2.1. Root systems and Weyl group

Let $\Phi $ be the set of all roots. Denote by $\Pi $ , $\Phi ^\pm $ , $\Phi _{{\bar 0}}$ and $\Phi _{\bar 1}$ the sets of simple, positive/negative, even and odd roots of $\mathfrak {g}$ , respectively. Define $\Phi _\epsilon ^\pm :=\Phi _\epsilon \cap \Phi ^\pm $ , $\epsilon =\bar {0},\bar {1}$ . Set $\Pi _{\bar 0}$ to be the simple system for $\Phi _{\bar 0}^+$ . For each $\alpha \in \Phi $ , we denote by $\mathfrak {g}_\alpha $ the root space of $\mathfrak {g}$ corresponding to $\alpha $ . The corresponding Borel subalgebra is $\mathfrak b=\mathfrak h\oplus \mathfrak n^+$ , with $\mathfrak n^+=\bigoplus _{\alpha \in \Phi ^+}\mathfrak {g}_\alpha $ .

Recall that we fixed a character $\zeta $ that gives rise to a Levi subalgebra $\mathfrak l_\zeta $ of $\mathfrak {g}$ as given in Section 1.2. Also, recall that the set of all simple roots of $\mathfrak l_\zeta $ is denoted as follows

$$ \begin{align*} \Pi_\zeta:=\{\alpha\in \Pi_{\bar 0}|~\zeta(\mathfrak g_\alpha)\neq 0\}. \end{align*} $$

We note that $\zeta ([\mathfrak n^+_{\bar 1}, \mathfrak n^+_{\bar 1}])=0$ , the latter implies that $\zeta $ extends trivially to a one-dimensional $\mathfrak n^+$ -module. We also denote by $\zeta $ the induced algebra homomorphism from $U(\mathfrak n^+)$ to ${\mathbb {C}}$ .

The Weyl group W is defined to be the Weyl group of $\mathfrak g_{\bar 0}$ , and it acts naturally on $\mathfrak {h}^\ast $ . For each $\alpha \in \Pi _{\bar 0}$ , let $s_\alpha \in W$ be the simple reflection corresponding to $\alpha $ . We consider the dot-action of W on $\mathfrak {h}^\ast $ ; that is, $w\cdot \lambda := w(\lambda +\rho _{\bar 0})-\rho _{\bar 0}$ , for any $w\in W$ and $\lambda \in \mathfrak {h}^\ast $ . Here, $\rho _{\bar 0}$ denotes the half-sum of all positive even roots. Also, we let $W_\zeta $ denote the Weyl group of $\mathfrak l_\zeta $ . We denote by $w_0$ and $w_0^\zeta $ the longest elements in W and $W_\zeta $ , respectively.

Fix a W-invariant nondegenerate bilinear form $(\cdot ,\cdot )$ on $\mathfrak {h}^\ast $ . For each $\alpha \in \Phi _{\bar 0}^+$ , we set $\alpha ^\vee :=2\alpha /(\alpha ,\alpha )$ . A weight $\lambda $ is called integral if $(\lambda ,\alpha ^\vee )\in {\mathbb Z}$ for all roots $\alpha \in \Phi _{\bar 0}$ . Let $\Lambda $ be the set of integral weights. For any given $\alpha \in \Phi _{\bar 0}^+$ , a weight $\lambda $ is called $\alpha $ -dominant (respectively, $\alpha $ -anti-dominant) if $(\lambda +\rho _{\bar 0},\alpha ^\vee )\notin {\mathbb Z}_{<0}$ (respectively, $(\lambda +\rho _{\bar 0},\alpha ^\vee )\notin {\mathbb Z}_{>0}$ ). We denote by ${\Lambda (\zeta )}$ the set of all integral weights that are $\alpha $ -anti-dominant, for all $\alpha \in \Pi _\zeta $ .

2.2. Induction and restriction functors

We denote by $U(\mathfrak {g})$ the universal enveloping algebra of $\mathfrak {g}$ and $Z(\mathfrak {g})$ its center. For any $\lambda \in \mathfrak {h}^\ast $ , we denote by $\chi _\lambda : Z(\mathfrak {g})\rightarrow {\mathbb {C}}$ the central character associated with $\lambda $ . Denote by $\mathfrak {g}\operatorname {-mod}\nolimits $ the category of all $\mathfrak {g}$ -modules. For any subalgebra $\mathfrak s\subseteq \mathfrak {g}$ , we have the exact restriction, induction and coinduction functors

(2.1) $$ \begin{align} \operatorname{Res}\nolimits_{\mathfrak s}^{\mathfrak{g}}(-): \mathfrak{g}\operatorname{-mod}\nolimits \rightarrow \mathfrak s\operatorname{-mod}\nolimits,~\operatorname{Ind}\nolimits_{\mathfrak s}^{\mathfrak{g}}(-), \mathrm{ Coind}_{\mathfrak s}^{\mathfrak{g}}(-): \mathfrak s\operatorname{-mod}\nolimits \rightarrow \mathfrak{g}\operatorname{-mod}\nolimits. \end{align} $$

In the case $\mathfrak {g}_{\bar 0}=\mathfrak s$ , we use the notations $\operatorname {Res}\nolimits (-):= \operatorname {Res}\nolimits _{\mathfrak {g}_{\bar 0}}^{\mathfrak {g}}(-)$ , $\operatorname {Ind}\nolimits (-):= \operatorname {Ind}\nolimits _{\mathfrak {g}_{\bar 0}}^{\mathfrak {g}}(-)$ and $\mathrm {Coind}(-):= \mathrm {Coind}_{\mathfrak {g}_{\bar 0}}^{\mathfrak {g}}(-)$ . By [Reference Bell and FarnsteinerBF, Theorem 2.2] (see also [Reference GorelikGo1]), $\operatorname {Ind}\nolimits (-)$ is isomorphic to $\mathrm {Coind}(-)$ up to tensoring with the one-dimensional $\mathfrak {g}_{\bar 0}$ -module $\wedge ^{\dim \mathfrak {g}_{\bar 1}}(\mathfrak {g}_{\bar 1})$ .

For any module M having a composition series, we will use $[M : L]$ to denote the Jordan-Hölder multiplicity of the simple module L in a composition series of M.

2.3. BGG category ${\mathcal {O}}$

We denote by ${\mathcal {O}}$ the BGG category with respect to the triangular decomposition (1.1). Namely, ${\mathcal {O}}$ consists of finitely-generated $\mathfrak {g}$ -modules that are semisimple over $\mathfrak {h}$ and locally finite over $\mathfrak n^+$ . We shall denote the corresponding BGG category of $\mathfrak {g}_{\bar 0}$ -modules by ${\mathcal {O}}_{\bar 0}$ . For $\lambda \in \mathfrak {h}^\ast $ , we denote by $M(\lambda )$ the Verma module of $\mathfrak {b}$ -highest weight $\lambda $ and by $L(\lambda )$ its unique simple quotient. Denote by $P(\lambda )$ the projective cover of $L(\lambda )$ in ${\mathcal {O}}$ . We will use the $0$ -subscript convention to denote the respective objects in ${\mathcal {O}}_{\bar 0}$ (e.g., $P_0(\lambda )$ is the projective cover of $L_0(\lambda )$ in ${\mathcal {O}}_{\bar 0}$ ).

Denote by ${\mathcal {O}}_{\mathbb Z}$ the integral BGG subcategory (i.e., the full subcategory of ${\mathcal {O}}$ consisting of all modules having integral weight spaces).

We denote by ${\mathcal {O}}(\mathfrak l_\zeta )$ the BGG category of $\mathfrak l_\zeta $ -modules with respect to the Borel subalgebra $\mathfrak l_\zeta \cap \mathfrak b$ . Similarly, we shall use the $\mathfrak l_\zeta $ -subscript to denote respective $\mathfrak l_\zeta $ -modules (e.g., we define the Verma module $M_{\mathfrak l_\zeta }(\lambda )$ and its simple quotient $L_{\mathfrak l_\zeta }(\lambda )$ in the category ${\mathcal {O}}(\mathfrak l_\zeta )$ ). Denote by $P_{\mathfrak l_\zeta }(\lambda )$ the projective cover of $L_{\mathfrak l_\zeta }(\lambda )$ in ${\mathcal {O}}(\mathfrak l_\zeta )$ . Similarly, we denote by ${\mathcal {O}}(\mathfrak l_\zeta )_{\mathbb Z}$ the integral BGG subcategory of ${\mathcal {O}}(\mathfrak l_\zeta )$ .

2.4. The category ${\mathcal {O}}^{\zeta \text {-pres}}$

A projective module in ${\mathcal {O}}$ (respectively, ${\mathcal {O}}_{\bar 0}$ ) is called admissible if any of its simple quotients are of the form $L(\lambda )$ (respectively, $L_0(\lambda )$ ) with $\lambda \in {\Lambda (\zeta )}$ . We denote by ${\mathcal {O}}^{\zeta \text {-pres}}$ the full subcategory of ${\mathcal {O}}_{\mathbb Z}$ consisting of modules M that have a two-step resolution of the form

(2.2) $$ \begin{align} &P_1\rightarrow P_2\rightarrow M \rightarrow 0, \end{align} $$

where $P_1,P_2$ are admissible projective modules in ${\mathcal {O}}$ . Similarly, we define the full subcategory ${\mathcal {O}}^{\zeta \text {-pres}}_{\bar 0}$ of ${\mathcal {O}}_{\bar 0}$ consisting of all modules that have $2$ -step resolutions by direct sums of projective modules of the form $P_0(\lambda )$ with $\lambda \in {\Lambda (\zeta )}$ .

