2.1 BGG category $\mathcal O$

Associated to the triangular decomposition (2-1) above, the BGG category $\mathcal O $ (respectively $\mathcal O_{\bar 0}$ ) is defined as the full subcategory of the category of finitely generated $\mathfrak {g}$ -modules (respectively $\mathfrak g_{\bar 0}$ -modules) which are semisimple over $\mathfrak h$ and locally $U(\mathfrak {n}^+)$ -finite (respectively $U(\mathfrak {n}^+_{\bar 0})$ -finite); see [Reference Bernstein, Gelfand and GelfandBGG2, Reference HumphreysHu].

For $\lambda \in \mathfrak {h}^\ast $ , the corresponding Verma modules over $\mathfrak g_{\bar 0}$ and $\mathfrak g$ are defined as

$$ \begin{align*}\Delta_{{\bar 0}}(\lambda)=U(\mathfrak{g}_{{\bar 0}})\otimes_{U(\mathfrak{b}_{{\bar 0}})}\mathbb{C}_{\lambda}\quad\text{and}\quad\Delta(\lambda) := U(\mathfrak g)\otimes_{U(\mathfrak b)} \mathbb{C}_{\lambda},\end{align*} $$

respectively. Here, $\mathbb {C}_{\lambda }$ is the one-dimensional module over the Borel subalgebra corresponding to $\lambda $ . The unique simple quotients of $\Delta _{\bar 0}(\lambda )$ and $\Delta (\lambda )$ are denoted by $L_{\bar 0}(\lambda )$ and $L(\lambda )$ , respectively. The corresponding dual Verma modules $\nabla _{\bar 0}(\lambda )$ and $\nabla (\lambda )$ are similarly defined as coinduced modules; see [Reference HumphreysHu, Section 3.2] and [Reference Chen and CoulembierCCC, Definition 3.2] for more details. We note that $L(\lambda ) \hookrightarrow \nabla (\lambda ).$ Also, we denote by $T_{\bar 0}(\lambda )$ and $T(\lambda )$ the tilting modules of highest weight $\lambda $ in $\mathcal O_{\bar 0}$ and $\mathcal O$ , respectively; see [Reference Mazorchuk, Gorelik and PapiMa3, Section 4.3] and [Reference Chen and CoulembierCCC, Theorem 3.5] for more details.

Let us denote by $\operatorname {Res}\nolimits :=\operatorname {Res}\nolimits _{\mathfrak g_{\bar 0}}^{\mathfrak {g}}$ , $\operatorname {Ind}\nolimits :=\operatorname {Ind}\nolimits _{\mathfrak g_{\bar 0}}^{\mathfrak {g}}$ and $\mathrm {Coind}:=\mathrm {Coind}_{\mathfrak g_{\bar 0}}^{\mathfrak {g}}$ the corresponding restriction, induction and coinduction functors, respectively. It is well known; see [Reference Bell and FarnsteinerBF, Theorem 2.2] and [Reference GorelikGor1], that there is an isomorphism of functors

$$ \begin{align*}\operatorname{Ind}\nolimits \cong \mathrm{Coind}( \Lambda^{\text{top}}\mathfrak g_{\bar 1}\otimes-).\end{align*} $$

2.2 Lie superalgebras of type I

A classical Lie superalgebra $\mathfrak g = \mathfrak g_{\overline 0}\oplus \mathfrak g_{{\bar 1}}$ is said to be of type I if $\mathfrak g$ has a compatible $\mathbb {Z}$ -grading $\mathfrak g= \mathfrak g_{-1}\oplus \mathfrak g_0 \oplus \mathfrak g_{1}$ such that $\mathfrak g_0 =\mathfrak g_{\overline 0}$ and $\mathfrak g_{\bar 1} = \mathfrak g_1 \oplus \mathfrak g_{-1}$ , where $\mathfrak g_{\pm 1}$ are $\mathfrak g_{\overline 0}$ -submodules of $\mathfrak g_{\bar 1}$ with $[\mathfrak g_{1},\mathfrak g_{ 1}] =[\mathfrak g_{-1},\mathfrak g_{-1}] =0$ . We set $\mathfrak g_{\geq 0}: = \mathfrak g_0 \oplus \mathfrak g_1$ and $\mathfrak g_{\leq 0}: = \mathfrak g_0 \oplus \mathfrak g_{-1}$ .

2.2.1 Triangular decompositions for $\mathfrak g$ of type I

For classical Lie superalgebras $\mathfrak g$ of type I considered in the present paper, we always assume that the triangular decomposition in (2-1) has the property

(2-2) $$ \begin{align} &\mathfrak b = \mathfrak b_{\bar 0} \oplus \mathfrak g_1, \end{align} $$

that is, $\mathfrak b_{\bar 1}=\mathfrak {g}_1$ .

In particular, we are interested in the following superalgebras from Kac’s list (see [Reference KacKa]):

(2-3) $$ \begin{align} &(\text{Type}\ {\bf A}): ~\mathfrak{gl}(m|n), \mathfrak{sl}(m|n) \text{ and } \mathfrak{sl}(n|n)/\mathbb{C}I_{n|n}; \end{align} $$

(2-4) $$ \begin{align} (\text{Type}\ {\bf C}) &: ~\mathfrak{osp}(2|2n); \nonumber\\ (\text{Type}\ {\bf P}) &: ~\mathfrak{pe} (n)\text{ and } [\mathfrak{pe} (n), \mathfrak{pe} (n)]; \kern37pt \end{align} $$

where $m>n \geq 1$ are integers (see, for example, [Reference MussonMu, Ch. 2 and 3] and [Reference Cheng and WangCW1, Section 1.1]). See Sections 3.2 and 6.1 for more details on $\mathfrak {pe} (n)$ and $\mathfrak {osp}(2|2n)$ .

2.2.2 Induced and coinduced modules

For a given object $V \in \mathcal O_{\bar 0}$ , we extend V trivially to a $\mathfrak g_{\geq 0}$ -module and define the Kac module

$$ \begin{align*}K(V):=\operatorname{Ind}\nolimits_{\mathfrak g_{\geq 0}}^{\mathfrak g}V.\end{align*} $$

This defines an exact functor $K({}_-): \mathcal O_{\bar 0} \rightarrow \mathcal O$ which is called the Kac functor. For $\lambda \in \mathfrak {h}^\ast $ , we define $K(\lambda ) := K(L_{\bar 0}(\lambda ))$ . Also, we note that $\Delta (\lambda ) \cong K(\Delta _{\bar 0}(\lambda ))$ and $\nabla (\lambda ):= \text {Coind}_{\mathfrak g_{\leq 0}}^{\mathfrak g} (\nabla _{\bar 0}(\lambda )).$

3.1 Kac functor and homomorphisms between Verma supermodules

Using the same argument as in the proof of [Reference Cheng, Mazorchuk and WangCP, Lemma 4.3], we have the following useful lemma.

The following is our first main result.

Proof. In the proof, we regard $\Delta _{\bar 0}(\lambda )$ and $\Delta _{\bar 0}(\mu )$ as direct summands of $\operatorname {Res}\nolimits \Delta (\lambda )$ and $\operatorname {Res}\nolimits \Delta (\mu )$ , respectively.

We first prove $(1)\Rightarrow (3)$ . Using the exactness of Kac functor, the characters of $\Delta (\lambda )$ and $\Delta (\mu )$ are given as follows:

$$ \begin{align*} &\text{ch}\Delta(\lambda) = \sum_{\zeta} [\Delta_{\bar 0}(\lambda): L_{\bar 0}(\zeta)] \text{ch}K(\zeta),\quad \text{ch}\Delta(\mu) = \sum_{\zeta} [\Delta_{\bar 0}(\mu): L_{\bar 0}(\zeta)] \text{ch}K(\zeta). \end{align*} $$

This implies that there exists $\zeta \in W\cdot \mu $ such that $[K(\zeta ):L(\underline \lambda )]>0$ because f is nonzero and soc $(\Delta (\lambda )) =L(\underline \lambda )$ by [Reference Chen and MazorchukCM, Theorem 51].

Because $d\in \mathcal Z(\mathfrak g_{\bar 0})$ , the element d acts on $\Delta _{\bar 0}(\lambda )$ and $\Delta _{\bar 0}(\mu )$ via some scalars, say $c_\lambda $ and $c_\mu $ , respectively. From Lemma 1, we conclude that $\zeta =\underline \lambda $ because $[K(\zeta ):L(\underline \lambda )]>0$ and $K(\underline \lambda )=L(\underline \lambda )$ . Therefore, $\mu \in W\cdot \lambda $ and hence $c_\lambda =c_\mu $ .

Applying the restriction functor to f, we obtain a $\mathfrak {g}_{\bar 0}$ -homomorphism

$$ \begin{align*}\operatorname{Res}\nolimits(f): \Lambda^{\bullet} \mathfrak g_{-1} \otimes \Delta_{\bar 0}(\mu) \rightarrow \Lambda^{\bullet} \mathfrak g_{-1}\otimes \Delta_{\bar 0}(\lambda).\end{align*} $$

We now consider the eigenspaces with respect to the operator d. Observe that for any $k\geq 0$ , the subspace $\Lambda ^{k} \mathfrak g_{-1}\otimes \Delta _{\bar 0}(\mu )$ of $\operatorname {Res}\nolimits \Delta (\mu )$ is the eigenspace for d corresponding to the eigenvalue $c_\mu +kD$ . Similarly, the subspace $\Lambda ^{k} \mathfrak g_{-1}\otimes \Delta _{\bar 0}(\lambda )$ of $\operatorname {Res}\nolimits \Delta (\lambda )$ is the eigenspace for d corresponding to the eigenvalue $c_\lambda +kD$ . Therefore, we have

(3-1) $$ \begin{align} 0\neq \operatorname{Res}\nolimits(f)(\Delta_{\bar 0}(\mu)) \subseteq \Delta_{\bar 0}(\lambda) \end{align} $$

because $f\neq 0$ and $c_\lambda =c_\mu .$ Consequently, we may conclude that

(3-2) $$ \begin{align} \operatorname{Res}\nolimits(f){|}_{\Delta_{\bar 0}(\mu)}: \Delta_{\bar 0}(\mu) \rightarrow \Delta_{\bar 0}(\lambda) \end{align} $$

is an embedding by [Reference DixmierDi, Theorem 7.6.6].

Let K be the kernel of f. If we assume that f is not an embedding, then we have $K \supseteq $ soc $(\Delta (\mu ))$ by [Reference Chen and MazorchukCCM, Theorem 51]. This gives

$$ \begin{align*} &K \supseteq \text{soc}(\Delta(\mu)) = U(\mathfrak g)\cdot (\Lambda^{\text{top}}\mathfrak g_{-1}\otimes \text{soc}(\Delta_{\bar 0}(\mu))), \end{align*} $$

which means that

$$ \begin{align*}\Lambda^{\text{top}}\mathfrak g_{-1}\otimes \operatorname{Res}\nolimits(f)(\text{soc} (\Delta_{\bar 0}(\mu)))=f(\Lambda^{\text{top}}\mathfrak g_{-1}\otimes \text{soc}(\Delta_{\bar 0}(\mu))) =0,\end{align*} $$

a contradiction to (3-1). Note that this proves the direction $(1)\Rightarrow (2)$ . We proceed, however, with the proof of the direction $(1)\Rightarrow (3)$ .

Observe that the Kac functor gives rise to a monomorphism

$$ \begin{align*}K({}_-): \text{Hom}_{\mathcal O_{\bar 0}}(\Delta_{\bar 0}(\mu),\Delta_{\bar 0}(\lambda)) \rightarrow \text{Hom}_{\mathcal O}(\Delta(\mu),\Delta(\lambda)).\end{align*} $$

It suffices to show that for given nonzero $f,g\in \text {Hom}_{\mathcal O}(\Delta (\mu ),\Delta (\lambda ))$ , there exists $\alpha \in {\mathbb C}$ such that $f=\alpha g.$ By (3-2), both f and g restrict to nonzero $\mathfrak g_{\bar 0}$ -homomorphisms $\operatorname {Res}\nolimits (f){|}_{\Delta _{\bar 0}(\mu )}$ and $\operatorname {Res}\nolimits (g){|}_{\Delta _{\bar 0}(\mu )}$ from $\Delta _{\bar 0}(\mu )$ to $\Delta _{\bar 0}(\lambda )$ , respectively. From [Reference DixmierDi, Theorem 7.6.6], we have that there exists $\alpha \in {\mathbb C}$ such that

$$ \begin{align*}\operatorname{Res}\nolimits(f){|}_{\Delta_{\bar 0}(\mu)} =\alpha \operatorname{Res}\nolimits(g){|}_{\Delta_{\bar 0}(\mu)}.\end{align*} $$

This proves the part $(1)\Rightarrow (3)$ because $\Delta (\mu )$ is generated by $\Delta _{\bar 0}(\mu )$ as a $\mathfrak g$ -module.

