in $M(\underline{a})$ that becomes a highest-weight vector for $L(\underline{a}^{\mathbb{D}})$ in a quotient of $M(\underline{a})$ , and we write $v_{\underline{a}}^{\mathbb{D},\infty}$ for any vector in $M^{k+\infty}(\lambda_{\underline{a},n})$ that becomes a highest-weight vector for $L^{k+\infty}(\lambda_{\underline{a}^{\mathbb{D}},n})$ in a quotient of $M^{k+\infty}(\lambda_{\underline{a},n})$ . Similarly, we write ${\sharp}{}{v_{\underline{a}}^{\mathbb{D},\infty}}$ for any vector in $K^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ that becomes a highest-weight vector for $L^{k|\infty}(\lambda^{\sharp}_{\underline{a}^{\mathbb{D}},n})$ in a quotient of $K^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ .

With this setup, we now proceed with the proof of Theorem 4.3.

1. (i) It suffices to show that $v^{\mathbb{D}'}_{\underline{a}}\in U(\mathfrak{gl}_n) v_{\underline{a}}^{\mathbb{D}}$ in $M(\underline{a})$ when $\mathbb{D}\subseteq \mathbb{D}'$ and $|\mathbb{D}'|=|\mathbb{D}|+1$ . Since the socle filtration on $M^{k+\infty}(\lambda_{\underline{a},n})$ is determined by the number of paths in the Dyck pattern, it suffices to prove that $v^{\mathbb{D}',\infty}_{\underline{a}}\in U(\mathfrak{gl}_{k+\infty}) v_{\underline{a}}^{\mathbb{D},\infty}$ in $M^{k+\infty}(\lambda_{\underline{a},n})$ . By super duality, this is equivalent to ${\sharp}{}{v^{\mathbb{D}',\infty}_{\underline{a}}}\in U(\mathfrak{gl}_{k|\infty}) {\sharp}{}{v_{\underline{a}}^{\mathbb{D},\infty}}$ in $K^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ , which follows from [Reference Su and ZhangSZ12, Theorem 5.18].

2. (ii) Suppose that $\mathbb{D}=\{(x,y)\}$ and that $v^{\mathbb{D}'}_{\underline{a}}\in U(\mathfrak{gl}_n) v_{\underline{a}}^{\mathbb{D}}$ in $M(\underline{a})$ . By super duality, we have that ${\sharp}{}{v^{\mathbb{D}',\infty}_{\underline{a}}}\in U(\mathfrak{gl}_{k|\infty}) {\sharp}{}{v_{\underline{a}}^{\mathbb{D},\infty}}$ in $K^{k|\infty}(\lambda_{\underline{a},n}^{\sharp})$ . We induct on $|\mathbb{D}'|$ . If $|\mathbb{D}'|=2$ , then since $\mathbb{D}=\{(x,y)\}$ is a ‘path without bridges’, it follows from [Reference Su and ZhangSZ12, Theorem 5.18] that $\mathbb{D}\subseteq \mathbb{D}'$ . If $|\mathbb{D}'| \gt 2$ , then by inductive hypothesis there exists $\mathbb{D}''$ with $\mathbb{D}\subseteq \mathbb{D}''$ , ${\sharp}{}{v^{\mathbb{D}'',\infty}_{\underline{a}}}\in U(\mathfrak{gl}_{k|\infty}) {\sharp}{}{v_{\underline{a}}^{\mathbb{D},\infty}}$ , ${\sharp}{}{v^{\mathbb{D}',\infty}_{\underline{a}}}\in U(\mathfrak{gl}_{k|\infty}) {\sharp}{}{v_{\underline{a}}^{\mathbb{D}'',\infty}}$ , and $|\mathbb{D}''|=|\mathbb{D}'|-1$ . Then, by [Reference Su and ZhangSZ12, Theorem 5.18], we have that either $\mathbb{D}''\subseteq \mathbb{D}'$ or $\mathbb{D}'$ is obtained from $\mathbb{D}''$ by replacing a path $P\neq \{(x,y)\}$ in $\mathbb{D}''$ with two paths. In either case, we have that $\{(x,y)\} \in \mathbb{D}'$ .