Fix $\alpha \in \Pi _\zeta $ . A $\mathfrak {g}$ -module M is said to be $\alpha $ -finite (respectively, $\alpha $ -free) if the action of non-zero root vectors in $\mathfrak {g}_{-\alpha }$ on M is locally finite (respectively, injective). For $\alpha \in \Pi _\zeta $ , we have $(\lambda +\rho ,\alpha ^\vee )=(\lambda +\rho _{\mathfrak l_\zeta },\alpha ^\vee )=(\lambda +\rho _{\bar 0},\alpha ^\vee )$ , where $\rho _{\mathfrak l_\zeta }$ denotes the half-sum of positive roots in $\mathfrak l_\zeta $ , and hence, $L(\lambda )$ is $\alpha $ -free if and only if $L_0(\lambda )$ is $\alpha $ -free if and only if $\lambda \in {\Lambda (\zeta )}$ , see, for example, [Reference Coulembier and MazorchukCM, Lemma 2.1].

2.5. Serre quotient category ${\overline {{\mathcal {O}}}}$

Let $\mathcal I_\zeta $ denote the Serre subcategory of ${\mathcal {O}}_{\mathbb Z}$ generated by simple modules $L(\lambda )$ with $\lambda \notin {\Lambda (\zeta )}$ ; that is, $L(\lambda )\in {\mathcal {O}}_{\mathbb Z}$ is $\alpha $ -finite for some $\alpha \in \Pi _\zeta $ . There is an associated abelian quotient category ${\overline {{\mathcal {O}}}}:={\mathcal {O}}_{\mathbb Z}/\mathcal I_\zeta $ in the sense of [Reference GabrielGa, Chapter III] which has the same objects as ${\mathcal {O}}_{\mathbb Z}$ and morphisms given by

(2.3) $$ \begin{align} &\operatorname{Hom}\nolimits_{{\overline{{\mathcal{O}}}}}(X,Y):=\lim_{\rightarrow}\operatorname{Hom}\nolimits_{{\mathcal{O}}}(X',Y/Y'), \end{align} $$

where the limit is taken over all $X'\subseteq X$ and $Y'\subseteq Y$ such that $X/X',Y'\in \mathcal I_\zeta $ . There is an exact canonical quotient functor $\pi :{\mathcal {O}}_{\mathbb Z}\rightarrow {\overline {{\mathcal {O}}}}$ , which is the identity on objects and is the canonical map from $\operatorname {Hom}\nolimits _{{\mathcal {O}}}(X,Y)$ to $\lim \limits _{\rightarrow }\operatorname {Hom}\nolimits _{{\mathcal {O}}}(X',Y/Y')$ . We refer to $\pi $ as the associated Serre quotient functor.

By [Reference Chen, Cheng and MazorchukCCM2, Lemma 12], the functor $\pi $ gives rise to an equivalence from $ {\mathcal {O}}^{{\zeta }\text {-pres}}$ to $\overline {{\mathcal {O}}}.$ In particular, ${\mathcal {O}}^{{\zeta }\text {-pres}}$ inherits an abelian category structure, and $\{\pi (L(\lambda ))|~\lambda \in {\Lambda (\zeta )}\}$ is an exhaustive list of simple objects in $\overline {{\mathcal {O}}}$ . Furthermore, $\pi (P(\lambda ))$ is the indecomposable projective cover of $\pi (L(\lambda ))$ in $\overline {{\mathcal {O}}}$ , for any $\lambda \in {\Lambda (\zeta )}$ .

With respect to the inherited abelian category structure, the induced functors $\operatorname {Ind}\nolimits (-)$ and $\operatorname {Res}\nolimits (-)$ between ${\mathcal {O}}^{\zeta \text {-pres}}_{\bar 0}$ and ${\mathcal {O}}^{\zeta \text {-pres}}$ are exact since $\operatorname {Ind}\nolimits (-)$ is also right adjoint to $\operatorname {Res}\nolimits (-)$ , up to an auto-equivalence on ${\mathcal {O}}^{\zeta \text {-pres}}_{\bar 0}$ .

2.6. The Whittaker category $\mathcal W(\zeta )$

Let $\lambda \in \mathfrak {h}^\ast $ . We set $\chi _\lambda ^{\mathfrak l_\zeta }:Z(\mathfrak l_\zeta )\rightarrow {\mathbb {C}}$ to be the central character associated with $\lambda $ . We recall from [Reference ChenCh1, Section 3.1] that the standard Whittaker module is defined as follows:

$$\begin{align*}M(\lambda, \zeta):= U(\mathfrak{g})\otimes_{U(\mathfrak p)} Y_\zeta(\lambda, \zeta),\end{align*}$$

where $Y_\zeta (\lambda , \zeta ):=U(\mathfrak l_\zeta )/(\text {Ker}\chi _\lambda ^{\mathfrak l_\zeta }) U(\mathfrak l_\zeta )\otimes _{U({\mathfrak n^+}\cap \mathfrak l_\zeta )}{\mathbb {C}}_\zeta $ denotes Kostant’s simple Whittaker modules from [Reference KostantKo]. We denote by $L(\lambda ,\zeta )$ the (simple) top of $M(\lambda , \zeta )$ . Furthermore, we have

$$ \begin{align*}{L}(\lambda, \zeta)\cong {L}(\mu, \zeta)\Leftrightarrow {M}(\lambda, \zeta)\cong {M}(\mu, \zeta)\Leftrightarrow W_\zeta \cdot \lambda =W_\zeta \cdot \mu,\end{align*} $$

for any $\mu \in \mathfrak {h}^\ast $ . We refer to [Reference ChenCh1, Theorem 6] for more details. In particular, we have $M(\lambda ,0)=M(\lambda )$ . We denote by $\mathcal N_0(\zeta )$ the corresponding Whittaker category of $\mathfrak {g}_{\bar 0}$ -modules and by $M_0(\lambda ,\zeta )$ the corresponding standard Whittaker module over $\mathfrak {g}_{\bar 0}$ .

Recall that $\mathcal N(\zeta )$ denotes the category of finitely generated $\mathfrak {g}$ -modules M that are locally finite over $Z(\mathfrak {g}_{\bar 0})$ and $U(\mathfrak n^+)$ such that $x-\zeta (x)$ acts on M locally nilpotently, for any $x\in \mathfrak n^+_{\bar 0}$ . The set $\{{L}(\lambda , \zeta )|\lambda \in \mathfrak {h}^*/W_\zeta \}$ is a complete and nonredundant set of representatives of the isomorphism classes of simple objects in ${\mathcal N}(\zeta )$ .

We consider the category $\mathcal W(\zeta )$ introduced in [Reference Chen, Cheng and MazorchukCCM2, Section 4.4.4], which is a deformation of the category ${\mathcal {O}}$ and contains the modules $M(\lambda ,\zeta )$ and $L(\lambda ,\zeta )$ for all $\lambda \in \Lambda $ ; see also [Reference ChenCh1, Theorem 16, Proposition 33]. The category $\mathcal W(\zeta )$ is defined as the full subcategory of $\mathcal N(\zeta )$ consisting of modules M that have a two step resolution of modules of the form $E\otimes \operatorname {Ind}\nolimits (M_0(\nu ,\zeta ))$ , where E is a finite-dimensional $\mathfrak {g}$ -module and $\nu $ is a dominant (i.e., $\nu $ is $\alpha $ -dominant for any $\alpha \in \Phi _{\bar 0}^+$ ) and integral weight such that its stabilizer subgroup of W under the dot-action is equal to $W_\zeta $ .

Let us conclude this section with some remarks on equivalences of categories. There is an equivalence between ${\mathcal {O}}^{\zeta \text {-pres}}$ and $\mathcal W(\zeta )$ ; see Section 7.1. As a consequence, we have equivalences ${\mathcal {O}}^{\zeta \text {-pres}}\cong \mathcal W(\zeta )\cong {\overline {{\mathcal {O}}}}$ . In the case that $\mathfrak {g}$ is one from Kac’s list (1.2), a description of objects in ${\mathcal {O}}^{\zeta \text {-pres}}$ and ${\overline {{\mathcal {O}}}}$ corresponding to Whittaker modules $M(\lambda ,\zeta )$ and $L(\lambda ,\zeta )$ can be found in Remark 30.

3.1. Backelin functor $\Gamma _\zeta $

Recall that the image $\Gamma _\zeta (M)$ of the exact Backelin functor $\Gamma _\zeta :{\mathcal {O}}\rightarrow \mathcal N(\zeta )$ on a module $M\in {\mathcal {O}}$ is defined as the submodule consisting of vectors in $\overline M:=\prod _{\lambda \in \mathfrak h^\ast } M_\lambda $ annihilated by a power of $x-\zeta (x)$ , for all $x\in \mathfrak n^+_{\bar 0}$ . Let $\Gamma _\zeta ^0:{\mathcal {O}}_{\bar 0}\rightarrow \mathcal N_0(\zeta )$ denote the Backelin functor for $\mathfrak {g}_{\bar 0}$ -modules from [Reference BackelinBac]. Similarly, we denote by $\Gamma _\zeta ^{\mathfrak l_\zeta }$ the Backelin functor for $\mathfrak l_\zeta $ -modules. We note that $\operatorname {Res}\nolimits (-)$ intertwines $\Gamma _\zeta $ and $\Gamma ^0_\zeta $ . The following proposition is crucial.

