2.1 Notations

Throughout the paper, we assume that m and n are positive integers with $m\ge 2$ . We keep the same notations as in [Reference Jang, Kwon and Uruno13, Section 2], which we briefly recall.

• • $\mathbb {I} = \{\, 1, \dots ,m , m+1, \dots , m+n \,\}$ with a usual linear order,

• • $\mathbb {I} = \mathbb {I}_{\overline 0} \cup \mathbb {I}_{\overline 1}$ , where $\mathbb {I}_{\overline 0} = \{\, 1, \dots , m \,\}$ and $\mathbb {I}_{\overline 1} = \{ \, m+1,\dots , m+n \,\}$ ,

• • $\epsilon =(\epsilon _a)_{a\in \mathbb {I}}$ : a sequence with $\epsilon _a=\varepsilon $ for $a\in \mathbb {I}_{\varepsilon }$ ( $\varepsilon ={\overline 0},{\overline 1}$ ),

• • $P = \bigoplus _{a\in \mathbb {I}}\mathbb {Z} \delta _a \oplus \mathbb {Z}\Lambda _0$ : a free abelian group,

• • $I=\{\,0,1, \ldots ,m,m+1,\dots , m+n-1\}$ ,

• • $\mathfrak {g}=\mathfrak {b}_{m|n}, \mathfrak {c}_{m|n}, \mathfrak {d}_{m|n}$ : an orthosymplectic Lie superalgebra associated to the Dynkin diagram with the set of simple roots $\Pi =\{\,\alpha _i\,|\,i\in I\,\}\subset P$ as follows:

  
  $$ \begin{align*} \alpha_i= \begin{cases} -\delta_{1} & \text{if }i=0,\\ \delta_{i}-\delta_{i+1} & \text{if }1 \le i \le m+n-1, \end{cases} \end{align*} $$
  
  $$ \begin{align*} \alpha_i= \begin{cases} -2\delta_{1} & \text{if }i=0,\\ \delta_i-\delta_{i+1} & \text{if }1\le i \le m+n-1, \end{cases} \end{align*} $$
  
  $$ \begin{align*} \alpha_i= \begin{cases} -\delta_{1}-\delta_{2} & \text{if }i=0,\\ \delta_i-\delta_{i+1} & \text{if }1\le i \le m+n-1. \end{cases} \end{align*} $$

• • $(\, \cdot \, |\, \cdot \, )$ : a symmetric bilinear form on P satisfying $(\delta _a|\delta _b)=(-1)^{\epsilon _a}r_{\mathfrak {g}}\delta _{ab}$ , $(\delta _a|\Lambda _0)=-\frac {r_{\mathfrak {g}}}{2}\ (a,b\in \mathbb {I})$ with $r_{\mathfrak {b}_{m|n}}=2$ , $r_{\mathfrak {c}_{m|n}}=r_{\mathfrak {d}_{m|n}}=1$ (note that $(\Lambda _0|\alpha _i)=\delta _{0i}$ for $i\in I$ ),

• • $\Phi ^+$ : the set of reduced positive roots,

• • $\Phi ^+_{\overline {0}}$ (resp. $\Phi ^+_{\overline {1}}$ ): the set of even (resp. odd) roots in $\Phi ^+$ ,

• • $\mathfrak {l}=\mathfrak {l}_{m|n}$ : the subalgebra of $\mathfrak {g}$ corresponding to $\Pi \setminus \{\alpha _0\}$ isomorphic to $\mathfrak {gl}_{m|n}$ ,

• • $\Phi ^+(\mathfrak {l})$ : the set of positive roots of $\mathfrak {l}$ ,

• • $\mathfrak {l}_{\overline 0}$ : the subalgebra of $\mathfrak {l}$ corresponding to even roots in $\Phi ^+(\mathfrak {l})$ ,

• • $\mathfrak {l}_{m|0}$ , $\mathfrak {l}_{0|n}$ : the subalgebras of $\mathfrak {l}_{\overline 0}$ corresponding to $\mathfrak {gl}_m$ and $\mathfrak {gl}_n$ , respectively,

• • $\mathfrak {u}$ : the subalgebra of $\mathfrak {g}$ generated by the root vectors corresponding to $\Phi ^+\setminus \Phi ^+(\mathfrak {l})$ ,

• • $\Phi ^+(\mathfrak {u})= \Phi ^+\setminus \Phi ^+(\mathfrak {l})$ .