Note that the implication $(3)\Rightarrow (2)$ is clear. It remains to prove the implication $(2)\Rightarrow (1)$ , so we now assume $(2)$ . We set $L(\nu ):=\text {soc}(K(\underline \lambda ))$ . Then we have the map $\Delta (\nu ) \twoheadrightarrow L(\nu ) \hookrightarrow K(\underline \lambda )= \Delta (\underline \lambda )$ . Because this map is an embedding by $(2)$ , it follows that $\Delta (\nu ) = L(\nu ) \hookrightarrow K(\underline \lambda )= \Delta (\underline \lambda )$ . By Lemma 1, from $[K(\underline \lambda ):L(\nu )]\neq 0$ and $K(\nu ) =L(\nu )$ , we deduce that $\underline \lambda = \nu $ . This gives $K(\underline \lambda ) =L(\underline \lambda )$ and completes the proof.

3.2 Example: the periplectic Lie superalgebras $\mathfrak {pe} (n)$

For positive integers $m,n$ , the general linear Lie superalgebra $\mathfrak {gl}(m|n)$ can be realized as the space of $(m+n) \times (m+n)$ complex matrices

$$ \begin{align*} \left( \begin{array}{cc} A & B\\ C & D\\ \end{array} \right), \end{align*} $$

where $A,B,C$ , and D are $m\times m, m\times n, n\times m$ , and $n\times n$ matrices, respectively. The Lie bracket of $\mathfrak {gl}(m|n)$ is given by the super commutator. Let $E_{ab}$ , for $1\leq a,b \leq m+n$ , be the elementary matrix in $\mathfrak {gl}(m|n)$ . Its $(a,b)$ -entry is equal to $1$ and all other entries are $0$ .

The standard matrix realization of the periplectic Lie superalgebra $\mathfrak {pe} (n)$ inside the general linear Lie superalgebra $\mathfrak {gl}(n|n)$ is given by

(3-3) $$ \begin{align} \mathfrak g= \mathfrak{pe}(n):= \left\{ \left( \begin{array}{cc} A & B\\ C & -A^t\\ \end{array} \right)\bigg\| ~ A,B,C\in {\mathbb C}^{n\times n},~B^t=B,\ \text{ and }\ C^t=-C \right\}. \end{align} $$

Throughout the present paper, we fix the Cartan subalgebra $\mathfrak h= \mathfrak h_{\bar 0} \subset \mathfrak g_{\overline 0}$ consisting of diagonal matrices. We denote the dual basis of $\mathfrak h^*$ by $\{\epsilon _1, \epsilon _2, \ldots , \epsilon _n\}$ with respect to the standard basis of $\mathfrak h$ defined as

$$ \begin{align*} \{H_i:=E_{i,i}-E_{n+i,n+i}\mid 1\leq i \leq n \}\subset \mathfrak{pe} (n), \end{align*} $$

where $E_{a,b}$ denotes the $(a,b)$ -matrix unit for $1\leq a,b \leq 2n$ . The sets $\Phi _{\bar 0}$ , $\Phi _{\bar 1}^\pm $ , and $\Phi _{\bar 1}$ of even, odd positive, odd negative, and odd roots are, respectively, given by

$$ \begin{align*} &\Phi_{\bar 0}=\{\epsilon_i-\epsilon_j\mid 1\le i\not=j\le n\}, \nonumber\\ &\Phi^+_{\bar 1}=\{\epsilon_i+\epsilon_j\mid 1\le i\le j\le n\} ,\nonumber\\ &\Phi^-_{\bar 1}= \{- \epsilon_i-\epsilon_j\mid 1\le i< j\le n\},\nonumber\\ &\Phi_{\bar 1}:=\Phi_{\bar 1}^+\cup \Phi_{\bar 1}^-. \end{align*} $$

The space $\mathfrak h^\ast = \bigoplus _{i=1}^n{\mathbb C} \epsilon _i$ is equipped with a natural bilinear form $\langle \epsilon _i , \epsilon _j \rangle =\delta _{i,j}$ , for any $1\leq i,j \leq n$ . The Weyl group $W=\mathfrak S_n$ is the symmetric group acting on $\mathfrak {h}^\ast $ . For any $\alpha \in \Phi _{\bar 0}$ , we let $s_\alpha $ denote the corresponding reflection in W.

We fix the Borel subalgebra $\mathfrak {b}_{{\bar 0}}$ of $\mathfrak {g}_{{\bar 0}}=\mathfrak {gl}(n)$ consisting of matrices in (3-3) with $B=C=0$ and A upper triangular.

The subalgebras $\mathfrak g_1$ and $\mathfrak g_{-1}$ are given by

$$ \begin{align*} \mathfrak g_1:= \bigg\{\begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix}\bigg|B^t=B\bigg\}\quad\mbox{and}\quad \mathfrak g_{-1}:= \bigg\{\begin{pmatrix} 0 & 0 \\ C & 0 \end{pmatrix}\bigg|C^t=-C\bigg\}. \end{align*} $$

We note that $\Phi _{\bar 1}^+$ and $\Phi _{\bar 1}^-$ are the sets of weights of the $\mathfrak g_0$ -modules $\mathfrak g_1$ and $\mathfrak g_{-1}$ , respectively.

The Borel subalgebra is defined as $\mathfrak b:=\mathfrak b_{\bar 0} \oplus \mathfrak {g}_{1}$ , while the reverse Borel subalgebra $\mathfrak b^r$ is defined as $\mathfrak b^r:=\mathfrak b_{\bar 0} \oplus \mathfrak g_{-1}.$ It can be shown that $\mathfrak b^r$ is still a Borel subalgebra in the sense of [Reference Mazorchuk, Gorelik and PapiMa3, Section 2.4] (see, for example, [Reference Chen and CoulembierCCC, Section 5]).

Furthermore, we let $L^{r}(\lambda )$ be the irreducible module in the corresponding ‘reversed’ category $\mathcal O$ of $\mathfrak b^r$ -highest weight $\lambda $ . We let $\lambda ^+\in \mathfrak {h}^\ast $ be uniquely determined by

(3-4) $$ \begin{align} &L(\lambda^+) =L^r(\lambda). \end{align} $$

It is known that the weight $\lambda ^+$ can be computed using odd reflections, and we refer to [Reference Penkov and SerganovaPS1, Section 2.2] and [Reference MussonMu, Theorem 3.6.10] for a treatment of odd reflections for the periplectic Lie superalgebras $\mathfrak {pe} (n)$ . In particular, the effect on the highest weight of a simple $\mathfrak {pe} (n)$ -module under odd reflection and inclusion was computed in [Reference Penkov and SerganovaPS1, Lemma 1].

Finally, we set $ X:=\bigoplus _{i=1}^n\mathbb {Z}\epsilon _i$ ; $\omega _k:=\epsilon _1+\cdots +\epsilon _k$ for any $1\leq k\leq n$ ; and also $\eta :=(1-n)(\epsilon _{1}+\cdots +\epsilon _{n})$ . Then we have $\Lambda ^{\text {top}}\mathfrak g_{-1}\cong {\mathbb C}_{\eta }$ as a $\mathfrak g_{\bar 0}$ -module.

Let $\lambda , \mu \in \mathfrak {h}^\ast $ be two integer weights. If there are positive roots

$$ \begin{align*}\alpha_1,\alpha_2,\ldots, \alpha_k\in \Phi_{\bar 0}^+,\end{align*} $$

such that $\lambda =s_{\alpha _k} s_{\alpha _{k-1}}\cdots s_{\alpha _1}\cdot \mu $ and, for each $1\leq q \leq k-1$ , we have

$$ \begin{align*}\langle s_{\alpha_q}s_{\alpha_{q-1}}\cdots s_{\alpha_1}\cdot \mu,~ \alpha_{q+1} \rangle \geq 0, \end{align*} $$

then we write $\mu \uparrow \lambda $ ; see [Reference DixmierDi, Section 7.6].

Remark 4. In [Reference KumarLLW], Liu et al. established a closed formula for a singular vector of weight $\lambda -\beta $ in the Verma module of highest weight $\lambda $ for Lie superalgebra $\mathfrak {gl}(m|n)$ when $\lambda $ is atypical with respect to an odd positive root $\beta $ . In particular, the authors proved the following identity:

(3-5) $$ \begin{align} &\text{Hom}(\Delta(\lambda -\beta),\Delta(\lambda))={\mathbb C}, \end{align} $$

where $\Delta $ terms denote Verma supermodules for $\mathfrak {g} =\mathfrak {gl}(m|n)$ . This identity was also known in [Reference Cheng and WangCW2] for the exceptional Lie superalgebra $D(2|1;\zeta )$ . By Proposition 3, it follows that the identity (3-5) fails to hold for $\mathfrak {pe} (n)$ .

Remark 5. The Borel subalgebras have been classified in [Reference Chen and CoulembierCCC, Section 5]. It would be interesting to determine for which other Borel subalgebras one has an analogue of Proposition 3. We note that Proposition 3 fails in the case of the reverse Borel subalgebra $\mathfrak b^r$ . To see this, we define Verma supermodules with respect to $\mathfrak b^r$ as

$$ \begin{align*}\Delta^r(\lambda):=U(\mathfrak g)\otimes_{\mathfrak b^r}{\mathbb C}_\lambda \cong \text{Ind}^{\mathfrak{g}}_{\mathfrak g_{\leq 0}}\Delta_{\bar 0}(\lambda),\end{align*} $$

which admit a filtration with subquotients $ \text {Ind}^{\mathfrak {g}}_{\mathfrak g_{\leq 0}}L_{\bar 0}(\zeta )$ , for $\zeta \in \mathfrak {h}^\ast $ . By [Reference Chen and MazorchukCM, Proposition 4.15], the modules $\text {Ind}^{\mathfrak {g}}_{\mathfrak g_{\leq 0}}L_{\bar 0}(\zeta )$ are never simple, for any $\zeta \in \mathfrak {h}^\ast $ . By a similar argument as used in the directions $(3)\Rightarrow (2)$ and $(2)\Rightarrow (1)$ in the proof of Proposition 9, it follows that Proposition 9 fails for these modules $\Delta ^r(\lambda )$ .

5.1 Finitistic dimension of various categories of supermodules

We denote by $\mathfrak g$ -Mod and $\mathfrak g_{\bar 0}$ -Mod the category of all $\mathfrak g$ -modules and the category of all $\mathfrak g_{\bar 0}$ -modules, respectively.

Let $\tilde {\mathcal C}$ and ${\mathcal C}$ be abelian subcategories of $\mathfrak g$ -Mod and $\mathfrak g_{\bar 0}$ -Mod, respectively. Assume that:

1. (i) $\operatorname {Ind}\nolimits , \mathrm {Coind}$ , and $\operatorname {Res}\nolimits $ restrict to well-defined functors between ${\mathcal C} $ and $\tilde {\mathcal C}$ ;

2. (ii) $\mathcal C$ has enough injective and projective modules;

3. (iii) $\text {gl.dim}\,\mathcal C <\infty $ .

The following theorem generalizes [Reference Coulembier and SerganovaCS, Theorem 4.1] (also see [Reference Coulembier and SerganovaCS, Lemma 4.3]) where the case of the superalgebra $\mathfrak {gl}(m|n)$ was considered.

Proof. Because $\operatorname {Ind}\nolimits $ is isomorphic to $\mathrm {Coind}$ , up to parity change, it follows that the functors $\operatorname {Ind}\nolimits $ , $\mathrm {Coind}$ , and $\operatorname {Res}\nolimits $ send injective (respectively projective) objects to injective (respectively projective) objects.