Proposition 4. For any $\lambda \in \mathfrak {h}^*$ , we have

(3.1) $$ \begin{align} &\Gamma_\zeta(M(\lambda))\cong M(\lambda,\zeta). \end{align} $$

In particular, for any $w\in W_\zeta $ , we have

$$ \begin{align*} \Gamma_\zeta(M(\lambda))=\Gamma_\zeta(M(w\cdot\lambda)). \end{align*} $$

Proof. By the PBW basis theorem, we may regard $\overline {M(\lambda )}$ as the vector space $U(\mathfrak u^-_{\bar 1})\otimes \overline {U(\mathfrak u^-_{\bar 0})M_{\mathfrak l_\zeta }(\lambda )}=U(\mathfrak u^-_{\bar 1})\otimes \overline {M_0(\lambda )}.$ Therefore, $\Gamma _\zeta (M(\lambda ))$ consists of vectors v in $U(\mathfrak u^-_{\bar 1})\otimes \overline {M_0(\lambda )}$ such that $x-\zeta (x)$ acts nilpotently on v, for each $x\in \mathfrak n^+_{\bar 0}$ . There is an embedding $U(\mathfrak u^-_{\bar 1})\otimes \Gamma ^0_\zeta (M_0(\lambda ))\hookrightarrow \Gamma _\zeta (M(\lambda ))$ induced by the inclusion $\Gamma ^0_\zeta (M_0(\lambda ))\hookrightarrow \overline {M_0(\lambda )}$ . Now by [Reference BackelinBac, Proposition 6.9], one has $\Gamma ^0_\zeta (M_0(\lambda ))=M_0(\lambda ,\zeta )$ , and hence, we get an inclusion

(3.2) $$ \begin{align} &U(\mathfrak u^-_{\bar 1}) X\hookrightarrow \Gamma_\zeta(M(\lambda)), \end{align} $$

such that $U(\mathfrak u^-_{\bar 1})X= U(\mathfrak u^-_{\bar 1})\otimes X$ , where $X\subset \overline {M_0(\lambda )}$ is a $\mathfrak {g}_{\bar 0}$ -submodule isomorphic to $M_0(\lambda ,\zeta )$ .

We claim that the inclusion (3.2) is an equality. We can argue as follows: For $M\in \mathcal N(\zeta )$ , we have $\operatorname {Res}\nolimits ^{\mathfrak {g}}_{\mathfrak l_\zeta }\Gamma _\zeta (M)=\Gamma ^{\mathfrak l_\zeta }_\zeta \operatorname {Res}\nolimits ^{\mathfrak {g}}_{\mathfrak l_\zeta }(M)$ . For $\lambda \in \mathfrak {h}^*$ , we now compute

$$ \begin{align*} \operatorname{Res}\nolimits^{\mathfrak{g}}_{\mathfrak l_\zeta}\Gamma_\zeta(M(\lambda))=& \operatorname{Res}\nolimits^{\mathfrak{g}}_{\mathfrak l_\zeta}\Gamma_\zeta(U(\mathfrak u^-)\otimes M_{\mathfrak l_\zeta}(\lambda)) = \operatorname{Res}\nolimits^{\mathfrak{g}}_{\mathfrak l_\zeta}U(\mathfrak u^-)\otimes \Gamma^{\mathfrak l_\zeta}_\zeta(M_{\mathfrak l_\zeta}(\lambda))\\ =& \operatorname{Res}\nolimits^{\mathfrak{g}}_{\mathfrak l_\zeta}U(\mathfrak u^-)\otimes M_{\mathfrak l_\zeta}(\lambda,\zeta) = \operatorname{Res}\nolimits^{\mathfrak{g}}_{\mathfrak l_\zeta}M(\lambda,\zeta). \end{align*} $$

The third identity above follows from [Reference Miličić and SoergelMSo, Lemma 5.12] and the fact that $U(\mathfrak u^-)$ is a weight module and at the same time a direct sum of finite-dimensional irreducible $\mathfrak l_\zeta $ -modules. Thus, we conclude that $U(\mathfrak u^-_{\bar 1})X \hookrightarrow \Gamma _{\zeta }(M(\lambda ))$ is an isomorphism.

We also provide an alternative proof of the subjectivity of the inclusion in (3.2) as follows. For any $M\in \mathcal N(\zeta )$ and $c\in {\mathbb {C}}$ , with respect to the operator H from (1.4), we denote by $M_c$ the generalized eigenspace of M, which lies in $\mathcal N_{\mathfrak l_\zeta }$ and so has finite length as an $\mathfrak l_\zeta $ -module. We note that $U(\mathfrak u^-_{\bar 1})X\hookrightarrow \Gamma _\zeta (M(\lambda ))$ is surjective if and only if the $\mathfrak l_\zeta $ -modules $(U(\mathfrak u^-_{\bar 1})X)_c$ and $ \Gamma _\zeta (M(\lambda ))_c$ have the same composition factors for any $c\in {\mathbb {C}}$ . Recall that $\operatorname {Res}\nolimits M(\lambda )$ has filtration subquotients of which are isomorphic to $M_{\bar 0}(\lambda +\gamma )$ with $\gamma \in \Phi (\wedge (\mathfrak u^-_{\bar 1}))$ . Here, $\gamma \in \Phi (\wedge (\mathfrak u^-_{\bar 1}))$ denotes the set of roots in the space $\wedge (\mathfrak u^-_{\bar 1})$ . Also, for any finite-dimensional weight module E over $\mathfrak l_\zeta $ , the tensor product $E\otimes Y_\zeta (\lambda ,\zeta )$ has filtration subquotients of which are $Y_\zeta (\lambda +\gamma ,\zeta )$ with $\gamma $ being weights of E by [Reference Miličić and SoergelMSo, Lemma 5.12]. Since the restriction functor intertwines $\Gamma _\zeta $ and $\Gamma _\zeta ^0$ , it follows that the $\mathfrak l_\zeta $ -modules $\Gamma _\zeta M(\lambda )_c$ , $(U(\mathfrak u^-_{\bar 0})\otimes U(\mathfrak u^-_{\bar 1})\otimes Y_\zeta (\lambda ,\mathfrak l_\zeta ))_c$ and $M(\lambda ,\zeta )_c$ have the same composition factors. As a consequence, $(U(\mathfrak u^-_{\bar 1})X)_c$ and $\Gamma _\zeta (M(\lambda ))_c$ have the same composition factors for any $c\in {\mathbb {C}}$ .

Recall again $H\in \mathfrak {h}$ from the parabolic decomposition (1.4). Since the inclusion in (3.2) is an equality, it follows that H acts on $U(\mathfrak u_{\bar 1}^-)X=\Gamma _\zeta (M(\lambda ))$ semisimply with the highest eigenvalue $\chi _{\lambda }^{\mathfrak l_\zeta }(H)$ . Let $V\subseteq \operatorname {Res}\nolimits _{\mathfrak l_\zeta }^{\mathfrak {g}_{\bar 0}} X$ isomorphic to $Y_\zeta (\lambda ,\zeta )$ , which is the $\chi _\lambda ^{\mathfrak l_\zeta }(H)$ -eigenspace in $\Gamma _\zeta (M(\lambda ))$ . Then $\mathfrak u^+ V =0$ . Thus, by Frobenius reciprocity, we have an isomorphism $M(\lambda ,\zeta )$ to $\Gamma _\zeta (M(\lambda ))$ . The conclusion follows.

3.2. Proof of Theorem 1

For a given $\mathfrak {g}$ -module M, we denote the subspace of Whittaker vectors associated to $\zeta $ by $Wh_\zeta (M)$ – namely,

(3.3) $$ \begin{align} &Wh_\zeta(M):=\{m\in M|~xm=\zeta(x)m,\text{ for\ any}\ x\in \mathfrak n^+\}. \end{align} $$

We have the following generalization of [Reference Batra and MazorchukBM, Lemma 37] to quasi-reductive Lie superalgebras with a similar proof.

A weight is called $W_\zeta $ -anti-dominant if it is $\alpha $ -anti-dominant for any root $\alpha $ in $\mathfrak l_\zeta \cap \mathfrak {n}^+$ . The following theorem completes the proof of Theorem 1 and determines homomorphisms from standard Whittaker modules to the full dual of Verma modules:

4.1. Relation with the category ${\mathcal {O}}(\mathfrak l_\zeta )$

Recall that ${\mathcal {O}}(\mathfrak l_\zeta )$ denotes the BGG category of $\mathfrak l_\zeta $ -modules with respect to the Borel subalgebra $\mathfrak l_\zeta \cap \mathfrak b$ . Also, ${\mathcal {O}}(\mathfrak l_\zeta )_{\mathbb Z}$ denotes the integral BGG subcategory of $\mathfrak l_\zeta $ -modules. In this subsection, we provide a characterization of objects in ${\mathcal {O}}^{\zeta \text {-pres}}$ in terms of projective-injective modules in ${\mathcal {O}}(\mathfrak l_\zeta )_{\mathbb Z}$ . In [Reference Konig and MazorchukKMa, Theorem 2], König and Mazorchuk have shown that objects in ${\mathcal {O}}^{\zeta \text {-pres}}_{\bar 0}$ are modules in ${\mathcal {O}}_{\bar 0}$ which, as $\mathfrak l_\zeta $ -modules, are presented by projective-injective modules in ${\mathcal {O}}(\mathfrak l_\zeta )_{\mathbb Z}$ . The following is a generalization to Lie superalgebras.

Before we prove Proposition 7, we need some preparatory results. Recall that a projective module $M\in {\mathcal {O}}$ is said to be admissible if M is a direct sum of projective modules of the form $P(\lambda )$ with $\lambda \in {\Lambda (\zeta )}$ . For any $M\in {\mathcal {O}}$ , we let $Tr(M)$ and $Tr_0(\operatorname {Res}\nolimits M)$ denote the sum of images of homomorphisms from admissible projective modules in ${\mathcal {O}}$ (respectively, in ${\mathcal {O}}_{\bar 0}$ ) to M (respectively, to $\operatorname {Res}\nolimits M$ ). Recall the coapproximation functor $\mathfrak j_0:{\mathcal {O}}_{\bar 0}\rightarrow {\mathcal {O}}^{\zeta \text {-pres}}_{\bar 0}$ with respect to admissible projective modules studied in [Reference Mazorchuk and StroppelMSt, Section 2.4]; see also [Reference AuslanderAu, Section 3]. In general, we define the coapproximation functor $\mathfrak j:{\mathcal {O}}\rightarrow {\mathcal {O}}^{\zeta \text {-pres}}$ for Lie superalgebras in the same fashion. Let $M\in {\mathcal {O}}$ with an epimorphism $f:P_M\twoheadrightarrow Tr(M)$ from a projective module $P_M\in {\mathcal {O}}$ . Then, $\mathfrak j(M)$ is defined to be the quotient module $P_M/Tr(\text {ker}(f)).$ This definition is independent of the choices made. We refer to [Reference AuslanderAu, Section 3] and [Reference Khomenko and MazorchukKM, Section 2.4] for more details.