2.2 Quantum orthosymplectic superalgebra

We also recall the following [Reference Jang, Kwon and Uruno13]:

• • $\Bbbk =\mathbb {Q}(v)$ and $\Bbbk ^{\times }=\Bbbk \setminus \left \{0\right \}$ where $v=q$ if $\mathfrak {g}=\mathfrak {b}, \mathfrak {c} $ and $v=q^{\frac {1}{2}}$ if $\mathfrak {g}=\mathfrak {d} $ with $ q^{\frac {1}{2}}$ an indeterminate,

• • ${\texttt {q}}_a=(-1)^{\epsilon _a} q^{(-1)^{\epsilon _a} r_{\mathfrak {g}}}\ (a\in \mathbb {I})$ ,

• • $q_i = q^{d_i}\ (i\in I\setminus \{m\})$ , where $d_i=|(\alpha _i|\alpha _i)|/2$ , and $q_m = q^{r_{\mathfrak g}}$ ,

• • $\mathbf {q}(\mu ,\nu ) = v^{\sum _{a\in \mathbb {I}} \ell ' \mu _a + \ell \nu _a } \prod _{a\in \mathbb {I}}\mathtt {q}_a^{\mu _a\nu _a}$ for $\mu =\sum _{a\in \mathbb {I}}\mu _a\delta _a +\ell \Lambda _0 $ and $\nu =\sum _{a\in \mathbb {I}}\nu _a\delta _a + \ell ' \Lambda _0$ (note that $\mathbf { q}(\Lambda _0,\alpha _i)=q_0^{\delta _{0 i}}$ ),

• • For $s\in \mathbb {Z}_{\ge 0}$ and $z\in q^{\mathbb {Z}_{>0}}$ , $[s]_z=\frac {z^s-z^{-s}}{z-z^{-1}}$ , and $[s]_z!=[s]_z [s-1]_z \cdots [1]_z (s\geq 1)$ with $[0]_z!=1$ , where we simply write $[s]$ when $z=q$ , and $[s]_i=[s]_{q_i}$ when $z=q_i$ ( $i\in I$ ),

• • ${{\mathcal U}}(\mathfrak {g})$ : the associative $\Bbbk $ -algebra with $1$ generated by $k_\mu , e_i, f_i$ for $\mu \in P$ and $i\in I$ satisfying

  $$ \begin{align*} \begin{aligned} k_{\mu}=1 \quad (\mu=0), \quad k_{\nu +\nu'}=k_{\nu}k_{\nu'} \quad (\nu, \nu' \in P), \qquad\qquad\\k_\mu e_i k_{-\mu}=\mathbf{q}(\mu,\alpha_i)e_i,\quad k_\mu f_i k_{-\mu}=\mathbf{q}(\mu,\alpha_i)^{-1}f_i\quad (i\in I, \mu\in P), \\e_if_j - f_je_i =\delta_{ij}\frac{k_{i} - k^{-1}_{i}}{q_i-q_i^{-1}}\quad (i,j\in I),\qquad\qquad\qquad\\e_m^2= f_m^2 =0,\qquad\qquad\qquad\qquad\qquad\qquad \end{aligned} \end{align*} $$
  where $k_i=k_{\alpha _i}$ for $i\in I$ , and
  $$ \begin{align*}\begin{aligned} x_i x_j - x_j x_i =0 \quad (i,j \in I\text{ and }(\alpha_i|\alpha_j) = 0), \qquad\qquad\qquad\qquad\quad\\x_i^2 x_j- (-1)^{\epsilon_i}[2]_i x_i x_j x_i + x_j x_i^2= 0 \quad (i\in I\setminus\{0, m\}, j\in I\setminus\{0\}, i\neq j\text{ and }(\alpha_i|\alpha_j)\neq 0), \\\begin{array}{ll} x_0^3 x_1- [3]_0 x_0^2 x_1 x_0 + [3]_0 x_0 x_1 x_0^2 - x_1 x_0^3 = 0\\ x_1^2 x_0- [2]_1 x_1 x_0 x_1 + x_0 x_1^2= 0 \end{array} \quad (\mathfrak{g} = \mathfrak{b}_{m|n}), \qquad\qquad\qquad\\\begin{array}{ll} x_0^2 x_1- [2]_0 x_0 x_1 x_0 + x_1 x_0^2= 0 \\ x_1^3 x_0- [3]_1 x_1^2 x_0 x_1 + [3]_1 x_1 x_0 x_1^2 - x_0 x_1^3 = 0 \end{array} \quad (\mathfrak{g} = \mathfrak{c}_{m|n}), \qquad\qquad\qquad\\\begin{array}{ll} x_0^2 x_2- [2]_0 x_0 x_2 x_0 + x_2 x_0^2= 0 \\ x_2^2 x_0- [2]_2 x_2 x_0 x_2 + x_0 x_2^2= 0 \end{array} \quad (\mathfrak{g} = \mathfrak{d}_{m|n}\ \text{with }m\ge 3),\qquad\qquad\qquad\\\begin{array}{ll} x_{m}x_{m'}x_{m}x_{m+1} - x_{m}x_{m+1}x_{m}x_{m'} + x_{m+1}x_{m}x_{m'}x_{m} \\ \qquad \qquad \qquad \qquad \qquad - x_{m'}x_{m}x_{m+1}x_{m} + [2]_{m} x_{m}x_{m'}x_{m+1}x_{m} =0 \\\hskip 6cm (m'=0,1 \text{ if }\mathfrak{g}={\mathfrak d}_{2|n}, \text{ and } m'=m-1 \text{ otherwise} ), \end{array} \end{aligned} \end{align*} $$
  where $x=e,f$ ,