The category $\tilde {\mathcal C}$ is a Frobenius extension of $\mathcal C$ in the sense of [Reference Coulembier and SerganovaCo, Definition 2.2.1]. Therefore, $\tilde {\mathcal C}$ has enough projective and injective objects by [Reference Coulembier and SerganovaCo, Proposition 2.2.1]. Moreover, every injective (respectively projective) object in $\tilde {\mathcal C}$ is a direct summand of an object induced from an injective (respectively projective) object of $\mathcal C$ .

Consequently, all injective objects in $\tilde {\mathcal C}$ have finite projective dimension. By an argument used in the proof of [Reference Mazorchuk, Gorelik and PapiMa3, Theorem 3] (see also [Reference Coulembier and SerganovaCo, Corollary 3.6.10]), we have that $\text {fin.dim}\, \tilde {\mathcal C}$ is the maximum of the projective dimensions of injective objects in $\tilde {\mathcal C}$ . Note also that the global dimension of $\mathcal C$ is the maximum of the projective dimensions of injective objects in $\mathcal C$ .

We observe that for any injective I in $\mathcal C$ and $Y\in \tilde {C}$ , it follows from the Frobenius reciprocity that

$$ \begin{align*}\operatorname{Ext}\limits^d_{\tilde{\mathcal C}}(\operatorname{Ind}\nolimits I, Y) =\operatorname{Ext}\limits^d_{\mathcal C}(I,\operatorname{Res}\nolimits Y), \end{align*} $$

which implies that $\text {fin.dim} \,\tilde {\mathcal C} \leq \text {gl.dim}\,\mathcal C$ .

Finally, let $I\in \mathcal C$ be injective. Then I is a direct summand of $\operatorname {Res}\nolimits \operatorname {Ind}\nolimits I\cong \Lambda ^{\bullet } \mathfrak g_{\bar 1} \otimes I$ , which implies that

$$ \begin{align*}{\text{pd}}\, I \geq {\text{pd}}\, \operatorname{Ind}\nolimits I \geq {\text{pd}} \,\operatorname{Res}\nolimits \operatorname{Ind}\nolimits I \geq {\text{pd}}\, I. \end{align*} $$

This shows that $\text {fin.dim} \,\tilde {\mathcal C} \geq \text {gl.dim}\,\mathcal C$ and the result follows.

Below we apply Theorem 12 to various examples.

Example 13. Let $\tilde {\mathcal C}= \mathfrak g$ -Mod and ${\mathcal C}= g_{\bar 0}$ -Mod. Condition (i) is obvious and conditions (ii) and (iii) are clear because $U(g_{\bar 0})$ is Noetherian. We have $\text {gl.dim}\,\mathcal C =\text {dim}\,\mathfrak g_{\bar 0}$ . Therefore, by Theorem 12, we have

$$ \begin{align*}\text{fin.dim}~\mathfrak g\text{-Mod} = \dim \mathfrak{g}_{\bar 0}.\end{align*} $$

Example 14. Consider the super analog $\mathcal C(\mathfrak g,\mathfrak {h})$ of the category $\mathcal C(\mathfrak g_{\bar 0},\mathfrak {h})$ of weight $\mathfrak g$ -modules from [Reference Zuckerman, Joseph, Melnikov and PenkovZu], that is, $\mathcal C(\mathfrak g,\mathfrak {h})$ , which is the full subcategory of $\mathfrak g$ -Mod consisting of $\mathfrak g$ -modules that are semisimple over $\mathfrak h$ .

For $\lambda \in \mathfrak {h}^\ast $ , let $I_\lambda $ be the left ideal of $U(\mathfrak g)$ generated by $h-\lambda (h)$ for $h\in \mathfrak h$ . Then the quotient of $U(\mathfrak g)$ by $I_\lambda $ is projective in $\mathcal C(\mathfrak {g},\mathfrak {h})$ . Also, every weight $\mathfrak g$ -module is a quotient of a direct sum of the $I_\lambda $ terms. Therefore, $\mathcal C(\mathfrak g_{\bar 0}, \mathfrak {h})$ has enough projective objects. By [Reference Zuckerman, Joseph, Melnikov and PenkovZu, Lemma 2.2], $\mathcal C(\mathfrak g_{\bar 0},\mathfrak h)$ has enough injective objects as well.

Let $M\in \mathcal C(\mathfrak g_{\bar 0},\mathfrak {h})$ . By [Reference MazorchukKu, Corollary 3.1.8], the trivial module ${\mathbb C}$ in $\mathcal C(\mathfrak g_{\bar 0},\mathfrak {h})$ has the following finite projective resolution in $\mathcal C(\mathfrak g_{\bar 0},\mathfrak {h})$ :

$$ \begin{align*}\cdots \rightarrow D_2\rightarrow D_1 \rightarrow D_0 \rightarrow {\mathbb C} \rightarrow 0, \end{align*} $$

where $D_p$ is the induced module $U(\mathfrak g_{\bar 0})\otimes _{U(\mathfrak h)}\Lambda ^p(\mathfrak g_{\bar 0}/\mathfrak h)$ for $p\geq 0$ . Tensoring with this resolution (over $\mathbb {C}$ ), we conclude that every $M\in \mathcal C(\mathfrak g_{\bar 0},\mathfrak h)$ has finite projective dimension.

In particular, we have

$$ \begin{align*} &\text{fin.dim}\, \mathcal C(\mathfrak{g},\mathfrak{h})= \text{fin.dim} \,\mathcal C(\mathfrak{g}_{\bar 0},\mathfrak{h}) = \text{dim}\mathfrak{g}_{\bar 0} -\text{dim}\mathfrak{h}. \end{align*} $$

Example 15. Let $\mathcal O^{\mathfrak {p}}$ be the parabolic BGG category of $\mathfrak g$ -modules in the sense of [Reference Mazorchuk, Gorelik and PapiMa3, Section 3]. By [Reference Coulembier and MazorchukCoM2], we have

(5-1) $$ \begin{align} &\text{fin.dim}\, \mathcal O^{\mathfrak{p}} = {\text{gl.dim}} \,\mathcal O_{\bar 0}^{\mathfrak{p}}=2\ell(w_0) -2\ell(w_0^{\mathfrak{p}}), \end{align} $$

where $w_0$ is the longest element in W and $w_0^{\mathfrak {p}}$ is the longest element in the Weyl group of the Levi subalgebra of $\mathfrak {p}$ .

Example 16. Consider $\mathfrak g=\mathfrak {pe} (n)$ with a reduced parabolic subalgebra $\mathfrak {p}$ . For $\lambda \in \mathfrak {h}^\ast $ , let $\text {Irr}\,\mathcal O_{\lambda }^{\mathfrak {p}}$ denote the set of all highest weights of irreducible modules in the parabolic block $\mathcal O_\lambda ^{\mathfrak {p}}$ . Recall the equivalence relation $\sim $ on $\mathfrak {h}^*$ defined in [Reference Chen and CoulembierChC, Section 5.2] which is transitively generated by

$$ \begin{align*}\begin{cases}\lambda\sim \lambda \pm2\epsilon_k \quad&\mbox{for }1\le k\le n;\\ \lambda\sim w \cdot \lambda \quad&\mbox{for }w\in W_{[\lambda]}. \end{cases}\end{align*} $$

Here $W_{[\lambda ]}$ denotes the integral Weyl group associated to $\lambda $ . It is generated by all reflections $s_\alpha $ with $\langle \lambda , \alpha \rangle \in {\mathbb Z}.$ It is well known that $W_{[\lambda ]}$ is the Weyl group of a certain semisimple Lie algebra (see, for example, [Reference HumphreysHu, Theorem 3.4]). We define $w_0^{\lambda }$ to be the longest element of $W_{[\lambda ]}$ . For $\lambda ,\nu \in \Sigma ^+_{\mathfrak {p}}$ , it follows from [Reference Cheng, Mazorchuk and WangCP, Theorem B] (see also [Reference Cheng, Mazorchuk and WangCP, Theorem 3.2]) that $\mathcal O_\lambda ^{\mathfrak {p}} =\mathcal O_\nu ^{\mathfrak {p}}$ if and only if $\lambda \sim \nu $ . As a consequence, there exists $\nu \in \text {Irr}\,\mathcal O_{\lambda }^{\mathfrak {p}}$ which is dominant and regular.

Let $\lambda \in \mathfrak {h}^\ast $ be a dominant weight such that the parabolic block $\mathcal O^{\mathfrak {p}}_\lambda $ (see Section 6.2 for the precise definition) is nonzero. Let $ (\mathcal O^{\mathfrak {p}}_{{\bar 0}})_{\lambda +X}$ denote the full subcategory of $\mathcal O^{\mathfrak {p}}_{{\bar 0}}$ consisting of all modules whose weights belong to the set $\lambda +X$ . Applying, if necessary, the equivalence established in [Reference Coulembier and SerganovaCMW, Section 2.3] (see also [Reference Chen and CoulembierChC, Section  4.3]), we can assume that $W_{[\lambda ]}$ is a parabolic subgroup of W. Then the problem of global dimension of $(\mathcal O^{\mathfrak {p}}_{{\bar 0}})_{\lambda +X}$ reduces to that of an integral block for a semisimple Lie algebra of type $\bf A$ by [Reference Coulembier and SerganovaCMW, Proposition 2.3]. By [Reference Coulembier and MazorchukCoM2, Theorem 5.2],

$$ \begin{align*} &\text{gl.dim} (\mathcal O^{\mathfrak{p}}_{{\bar 0}})_{\lambda +X} =2\ell(w_0^{\lambda}) -2\ell(w_0^{\mathfrak{p}}). \end{align*} $$

Take $\nu \in \text {Irr}\,\mathcal O_{\lambda }^{\mathfrak {p}}$ dominant and regular. In this case, ${\text {pd}}\, I_{\bar 0}^{\mathfrak {p}}(\nu ) =2 \ell (w_0^{\lambda }) -2\ell (w_0^{\mathfrak {p}})$ by [Reference Coulembier and MazorchukCoM2, Theorem 4.1 and Proposition 6.9]. By Theorem 12, thus

$$ \begin{align*} &\text{fin.dim}\, \mathcal O_\lambda^{\mathfrak{p}} =2\ell(w_0^{\lambda}) - 2\ell(w_0^{\mathfrak{p}}). \end{align*} $$

5.2 The parabolic BGG category $\mathcal O^{\mathfrak {p}}$

In this section, we let $\mathfrak g$ be an arbitrary classical Lie superalgebra. We let $\langle {}_-,{}_-\rangle $ denote the usual bilinear form on $\mathfrak h^\ast $ .

In the remaining parts of the present paper, we use the notion of parabolic decomposition as defined in [Reference Mazorchuk, Gorelik and PapiMa3, Section 2.4] and consider the reduced parabolic subalgebra $\mathfrak {p}$ from [Reference Chen and CoulembierCCC, Section 1.4]. Namely, we assume that $\mathfrak b\subseteq \mathfrak {p}\subseteq \mathfrak g$ where there is a purely even Levi subalgebra $\mathfrak l\subseteq \mathfrak {p}$ . The even part $\mathfrak p_{\bar 0}$ of $\mathfrak p$ is a parabolic subalgebra of $\mathfrak g_{\bar 0}$ arising from a parabolic decomposition of $\mathfrak g_{\bar 0}$ . In what follows, we always assume that $\mathfrak {p}_{\bar 1} =\mathfrak {g}_1$ in the case when $\mathfrak g$ is of type I.

The parabolic category $\mathcal O^{\mathfrak {p}}$ (respectively $\mathcal O^{\mathfrak {p}_{\bar 0}}_{\bar 0}$ ) is the full subcategory of $\mathcal O$ (respectively $\mathcal O_{\bar 0}$ ) consisting of all finitely generated $\mathfrak g$ -modules (respectively $\mathfrak g_{\bar 0}$ -modules) on which $\mathfrak {p}$ (respectively $\mathfrak {p}_{\bar 0}$ ) acts locally finitely. For simplicity of notation, we define $\mathcal O^{\mathfrak {p}}_{\bar 0}:= \mathcal O^{\mathfrak {p}_{\bar 0}}_{\bar 0}$ , and, in what follows, we omit ${\bar 0}$ from $(-)^{\mathfrak {p}_{\bar 0}}$ or $(-)_{\mathfrak {p}_{\bar 0}}$ when defining notation for $\mathfrak {g}_{\bar 0}$ -modules.