We need the following auxiliary lemma.

4.2. Standard and proper standard objects.

Recall that $P(\lambda )$ and $P_0(\lambda )$ denote the projective covers of $L(\lambda )$ and $L_0(\lambda )$ in ${\mathcal {O}}$ and ${\mathcal {O}}_{\bar 0}$ , respectively. For $\lambda \in \Lambda (\zeta )$ , we define the proper standard object $\overline \Delta (\lambda )$ for ${\mathcal {O}}^{{\zeta }\text {-pres}}$ to be $P(\lambda )/ Tr(K_\lambda )$ , where $K_\lambda $ is the kernel of the canonical quotient $P(\lambda )\rightarrow M(\lambda )$ . Note that the proper standard objects indeed lie in ${\mathcal {O}}^{{\zeta }\text {-pres}}$ and furthermore,

(4.1) $$ \begin{align} [\overline{\Delta}(\lambda)/M(\lambda):L(\gamma)]=0, \text{ for } \gamma\in{\Lambda(\zeta)}. \end{align} $$

We also define the standard object $\Delta (\lambda )$ for ${\mathcal {O}}^{{\zeta }\text {-pres}}$ by

(4.2) $$ \begin{align} &\Delta(\lambda):= \operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak{g}} P_{\mathfrak l_\zeta}(\lambda). \end{align} $$

Here, $P_{\mathfrak {l}_\zeta }(\lambda )$ is regarded as a $\mathfrak {p}$ -module by letting $\frak u^+$ act trivially. Similarly, we define $\Delta _0(\lambda ):= \operatorname {Ind}\nolimits _{\mathfrak p_{\bar 0}}^{\mathfrak {g}_{\bar 0}} P_{\mathfrak l_\zeta }(\lambda ). $ It follows from Proposition 7 that $\Delta (\lambda )$ is an object in ${\mathcal {O}}^{\zeta \text {-pres}}$ . We will provide an alternative proof in Corollary 15 below.

We set $\mathcal F(\Delta )$ to be the full subcategory of ${\mathcal {O}}$ consisting of modules which admit a $\Delta $ -flag. Similarly, we define $\mathcal F(\Delta _0)\subset {\mathcal {O}}_{\bar 0}$ .

Proof. It is sufficient to prove this for $V=\Delta _0(\lambda )$ with $\lambda \in {\Lambda (\zeta )}$ . We note that

(4.3)

We claim that the $\mathfrak p$ -module $\operatorname {Ind}\nolimits ^{\mathfrak p}_{\mathfrak p_{\bar 0}}P_{\mathfrak l_\zeta }(\lambda )$ has a filtration any subquotient S of which is of the form $\operatorname {Res}\nolimits _{\mathfrak l_\zeta }^{\mathfrak p}S\cong P_{\mathfrak l_\zeta }(\gamma )$ with $\gamma \in {\Lambda (\zeta )}$ and $\mathfrak u^+S=0$ . To see this, we recall the grading operator $H\in \mathfrak h$ introduced in (1.4).

Set $E:=U(\mathfrak u^+_{\bar 1})$ , which is an $\mathfrak l_\zeta $ -submodule of $U(\mathfrak g)$ under the adjoint action. For each complex number $c\in {\mathbb {C}}$ , we let $E_c$ denote the c-eigenspace of H under the adjoint action. Since H lies in the center of $\mathfrak l_\zeta $ , it follows that each $E_c$ is an $\mathfrak l_\zeta $ -submodule of E. Then, E decomposes into a sum of $\mathfrak l_\zeta $ -submodules

(4.4) $$ \begin{align} &E=\oplus_{i=1}^kE_{c_i}, \end{align} $$

for some $c_1,\ldots ,c_k\in {\mathbb {C}}$ such that $\text {Re}(c_1)<\text {Re}(c_2)<\cdots <\text {Re}(c_k)$ .

We may note that

(4.5) $$ \begin{align} &\operatorname{Res}\nolimits_{\mathfrak l_\zeta}^{\mathfrak p}\operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda)\cong E\otimes P_{\mathfrak l_\zeta}(\lambda) = \bigoplus_{i=1}^k (E_{c_i} \otimes P_{\mathfrak l_\zeta}(\lambda)) \end{align} $$

is a direct sum of projective-injective $\mathfrak l_\zeta $ -modules in the category ${\mathcal {O}}(\mathfrak l_\zeta )$ , each of which is of the form $P_{\mathfrak l_\zeta }(\gamma )$ with $\gamma \in {\Lambda (\zeta )}$ . Furthermore, we define a filtration

(4.6) $$ \begin{align} &0=\operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda)_{k+1}\subset \operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda)_k \subset \operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda)_{k-1}\subset \cdots \subset \operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda)_1 = \operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda). \end{align} $$

by letting

(4.7) $$ \begin{align} &\operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda)_j:= \bigoplus_{i=j}^k (E_{c_i} \otimes P_{\mathfrak l_\zeta}(\lambda)), \end{align} $$

for $j=1,\ldots , k$ . Fix $1\leq j\leq k$ . We note that both $\mathfrak u_{\bar 1}^+E_{c_j}$ and $[\mathfrak u_{\bar 0}^+, E_{c_j}]$ are subspaces of $ \bigoplus _{i>j}E_{c_i}.$ Consequently, we have

(4.8) $$ \begin{align} &\mathfrak u^+(\operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda))_j \subset (\operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p}P_{\mathfrak l_\zeta}(\lambda))_{j+1}. \end{align} $$

Therefore, the filtration (4.6) gives rise to the desired filtration. This completes the proof.

The following is a generalization of [Reference Mazorchuk and StroppelMSt, Corollary 2.14], where the case of Lie algebras was considered. Here, we provide an alternative proof.

Proof. Let $M\in \mathcal F(\Delta )$ . Set N to be a direct summand of M. We shall show that $N\in \mathcal F(\Delta )$ . Let

(4.9) $$ \begin{align} &0=M^{k+1}\subset M^k\subset M^{k-1}\subset \cdots\subset M^1 =M \end{align} $$

be a $\Delta $ -flag.

We decompose $\operatorname {Res}\nolimits ^{\mathfrak {g}}_{\mathfrak l_\zeta } M$ into a direct sum

(4.10) $$ \begin{align} &M=\bigoplus_{c\in {\mathbb{C}}} M_c, \end{align} $$

where $M_c$ is the c-eigenspace with respect to the action of the grading operator H from (1.4). By Lemmas 11 and 12, each $M_c$ is a projective-injective module in ${\mathcal {O}}(\mathfrak l_\zeta )$ . Also, there is a complex number $d\in {\mathbb {C}}$ such that $M_d\neq 0$ and $M_c=0$ for any $c\in {\mathbb {C}}$ with $\text {Re}(c)>\text {Re}(d)$ .

Note that $U(\mathfrak {g})V=U(\mathfrak u^-)V$ , for any direct summand V of $M_d$ . We claim that both $U(\mathfrak {g})V$ and $M/(U(\mathfrak {g})V)$ lie in $\mathcal F(\Delta )$ . We shall prove by induction on the length k. The conclusion for the case $k=1$ holds since $U(\mathfrak {g})M_d=M$ in this case.

Since (4.9) is a $\Delta $ -flag, there is $1\leq i< k+1$ with the canonical quotient $p: M^{i}\twoheadrightarrow M^{i}/M^{i+1}$ such that

(4.11) $$ \begin{align} &(M^{i}/M^{i+1})_d\cong (\operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak{g}}{P_{\mathfrak l_\zeta}}(\lambda))_d\cong {P_{\mathfrak l_\zeta}}(\lambda), \end{align} $$

for some $\lambda \in {\Lambda (\zeta )}.$ Consider the following split short exact sequence induced by p and the inclusion $M^{i+1}\hookrightarrow M^i$

$$\begin{align*}0\rightarrow (M^{i+1})_d\rightarrow (M^i)_d \rightarrow (M^{i}/M^{i+1})_d\rightarrow 0\end{align*}$$

in ${\mathcal {O}}(\mathfrak l_\zeta )$ . We get an $\mathfrak l_\zeta $ -submodule S of $M_d$ such that $p|_S: S\rightarrow {P_{\mathfrak l_\zeta }}(\lambda )$ is an isomorphism of $\mathfrak l_\zeta $ -modules. Since $U(\mathfrak {g})S=U(\mathfrak u^-)S$ , as observed above, it follows that p restricts to a $\mathfrak {g}$ -module isomorphism from $U(\mathfrak g)S$ to $M^{i}/M^{i+1}\cong \operatorname {Ind}\nolimits _{\mathfrak p}^{\mathfrak {g}}{P_{\mathfrak l_\zeta }}(\lambda ).$ Consequently, we get a split short exact sequence

(4.12) $$ \begin{align} &0\rightarrow M^{i+1}\rightarrow M^i \rightarrow M^{i}/M^{i+1}\rightarrow 0 \end{align} $$

in ${\mathcal {O}}$ .

Put $K:=U(\mathfrak {g})S$ , and so $M^{i}=K\oplus M^{i+1}$ . Consider the exact sequence

(4.13) $$ \begin{align} &0\rightarrow M^{i+1}\rightarrow M/K \rightarrow M/M^{i}\rightarrow 0. \end{align} $$

Since both $M^{i+1}$ and $M/M^i$ lie in $\mathcal F(\Delta )$ , it follows that $M/K$ is an object in $\mathcal F(\Delta )$ . By induction, it follows that $U(\mathfrak g)M_d=U(\mathfrak u^-)M_d\cong \operatorname {Ind}\nolimits _{\mathfrak p}^{\mathfrak {g}} M_d$ is a direct sum of standard objects $\Delta (\mu )$ , for $\mu \in {\Lambda (\zeta )}$ , and $M/(U(\mathfrak {g})M_d)$ lies in $\mathcal F(\Delta )$ . Let V be a direct summand of $M_d$ . Then $U(\mathfrak {g})V\cong \operatorname {Ind}\nolimits _{\mathfrak p}^{\mathfrak {g}} V\in \mathcal F(\Delta )$ and $U(\mathfrak g)(M/V)\cong \operatorname {Ind}\nolimits _{\mathfrak p}^{\mathfrak {g}} (M/V)\in \mathcal F(\Delta )$ . Consequently, we have $M/(U(\mathfrak {g})V)\in \mathcal F(\Delta )$ , as desired.