Let $\underline{a},\underline{b}\in \mathcal{R}_{k,n}$ with $\underline{b}=\underline{a}^{(\{(x,y)\})}$ , where (x,y) is a corner of $\underline{a}$ . By [Reference Boe and EnrightBE86], there is a unique (up to nonzero scalar) nonzero map $\phi^{\underline{a}}_{\underline{b}}:N(\underline{a})\to N(\underline{b})$ . In particular, $\phi^{\underline{a}}_{\underline{b}}$ appears in the global Grothendieck–Cousin complex (Corollary 3.7), and thus is strict with respect to the weight filtration $W_{\bullet}$ .

1. (i) If $\mathbb{D}'=\mathbb{D}\cup (\{(x,y)\})$ , then $(\operatorname{{\rm gr}}^W_p\phi^{\underline{a}}_{\underline{b}})(L(\underline{a}^{\mathbb{D}'}))\cong L(\underline{b}^{\mathbb{D}})$ .

2. (ii) If $\mathbb{D}'\neq \mathbb{D}\cup (\{(x,y)\})$ , then $(\operatorname{{\rm gr}}^W_p\phi^{\underline{a}}_{\underline{b}})(L(\underline{a}^{\mathbb{D}'}))=0$ .

1. (i) The image of $\psi^{\underline{a}}_{\underline{b}}$ is the submodule of $M(\underline{a})$ generated by $v_{\underline{a}}^{\{(x,y)\}}$ . If $\mathbb{D}'$ is $\underline{a}$ -admissible, then by Theorem 4.3(i), we have that $\psi^{\underline{a}}_{\underline{b}}(v_{\underline{b}}^{\mathbb{E}})=v_{\underline{a}}^{\mathbb{D}'}$ for some $\mathbb{E}$ with $\underline{b}^{\mathbb{E}}=\underline{c}$ . Since $L(\underline{c})$ is a composition factor of $M(\underline{b})$ with multiplicity one, we conclude that $\mathbb{E}=\mathbb{D}$ , as required.

2. (ii) If $\mathbb{D}'\neq \mathbb{D}\cup (\{(x,y)\})$ and $\psi^{\underline{a}}_{\underline{b}}(v_{\underline{b}}^{\mathbb{D}})\neq 0$ , then $\psi^{\underline{a}}_{\underline{b}}(v_{\underline{b}}^{\mathbb{D}})=v_{\underline{a}}^{\mathbb{E}'}$ , where $\{(x,y)\}\in \mathbb{E}'$ by Theorem 4.3(ii). Since $\mathbb{D}'\neq \mathbb{E}'$ and $L(\underline{c})$ has multiplicity one as a composition factor of $M(\underline{a})$ , we obtain a contradiction.

4.3 Cohomology of the Grothendieck–Cousin complex

The argument in this section is based on [Reference HuangHua20, § 3], though functoriality and strictness of the weight filtration simplifies things in our setting (see §4.4). We fix $\underline{a}\in \mathcal{R}_{k,n}$ .

Lemma 4.5. Let $\underline{b}\subseteq \underline{a}$ , and let $\mathbb{D}\in {Dyck}(\underline{b})$ . We define the following set of boxes:

$$\mathcal{I}(\underline{a},\underline{b},\mathbb{D})=\big\{(i,j)\in \underline{a} / \underline{b} \mid \mbox{$\underline{b}\cup (i,j)$ {is a partition}, $\mathbb{D}\cup \{(i,j)\}\in {Dyck}(\underline{b} \cup (i,j))$}\big\}.$$

For all $J\subseteq \mathcal{I}(\underline{a},\underline{b},\mathbb{D})$ we have that $\underline{b} \cup J$ is a partition and $\mathbb{D}\cup J\in {\rm Dyck}(\underline{b}\cup J)$ .