For any $\lambda \in \mathfrak {h}^\ast $ , let $L^{\mathfrak l}(\lambda )$ denote the irreducible $\mathfrak l$ -module with highest weight $\lambda $ . We define the set of $\mathfrak l$ -dominant weights (called $\mathfrak {p}$ -dominant weights in [Reference Chen and CoulembierCCC, Section 3]) as follows:

$$ \begin{align*} &\Sigma^+_{\mathfrak{p}}:=\{\lambda\in \mathfrak h^\ast\mid \langle\lambda, \alpha\rangle \in \mathbb Z_{\geq 0}, \text{ for all } \alpha \in \Phi^+(\mathfrak l) \}. \end{align*} $$

The corresponding parabolic category $\mathcal O^{\mathfrak {p}}$ is the Serre subcategory of $\mathcal O$ generated by $\{L(\lambda ) \, | \, \lambda \in \Sigma _{\mathfrak {p}}^+\}$ .

Let $\mathcal O^{\mathfrak {p}}_\lambda $ denote the (indecomposable) block in $\mathcal O^{\mathfrak {p}}$ containing $L(\lambda )$ . In particular, $\mathcal O_\lambda :=\mathcal O_\lambda ^{\mathfrak b}$ is the (indecomposable) block containing $L(\lambda )$ in category $\mathcal O$ .

For $\lambda \in \Sigma _{\mathfrak {p}}^+$ , we define the parabolic Verma $\mathfrak g_{\bar 0}$ -module and the parabolic Verma $\mathfrak g$ -module, respectively, as

$$ \begin{align*} &\Delta^{\mathfrak{p}}_{\bar 0}(\lambda) : = \operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak g_{\bar 0}}L^{\mathfrak l}(\lambda)\quad\text{and}\quad \Delta^{\mathfrak{p}}(\lambda):= \operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak g}L^{\mathfrak l}(\lambda). \end{align*} $$

Similarly, we can define the dual parabolic Verma modules $\nabla ^{\mathfrak {p}}_{\bar 0}(\lambda )$ and $\nabla ^{\mathfrak {p}}(\lambda )$ in $\mathcal O^{\mathfrak {p}}_{\overline 0}$ and $\mathcal O^{\mathfrak {p}}$ , respectively (see [Reference Chen and CoulembierCCC, Definition 3.2]). We have

$$ \begin{align*}\Delta^{\mathfrak{p}}(\lambda) \twoheadrightarrow L(\lambda)\quad\text{and}\quad L(\lambda) \hookrightarrow \nabla^{\mathfrak{p}}(\lambda).\end{align*} $$

Let $P^{\mathfrak {p}}(\lambda )$ be the projective cover of $L(\lambda )$ in $\mathcal O^{\mathfrak {p}}$ and $I^{\mathfrak {p}}(\lambda )$ be the injective envelope of $L(\lambda )$ in $\mathcal O^{\mathfrak {p}}$ . For $\mathcal O^{\mathfrak {p}} =\mathcal O$ , we denote the corresponding objects by $P(\lambda )$ and $I(\lambda )$ . Similarly, we define the indecomposable projectives $P_{\bar 0}(\lambda )$ , $P_{\bar 0}^{\mathfrak {p}}(\lambda )$ and injectives $I_{\bar 0}(\lambda )$ , $I^{\mathfrak {p}}_{\bar 0}(\lambda )$ in $\mathcal O_{\bar 0}$ and $\mathcal O_{\bar 0}^{\mathfrak {p}}$ , respectively. Also, we denote by $T^{\mathfrak {p}}(\lambda )$ and $T_{\bar 0}^{\mathfrak {p}}(\lambda )$ the tilting modules of highest weight $\lambda $ in $\mathcal O^{\mathfrak {p}}$ and $\mathcal O_{\bar 0}^{\mathfrak {p}}$ , respectively. Finally, we note that $\operatorname {Ind}\nolimits : \mathcal O_{\bar 0}^{\mathfrak {p}}\rightarrow \mathcal O^{\mathfrak {p}}$ and $\operatorname {Res}\nolimits : \mathcal O^{\mathfrak {p}}\rightarrow \mathcal O_{\bar 0}^{\mathfrak {p}}$ are well defined.

Also, we note that

$$ \begin{align*}\Delta^{\mathfrak{p}}_\lambda= K(\Delta^{\mathfrak{p}}_{\bar 0}(\lambda)),~\nabla^{\mathfrak{p}}(\lambda):= \operatorname{Ind}\nolimits_{\mathfrak g_{\bar 0}+\mathfrak g _{-1}}^{\mathfrak g}(\nabla^{\mathfrak{p}}_{\bar 0}(\lambda) \otimes \Lambda^{\text{top}}\mathfrak g_1^\ast)\cong \mathrm{Coind}_{\mathfrak p_{\bar 0}+\mathfrak g_{-1}}^{\mathfrak g}L^{\mathfrak l}(\lambda),\end{align*} $$

in the case when $\mathfrak g$ is of type I.

We define $w_0^{\mathfrak {p}}$ to be the longest element in the Weyl group of $\mathfrak l$ . Finally, we set $w_0:=w_{0}^{\mathfrak b}$ .

5.3 Projective dimension of modules in $\mathcal O^{\mathfrak {p}}$

In this subsection, we suppose that $\mathfrak g$ is of type I.

5.3.1 Preliminary results

The following lemma is the parabolic version of [Reference Coulembier and SerganovaCS, Lemma 4.3].

Lemma 17.

1. (1) For $M\in \mathcal O^{\mathfrak {p}}$ , we have pd $\, M \geq $ pd $\, \operatorname {Res}\nolimits M$ .

2. (2) For $N\in \mathcal O^{\mathfrak {p}}_{\bar 0}$ , we have pd $\, N = $ pd $\, \operatorname {Ind}\nolimits N =$ pd $\, \mathrm {Coind} N$ .

Proof. As already mentioned, $\operatorname {Res}\nolimits $ , $\operatorname {Ind}\nolimits $ , and $\mathrm {Coind}$ send projective resolutions to projective resolutions. Also, N is a direct summand of $\operatorname {Res}\nolimits \operatorname {Ind}\nolimits N$ . This implies all statements.

Corollary 18. For any weight $\zeta \in \Sigma ^+_{\mathfrak {p}},$ we have

$$ \begin{align*}{\mathrm{pd}}\, \Delta^{\mathfrak{p}}(\zeta) \geq {\mathrm{pd}} \, \Delta_{\bar 0}^{\mathfrak{p}}(\zeta) \quad\text{and}\quad {\mathrm{pd}}\, \nabla^{\mathfrak{p}}(\zeta) \geq {\mathrm{pd}} \, \nabla_{\bar 0}^{\mathfrak{p}}(\zeta).\end{align*} $$

Proof. Noting that the modules $\Delta _{\bar 0}^{\mathfrak {p}}(\zeta )$ and $\nabla _{\bar 0}^{\mathfrak {p}}(\zeta )$ are direct summands of $\operatorname {Res}\nolimits \Delta ^{\mathfrak {p}}(\zeta )$ and $\operatorname {Res}\nolimits \nabla ^{\mathfrak {p}}(\zeta )$ , respectively, the statement follows from Lemma 17.

The following result is a consequence of the combination of [Reference Coulembier and SerganovaCo, Corollary 3.2.5] and [Reference Coulembier and SerganovaCo, Theorem 8.2.1]. We provide more details in the proof.

Proposition 19. Suppose that $\mathfrak g$ is of type I. Then, for each $\lambda \in \Sigma ^+_{\mathfrak {p}}$ , we have

$$ \begin{align*}{\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}}\, I^{\mathfrak{p}}(\lambda) = {\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}\, I^{\mathfrak{p}}_{\bar 0}(\lambda).\end{align*} $$

Proof. The proof of [Reference Chen and CoulembierCCC, Corollary 4.7] shows that $L_{\bar 0}(\lambda )$ is a quotient of $\operatorname {Res}\nolimits L(\lambda )$ . Therefore,

$$ \begin{align*} &\text{Hom}_{\mathcal O^{\mathfrak{p}}}(L(\lambda), \mathrm{Coind}\, I^{\mathfrak{p}}_{\bar 0} (\lambda)) = \text{Hom}_{\mathcal O_{\bar 0}^{\mathfrak{p}}}(\operatorname{Res}\nolimits L(\lambda), I^{\mathfrak{p}}_{\bar 0} (\lambda)) = [\operatorname{Res}\nolimits L(\lambda): L_{\bar 0}(\lambda)]\neq 0. \end{align*} $$

This means that $L(\lambda ) \hookrightarrow \mathrm {Coind} I_{\bar 0}^{\mathfrak {p}}(\lambda )$ and so $I^{\mathfrak {p}}(\lambda )$ must be a direct summand of $\mathrm {Coind} I_{\bar 0}^{\mathfrak {p}}(\lambda )$ . Consequently ${\text {pd}}\, I^{\mathfrak {p}}(\lambda ) \leq {\text {pd}}\, I_{\bar 0}^{\mathfrak {p}}(\lambda ).$

Next, we claim that $L_{\bar 0}(\lambda )$ is a direct summand of the socle of $\operatorname {Res}\nolimits I^{\mathfrak {p}}(\lambda )$ . We start by observing that

$$ \begin{align*} &\operatorname{Ind}\nolimits L_{\bar 0}(\lambda) =\operatorname{Ind}\nolimits^{\mathfrak{g}}_{\mathfrak g_{\geq 0}} \operatorname{Ind}\nolimits^{\mathfrak{g}_{\geq 0}}_{\mathfrak g_{\bar 0}} L_{\bar 0}(\lambda) \cong \operatorname{Ind}\nolimits^{\mathfrak{g}}_{\mathfrak g_{\geq 0}} (\Lambda^{\bullet} \mathfrak g_{1}\otimes L_{\bar 0}(\lambda)). \end{align*} $$

This implies that

$$ \begin{align*} &\text{Hom}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}(L_{\bar 0}(\lambda), \operatorname{Res}\nolimits I^{\mathfrak{p}} (\lambda)) = \text{Hom}_{\mathcal O^{\mathfrak{p}}}(\operatorname{Ind}\nolimits L_{\bar 0}(\lambda), I^{\mathfrak{p}} (\lambda))\\ &\quad = [\operatorname{Ind}\nolimits L_{\bar 0}(\lambda): L(\lambda)]\geq[\operatorname{Ind}\nolimits^{\mathfrak g}_{\mathfrak g_{\geq 0}} L_{\bar 0}(\lambda): L(\lambda)] \neq 0. \end{align*} $$

Taking Lemma 17 into account, ${\text {pd}}\, I_{\bar 0}^{\mathfrak {p}}(\lambda )\leq {\text {pd}}\, \operatorname {Res}\nolimits I^{\mathfrak {p}}(\lambda ) \leq {\text {pd}} \,I^{\mathfrak {p}}(\lambda )$ . This completes the proof.

The following result is also a consequence of the combination of [Reference Coulembier and SerganovaCo, Corollary 3.2.5] and [Reference Coulembier and SerganovaCo, Theorem 8.2.1]. We provide more details in the proof.