Finally, let $M=N\oplus N'$ . Then, either $N_d\neq 0$ or $N^{\prime }_d\neq 0$ . Without loss of generality, we may assume that $N_d\neq 0$ , and then $M/(U(\mathfrak {g})N_d) = (N/U(\mathfrak {g})N_d)\oplus N'$ has a $\Delta $ -flag by the argument above. By induction on the length of M, it follows that $N/(U(\mathfrak {g})N_d)$ and $N'$ have $\Delta $ -flags, and hence N as well. This completes the proof.

4.3. Stratified structure and BGG reciprocity

We denote by $\leq $ the partial order on $\mathfrak h^\ast $ induced by the triangular decomposition (1.1). Namely, it is the transitive closure of the relations

$$ \begin{align*}\begin{cases} \lambda-\alpha \le\lambda, &\text{for a positive root}\ \alpha,\\ \lambda+\alpha \le\lambda, &\text{for a negative root}\ \alpha. \end{cases}\end{align*} $$

Recall the Serre quotient category ${\overline {{\mathcal {O}}}}$ and the canonical quotient functor $\pi :{\mathcal {O}}_{\mathbb Z}\rightarrow {\overline {{\mathcal {O}}}}$ associated to $\zeta $ from Section 2.5. Lemma 10, combined with Proposition 13, has the following consequence. Namely, the category ${\mathcal {O}}^{{\zeta }\text {-pres}}$ is stratified.

Set $Q(\lambda )$ to be the quotient of $P(\lambda )$ modulo the sum of the images of all homomorphisms $P(\mu )\rightarrow P(\lambda )$ , where $\mu \in {\Lambda (\zeta )}$ and $\mu>\lambda $ . Observe that $Q(\lambda )\in {\mathcal {O}}^{{\zeta }\text {-pres}}.$

For any $M\in \mathcal F(\Delta )$ , we denote by $(M : \Delta (\lambda ))$ the multiplicity of a standard object $\Delta (\lambda )$ in a $\Delta $ -flag of M. The following corollary determines the multiplicities of standard objects of the projective cover in ${\mathcal {O}}^{{\zeta }\text {-pres}}.$

Corollary 16 (BGG reciprocity).

For any $\lambda ,\mu \in {\Lambda (\zeta )}$ , we have

$$ \begin{align*} &( P(\lambda):\Delta(\mu)) = [\overline \Delta(\mu): L(\lambda)] = [M(\mu):L(\lambda)]. \end{align*} $$

Proof. Let $\gamma \in \mathfrak {h}^\ast $ be an integral weight with $\gamma \not \in {\Lambda (\zeta )}$ . Then $\operatorname {Ind}\nolimits _{\mathfrak p}^{\mathfrak {g}} L_{\mathfrak l_\zeta }(\gamma )$ is $\alpha $ -finite for some $\alpha \in \Pi _\zeta $ , which implies that $[\operatorname {Ind}\nolimits _{\mathfrak p}^{\mathfrak {g}} L_{\mathfrak l_\zeta }(\gamma ): L(\lambda )]=0$ since $L(\lambda )$ is $\alpha $ -free; see also Section 2.4. Therefore, for any $w\in W_\zeta $ , we have $[M(w\cdot \mu )/M(\mu ):L(\lambda )]=0$ and hence

(4.15) $$ \begin{align} & [M(w\cdot \mu): L(\lambda)] = [M(\mu):L(\lambda)]. \end{align} $$

By BGG reciprocity for category ${\mathcal {O}}$ , we have $(P(\lambda ): M(w\cdot \mu ))=(P(\lambda ):M(\mu ))$ . Finally, noting that the multiplicity of $\overline \Delta (\mu )$ in a $\overline \Delta $ -flag of $\Delta (\mu )$ is $|W_\zeta \cdot \mu |$ , the conclusion follows.

4.4. Tilting modules

In this subsection, we assume that $\mathfrak {g}$ is a basic Lie superalgebra (i.e., $\mathfrak {g}$ is one from (1.2) with $\mathfrak {g}\neq \mathfrak p(n)$ ). There is a simple-preserving duality D on ${\mathcal {O}}$ ; that is, D is a contravariant auto-equivalence such that $D^2\cong {\text {Id}}$ and $D(L(\lambda ))\cong L(\lambda )$ , for each $\lambda \in \mathfrak h^\ast $ .

For $\lambda \in {\Lambda (\zeta )}$ , we define the co-standard object $\nabla (\lambda )$ for ${\mathcal {O}}^{{\zeta }\text {-pres}}$ by

(4.16) $$ \begin{align} &\nabla(\lambda):= D \operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak{g}} P_{\mathfrak l_\zeta}(\lambda). \end{align} $$

It follows from Proposition 7 that $\nabla (\lambda )\in {\mathcal {O}}^{\zeta \text {-pres}}$ ; see also [Reference Mazorchuk and StroppelMSt, Corollary 2.11]. Similarly, we define co-standard object $\nabla _0(\lambda )$ for ${\mathcal {O}}^{\zeta \text {-pres}}_{\bar 0}$ . If ${\mathcal {O}}^{{\zeta }\text {-pres}}={\mathcal {O}}$ , then we arrive at the usual definition of dual Verma modules.

A module $M\in {\mathcal {O}}^{\zeta \text {-pres}}$ is referred to as a tilting module if M has both a $\Delta $ -flag and a $\nabla $ -flag. In the case that ${\mathcal {O}}^{{\zeta }\text {-pres}}={\mathcal {O}}$ , for each $\lambda \in \mathfrak h^\ast $ , there is a unique indecomposable tilting module of highest weight $\lambda $ , which we denote by $T^{{\mathcal {O}}}(\lambda )$ ; namely, $T^{{\mathcal {O}}}(\lambda )$ has both a Verma flag and a dual Verma flag; see, for example, [Reference BrundanBr1]. We also denote by $T^{{\mathcal {O}}_{\bar 0}}(\lambda )$ the indecomposable tilting module of highest weight $\lambda $ in ${\mathcal {O}}_{\bar 0}$ .

In the case that $\mathfrak g=\mathfrak g_{\bar 0}$ , the classification of indecomposable tilting modules in ${\mathcal {O}}^{\zeta \text {-pres}}_{\bar 0}$ is given in [Reference Futorny, König and MazorchukFKM, Section 6]; see also [Reference Mazorchuk and StroppelMSt, Section 5.4]. In particular, the list $\{T^{{\mathcal {O}}_{\bar 0}}(w_0^\zeta \cdot \lambda )|~\lambda \in {\Lambda (\zeta )}\}$ forms a complete set of non-isomorphic indecomposable tilting modules in ${\mathcal {O}}^{\zeta \text {-pres}}_{\bar 0}$ , and each $T_0(\lambda ):= T^{{\mathcal {O}}_{\bar 0}}(w_0^\zeta \cdot \lambda )$ is the unique indecomposable tilting module containing $\Delta _0(\lambda )$ such that $T_0(\lambda )/\Delta _0(\lambda )$ has a $\Delta _0$ -flag. Here, we recall that $w_0^\zeta $ denotes the longest element in $W_\zeta $ . We set

(4.17) $$ \begin{align} &T(\lambda):=T^{{\mathcal{O}}}(w_0^\zeta\cdot\lambda), \end{align} $$

for each $\lambda \in {\Lambda (\zeta )}$ . Below we give a classification of indecomposable tilting objects in ${\mathcal {O}}^{{\zeta }\text {-pres}}$ . We remark that the type I case was considered in [Reference Chen, Cheng and MazorchukCCM2, Section 6.1].

Proof. Let $\lambda \in \Lambda (\zeta )$ , and set $\mu =\lambda -2\rho _1\in \Lambda (\zeta )$ , where $\rho _1$ denotes the half-sum of roots in $\mathfrak n^+_{\bar 1}$ . We have

$$ \begin{align*} \operatorname{Ind}\nolimits\Delta_0(\mu)=\operatorname{Ind}\nolimits_{\mathfrak{g}_{\bar 0}}^{\mathfrak{g}}\operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak{g}_{\bar 0}} P_{\mathfrak l_\zeta}(\mu) \cong \operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak{g}}\operatorname{Ind}\nolimits_{\mathfrak p_{\bar 0}}^{\mathfrak p} P_{\mathfrak l_\zeta}(\mu) \cong \operatorname{Ind}\nolimits_{\mathfrak p}^{\mathfrak{g}}\left(U(\mathfrak n^+_{\bar 1})\otimes P_{\mathfrak l_\zeta}(\mu)\right). \end{align*} $$

Now, $P_{\mathfrak l_\zeta }(\mu )$ has an ${\mathfrak l_\zeta }$ -Verma flag with highest term $M_{\mathfrak l_\zeta }(w_0^\zeta \cdot \mu )$ , and hence, $U(\mathfrak n^+_{\bar 1})\otimes P_{\mathfrak l_\zeta }(\mu )$ has a ${\mathfrak l_\zeta }$ -Verma flag with highest term $M_{\mathfrak l_\zeta }(w_0^\zeta \cdot \mu +2\rho _{1})=M_{\mathfrak l_\zeta }(w_0^\zeta \cdot \left (\mu +2\rho _{1}\right ))=M_{\mathfrak l_\zeta }(w_0^\zeta \cdot \lambda )$ . Therefore, $\operatorname {Ind}\nolimits \Delta _0(\mu )$ has a Verma flag with highest term $M(w_0^\zeta \cdot \lambda )$ . This implies that $\operatorname {Ind}\nolimits T_0(\mu )$ has a Verma flag with highest term $M(w_0^\zeta \cdot \lambda )$ and hence, by Lemma 10, a $\Delta $ -flag with $\Delta (\lambda )$ as a highest term. Therefore, the indecomposable direct summand of $\operatorname {Ind}\nolimits T_0(\mu )$ containing $\Delta (\lambda )$ is (isomorphic to) $T(\lambda )$ . By Proposition 13, $T(\lambda )$ has a $\Delta $ -flag. Also, by Lemma 3, $\operatorname {Ind}\nolimits T_0(\mu )\in {\mathcal {O}}^{\zeta \text {-pres}}$ , and therefore, $T(\lambda )$ lies in ${\mathcal {O}}^{\zeta \text {-pres}}$ . Since $T(\lambda )$ is self-dual, it follows that $T(\lambda )$ has both a $\Delta $ -flag and a $\nabla $ -flag. Therefore, $T(\lambda )$ is an indecomposable tilting module in ${\mathcal {O}}^{\zeta \text {-pres}}.$

Now, if T is an indecomposable tilting module in ${\mathcal {O}}^{\zeta \text {-pres}}$ with a $\Delta $ -flag starting at $\Delta (\lambda )$ , then by uniqueness of tilting module (see, for example, [Reference SoergelSoe, Proposition 5.6]), we have $T\cong T(\lambda )$ . This completes the proof.