Let $GC^{\bullet}_{\underline{a}}$ denote the global Grothendieck–Cousin complex for $\underline{a}$ , so that for $j,p\in \mathbb{Z}$ we have

(4.5) \begin{equation}GC_{\underline{a}}^j=\bigoplus_{\underline{b}\subseteq \underline{a},\;|\underline{b}|=d_X-j}N(\underline{b}),\quad \operatorname{{\rm gr}}^W_p(GC^j_{\underline{a}})=\bigoplus_{\underline{b}\subseteq \underline{a},\;|\underline{b}|=d_X-j} \bigoplus_{\mathbb{D}\in \mathcal{Z}_p(\underline{b})} L(\underline{b}^{\mathbb{D}}).\end{equation}

Using Corollary 4.4, we break $\operatorname{{\rm gr}}^W_p GC^{\bullet}_{\underline{a}}$ into a direct sum of Koszul subcomplexes.

Theorem 4.6. Given $p,q\in \mathbb{Z}$ , we define

$$\mathcal{Y}_p(\underline{a};q)=\left\{(\underline{b},\mathbb{D})\middle\vert \begin{array}{l} \mbox{$b\subseteq \underline{a}$ and $q=d_{X}-|\underline{b}|$, }\\ \mbox{$\mathbb{D}\in {\rm Dyck}(\underline{b})$,} \\ \mbox{$|D|\geqslant 3$ for all $D\in \mathbb{D}$ and $|\mathbb{D}|=p-q-d_{X}$ } \end{array} \right\}.$$

The following is true about the complex $\operatorname{{\rm gr}}^W_pGC^{\bullet}_{\underline{a}}$ .

1. (i) We have that $\operatorname{{\rm gr}}^W_pGC^{\bullet}_{\underline{a}}$ is a direct sum of subcomplexes

   $$\operatorname{{\rm gr}}^W_p(GC^{\bullet}_{\underline{a}})=\bigoplus_{0\leqslant q\leqslant |\underline{a}|}\bigoplus_{(\underline{b},\mathbb{D})\in \mathcal{Y}_p(\underline{a};q)} \mathcal{K}(\underline{a},\underline{b},\mathbb{D}),$$
   where $\mathcal{K}(\underline{a},\underline{b},\mathbb{D})$ is a complex concentrated in degrees $q-|\mathcal{I}(\underline{a},\underline{b},\mathbb{D})|,\ldots, q$ , with
   $$\mathcal{K}(\underline{a},\underline{b},\mathbb{D})_j=\bigoplus_{J\subseteq \mathcal{I}(\underline{a},\underline{b},\mathbb{D}),\; |J|=q-j} L((\underline{b}\cup J)^{\mathbb{D}\cup J}),\quad q-|\mathcal{I}(\underline{a},\underline{b},\mathbb{D})|\leqslant j \leqslant q.$$

2. (ii) For $0\leqslant q\leqslant d_{X}$ , we have

   $$\mathcal{H}^q(\operatorname{{\rm gr}}^W_p(GC^{\bullet}_{\underline{a}})) =\bigoplus_{\substack{(\underline{b},\mathbb{D})\in \mathcal{Y}_p(\underline{a};q)\\ \mathcal{I}(\underline{a},\underline{b},\mathbb{D})=\emptyset}} L(\underline{b}^{\mathbb{D}}).$$

We note that all summands of $\mathcal{K}(\underline{a},\underline{b},\mathbb{D})$ are isomorphic to $L(\underline{b}^{\mathbb{D}})$ .

By Corollary 3.7, the morphism $\partial^j:GC^j_{\underline{a}}\to GC^{j+1}_{\underline{a}}$ induces a nonzero morphism from $N(\underline{x})$ to $N(\underline{x}\setminus k)$ for all $k\in J$ (and such a morphism is unique up to a nonzero scalar [Reference Boe and EnrightBE86]). By Corollary 4.4, it follows that $\partial^j$ maps $L(\underline{x}^{\mathbb{X}})$ diagonally to

$$\bigoplus_{k\in J} L((\underline{x}\setminus k)^{\mathbb{X}\setminus k}).$$

This proves part (i). To see part (ii), note that $\mathcal{K}(\underline{a},\underline{b},\mathbb{D})$ is a Koszul complex (see [Reference HuangHua20, Lemma 3.6]) and hence is exact unless it has only one term, which is equivalent to $\mathcal{I}(\underline{a},\underline{b},\mathbb{D})=\emptyset$ .