Proposition 20. Suppose that $\mathfrak g$ is of type I. Then, for each $\lambda \in \Sigma ^+_{\mathfrak {p}}$ ,

$$ \begin{align*}{\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}}\, T^{\mathfrak{p}}(\lambda) = {\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}\, T^{\mathfrak{p}}_{\bar 0}(\lambda).\end{align*} $$

Proof. Define $\xi _{\pm }\in \mathfrak {h}^\ast $ such that ${\mathbb C}_{\xi _\pm }\cong \Lambda ^{\text {top}} \mathfrak g_{\pm 1}$ , as $\mathfrak g_{\bar 0}$ -modules. We note that

$$ \begin{align*} &\operatorname{Ind}\nolimits \Delta_{\overline 0}^{\mathfrak{p}}(\lambda) \cong \operatorname{Ind}\nolimits_{\mathfrak g_{\overline 0}}^{\mathfrak g} \operatorname{Ind}\nolimits_{\mathfrak p_{\overline 0}}^{\mathfrak g_{\overline 0}}L_{\mathfrak l}(\lambda) \cong \operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak g} \operatorname{Ind}\nolimits_{\mathfrak p_{\overline 0}}^{\mathfrak p}L_{\mathfrak l}(\lambda). \end{align*} $$

This means that $\operatorname {Ind}\nolimits \Delta _{\overline 0}^{\mathfrak {p}}(\lambda ) $ has a $\Delta ^{\mathfrak {p}}$ -flag starting at $\Delta ^{\mathfrak {p}}(\lambda +\xi _+)$ . Therefore, the module $T^{\mathfrak {p}}(\lambda +\xi _+)$ is a direct summand of the module $\operatorname {Ind}\nolimits T_{\bar 0}^{\mathfrak {p}}(\lambda )$ . Using Lemma 17, we conclude that

$$ \begin{align*}{\text{pd}}\, T^{\mathfrak{p}}(\lambda) \leq {\text{pd}}\, \operatorname{Ind}\nolimits T^{\mathfrak{p}}_{\bar 0} (\lambda -\xi_+) ={\text{pd}} \,T^{\mathfrak{p}}_{\bar 0}(\lambda-\xi_+)={\text{pd}} \,T^{\mathfrak{p}}_{\bar 0}(\lambda).\end{align*} $$

Here the last equality follows by tensoring with $\Lambda ^{\text {top}} \mathfrak g_{\pm 1}$ .

Next, we claim that $ T_{\bar 0}^{\mathfrak {p}}(\lambda +\xi _-) \hookrightarrow \operatorname {Res}\nolimits T^{\mathfrak {p}}(\lambda )$ . To see this, we note that

$$ \begin{align*} &\Delta^{\mathfrak{p}}_{\bar 0}(\lambda +\xi_-) \hookrightarrow \Lambda^{\bullet} \mathfrak g_{-1}\otimes \Delta^{\mathfrak{p}}_{\bar 0}(\lambda) \cong \operatorname{Res}\nolimits \Delta^{\mathfrak{p}}(\lambda) \hookrightarrow \operatorname{Res}\nolimits T^{\mathfrak{p}}(\lambda). \end{align*} $$

This implies that there is an indecomposable direct summand of $\operatorname {Res}\nolimits T^{\mathfrak {p}}(\lambda )$ that has a $\Delta ^{\mathfrak {p}}_{\bar 0}$ -flag starting at $\Delta ^{\mathfrak {p}}_{\bar 0}(\lambda +\xi _-)$ . In particular, $T_{\bar 0}^{\mathfrak {p}}(\lambda +\xi _-)$ is a direct summand of $\operatorname {Res}\nolimits T^{\mathfrak {p}}(\lambda )$ . Consequently,

$$ \begin{align*}{\text{pd}} \, T_{\bar 0}^{\mathfrak{p}}(\lambda)= {\text{pd}}\, T_{\bar 0}^{\mathfrak{p}}(\lambda+\xi_-) \leq {\text{pd}}\, \operatorname{Res}\nolimits T^{\mathfrak{p}}(\lambda)\leq {\text{pd}}\, T^{\mathfrak{p}}(\lambda)\end{align*} $$

by Lemma 17. The result of the proposition follows.

5.3.2 Parabolic dimension shift

This subsection is devoted to the following formula for the projective dimension.

Theorem 21. Suppose that $M\in \mathcal O^{\mathfrak {p}}$ with ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}M<\infty $ . Then,

$$ \begin{align*} &{\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}}M = {\mathrm{pd}}_{\mathcal O}M -2\ell(w_0^{\mathfrak{p}}). \end{align*} $$

Proof. We first claim that

$$ \begin{align*}{\text{pd}}_{\mathcal O^{\mathfrak{p}}}M < \infty \Rightarrow {\text{pd}}_{\mathcal O}M < \infty. \end{align*} $$

To see this, we let Q be a projective module in $\mathcal O^{\mathfrak {p}}$ . The adjoint pair $\operatorname {Ind}\nolimits $ and $\operatorname {Res}\nolimits $ of functors between $\mathcal O_{\bar 0}^{\mathfrak {p}}$ and $\mathcal O^{\mathfrak {p}}$ gives an epimorphism $\operatorname {Ind}\nolimits \operatorname {Res}\nolimits Q \rightarrow Q$ , and so Q is a direct summand of $\operatorname {Ind}\nolimits \operatorname {Res}\nolimits Q$ . Using Frobenius reciprocity, it follows that

$$ \begin{align*}\text{Ext}_{\mathcal O}^d(\operatorname{Ind}\nolimits \operatorname{Res}\nolimits Q, N)\cong \text{Ext}_{\mathcal O_{\bar 0}}^d( \operatorname{Res}\nolimits Q, \operatorname{Res}\nolimits N)=0,\end{align*} $$

for any $d>{\text {pd}}_{\mathcal O_{\bar 0}}\operatorname {Res}\nolimits Q$ and $N\in \mathcal O$ . As a consequence, ${\text {pd}}_{\mathcal O}Q<\infty .$

Now suppose that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}M < \infty $ . This means that M has a finite projective resolution in $\mathcal O^{\mathfrak {p}}$ :

$$ \begin{align*} 0 \rightarrow Q^d\xrightarrow{f^d} Q^{d-1}\xrightarrow{f^{d-1}} \cdots \xrightarrow{f^2} Q^1\xrightarrow{f^1} Q^0 \xrightarrow{f^0} M \rightarrow 0.\end{align*} $$

We define $K^s$ to be the kernel of the map $f^s$ for each s. Then the inequality ${\text {pd}}_{\mathcal O}K^{s}<\infty $ together with the exactness of $0\rightarrow K^s \rightarrow Q^{s-1} \rightarrow K^{s-1}\rightarrow 0$ imply that ${\text {pd}}_{\mathcal O}K^{s-1}<\infty $ . Consequently, we have ${\text {pd}}_{\mathcal O}M < \infty $ .

Now suppose that both ${\text {pd}}_{\mathcal O}M$ and ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}M$ are finite. Then, by [Reference Coulembier and SerganovaCo, Corollary 3.2.5] and [Reference Coulembier and SerganovaCo, Theorem 7.2.1(iii)],

$$ \begin{align*}{\text{pd}}_{\mathcal O}M={\text{pd}}_{\mathcal O_{\bar 0}}\operatorname{Res}\nolimits M \quad\text{and}\quad{\text{pd}}_{\mathcal O^p}M={\text{pd}}_{\mathcal O^{\mathfrak{p}}_0}\operatorname{Res}\nolimits M.\end{align*} $$

The proof now follows from [Reference Coulembier and MazorchukCoM2, Theorem 4.1].

Corollary 22. Let $M \in \mathcal O^{\mathfrak {p}}$ with ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}} M<\infty .$

1. (1) M is projective in $\mathcal O^{\mathfrak {p}}$ if and only if ${\mathrm {pd}}_{\mathcal O}M =2\ell (w_0^{\mathfrak {p}})$ .

2. (2) If $\mathfrak {p}=\mathfrak {g}$ , then ${\mathrm {pd}}_{\mathcal O}M =2\ell (w_0)$ .

Proof. Claim $(1)$ follows directly from Theorem 21.

If $\mathfrak {p}=\mathfrak {g}$ , then $\mathcal O^{\mathfrak {p}}$ is the category of finite dimensional modules. In this case, we have $\text {fin.dim}\,\mathcal O^{\mathfrak {p}} = 2\ell (w_0) - 2\ell (w_0^{\mathfrak {p}}) =0$ by (5-1), which implies that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}M=0$ . Claim $(2)$ now follows from Theorem 21.

5.4 Injective dimension of modules in $\mathcal O^{\mathfrak {p}}$

In this subsection, unless mentioned otherwise, we suppose that $\mathfrak {g}$ is of type I.

Let $\mu $ be a dominant integral weight whose singularity determines the parabolic subalgebra ${\mathfrak {p}}_{\bar 0}$ , namely, the Levi subalgebra of ${\mathfrak {p}}_{\bar 0}$ is generated by $\mathfrak h$ and ${\mathfrak {g}}_\alpha $ with roots satisfying $\langle \mu +\rho _{\bar 0}, \alpha \rangle =0$ . Following [Reference Chen and CoulembierCCC, Section 1.4], there is a parabolic subalgebra $\hat {\mathfrak {p}}$ (in the sense of [Reference Mazorchuk, Gorelik and PapiMa3, Section 2.4]) of $\mathfrak g$ such that $\hat {\mathfrak {p}}_{\bar 0}$ is determined by $\hat \mu := -w_0\mu $ . For instance, if $\mathfrak g$ is of type I, then $\hat {\mathfrak {p}}= \hat {\mathfrak {p}}_{\bar 0} \oplus {\mathfrak g}_{-1}$ . Also, we have $\hat {\mathfrak b} ={\mathfrak b}^r$ for $\mathfrak {pe} (n)$ from Section 3.2.

We recall, see [Reference Chen and CoulembierCCC, Section 1.3], that there is a duality (that is, an antiequivalence) $\bf D: \mathcal O^{\mathfrak {p}}\rightarrow \mathcal O^{\hat {\mathfrak {p}}}$ such that

(5-2) $$ \begin{align} &\mathbf{D} L^{\mathfrak{p}}(\lambda) =L^{\hat {\mathfrak{p}}}(-w_0\lambda), ~\mathbf{D}\Delta^{\mathfrak{p}}(\lambda) = \nabla^{\hat {\mathfrak{p}}}(-w_0\lambda), \text{ and }\mathbf{D}T^{\mathfrak{p}}(\lambda) = T^{\hat {\mathfrak{p}}}(-w_0\lambda). \end{align} $$

We refer to [Reference Chen and CoulembierCCC, Proposition 3.4 and Lemma 3.6] for more details. Applying the results from Section 5.3, we obtain the following proposition.

Proof. Applying $\bf D$ ,

$$ \begin{align*} &{\text{id}}_{\mathcal O^{\mathfrak{p}}} M ={\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}}\mathbf{D}M. \end{align*} $$

Suppose that ${\text {pd}}_{\mathcal O^{\hat {\mathfrak {p}}}}\mathbf {D}M <\infty $ . We note that $\operatorname {Res}\nolimits $ intertwines $\bf D$ and the usual duality on $\mathcal O^{\mathfrak {p}}_{\bar 0}$ which we also denote by $\bf D$ , abusing notation (see the proof of [Reference Chen and CoulembierCCC, Theorem 3.7]). Therefore, from [Reference Coulembier and SerganovaCo, Corollary 3.2.5] and [Reference Coulembier and SerganovaCo, Theorem 7.2.1], it follows that

$$ \begin{align*}{\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}}\mathbf{D}M= {\text{pd}}_{\mathcal O_{\bar 0}^{\hat {\mathfrak{p}}}}\operatorname{Res}\nolimits\mathbf{D}M={\text{pd}}_{\mathcal O_{\bar 0}^{\hat {\mathfrak{p}}}}\mathbf{D}\operatorname{Res}\nolimits M={\text{id}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} \operatorname{Res}\nolimits M.\end{align*} $$

This proves part $(1)$ and (5-3).