Corollary 18. (Ringel duality) For $\lambda ,\mu \in {\Lambda (\zeta )}$ , we have

(4.18) $$ \begin{align} (T(\lambda):\Delta(\mu)) = [\overline{\Delta}(-w_0^\zeta\cdot\mu): L(-w_0^\zeta\cdot\lambda)] = (P(-w_0^\zeta\cdot\lambda):\Delta(-w_0^\zeta\cdot\mu)). \end{align} $$

Proof. Let $\lambda ,\mu \in {\Lambda (\zeta )}$ . By (4.15), the BGG reciprocity and the Ringel-Soergel duality in ${\mathcal {O}}$ [Reference BrundanBr1] (see also [Reference Chen, Cheng and CoulembierCCC, Theorem 3.7]), we have

$$ \begin{align*} (T^{{\mathcal{O}}}(w_0^\zeta\cdot\lambda):M(\mu))= (T^{{\mathcal{O}}}(w_0^\zeta\cdot\lambda):M(w\cdot\mu)),\quad\forall w\in W_\zeta. \end{align*} $$

Thus, it follows from Proposition 17 that

$$ \begin{align*} (T(\lambda):\Delta(\mu))=&(T^{{\mathcal{O}}}(w_0^\zeta\cdot\lambda):M(w_0^\zeta\cdot\mu))\\ =&[M(-w_0^\zeta\cdot\mu):L(-w_0^\zeta\cdot\lambda)] = [\overline{\Delta}(-w_0^\zeta\cdot\mu): L(-w_0^\zeta\cdot\lambda)]. \end{align*} $$

The second identity in (4.18) follows from Corollary 16.

5.1. Quantum symmetric pair $(U,U^\imath )$

Let U denote the infinite-rank quantum group of type A with Chevalley generators $\{E_{\alpha _i},F_{\alpha _i},K^{\pm 1}_{\alpha _i}|i\in {\mathbb Z}\}$ (i.e., U is the associative algebra over $\mathbb {Q}(q)$ generated by these generators subject to certain well-known relations; see, for example, [Reference Cheng, Lam and WangCLW, Section 2.1]). The anti-linear ( $q\rightarrow q^{-1}$ ) map $\psi $ preserving $E_{\alpha _i}$ and $F_{\alpha _i}$ and sending $K_{\alpha _i}$ to $K^{-1}_{\alpha _i}$ induces the well-known bar involution on U. The algebra U is a Hopf algebra with co-multiplication $\Delta $ [Reference KashiwaraKas]:

$$ \begin{align*} &\Delta(E_{\alpha_i})=1\otimes E_{\alpha_i}+E_{\alpha_i}\otimes K^{-1}_{\alpha_i},\\ &\Delta(F_{\alpha_i})=F_{\alpha_i}\otimes 1+K_{\alpha_i}\otimes F_{\alpha_i},\\ &\Delta(K_{\alpha_i})=K_{\alpha_i}\otimes K_{\alpha_i}. \end{align*} $$

Now let $U^\imath $ be the associative algebra over $\mathbb {Q}(q)$ generated by $\{e_{\alpha _i},f_{\alpha _i},k^{\pm 1}_{\alpha _i},t|i>0\}$ subject to the relations in [Reference Bao and WangBW, Chapter 2.1]. Then, the anti-linear map $\psi ^\imath $ that preserves the generators $e_{\alpha _i},f_{\alpha _i},t$ , and that sends $k_{\alpha _i}$ to $k^{-1}_{\alpha _i}$ , for $i>0$ , defines an anti-linear involution on $U^\imath $ .

According to [Reference Bao and WangBW, Proposition 2.2], there is an embedding of associative algebras $\imath :U^\imath \rightarrow U$ given by

$$ \begin{align*} \imath(k_{\alpha_i})=K_{\alpha_i}K^{-1}_{\alpha_{-i}},\quad \imath(t)=E_{\alpha_0}+qF_{\alpha_0}K^{-1}_{\alpha_0}+K^{-1}_{\alpha_0},\\ \imath(e_{\alpha_i})=E_{\alpha_i}+K^{-1}_{\alpha_i}F_{\alpha_{-i}},\quad \imath(f_{\alpha_i})=F_{\alpha_i}K^{-1}_{\alpha_{-i}}+E_{\alpha_{-i}}, \end{align*} $$

so that $U^\imath $ may be regarded as a subalgebra of U. We note that $U^\imath $ is not a Hopf subalgebra, but rather a right coideal subalgebra of U. The pair $(U,U^\imath )$ forms a quantum symmetric pair, in the sense of [Reference LetzterLe2], corresponding to a quasi-split symmetric pair of Satake type AIII. The subalgebra $U^\imath $ is referred to as an $\imath $ -quantum group.

We observe that the involutions $\psi $ and $\psi ^\imath $ are not compatible with the embedding $\imath $ . However, there is a remarkable element $\Upsilon $ [Reference Bao and WangBW, Chapter 2.3] that lies in a certain topological completion of U and that intertwines these involutions in the following sense:

(5.1) $$ \begin{align} \imath(\psi^\imath(u))\Upsilon=\Upsilon\psi(\imath(u)),\quad \imath(u)\Upsilon=\Upsilon\psi(\imath(\psi^\imath(u))),\quad u\in U^\imath. \end{align} $$

This element $\Upsilon $ , referred to as the intertwiner, plays a crucial role in the Bao-Wang $\imath $ -canonical basis theory of quantum symmetric pairs.

Following [Reference Bao and WangBW], a U-module equipped with an anti-linear bar involution compatible with the bar involution $\psi $ will be called a U-involutive module. We shall denote such an involution on the module also by $\psi $ . Similarly, a $U^\imath $ -involutive $U^\imath $ -module is defined.

Given a U-involutive module with involution $\psi $ , we can define

(5.2) $$ \begin{align} \psi^\imath(m):=\Upsilon\left(\psi(m)\right),\quad m\in M. \end{align} $$

It follows from (5.1) that M is $U^\imath $ -involutive with involution $\psi ^\imath $ . We note that the element $\Upsilon $ is invertible in a completion of U.

5.2. Fock space $\mathbb {T}^{m|n}$

Let $\mathbb {V}$ be the natural U-module with linear basis $v_r$ , . The action of U is given by

Let $\mathbb {W}$ be the restricted dual with normalized dual basis and U-action given by

A fundamental property of Lusztig’s canonical basis theory for (finite rank) quantum groups is that tensor products of involutive modules can be made into involutive modules [Reference LusztigLu3, Chapter 27] by means of the quasi- $\mathcal R$ -matrix $\Theta $ [Reference LusztigLu3, Chapter 4]. (For the infinite-rank quantum group U, such a $\Theta $ is constructed in [Reference Cheng, Lam and WangCLW, Section 3.1] based on [Reference LusztigLu3].) To be precise, for U-involutive U-modules M and N with involutions $\psi _M$ and $\psi _N$ , respectively, the U-module $M\otimes N$ is U-involutive with involution:

$$ \begin{align*}\psi(m\otimes n):=\Theta(\psi_M(m)\otimes\psi_N(n)).\end{align*} $$

The U-modules $\mathbb {V}$ and $\mathbb {W}$ are both U-involutive with anti-linear involutions $\psi _{\mathbb {V}}$ and $\psi _{\mathbb {W}}$ determined by $\psi _{\mathbb {V}}(v_r)=v_r$ and $\psi _{\mathbb {W}}(w_r)=w_r$ , , respectively. It follows, therefore, that the U-module $\mathbb {T}^{m|n}:=\mathbb {V}^{\otimes m}\otimes \mathbb {W}^{\otimes n}$ , or rather a certain topological completion $\widehat {\mathbb {T}}^{m|n}$ , is U-involutive. Using the formula (5.2), we can therefore make $\widehat {\mathbb {T}}^{m|n}$ into a $U^\imath $ -involutive $U^\imath $ -module. We shall denote the involution on $\widehat {\mathbb {T}}^{m|n}$ also by $\psi ^\imath $ so that we have $\psi ^\imath (uv)=\psi ^\imath (u)\psi ^\imath (v)$ , for $u\in U^\imath $ and $v\in \widehat {\mathbb {T}}^{m|n}$ .

5.3. $\imath $ -Canonical bases

Let $\mathbb {I}^{m|n}$ denote the set of functions from $\{1,2,\cdots ,m,\bar {1},\cdots ,\bar {n}\}$ to . For each $f\in \mathbb {I}^{m|n}$ , we define

$$ \begin{align*} M_f:=v_{f(1)}\otimes \cdots\otimes v_{f(m)}\otimes w_{f(\bar{1})}\otimes \cdots\otimes w_{f(\bar{n})}. \end{align*} $$

The set $\{M_f|f\in \mathbb {I}^{m|n}\}$ is a topological basis for $\widehat {\mathbb {T}}^{m|n}$ .