In the following table, the column indices are the cohomological degree and the row indices indicate the graded pieces of the weight filtration.

To create the sets $\mathcal{Y}_p(\underline{a};q)$ , we select all composition factors of $N(\underline{b})$ in cohomological degree q that were obtained by removing Dyck patterns for which all paths have length $\geqslant 3$ (including the empty pattern):

Therefore, we obtain direct-sum decompositions:

For each pair $(\underline{b},\mathbb{D})$ as above, we have $\mathcal{I}(\underline{a},\underline{b},\mathbb{D})=\emptyset$ if and only if $(\underline{b},\mathbb{D})$ is one of the following:

These correspond to the Koszul complexes with one term. In particular, we obtain a non-split exact sequence

and $H^q_{Z_{\underline{a}}}(X,\mathcal{O}_X)=0$ for $q\neq 1$ . We note that $Z_{(0)}$ is the singular locus of $Z_{\underline{a}}$ .

We provide a combinatorial bijection relating Theorems 4.6 and 1.1.

Lemma 4.8. There is a bijection between the sets

$$\{(\underline{b},\mathbb{D})\in \mathcal{Y}_p(\underline{a};q)\mid \mathcal{I}(\underline{a},\underline{b},\mathbb{D})=\emptyset\}\quad \leftrightarrow \quad \mathcal{A}_p(\underline{a};q),$$

given by sending $(\underline{b},\mathbb{D}=(D_1,\ldots, D_{p-q-d_{X}}))$ to

$$\mathbb{D}'=(D_1,\ldots, D_{p-q-d_{X}};\mathbb{B}),$$

where $\mathbb{B}$ is the set of boxes in $\underline{a} /\underline{b}$ .

The map is clearly injective, as $\mathbb{B}$ completely determines $\underline{b}$ , and $(D_1,\ldots, D_{p-q-d_{X}})$ completely determines $\mathbb{D}$ . To show surjectivity, let $\mathbb{D}'=(D_1,\ldots, D_{p-q-d_{X}};\mathbb{B})\in \mathcal{A}_p(\underline{a};q)$ , $\underline{b}=\underline{a}\setminus \mathbb{B}$ , and $\mathbb{D}=(D_1,\ldots, D_{p-q-d_{X}})$ . We need to show that $\mathcal{I}(\underline{a},\underline{b},\mathbb{D})=\emptyset$ . All boxes of $\underline{a}/\underline{b}$ belong to the heads or tails of paths in $\mathbb{D}$ . Thus, for every box (i,j) in $\mathbb{B}$ such that $\underline{b}\cup (i,j)$ is a partition, there exists a path $D\in \mathbb{D}$ such that (i,j) belongs directly to the head or tail of D. In particular, the covering condition fails for D and (i,j). Thus, (i,j) does not belong to $\mathcal{I}(\underline{a},\underline{b},\mathbb{D})$ , and thus $\mathcal{I}(\underline{a},\underline{b},\mathbb{D})=\emptyset$ .

Combining Corollary 3.7, Theorem 4.6(ii), and Lemma 4.8, we obtain the description of $[\mathscr{H}^q_{Z_{\underline{a}}}(\mathcal{O}_X)]$ and $[H^q_{Z_{\underline{a}}}(X,\mathcal{O}_X)]$ in Theorem 1.1. Via Theorem 3.1, the following result implies the assertion in Theorem 1.1 about the graded pieces of the weight filtration.