To prove part $(2)$ , we note that from Theorem 21 and the proof of [Reference Mazorchuk, Gorelik and PapiMa3, Theorem  3], we obtain ${\text {id}}_{\mathcal O^{\mathfrak {p}}} M<\infty \Rightarrow {\text {id}}_{\mathcal O}M <\infty $ . Therefore, we may assume that both ${\text {id}}_{\mathcal O} M$ and ${\text {id}}_{\mathcal O^{\mathfrak {p}}}M$ are finite. From (5-3) and [Reference Coulembier and MazorchukCoM2, Theorem 4.1],

$$ \begin{align*} &{\text{id}}_{\mathcal O^{\mathfrak{p}}} M = {\text{id}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} \operatorname{Res}\nolimits M = {\text{id}}_{\mathcal O_{\bar 0}} \operatorname{Res}\nolimits M-2\ell(w_0^{\mathfrak{p}}) ={\text{id}}_{\mathcal O} M -2\ell(w_0^{\mathfrak{p}}). \end{align*} $$

This proves (5-4) and part $(2)$ . An alternative proof of part $(2)$ arises from the following observation:

$$ \begin{align*} {\text{pd}}_{\mathcal O^{\mathfrak{p}}}M = {\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}}\mathbf{D}M &= {\text{pd}}_{\mathcal O}\mathbf{D}M -2\ell(w_0^{\hat {\mathfrak{p}}}) \\ &= {\text{pd}}_{\mathcal O}\mathbf{D}M -2\ell(w_0^{\mathfrak{p}}) = {\text{pd}}_{\mathcal O^{\mathfrak{p}}}\mathbf{D}M={\text{id}}_{\mathcal O^{\mathfrak{p}}}M. \end{align*} $$

It remains to prove part $(3)$ . By (5-2),

$$ \begin{align*}{\text{id}}_{\mathcal O^{\mathfrak{p}}} P^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}} I^{\hat {\mathfrak{p}}}(-w_0\lambda),~ {\text{id}}_{\mathcal O^{\mathfrak{p}}} T^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O^{\hat {\mathfrak{p}}}} T^{\hat {\mathfrak{p}}}(-w_0\lambda).\end{align*} $$

It follows that

$$ \begin{align*}{\text{id}}_{\mathcal O^{\mathfrak{p}}} P^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O_{\bar 0}^{\hat {\mathfrak{p}}}} I_{\bar 0}^{\hat {\mathfrak{p}}}(-w_0\lambda) ={\text{pd}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} I_{\bar 0}^{ \mathfrak{p}}(\lambda)={\text{id}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} P_{\bar 0}^{ \mathfrak{p}}(\lambda), \end{align*} $$

$$ \begin{align*} {\text{id}}_{\mathcal O^{\mathfrak{p}}} T^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O_{\bar 0}^{\hat {\mathfrak{p}}}} T^{\hat {\mathfrak{p}}}_{\bar 0}(-w_0\lambda)= {\text{pd}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} T_{\bar 0}^{ \mathfrak{p}}(\lambda), \end{align*} $$

by Proposition 19, Theorem 20, and [Reference SoergelSo, Theorem 11].

5.5 A sufficient condition for infiniteness of projective dimension

We continue to suppose that $\mathfrak g$ is of type I. We recall the definition of the associated variety from [Reference Duflo and SerganovaDS]. Let $M\in \mathfrak {g}$ -Mod and $x\in \mathfrak g_{\overline 1}$ with $[x,x]=0$ . Define

$$ \begin{align*}M_x:= \text{ker}_x/xM,\end{align*} $$

where $\text {ker}_x$ is the kernel of the map $x: M\rightarrow M$ sending m to $xm$ , where $m\in M.$ Then the associated variety $X_M$ of M is defined as

$$ \begin{align*}X_M:=\{x\in \mathfrak{g}_{\bar 1}\mid [x,x]=0 \text{ and } M_x\neq 0\}.\end{align*} $$

The following statement is obtained [Reference Duflo and SerganovaDS, Lemma 2.2(i)].

Proof. Let $c\in C$ be such that $xc=0$ . We want to show that there exists $b\in B$ such that $c=xf(b)$ , which implies that $C_x =0$ . To see this, we first pick $d\in B$ such that $f(d)=c$ . Then $f(xd)=0$ implies that $xd\in A$ .

Now $x(xd) =x^2d =0$ (owing to $x^2=0$ ) implies $xd\in xA$ and so there is $a\in A$ such that $xd =xa$ because $A_x =0$ . Finally, we note that $x(d-a)=0$ implies that $d-a \in xB$ because $B_x=0$ . Therefore, there is $b\in B$ such that $d-a =xb$ . We now apply f and obtain

$$ \begin{align*}c=f(d)=f(d-a)=xf(b),\end{align*} $$

as desired.

The following observation is known (see, for example, [Reference Coulembier and SerganovaCS, Section 4 and proof of Theorem 4.1] and [Reference Duflo and SerganovaDS, Lemma 2.2]).

Proof. Suppose that there exists a finite projective resolution of M in $\tilde {\mathcal C}$ :

$$ \begin{align*}0 \xrightarrow{f^{d+1}} P^d \xrightarrow{f^d} \cdots \xrightarrow{} P^2\xrightarrow{f^2} P^1\xrightarrow{f^1} P^0 \rightarrow M \xrightarrow{f^0} 0.\end{align*} $$

Let $0\leq s\leq d$ . By [Reference Coulembier and SerganovaCo, Proposition 2.2.1], every projective module $P^s$ is a direct summand of a module induced from $\mathcal C$ to $\tilde {\mathcal C}$ . By Lemma 24, we thus have $X_{P^s} = 0$ .

Define $K^s$ to be the kernel of $f^s$ . For any s, we have an exact sequence

$$ \begin{align*}0\rightarrow K^s \rightarrow P^{s-1} \rightarrow K^{s-1}\rightarrow 0.\end{align*} $$

In particular, the fact that $K^d=P^d$ implies that $K^{d-1}_x=0$ by Lemma 25. By induction on s, we may conclude that $M_x=K^0_x =0$ . The result follows.

The following corollary addresses infiniteness of projective dimension for $\mathfrak {p}$ -highest weight modules.

We are going to apply Corollary 29 to the periplectic Lie superalgebra $\mathfrak {pe} (n)$ later in Proposition 36.

6.1 Example I: projective dimension of modules for $\mathfrak {osp}(2|2n)$

The orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$ is the following subsuperalgebra of $\mathfrak {gl}(2|2n)$ :

(6-1) $$ \begin{align} \mathfrak{osp}(2\vert 2n)= \left\{ \left( \begin{array}{cccc} c &0 & x &y\\ 0 & -c& v & u\\ -u^t& -y^t & a &b\\ v^t &x^t & c& -a^t \\ \end{array} \right): \begin{array}{c} c\in {\mathbb C};\,\, x,y,v,u\in {\mathbb C}^{n};\\ a,b,c\in {\mathbb C}^{n^2};\\ b=b^t,\,\, c=c^t. \end{array} \right\}. \end{align} $$

The even part $\mathfrak {osp}(2|2n)_{\bar 0}$ of $\mathfrak {osp}(2|2n)$ is isomorphic to ${\mathbb C} \oplus \mathfrak {sp}(2n)$ and the odd part is isomorphic to ${\mathbb C}^2\otimes {\mathbb C}^{2n}$ . We refer the reader to [Reference Cheng and WangCW1, Section 1.1.3] and [Reference MussonMu, Section  2.3] for more details. Recall from Section 3.2 that we denote by $E_{ab}$ the elementary matrices. We now define

$$ \begin{align*}H_\epsilon:=E_{11}-E_{22},~H_i:=E_{2+i,2+i}-E_{2+i+n,2+i+n}, \quad \text{ for }1\leq i\leq n. \end{align*} $$

The Cartan subalgebra $\mathfrak h$ of $\mathfrak {osp}(2|2n)$ is spanned by $H_\epsilon $ and $H_1,H_2,\ldots , H_n$ . We let $\{\epsilon ,\delta _1,\delta _2,\ldots ,\delta _n\}$ be the basis of $\mathfrak h^\ast $ dual to $\{H_\epsilon ,H_1,H_2,\ldots , H_n\}$ . The bilinear form $({}_-,{}_-):\mathfrak h^\ast \times \mathfrak h^\ast \rightarrow {\mathbb C}$ is given by $(\epsilon ,\epsilon ) =1$ and $(\delta _i,\delta _j) =-\delta _{i,j}$ , for $1\leq i,j\leq n.$

We define the set $\Phi ^+_{\bar 0}$ of even positive roots and the set $\Phi ^+_{\bar 1}$ of odd positive roots by

$$ \begin{align*} &\Phi^+_{\bar 0}:=\{\delta_i \pm\delta_j, 2\delta_p\mid 1\leq i< j <n, ~1\leq p \leq n \},\nonumber\\ &\Phi_{\bar 1}^+:=\{\epsilon \pm \delta_p\mid 1\leq p\leq n\}. \end{align*} $$

This gives rise to a triangular decomposition $\mathfrak g=\mathfrak n^+\oplus \mathfrak h \oplus \mathfrak n^-$ with Borel subalgebra $\mathfrak b:= \mathfrak b_{\bar 0} \oplus \mathfrak g_1.$

6.1.1 Simple supermodules over $\mathfrak {osp}(2|2n)$

The aim of this subsection is to study the projective dimension of simple supermodules over arbitrary basic classical Lie superalgebras of type I including $\mathfrak {osp}(2|2n)$ .

We recall the notation of typical weights for basic classical Lie superalgebras of type I; see, for example, [Reference GorelikGor2, Section 2.4]. Let $X^{-}\in \Lambda ^{\max }\mathfrak g_{-1}$ and $\quad X^{+}\in \Lambda ^{\max }\mathfrak g_{1}$ be nonzero vectors. Then by [Reference Chen and MazorchukCM, Lemma 6.5] (also, see [Reference GorelikGor1, Section 4]) it follows that

$$ \begin{align*} &X^{+} X^- = \Omega + \sum_i x_i r_i y_i, \end{align*} $$

for some $x_i \in \Lambda (\mathfrak g_{ -1})\backslash \mathbb {C}$ , $y_i \in \Lambda (\mathfrak g_{1})\backslash \mathbb {C}$ , $r_i\in U(\mathfrak g_{{\overline 0}})$ , and $\Omega \in Z(\mathfrak g_{\overline 0})$ . A weight $\lambda \in \mathfrak {h}^\ast $ is typical if its evaluation at the Harish-Chandra projection of $\Omega $ is nonzero. It was shown in [Reference Kac, Bleuler, Petry and ReetzKa2, Theorem 1] that the typicality of $\lambda $ is equivalent to the simplicity of the Kac module of highest weight $\lambda $ . A more general definition of typical Kac modules can be found in [Reference Chen and MazorchukCM, Section 6.2].

The following proposition shows that all simple supermodules of atypical blocks over basic classical Lie superalgebras have infinite projective dimension in $\mathcal O$ . This generalizes [Reference Coulembier and SerganovaCS, Proposition 5.14] where the case of the superalgebra $\mathfrak {gl}(m|n)$ was considered.

Proposition 31. Let $\mathfrak g$ be a basic classical Lie superalgebra of type I. Suppose that $\lambda $ is atypical. Then the associated variety $X_{L(\lambda )}$ is nonzero. In particular, if $L(\lambda ) \in \mathcal O^{\mathfrak {p}}$ for some $\mathfrak {p}$ , then ${\textrm {pd}}_{\mathcal O^{\mathfrak {p}}}L(\lambda )=\infty .$

Before we prove Proposition 31, we need to recall the notion of odd reflection for basic classical Lie superalgebras; see, for example, [Reference Penkov and SerganovaPS1, Lemma 1], [Reference Cheng and WangCW1, Section 1.4] and [Reference MussonMu, Section 3] for more details.

Because $\mathfrak g$ is contragradient, the set of roots of $\mathfrak g_{-1}$ can be obtained from the set of roots of $\mathfrak g_{1}$ by multiplying with the scalar $-1$ . Set $d:=\text {dim}\mathfrak g_1=\text {dim}\mathfrak {g}_{-1}$ . By [Reference Cheng and WangCW1, Lemma 1.30 and Proposition 1.32] (also, see [Reference MussonMu, Theorem 3.13]), the positive root system given by the Borel subalgebra $\mathfrak b_{\bar 0}\oplus \mathfrak g_{-1}$ can be obtained from that of $\mathfrak b$ via a sequence of odd reflections. Namely, we can fix an ordering of positive odd roots

(6-2) $$ \begin{align} &\{\alpha_1, \alpha_2, \ldots, \alpha_d\} = \Phi^+_{\bar 1} \end{align} $$

such that the following conditions are satisfied.

1. (1) For any $1\leq i\leq d$ , the set $R^{i} =\Phi ^+_{\bar 0} \cup \{-\alpha _1,\ldots ,-\alpha _{i}\} \cup \{\alpha _{i+1},\ldots , \alpha _d\}$ forms a positive root system.

2. (2) We set $R^{-1}:=R^0$ . Then $\alpha _i$ is a simple root in $R^{i-1}$ and $-\alpha _i$ is a simple root in $R_{i}$ , for any $1\leq i\leq d$ .