We have a bijection between $\mathbb {I}_{m|n}$ and $\Lambda ^{\texttt {int}}$ , the set of integer weights for the Lie superalgebra $\mathfrak {osp}(2m+1|2n)$ (see (6.1)). This bijection allows us to transfer notions that make sense for weights such as typical, dominant, anti-dominant, Bruhat order $\succeq $ etc. to corresponding notions on $\mathbb {I}_{m|n}$ . The precise bijection is not important for the subsequent discussion in this section. In this section, we shall only need the fact that the Bruhat order on weights induces a partial order $\succeq $ on $\mathbb {I}_{m|n}$ .

As we have seen earlier, $\mathbb {T}^{m|n}$ is a $U^\imath $ -involutive $U^\imath $ -module. Recall that we have a right action of the Hecke algebra $\mathcal H_{B_m}$ (corresponding to the Weyl group $W_{B_m}$ of the Lie algebra of type $B_m$ ) on $\mathbb {V}^{\otimes m}$ (see, for example, [Reference Bao and WangBW, (5.2)] for the explicit action). According to [Reference Bao and WangBW, Theorem 5.8], this action commutes with the action of $U^\imath $ on $\mathbb {V}^{\otimes m}$ . There is also a right action of the Hecke algebra $\mathcal H_{A_n}$ (corresponding to the symmetric group $\mathfrak S_n$ ) on $\mathbb {W}^{\otimes n}$ (see, for example, [Reference Cheng, Lam and WangCLW, (2.9)]) that commutes with the action of U by [Reference JimboJi, Proposition 3]. We recall that Hecke algebras have natural anti-linear bar involutions determined by $\overline {H}_i:=H_i^{-1}$ , where the $H_i$ are the usual generators satisfying $(H_i-q^{-1})(H_i+q)=0$ and the corresponding braid relations. Furthermore, in [Reference Bao and WangBW, Theorem 5.8] and [Reference JimboJi, Proposition 3], it is shown that the involutive module structures on $\mathbb {V}^{\otimes m}$ and $\mathbb {W}^{\otimes n}$ are compatible with the bar involutions on the respective Hecke algebras. We have the following.

Proposition 19. The action of $U^\imath $ and that of $\mathcal H_{B_m}\times \mathcal H_{A_n}$ on $\mathbb {T}^{m|n}$ commute with each other. Furthermore, these actions are compatible with their respective involutions; that is, we have for $u\in U^\imath $ , $t\in \mathbb {T}^{m|n}$ , and $H\in \mathcal H_{B_m}\times \mathcal H_{A_n}$ :

$$ \begin{align*} \psi^\imath(u t H)=\psi^\imath(u)\psi^\imath(t) \overline{H}. \end{align*} $$

Proof. Let $x\in \mathbb V^{\otimes m}$ , $y\in \mathbb W^{\otimes n}$ , $u\in U^\imath $ , $h_1\in \mathcal H_{B_m}$ and $h_2\in \mathcal H_{A_n}$ . We have $\Delta (u)=\sum _{i}u_1^i\otimes u_2^i$ , with $u_1^i\in U^\imath $ and $u_2^i\in U$ . We have

$$ \begin{align*} \left(\Delta(u)(x\otimes y)\right)(h_1\otimes h_2)&=\left(\sum_i u_1^i x\otimes u_2^i y\right)(h_1\otimes h_2) =\sum_i \left(u_1^i x\right)h_1\otimes \left(u_2^i y\right)h_2\\ &= \sum_iu_1^i \left(xh_1\right)\otimes u_2^i \left(yh_2\right) = \Delta(u)\left((x\otimes y)(h_1\otimes h_2)\right), \end{align*} $$

where the penultimate identity above follows from [Reference Bao and WangBW, Theorem 5.8] and [Reference JimboJi, Proposition 3]. This implies that the actions of $U^\imath $ and $\mathcal H_{B_m}\otimes \mathcal H_{A_n}$ on $\mathbb {T}^{m|n}$ commute.

By [Reference Bao and WangBW, Proposition 3.7 and Remark 3.14], the involution $\psi ^\imath $ on $\mathbb {T}^{m|n}$ can be written in the following form as an involution on the tensor product of the $U^\imath $ -involutive module $\mathbb V^{\otimes m}$ and the U-involutive module $\mathbb W^{\otimes n}$ using the quasi- $\mathcal K$ -matrix $\Theta ^\imath $ :

(5.3) $$ \begin{align} \psi^\imath(x\otimes y)=\Theta^\imath(\psi^\imath(x)\otimes\psi(y)), \end{align} $$

where $\Theta ^\imath :=\Delta (\Upsilon )\Theta (\Upsilon ^{-1}\otimes 1)$ . It is shown in [Reference Bao and WangBW, Chapter 3.3] that $\Theta ^\imath $ is an element in (the completion of) $U^\imath \otimes U$ . We shall write $\Theta ^\imath =\sum _i \theta _1^i\otimes \theta _2^i$ , with $\theta _1^i\in U^\imath $ and $\theta _2^i\in U$ .

Now, for $u\in U^\imath $ , we have clearly $\psi ^\imath (u\cdot x\otimes y)=\psi ^\imath (u)\cdot \psi ^\imath (x\otimes y)$ , since $\mathbb {T}^{m|n}$ is a $U^\imath $ -involutive module by [Reference Bao and WangBW, Proposition 3.13].

However, for x, y, $h_1$ and $h_2$ as above, we have, again using [Reference Bao and WangBW, Theorem 5.8] and [Reference JimboJi, Proposition 3], the following:

$$ \begin{align*} \psi^\imath\left(\left(x\otimes y\right)\left(h_1\otimes h_2\right)\right)&=\Theta^\imath\left(\psi^\imath(xh_1)\otimes\psi(yh_2)\right) = \Theta^\imath\left(\psi^\imath(x)\overline{h}_1\otimes\psi(y)\overline{h}_2\right)\\ =\sum_i &\theta_1^i(\psi^\imath(x)\overline{h}_1)\otimes\theta_2^i(\psi(y)\overline{h}_2) = \sum_i \left(\theta_1^i\psi^\imath(x)\right)\overline{h}_1\otimes\left(\theta_2^i\psi(y)\right)\overline{h}_2\\ =\Theta^\imath&\left(\psi^\imath(x)\otimes\psi(y)\right)(\overline{h}_1\otimes\overline{h}_2) = \psi^\imath\left(x\otimes y\right)(\overline{h}_1\otimes\overline{h}_2). \end{align*} $$

This concludes the proof.

One has the following important property of the involution $\psi ^\imath :\widehat {\mathbb {T}}^{m|n}\rightarrow \widehat {\mathbb {T}}^{m|n}$ [Reference Bao and WangBW, Lemma 9.7]:

$$ \begin{align*} \psi^\imath(v_f)\in v_f+\sum_{g\prec f}{\mathbb Z}[q,q^{-1}]v_g\subseteq \widehat{\mathbb{T}}^{m|n}. \end{align*} $$

Since the partial order $\preceq $ satisfies the finite interval property by [Reference Bao and WangBW, Lemma 8.4], we can apply [Reference LusztigLu3, Lemma 24.2.1] and conclude the existence of $\psi ^\imath $ -invariant bases:

Proposition 20. There exist unique $\psi ^\imath $ -invariant bases $\{T_f|f\in \mathbb {I}^{m|n}\}$ and $\{L_f|f\in \mathbb {I}^{m|n}\}$ of the form

$$ \begin{align*} &T_f=M_f+\sum_{g\prec f}t_{gf}(q)M_g,\quad t_{gf}(q)\in q{\mathbb Z}[q],\\ &L_f=M_f+\sum_{g\prec f}\ell_{gf}(q)M_g,\quad\ell_{gf}(q)\in q^{-1}{\mathbb Z}[q^{-1}]. \end{align*} $$

We set $t_{ff}=\ell _{ff}=1$ and $t_{gf}=\ell _{gf}=0$ , for $g\not \preceq f$ .

5.4. q-symmetric tensors

For a finite Weyl group W of a simple Lie algebra, we recall the well-known Chevalley-Solomon formula [Reference SolomonSol, Corollary 2.3]:

$$ \begin{align*} \sum_{\tau\in W}q^{\ell(\tau)}=\prod_{i=1}^r\left(1+q+\cdots+q^{e_i}\right), \end{align*} $$

where r is the rank of W and $e_1,\cdots ,e_r$ are the exponents of the corresponding Lie algebra. This implies the following formula:

$$ \begin{align*} \sum_{\tau\in W}q^{\ell(w_0)-\ell(\tau)}=\prod_{i=1}^r[e_i+1], \end{align*} $$

where $[s]$ is the quantum integer $\frac {q^s-q^{-s}}{q-q^{-1}}$ , for $s\in \mathbb {N}$ . We shall write $[W]:=\prod _{i=1}^r[e_i+1]$ . Now, suppose that W is a Weyl group of a reductive Lie algebra so that $W=W^1\times \cdots \times W^t$ , where each $W^j$ , $1\le j\le t$ , is a Weyl group of a simple Lie algebra. We shall use the following notation:

$$ \begin{align*} [W]:=\prod_{j=1}^t[W^j]. \end{align*} $$

Let $W_\zeta $ be a (parabolic) subgroup of $W_{B_m}\times \mathfrak {S}_n$ generated by simple reflections. Indeed, $W_\zeta $ is a product of Weyl groups of the form $W_\zeta = W^1\times \cdots \times W^s\times W^{s+1}\times \cdots \times W^{s+t}$ , where $W^1$ is either of type A or B, and the remaining $W^i$ s are all of type A. Furthermore, the sum of the ranks of the $W^i$ s, $i=1,\cdots ,s$ (respectively, $i=s+1,\cdots ,s+t$ ) equals m (respectively, equals n). Let $\mathcal H_\zeta $ be the Hecke subalgebra of the Hecke algebra $\mathcal {H}_{B_m}\times \mathcal H_{A_n}$ associated with $W_\zeta $ . Let