Corollary 4.9. For all $\underline{a}\in \mathcal{R}_{k,n}$ , $0\leqslant q\leqslant d_X$ , and $p\in \mathbb{Z}$ , we have that

$$\mathcal{H}^q(\operatorname{{\rm gr}}^W_p\mathcal{GC}^{\bullet}_{\underline{a}})\cong \bigoplus_{\mathbb{D}\in \mathcal{A}_p(\underline{a};q)} IC^H_{Z_{\underline{a}^{\mathbb{D}}}}\left(\frac{(|\underline{a}^{\mathbb{D}}|-p)}{2}\right).$$

Remark 4.10. One could attempt to carry out a similar argument for the remaining Hermitian pairs:

$$(\mathsf{B}_n,\mathsf{B}_{n-1}),\quad (\mathsf{C}_n,\mathsf{A}_{n-1}),\quad (\mathsf{D}_n,\mathsf{A}_{n-1}),\quad (\mathsf{D}_n,\mathsf{D}_{n-1}),\quad (\mathsf{E}_6,\mathsf{D}_5),\quad (\mathsf{E}_7,\mathsf{E}_6).$$

The corresponding parabolic Verma modules are again rigid and multiplicity free [Reference Collingwood, Irving and SheltonCIS88], and the composition factors are described by parabolic Kazhdan–Lusztig polynomials [Reference Casian and CollingwoodCC87] (see also [Reference BoeBoe88, Reference BrentiBre09]). Unfortunately, the submodule lattices are not known in full generality, so that we do not currently have analogues of Theorem 4.3 for all Hermitian pairs. One may use our techniques in any setting where the relevant parabolic Verma modules have Loewy length at most 3 (because then the submodule lattice is determined by the Loewy filtration), which covers the cases $(\mathsf{B}_n,\mathsf{B}_{n-1})$ and $(\mathsf{D}_n,\mathsf{D}_{n-1})$ . In the cases $(\mathsf{E}_6,\mathsf{D}_5)$ and $(\mathsf{E}_7,\mathsf{E}_6)$ , many of the parabolic Verma modules have Loewy length at most 3 (see [Reference Collingwood, Irving and SheltonCIS88, Tables 7.1 and 7.2]), so our techniques are applicable to some of the Schubert varieties. In the major cases $(\mathsf{C}_n,\mathsf{A}_{n-1})$ and $(\mathsf{D}_n,\mathsf{A}_{n-1})$ , the weight filtrations are dictated by shifted Dyck patterns [Reference BrentiBre09]. We expect a combinatorial formula for local cohomology in these cases similar to ours, via ‘augmented shifted Dyck patterns’. Formulating and proving such a result would allow one to calculate the weight filtration on local cohomology with support in determinantal varieties of symmetric matrices and Pfaffian varieties of skew-symmetric matrices as in §4.5, upon restricting to the opposite big cell. The composition factors are known [Reference Raicu and WeymanRW16], but the Hodge and weight filtrations are not.

4.4 Equivalence to syzygies of determinantal thickenings

A determinantal thickening is a $\operatorname{{\rm GL}}_m(\mathbb{C})\times \operatorname{{\rm GL}}_n(\mathbb{C})$ invariant ideal in the polynomial ring $S=\operatorname{{\rm Sym}}(\mathbb{C}^m\otimes \mathbb{C}^n)\cong \mathbb{C}[x_{i,j}]_{1\leqslant i\leqslant m,1\leqslant j\leqslant n}$ on the space of $m\times n$ matrices, with $m\geqslant n$ (see [Reference de Concini, Eisenbud and ProcesidCEP80]). Given a partition $\lambda=(\lambda_1\geqslant \cdots \geqslant \lambda_n\geqslant 0)$ , there is a basic thickening $I_{\lambda}\subseteq S$ generated by the subrepresentation $\mathbb{S}_{\lambda}\mathbb{C}^m\otimes \mathbb{S}_{\lambda}\mathbb{C}^n\subseteq S$ . Via the BGG correspondence, the linear strands of the free resolutions of all determinantal thickenings admit actions of the general linear Lie superalgebra $\mathfrak{gl}(m|n)$ (see [Reference Raicu and WeymanRW19] and references therein). In [Reference Raicu and WeymanRW19], Raicu and Weyman make a conjecture on the structure of the linear strands of the basic thickenings $I_{\lambda}$ , in terms of admissible augmented Dyck patterns [Reference Raicu and WeymanRW19, Conjecture 4.1], and prove it for the first strand [Reference Raicu and WeymanRW19, Theorem 5.1]. Huang proves their full conjecture in [Reference HuangHua20, Theorem 3.7].