Let us fix root vectors $X_i\in \mathfrak g_1\cap \mathfrak g^{\alpha _i}$ and $Y_i \in \mathfrak g_{-1}\cap \mathfrak g^{-\alpha _i}$ for $1\leq i \leq n$ . Also, let $\mathfrak b^{(i)}$ denote the Borel subalgebra corresponding to $R^i$ , namely, $\mathfrak b^{(i)}_{\bar 0} = \mathfrak b_{\bar 0}$ and $\mathfrak b^{(i)}_{\bar 1}$ is generated by all root vectors of odd roots in $R^i$ . By [Reference Cheng and WangCW1, Lemma 1.40], we have the following lemma.

Lemma 32. Suppose that $L\in \mathcal O$ is a simple supermodule having $\mathfrak b^{(i)}$ -highest weight $\mu \in \mathfrak {h}^\ast $ . Let $v\in L$ be a $\mathfrak b^{(i)}$ -highest weight vector.

1. (1) If $Y_{i+1} v=0$ , then v is a $\mathfrak b^{(i+1)}$ -highest weight vector of $L.$

2. (2) If $Y_{i+1} v\neq 0$ , then $Y_{i+1} v$ is a $\mathfrak b^{(i+1)}$ -highest weight vector of L.

Example 33. Let $\mathfrak g= \mathfrak {osp}(2|2n)$ . We start with the simple system (see [Reference Cheng and WangCW1, Section 1.3.4])

$$ \begin{align*}\{\epsilon-\delta_1,~\delta_2-\delta_3,\ldots,\delta_{n-1}-\delta_n,~ 2\delta_n\}.\end{align*} $$

Then, by a direct computation using [Reference Cheng and WangCW1, Equation (1.44) of Lemma 1.30], the ordering (6-2) in this case can be chosen as follows:

$$ \begin{align*} &\epsilon -\delta_1,~\epsilon-\delta_2,\ldots, \epsilon- \delta_n,~\epsilon +\delta_n,~\epsilon +\delta_{n-1},\ldots,\epsilon+\delta_1. \end{align*} $$

Now we are in a position to prove Proposition 31.

Proof of Proposition 31. Let $v\in L(\lambda )$ and $v'\in K(\lambda )$ be highest weight vectors. Because $\lambda $ is atypical, the kernel of the natural epimorphism $\phi : K(\lambda )\rightarrow L(\lambda )$ is nonzero. By [Reference Chen and MazorchukCM, Lemma 3.2], the kernel of $\phi $ contains $Y_dY_{d-1}\cdots Y_1v'$ and so $Y_dY_{d-1}\cdots Y_1v= 0$ . This means that there is $0\leq \ell < d$ such that $Y_{\ell } Y_{\ell - 1}\cdots Y_1 v\neq 0$ and $Y_{\ell +1} Y_{\ell }\cdots Y_1 v =0$ . By Lemma 32, $Y_{\ell } Y_{\ell - 1}\cdots Y_1 v$ is a highest weight vector of L with respect to both $\mathfrak b^{\ell }$ and $\mathfrak b^{\ell +1}$ . Therefore, the weight spaces corresponding to the weights $\lambda -\alpha _1 +\cdots -\alpha _{\ell }\pm \alpha _{\ell +1}$ of $L(\lambda )$ are zero. In particular, the vector $Y_{\ell } Y_{\ell - 1}\cdots Y_1 v\not \in Y_{\ell +1} L(\lambda )$ . This means that $L(\lambda )_{Y_{\ell +1}}\neq 0$ and the result follows from Corollary 26.

6.1.2 (Dual) Verma supermodules over $\mathfrak {osp}(2|2n)$

In this subsection, we assume that $\mathfrak g=\mathfrak {osp}(2|2n)$ for some positive integer n. The following proposition shows that atypical Verma supermodules have infinite projective dimension. We recall that $\Sigma ^+_{\mathfrak {p}}$ denotes the set of highest weights of simple modules in $\mathcal O^{\mathfrak {p}}$ .

Proposition 34. Suppose that $\lambda \in \Sigma _{\mathfrak {p}}^+$ . Then the following conditions are equivalent:

1. (1) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}\Delta ^{\mathfrak {p}}(\lambda ) <\infty $ ;

2. (2) ${\mathrm {pd}}_{\mathcal O}\Delta ^{\mathfrak {p}}(\lambda ) <\infty $ ;

3. (3) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}\Delta ^{\mathfrak {p}}(\lambda ) = {\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}_{\bar 0}}\Delta ^{\mathfrak {p}}(\lambda ) = {\mathrm {pd}}_{\mathcal O}\Delta ^{\mathfrak {p}}(\lambda ) -2\ell (w_0^{\mathfrak {p}})$ ;

4. (4) $\lambda $ is typical.

Proof. We first note that the implication $(1)\Rightarrow (3)$ follows from [Reference Coulembier and SerganovaCo, Corollary 3.2.5] and [Reference Coulembier and SerganovaCo, Lemma 6.3.2]. Therefore, we have $(1)\Leftrightarrow (3)$ . From [Reference GorelikGor2, Theorems 1.3.1 and 1.4.1], it follows that $(4)\Rightarrow (1), (2)$ . Therefore, it remains to show that if $\lambda $ is atypical, then the associated variety of $\Delta ^{\mathfrak {p}}(\lambda )$ is nonzero.

We first define an ordering of roots of $\mathfrak g_{-1}$ (that is, roots in $\Phi _{\bar 1} \backslash \Phi ^+_{\bar 1}$ ) as follows:

$$ \begin{align*} \alpha_1:&=-\epsilon -\delta_1<\alpha_2:=-\epsilon-\delta_2<\cdots< \alpha_{n}:=-\epsilon- \delta_n \end{align*} $$

$$ \begin{align*}&< \alpha_{n+1}:=-\epsilon +\delta_n <\alpha_{n+2}:=-\epsilon +\delta_{n-1}<\cdots<\alpha_{2n}:=-\epsilon+\delta_1. \end{align*} $$

By (6-1), for each $1\leq i \leq n$ , there exist nonzero vectors $X_i \in \mathfrak {g}^{-\alpha _i}$ and $Y_{i}\in \mathfrak {g}^{\alpha _i}$ such that

$$ \begin{align*} [X_i,Y_i] = H_\epsilon-H_{i},\quad\text{for }1\leq i\leq n, \end{align*} $$

$$ \begin{align*}[X_{n+i},Y_{n+i}] = -H_\epsilon-H_{n-i+1},\quad\text{for }1\leq i\leq n. \end{align*} $$

We observe the following crucial fact:

$$ \begin{align*} &\mathfrak n^+_{\bar 0} Y_i \subset \bigoplus_{i< j}{\mathbb C} Y_j,~[X_i,Y_j] \subset \mathfrak n^+_{\bar 0}, \end{align*} $$

for any $1\leq i<j\leq 2n.$ As a consequence, we have

(6-3) $$ \begin{align} X_iY_{j}\cdots Y_{2n}v =0, \end{align} $$

for any $1\leq i<j\leq 2n$ .

Let $\rho : =-n \epsilon + \sum _{1\leq i\leq n} (n-i+1) \epsilon _i$ be the Weyl vector, that is, $\rho $ is the half of the sum of roots in $\Phi _{\bar 0}^+$ minus the half of the sum of roots in $\Phi ^+_{\bar 1}$ . Because $\lambda $ is atypical,

$$ \begin{align*} \prod_{\alpha\in\Phi^+_{\bar{1}}}(\lambda+\rho,\alpha) =0, \end{align*} $$

by [Reference GorelikGor1, Section 4.2]. Therefore, there exists $\alpha = \epsilon +c\delta _i$ such that $(\lambda +\rho ,\alpha ) =0$ and $c=\pm 1.$ We let v be a highest weight vector of $\Delta ^{\mathfrak {p}}(\lambda )$ . Also, we consider the element $\lambda = \lambda _\epsilon \epsilon +\sum _{1\leq i\leq n} \lambda _i \delta _i$ .

Suppose that $c=1$ . By (6-3) and direct computation, we get $X_iY_{i+1}\cdots Y_{2n}v =0$ and

$$ \begin{align*} &X_iY_iY_{i+1}\cdots Y_{2n}v \\[3pt] &\quad= (H_\epsilon -H_i)Y_{i+1}\cdots Y_{2n}v \\[3pt] &\quad= (\lambda_\epsilon -\lambda_i-2n+i -1) Y_{i+1}\cdots Y_{2n}v \\[3pt] &\quad=(\lambda+\rho, \alpha)Y_{i+1}\cdots Y_{2n}v =0. \end{align*} $$

Because the weight subspace of $\Delta ^{\mathfrak {p}}(\lambda )$ of weight $\lambda -\sum _{p\leq q\leq 2n}\alpha _q$ is one-dimensional and spanned by $Y_{p}Y_{p+1}\cdots Y_{2n}v$ , for any $1\leq p\leq 2n$ , we may conclude that $Y_{i+1}\cdots Y_{2n}v \notin X_i\Delta ^{\mathfrak {p}}(\lambda )$ , for otherwise $Y_{i+1}\cdots Y_{2n}v=0$ , which is a contradiction. Consequently, $X_{\Delta ^{\mathfrak {p}}(\lambda )}\neq 0$ . Hence, in this case, the fact that the associated variety of $\Delta ^{\mathfrak {p}}(\lambda )$ is nonzero follows from Corollary 26.

Suppose that $c=-1$ . Set $\overline i:=n-i+1$ for each $1\leq i\leq n$ . Similarly, using (6-3) and direct computation, we obtain $X_{n+\overline i}Y_{n+\overline i +1 +1}Y_{2n}v=0$ and

$$ \begin{align*} &X_{n+\overline i}Y_{n+\overline i}Y_{n+\overline i+1}\cdots Y_{2n}v \\[3pt] &\quad= -(H_\epsilon +H_i)Y_{n+\overline i+1}\cdots Y_{2n}v \\[3pt] &\quad=(-1) (\lambda_\epsilon +\lambda_i -i+1) Y_{n+\overline i+1}\cdots Y_{2n}v \\[3pt] &\quad=-(\lambda+\rho, \alpha)Y_{n+\overline i+1}\cdots Y_{2n}v =0. \end{align*} $$

As above, this implies that $Y_{n+\overline i+1}\cdots Y_{2n}v \notin X_{n+\overline i}\Delta \mathfrak {p}(\lambda )$ and yields $X_{\Delta ^{\mathfrak {p}}(\lambda )}\neq 0$ . Again, the fact that the associated variety of $\Delta ^{\mathfrak {p}}(\lambda )$ is nonzero now follows from Corollary 26. This completes the proof.

Corollary 35. Suppose that $\lambda \in \Sigma _{\mathfrak {p}}^+$ . Then the following are equivalent:

1. (1) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}\nabla ^{\mathfrak {p}}(\lambda ) <\infty $ ;

2. (2) ${\mathrm {pd}}_{\mathcal O}\nabla ^{\mathfrak {p}}(\lambda ) <\infty $ ;

3. (3) ${\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}}\nabla ^{\mathfrak {p}}(\lambda ) = {\mathrm {pd}}_{\mathcal O^{\mathfrak {p}}_{\bar 0}}\nabla ^{\mathfrak {p}}(\lambda ) = {\mathrm {pd}}_{\mathcal O}\nabla ^{\mathfrak {p}}(\lambda ) -2\ell (w_0^{\mathfrak {p}})$ ;

4. (4) $\lambda $ is typical.

Proof. The implication $(1)\Rightarrow (3)$ follows from [Reference Coulembier and SerganovaCo, Corollary 3.2.5] and [Reference Coulembier and SerganovaCo, Lemma 6.3.2]. Therefore, the statements in $(1)$ and $(3)$ are equivalent. The numbers in part (3) are finite by [Reference Bernstein, Gelfand and GelfandBGG2, Section 7].

We recall that $\mathfrak g= \mathfrak {osp}(2|2n)$ is one of the contragredient Lie superalgebras in [Reference MussonMu, Theorem 5.1.5]. Therefore, $\mathcal O^{\mathfrak {p}}$ admits a simple-preserving duality (see, for example, [Reference MussonMu, Section 13.7]) which sends dual Verma supermodules to Verma supermodules. By [Reference Mazorchuk, Gorelik and PapiMa3, Theorem 3], it follows from Proposition 34 that the statements in $(1),(2)$ , and $(4)$ are equivalent.