(5.4) $$ \begin{align} S_\zeta:=\sum_{\sigma\in W_\zeta} q^{\ell(w_0^\zeta)-\ell(\sigma)}H_\sigma = \prod_{i=1}^{s+t}S_i, \end{align} $$

where $w_0^\zeta $ is the longest element in $W_\zeta $ , $S_i=\sum _{\tau \in W^i} q^{\ell (w_0^i)-\ell (\tau )}H_\tau $ , and $w_0^i$ is the longest element in $W^i$ . The bar involution on $\mathcal {H}_{B_m}\times \mathcal H_{A_n}$ restricts to the bar involution on $\mathcal {H}_\zeta $ , and we have that

$$ \begin{align*} \overline{S_i}=S_i\quad \text{and}\quad S_i H_\tau=q^{-\ell(\tau)}S_i=H_\tau S_i ,\ \tau\in W^i,\\ \overline{S_\zeta}=S_\zeta\quad \text{and}\quad S_\zeta H_\sigma=q^{-\ell(\sigma)}S_\zeta=H_\sigma S_\zeta ,\ \sigma\in W_\zeta. \end{align*} $$

Consider the following $\mathbb {Q}(q)$ -linear subspace of $\mathbb {T}^{m|n}$ :

(5.5) $$ \begin{align} \mathbb T^{m|n}_\zeta:=\mathbb T^{m|n} S_\zeta. \end{align} $$

Then $\mathbb T^{m|n} S_\zeta $ is a $U^\imath $ -submodule by Proposition 19. Let $\mathbb {I}^{m|n}_{\zeta -}$ denote the set of elements in $\mathbb {I}^{m|n}$ that are anti-dominant with respect to $W_\zeta $ . For $f\in {\mathbb Z}^{m|n}_{\zeta -}$ , we let $W_f:=\{\sigma \in W_\zeta |f\cdot \sigma =f\}$ be the stabilizer subgroup inside $W_\zeta $ and $W^f$ the set of shortest length representatives of the coset $W_f\backslash W_\zeta $ .

Define the following sets of monomial bases for $\mathbb T^{m|n}_\zeta $ :

(5.6) $$ \begin{align} \widetilde{N}_f:=M_f S_\zeta;\qquad \widetilde{M}_f:=\frac{1}{[W_f]}M_f S_\zeta,\quad f\in\mathbb{I}^{m|n}_{\zeta-}. \end{align} $$

From the formulas of the action of the Hecke algebra on $M_f$ , it follows that the basis elements $\widetilde {N}_f$ lie in the ${\mathbb Z}[q,q^{-1}]$ -span of $\{M_g|g\in \mathbb {I}^{m|n}\}$ . The $\widetilde {M}_f$ s also lie in the ${\mathbb Z}[q,q^{-1}]$ -span of $\{M_g|g\in \mathbb {I}^{m|n}\}$ , which can be seen as follows. Suppose that the Hecke algebra of $W^i$ acts on $\mathbb V^{\otimes m_i}$ , for $i=1,\cdots ,s$ , and the Hecke algebra of $W^j$ acts on $\mathbb W^{\otimes n_{j}}$ , $j=s+1,\cdots ,s+t$ . We have $\sum _{i}m_i=m$ and $\sum _{j}n_j=n$ . The element

$$ \begin{align*} \widetilde{M}_{f_i}=\frac{1}{[W_f\cap W^i]}M_{f_i}S_i, \end{align*} $$

where $f_i$ is f restricted to the interval corresponding to $W^i$ , is an $\imath $ -canonical basis element for $i=1$ and a canonical basis element for $i>1$ . Thus, by [Reference Bao and WangBW, Theorem 4.20] in the case of $i=1$ and [Reference LusztigLu3, Theorem 27.3.2] in the case $i>1$ , it lies in the ${\mathbb Z}[q,q^{-1}]$ -span of corresponding $M_{g_i}$ , where the notation of $g_i$ is self-explanatory. Now, we have

$$ \begin{align*} \widetilde{M}_f=\widetilde{M}_{f_1}\otimes\cdots\otimes\widetilde{M}_{f_{s+t}}, \end{align*} $$

and hence, it lies in the ${\mathbb Z}[q,q^{-1}]$ -span of $M_{g}$ , $g\in \mathbb {I}^{m|n}$ .

For $f\in {\mathbb Z}^{m|n}_{\zeta -}$ , we also define

(5.7) $$ \begin{align} N_f:=\frac{[W_\zeta]}{[W_f]}\widetilde{N}_f. \end{align} $$

Suppose that $f\in \mathbb {I}^{m|n}$ , but not necessarily in $\mathbb {I}^{m|n}_{\zeta -}$ . Let $\tau \in W_\zeta $ be of shortest length such that $f\cdot \tau \in \mathbb {I}^{m|n}_{\zeta -}$ . Then we have

(5.8) $$ \begin{align} M_f S_\zeta= M_{f\cdot\tau} H_{\tau^{-1}} S_\zeta = q^{-\ell(\tau)} M_{f\cdot\tau}S_\zeta = q^{-\ell(\tau)} \widetilde{N}_{f\cdot\tau}. \end{align} $$

Similarly, we have the identity $\frac {1}{[W_f]}M_f S_\zeta =q^{-\ell (\tau )}\widetilde {M}_{f\cdot \tau }$ .

Proof. For $f\in \mathbb {I}^{m|n}_{\zeta -}$ , suppose that $\psi ^\imath (M_f) = M_f+\sum _{g\prec f}r_{gf}M_g$ with $r_{gf}\in {\mathbb Z}[q,q^{-1}]$ . Then, by Proposition 19 and (5.8), we have

$$ \begin{align*} \psi^\imath({\widetilde{N}}_f)&=\psi^\imath({M_f S_\zeta})=\psi^\imath({M}_f) S_\zeta = M_fS_\zeta+\sum_{g\prec f}r_{gf}M_gS_\zeta =\widetilde{N}_f + \sum_{g\prec f; g\in\mathbb{I}^{m|n}_{\zeta -}}r^{\prime}_{gf} \widetilde{N}_g. \end{align*} $$

Since $r_{gf}\in {\mathbb Z}[q,q^{-1}]$ , we have $r^{\prime }_{fg}\in {\mathbb Z}[q,q^{-1}]$ by (5.8). This proves Part (1).

We know that each $\widetilde {M}_f$ is a decomposable tensor, where the first factor is either an $\imath $ -canonical or canonical basis element and the remaining ones are canonical basis elements in the respective tensor powers of either $\mathbb V$ or $\mathbb W$ . Recalling the definition of bar involution $\psi ^\imath $ on a tensor product of a $U^\imath $ -module and a U-module from (5.3), we can now apply [Reference Bao, Wang and WatanabeBWW, (2.4)] and conclude that $\psi ^\imath \left ({\widetilde {M}}_f\right )$ lies in the ${\mathbb Z}[q,q^{-1}]$ -space of the ${\widetilde {M}}_g$ s. This gives Part (2).

Part (3) now follows from Part (2).

Now we can apply [Reference LusztigLu3, Lemma 24.2.1] and obtain the following.

Proposition 22. The $\mathbb {Q}(q)$ -vector space $\widehat {\mathbb T}^{m|n}_\zeta $ has unique $\psi ^\imath $ -invariant topological bases of the form

$$ \begin{align*} \{\mathcal{ T}^{\prime}_f|f\in\mathbb{I}^{m|n}_{\zeta-}\},\quad\{\mathcal{ T}_f|f\in\mathbb{I}^{m|n}_{\zeta-}\}\quad\text{and}\quad\{\mathcal{ L}_f|f\in\mathbb{I}^{m|n}_{\zeta-}\} \end{align*} $$

such that

$$ \begin{align*} \mathcal T^{\prime}_f=\widetilde{M}_f+\sum_{g\prec f}\texttt{t}^{\prime}_{gf}(q)\widetilde{M}_g, \quad \mathcal T_f={N}_f+\sum_{g\prec f}\texttt{t}_{gf}(q){N}_g, \quad {\mathcal{L}}_f=\widetilde{N}_f+\sum_{g\prec f}\texttt{l}_{gf}(q)\widetilde{N}_g, \end{align*} $$

with $\texttt {t}^{\prime }_{gf}(q),\texttt {t}_{gf}{(q)}\in q{\mathbb Z}[q]$ , and $\texttt {l}_{gf}(q)\in q^{-1}{\mathbb Z}[q^{-1}]$ , for $g\prec f$ in ${\mathbb Z}^{m|n}_{\zeta -}$ .

As usual, we adopt the convention $\texttt {t}^{\prime }_{ff}(q)=\texttt {t}_{ff}(q)=\texttt {l}_{ff}(q)=1$ , $\texttt {t}^{\prime }_{gf}=\texttt {t}_{gf}=\texttt {l}_{gf}=0$ for $g\npreceq f$ .

5.5. $\imath $ -Canonical Bases on q-symmetric tensors

This section compares the bases $\{T_f\}$ and $\{L_f\}$ with the bases $\{\mathcal T_f\}$ and $\{{\mathcal {L}}_f\}$ . Since this part is analogous to [Reference Chen, Coulembier and MazorchukCCM1, Section 3.7] and follows along the same line of arguments, we shall be brief and only summarize the main results.

Proposition 23. Let $f,g\in \mathbb {I}^{m|n}_{\zeta -}$ and $w_0^\zeta $ be the longest element in $W_\zeta $ . We have

$$ \begin{align*} \mathcal T^{\prime}_f = T_{fw_0^\zeta},\qquad \mathcal T^{\prime}_fS_\zeta=\mathcal T_f,\qquad L_fS_\zeta = \mathcal{L}_f. \end{align*} $$

In particular, we have $\texttt {t}_{gf}=\texttt {t}^{\prime }_{gf} =t_{g\cdot w_0^\zeta ,f\cdot w_0^\zeta }$ and $\texttt {l}_{gf} = \sum _{x\in W^g}\ell _{g\cdot x,f}q^{-\ell (x)}$ , where $W^g$ is the set of shortest length representatives $W_f\cap W_\zeta \backslash W_\zeta $ .