Via super duality (see [Reference Su and ZhangSZ12, §3.3] or the proof of Theorem 4.3 above), our Theorem 1.1 is equivalent to [Reference HuangHua20, Theorem 3.7]. Indeed, over infinite-dimensional Lie (super)algebras, our complex $GC^{\bullet}_{\underline{a}}$ is identified with dual of the complex $\tilde{\mathbf{R}}(I_{\underline{a}^c})$ in [Reference HuangHua20]. The cohomology of $GC^{\bullet}_{\underline{a}}$ calculates local cohomology for us, while the cohomology of $\tilde{\mathbf{R}}(I_{\underline{a}^c})$ calculates the linear strands of the free resolution of $I_{\underline{a}^c}$ . This explains the appearance of admissible augmented Dyck patterns in both works. Our argument above is closely modeled after that of Huang. There are two advantages to our treatment, however. First, from our point of view, it is clear that the weight filtration is functorial and that $GC^{\bullet}_{\underline{a}}$ is a filtered complex, whereas it is not immediate that the Loewy filtration in [Reference HuangHua20] is compatible with the morphisms in $\tilde{\mathbf{R}}(I_{\underline{a}^c})$ . Second, strictness of the weight filtration immediately implies that the spectral sequence for $(GC^{\bullet}_{\underline{a}}, W_{\bullet})$ degenerates, whereas it requires justification for $\tilde{\mathbf{R}}(I_{\underline{a}^c})$ .

It would be interesting to use duality results for other Lie (super)algebras (see [Reference Cheng and WangCW12]) to relate local cohomology on other flag varieties to linear strands of free resolutions of interesting classes of ideals. A natural case to investigate next would be the Hermitian pairs $(\mathsf{C}_n,\mathsf{A}_{n-1})$ , $(\mathsf{D}_n,\mathsf{A}_{n-1})$ .

4.5 Application to local cohomology supported in determinantal varieties

Let $\mathscr{X}=\mathbb{C}^{m\times n}$ be the space of $m\times n$ matrices over the complex numbers with $m\geqslant n$ , and for $0\leqslant p \leqslant n$ let $\mathscr{Z}_p\subseteq \mathscr{X}$ denote the determinantal variety of matrices of rank $\leqslant p$ . In recent past, a great deal of information has been learned about the local cohomology modules $H^j_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}})$ (see [Reference Raicu, Weyman and WittRWW14, Reference Raicu and WeymanRW14, Reference RaicuRai16, Reference Raicu and WeymanRW16, Reference Lõrincz and WaltherLW19, Reference Lõrincz and RaicuLR20, Reference Perlman and RaicuPR21, Reference PerlmanPer24]).

In this section, we use Theorem 1.1 to recover the composition factors and weight filtration on $H^j_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}})$ , obtained in full generality in [Reference Raicu and WeymanRW16] and [Reference PerlmanPer24]. We set $D_p=\mathcal{L}(\mathscr{Z}_p,\mathscr{X})$ for the intersection cohomology $\mathcal{D}$ -module associated to the trivial local system on $\mathscr{Z}_p\setminus \mathscr{Z}_{p-1}$ .

1. (i) [Reference Raicu and WeymanRW16, Main Theorem]. We have the following in the Grothendieck group of $\mathcal{D}$ -modules on $\mathscr{X}$ :

   $$\sum_{j\geqslant 0} \big[H^j_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}})\big]\cdot q^j=\sum_{s=0}^p [D_s]\cdot q^{(n-p)^2+(n-s)\cdot (m-n)}\cdot \binom{n-s-1}{p-s}_{q^2}.$$

2. (ii) [Reference PerlmanPer24, Theorem 1.1]. Each copy of $D_s$ in $H^j_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}})$ underlies a pure Hodge module of weight $mn+p-s+j$ .