6.2 Example II: projective dimension of modules for $\mathfrak {pe} (n)$

In this section, we assume that $\mathfrak g=\mathfrak {pe} (n)$ , for some positive integer n. It is known from [Reference Chen and CoulembierCCC, Section 5.5] that every parabolic category is equivalent to the parabolic category given by a parabolic subalgebra $\mathfrak p$ with $\mathfrak p_{\bar 1}=\mathfrak g_1$ . We recall that $\Sigma ^+_{\mathfrak {p}}$ denotes the set of highest weights of simple modules in $\mathcal O^{\mathfrak {p}}$ .

Proposition 36. Suppose that $M\in \mathcal O^{\mathfrak {p}}$ is a nonzero quotient of $\Delta ^{\mathfrak {p}}(\lambda )$ , for some $\lambda \in \Sigma ^+_{\mathfrak {p}}$ . Then the associated variety $X_M$ is nonzero. In particular,

$$ \begin{align*}{\mathrm{pd}}_{\mathcal O}M ={\mathrm{pd}}_{\mathcal O^{\mathfrak{p}}}M =\infty.\end{align*} $$

Especially, the modules $L(\lambda ),~K(\lambda )$ and $\Delta ^{\mathfrak {p}}(\lambda )$ all have infinite projective dimension in both $\mathcal O$ and $\mathcal O^{\mathfrak {p}}$ .

We recall from [Reference SerganovaSe2] that a weight $\lambda =\sum _{1\leq i\leq n}\lambda _i\epsilon _i\in \mathfrak {h}^\ast $ is called atypical if

$$ \begin{align*} \prod_{1\leq i\neq j \leq n}(\lambda_i -\lambda_j +j-i-1) =0. \end{align*} $$

We describe the projective dimension of costandard objects in the following theorem.

Proof. From [Reference Coulembier and SerganovaCo, Corollary 3.2.5] and [Reference Coulembier and SerganovaCo, Lemma 6.3.2], it follows that

$$ \begin{align*}{\text{pd}}_{\mathcal O^{\mathfrak{p}}}{\nabla}^{\mathfrak{p}}_{\mathcal O^{\mathfrak{p}}}(\lambda) = {\text{pd}}_{\mathcal O^{\mathfrak{p}}_{\bar 0}}{\nabla}^{\mathfrak{p}}_{\bar 0}(\lambda),\end{align*} $$

if ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}\nabla ^{\mathfrak {p}}(\lambda )<\infty .$ Therefore, it remains to show that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}{\nabla }^{\mathfrak {p}}(\lambda )<\infty $ if and only if $\lambda $ is typical.

We first suppose that $\lambda $ is typical. Then, by [Reference Cheng, Mazorchuk and WangCP, Theorem 4.6],

$$ \begin{align*}(T^{\mathfrak{p}}(\lambda): \nabla^{\mathfrak{p}}(\mu)) = (T^{\mathfrak{p}}_{\bar 0}(\lambda): \nabla^{\mathfrak{p}}_{\bar 0}(\mu)),\end{align*} $$

for any $\mu \in \Sigma _{\mathfrak {p}}^+$ . We start by fixing $\lambda $ such that $T^{\mathfrak {p}}(\lambda ) = \nabla ^{\mathfrak {p}}(\lambda )$ . That is, let $J^{\mathfrak {p}}_\lambda $ be the intersection of the set of shortest representatives in $W^{\mathfrak {p}} \backslash W$ and the set of longest representatives in $ W/W_\lambda $ . We choose $\lambda $ that is a minimal element in $J^{\mathfrak {p}}_\lambda \cdot \lambda $ . We want to show that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}} \nabla (\mu ) <\infty $ , for any $\mu \in J^{\mathfrak {p}}_\lambda \cdot \lambda $ .

We first note that

$$ \begin{align*} &{\text{pd}}_{\mathcal O^{\mathfrak{p}}}\nabla^{\mathfrak{p}}(\lambda) ={\text{pd}}_{\mathcal O^{\mathfrak{p}}} T^{\mathfrak{p}}(\lambda) = {\text{pd}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} T^{\mathfrak{p}}_{\bar 0}(\lambda) = {\text{pd}}_{\mathcal O_{\bar 0}^{\mathfrak{p}}} \nabla^{\mathfrak{p}}_{\bar 0}(\lambda), \end{align*} $$

by Theorem 20. Assume that $x\in J^{\mathfrak {p}}_\lambda $ is such that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}}\nabla (y\cdot \lambda )<\infty ,$ for any $y\in J^{\mathfrak {p}}_\lambda $ with $x\cdot \lambda> y\cdot \lambda .$ By [Reference Chen and CoulembierCCC, Theorem 3.5], there is a submodule K of $T^{\mathfrak {p}}(x\cdot \lambda )$ such that we have the short exact sequence

$$ \begin{align*}0\rightarrow K\rightarrow T^{\mathfrak{p}}(x\cdot \lambda) \rightarrow \nabla^{\mathfrak{p}}(x\cdot \lambda) \rightarrow 0,\end{align*} $$

where $(K:\nabla ^{\mathfrak {p}}(z\cdot \lambda ))>0$ only if $z\in J^{\mathfrak {p}}_\lambda $ and $x\cdot \lambda>z\cdot \lambda $ . From our assumption, we have ${\text {pd}} K<\infty $ . Because ${\text {pd}}_{\mathcal O^{\mathfrak {p}}} T^{\mathfrak {p}}(x\cdot \lambda )<\infty $ , it follows that ${\text {pd}}_{\mathcal O^{\mathfrak {p}}} \nabla ^{\mathfrak {p}}(x\cdot \lambda )<\infty $ . This completes the proof of the implication $(3)\Rightarrow (1).$

Before proving the direction $(1)\Rightarrow (3)$ , we make the following observations.

For any positive odd root $\alpha = \epsilon _i +\epsilon _j$ , we let $X_\alpha \in \mathfrak {g}_1$ be a root vector for $\alpha $ . If $\alpha =\epsilon _i +\epsilon _j$ with $i< j$ , we can pick root vectors $Y_{\alpha }\in \mathfrak {g}_{-1}$ for odd roots $-\alpha $ such that $[Y_{\alpha }, X_{\alpha }] =H_i-H_j$ .

Define a total ordering $\preceq $ on the set $\Phi _{\bar 1}^+:=\{\epsilon _i+\epsilon _j\mid 1\leq i<j\leq n\}$ by letting

$$ \begin{align*}\epsilon_i +\epsilon_{j+1}\prec \epsilon_i+\epsilon_{j},~2\epsilon_{i}\prec \epsilon_{i-1} +\epsilon_n,\end{align*} $$

for $1< i\leq n$ and $1\leq j<n$ . For any $1\leq i<j\leq n$ and any subset

$$ \begin{align*}I\subset \hat I:=\{\epsilon_i+\epsilon_t\mid i\leq t\leq j-1\},\end{align*} $$

we define

$$ \begin{align*} &S_{i,j,I}:=\{\epsilon_s+\epsilon_t\mid s<t,~1\leq s\leq i-1,~1\leq t\leq j\}\cup I. \end{align*} $$

Let $\lambda \in \mathfrak {h}^\ast $ and $v_\lambda \in \nabla ^{\mathfrak {p}}_{\bar 0}(\lambda )\subset {\nabla }^{\mathfrak {p}}(\lambda )$ be a highest weight vector. Set

$$ \begin{align*}X^{i,j,I}:=\prod_{\alpha \in S_{i,j,I}}X_\alpha\otimes v_\lambda \in {\nabla}^{\mathfrak{p}}(\lambda).\end{align*} $$

We observe:

1. (a) for any $\alpha \in \Phi _{\bar 1}^+,$ we have

   $$ \begin{align*}[X_\alpha,\mathfrak n_{\bar 0}^+] \subseteq \bigoplus_{\beta\in\Phi_{\bar 1}^+, ~\alpha \prec \beta}{\mathbb C} X_\beta;\end{align*} $$

2. (b) $[Y_{\epsilon _i+\epsilon _j}, X_{\epsilon _k+\epsilon _i}]\in \mathfrak n_{\bar 0}^+$ has weight $\epsilon _k-\epsilon _j$ , for $1\leq k\leq i$ ;

3. (c) $[Y_{\epsilon _i+\epsilon _j}, X_{\epsilon _i+\epsilon _k}]\in \mathfrak n_{\bar 0}^+$ has weight $\epsilon _k-\epsilon _j$ , for $i<k<j$ ;

4. (d) $[Y_{\epsilon _i+\epsilon _j}, X_{\epsilon _k+\epsilon _j}]\in \mathfrak n_{\bar 0}^+$ has weight $\epsilon _k-\epsilon _i$ , for $1\leq k<i$ .

As a consequence, from (a)–(d) above, it follows that

$$ \begin{align*} &Y_{\epsilon_i +\epsilon_j} X^{i,j,I}=0. \end{align*} $$

We can now proceed with the proof of $(1)\Rightarrow (3)$ . Suppose that $\lambda $ is integral and atypical with $\lambda _i -\lambda _j +j-i=\pm 1$ , for some $i<j$ , satisfying ${\text {pd}}{\nabla }^{\mathfrak {p}}(\lambda )<\infty $ . Then, by Corollary 26, the associated variety $X_{\nabla ^{\mathfrak {p}}(\lambda )}$ is zero.

We have to consider the following two cases.

Case 1. Suppose that $\lambda _i -\lambda _j +j-i+1=0$ .

Let $x:=X^{i,j,\hat I}$ and $y:=Y_{\epsilon _i+\epsilon _j}$ . Then $X_{\nabla ^{\mathfrak {p}}(\lambda )}=0$ implies that ${\nabla }^{\mathfrak {p}}(\lambda )_y=0$ . Therefore, $x\in y{\nabla }^{\mathfrak {p}}(\lambda )$ .

Assume that $m\in {\nabla }^{\mathfrak {p}}(\lambda )$ is a weight vector such that $yx=0$ and $x=ym$ . Then $m \in {\mathbb C} X_{\epsilon _i+\epsilon _j}X^{i,j,\hat I}.$ This implies that

$$ \begin{align*}x=ym\in {\mathbb C} Y_{\epsilon_i+\epsilon_j}X_{\epsilon_i+\epsilon_j}X^{i,j,\hat I} = {\mathbb C}(H_i -H_j)X^{i,j,\hat I} = \lambda_i -\lambda_j +j-i+1 =0, \end{align*} $$

a contradiction.

Case 2. Suppose that $\lambda _i -\lambda _j +j-i-1=0$ . Set $I:= \hat I \backslash \{2\epsilon _i\}$ .

Let $x:=X^{i,j, I}$ and $y:=Y_{\epsilon _i+\epsilon _j}$ . Then $X_{\nabla ^{\mathfrak {p}}(\lambda )}=0$ implies that ${\nabla }^{\mathfrak {p}}(\lambda )_y=0$ . Again, we have $x\in y{\nabla }^{\mathfrak {p}}(\lambda )$ .

Assume that $m\in {\nabla }^{\mathfrak {p}}(\lambda )$ is a weight vector such that $yx=0$ and $x=ym$ . Observe that

$$ \begin{align*}(H_i-H_j)m = (\lambda_i-\lambda_j +j-i-1)m=0,\end{align*} $$

and

$$ \begin{align*}m = X^{i,j,\hat I}\otimes m_j+ \sum_{i\leq t< j} X^{i,j,\hat I\backslash \{\epsilon_i+\epsilon_t\}\cup \{\epsilon_i+\epsilon_j\}}\otimes m_t,\end{align*} $$

for some $m_j, m_t \in {\nabla }_{\bar 0}(\lambda ).$ By (a)–(d), for $i\leq t <j$ , we have

$$ \begin{align*}yX^{i,j,\hat I}\otimes m_j =0,\end{align*} $$

$$ \begin{align*}y X^{i,j,\hat I\backslash \{\epsilon_i+\epsilon_t\}\cup \{\epsilon_i+\epsilon_j\}}\otimes m_t =(H_i-H_j)X^{i,j,\hat I\backslash \{\epsilon_i+\epsilon_t\}}\otimes m_t=0.\end{align*} $$

The obtained contradiction completes the proof.