Let $X=\operatorname{{\rm Gr}}(n,m+n)$ and let $U\subseteq X$ be the opposite big cell with respect to the fixed flag $\mathcal{V}_{\bullet}$ . Then, $U\cong \mathscr{X}$ and we have that $Z_{\underline{p}}|_U=\mathscr{Z}_p$ , where $\underline{p}=(m^p,p^{n-p})$ is the partition obtained from the $n\times m$ rectangle by removing the bottom-right $(n-p)\times (m-p)$ rectangle (see [Reference Lakshmibai and RaghavanLR08, Chapter 5]). For instance, when $m=5$ and $n=4$ , the following Young diagrams correspond to Schubert varieties in $\operatorname{{\rm Gr}}(4,9)$ that restrict to the determinantal varieties in $\mathbb{C}^{5\times 4}$ .

Furthermore, via this identification, we have that

\begin{equation}\mathcal{L}(O_{\underline{p}},X)|_U=D_p,\quad IC^H_{Z_{\underline{p}}}|_U\cong IC^H_{\mathscr{Z}_p},\quad \mathscr{H}^j_{Z_{\underline{p}}}(\mathcal{O}_X^H)|_U\cong H^j_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}}^H).\end{equation}

We observe that the only elements of Dyck $^{\bullet}(\underline{p})$ consisting of paths of length $\geqslant 3$ are obtained by removing hooks supported on the inside corner of $\underline{p}$ . For instance, continuing the above example, we obtain the following elements of Dyck $^{\bullet}(\underline{2})$ when $m=5$ and $n=4$ .

In particular, by Theorem 1.1, the multiplicity of $D_s$ in $H^{(m-p)(n-p)+j}_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}})$ is the number of ways to remove $p-s$ hooks from $\underline{p}$ with j bullets. Since the said hooks must start and end on the same antidiagonal, and $m\geqslant n$ , there will always be a $(p-s)\times (m-n)$ rectangle of bullets in rows $s+1,\ldots, p$ of $\underline{p}$ and columns $n+1,\ldots, m$ (these are the bullets depicted in the fifth columns of the above patterns). Thus, we may reduce to the case $m=n$ , and assume that $\underline{p}$ and its Dyck patterns are symmetric with respect to the operation of transposing the diagram.

Let $m=n$ and let $0\leqslant p\leqslant n$ . Let $\mathbb{D}\in {\rm Dyck}^{\bullet}(\underline{p})$ consist of paths of length $\geqslant 3$ and set $\underline{s}=\underline{p}^{\mathbb{D}}$ for $0\leqslant s\leqslant p$ . The union of the support of the paths in $\mathbb{D}$ will contain all boxes (x,y) such that $s+1\leqslant x,y\leqslant p+1$ (except the box $(p+1,p+1)$ , which lies just outside the inside corner of $\underline{p}$ ). Due to the covering condition for Dyck patterns, the number of bullets in row $s+i$ must be less than or equal to the number of bullets in row $s+i+1$ for $1\leqslant i\leqslant p-1$ . Thus, there is a partition shape of bullets in rows $s+1,\ldots, p$ , and the transpose partition of bullets in columns $s+1,\ldots, p$ .

Here is a picture when $m=n=9$ , $p=4$ , and $s=1$ .

The partition formed by the bullets is (4,3,1). We conclude that, for $m=n$ , the multiplicity of $D_s$ in $H^{(n-p)^2+2j}_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}})$ is the number of partitions of size j that fit inside a $(p-s)\times (n-p-1)$ rectangle, which is the coefficient of $q^j$ in the Gaussian binomial coefficient $({n-s-1\atop p-s})_q$ .

Putting it all together, we obtain, for $m\geqslant n$ , the following multiplicity formula:

$$\sum_{j \geqslant 0}\big[H^j_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}}): D_s \big]\cdot q^j=q^{(m-p)(n-p)+(p-s)(m-n)}\cdot \Big({n-s-1\atop p-s}\Big)_{q^2}.$$

Since $(m-p)(n-p)+(p-s)(m-n)=(n-p)^2+(n-s)(m-n)$ , we recover Theorem 4.11(i).

By Theorem 1.1, the weight of $D_s=\mathcal{L}(\underline{p}^{\mathbb{D}})|_U$ in $H^j_{\mathscr{Z}_p}(\mathscr{X},\mathcal{O}_{\mathscr{X}}^H)$ is $mn+|\mathbb{D}|+j=mn+p-s+j$ , recovering Theorem 4.11(ii).