• • $\Delta $ : the comultiplication on ${\mathcal U}(\mathfrak {g})$ given by $\Delta (k_\mu )=k_\mu \otimes k_\mu $ , $\Delta (e_i)= 1\otimes e_i + e_i\otimes k_i^{-1}$ , $\Delta (f_i)= f_i\otimes 1 + k_i\otimes f_i$ for $\mu \in P$ and $i\in I$ ,

• • $e^{\prime }_i, e^{\prime \prime }_i$ : $\Bbbk $ -linear maps on ${\mathcal U}(\mathfrak {g})^-$ defined by

  (2.1) $$ \begin{align} \begin{aligned} e^{\prime}_i(f_j)&=\delta_{ij},\quad e^{\prime}_i(uv)=e^{\prime}_i(u)v + \mathbf{q}{(\alpha_i,|u|)}ue^{\prime}_i(v), \\ e^{\prime\prime}_i(f_j)&=\delta_{ij},\quad e^{\prime\prime}_i(uv)=e^{\prime\prime}_i(u)v + \mathbf{q}{(\alpha_i,|u|)}^{-1}ue^{\prime\prime}_i(v), \end{aligned} \end{align} $$
  for $i,j\in I$ and $u,v\in {\mathcal U}(\mathfrak {g})^-$ with u homogeneous.

We also consider the following subalgebras of ${\mathcal U}(\mathfrak {g})$ :

• • ${\mathcal U}(\mathfrak {g})^+= \langle \, e_i \,|\,i\in I\,\rangle $ , ${\mathcal U}(\mathfrak {g})^- = \langle \, f_i \,|\,i\in I\,\rangle $ , ${\mathcal U}(\mathfrak {g})^0 = \langle \, k_\mu \,|\,\mu \in P\,\rangle $ ,

• • ${\mathcal U}(\mathfrak {l})=\langle \, k_\mu , e_i, f_i \,|\,\mu \in P,\ i\in I\setminus \{0\}\,\rangle $ ,

• • ${\mathcal U}(\mathfrak {l}_{\overline 0})=\langle \, k_\mu , e_i, f_i \,|\,\mu \in P,\ i\in I\setminus \{0,m\}\,\rangle $ ,

• • ${\mathcal U}(\mathfrak {l}_{m|0})=\langle \, k_\mu , e_i, f_i \,|\,\mu \in P,\ i=1,\dots ,m-1 \,\rangle $ ,

• • ${\mathcal U}(\mathfrak {l}_{0|n})=\langle \, k_\mu , e_i, f_i \,|\,\mu \in P,\ i=m+1,\dots ,m+n-1 \,\rangle $ ,

• • ${\mathcal U}(\mathfrak {l}_\ast )^\pm = {\mathcal U}(\mathfrak {l}_\ast )\cap {\mathcal U}(\mathfrak {g})^\pm $ , where $\mathfrak {l}_\ast = \mathfrak {l}, \mathfrak {l}_{\overline 0}, \mathfrak {l}_{m|0}, \mathfrak {l}_{0|n}$ .

Note that ${\mathcal U}(\mathfrak {l}_{\overline 0})$ is isomorphic to $U_q(\mathfrak {gl}_m)\otimes U_{-q^{-1}}(\mathfrak {gl}_n)$ as a $\Bbbk $ -algebra up to the Cartan part, where $U_q(\mathfrak {gl}_m)$ is the usual quantized enveloping algebra associated to $\mathfrak {gl}_m$ . Recall that ${\mathcal U}(\mathfrak {g})^\pm $ is naturally graded by $Q^+:=\sum _{i\in I}\mathbb {Z}_{\ge 0}\alpha _i$ and $Q^-:=-Q^+$ .

Let $\mathfrak {a}$ denote either $\mathfrak {g}$ or $\mathfrak {l}_\ast $ above. For a ${\mathcal U}(\mathfrak a)$ -module V and $\mu \in P$ , let $V_\mu = \{\,u\in V\,|\,k_{\nu } u= \mathbf {q}(\mu ,\nu ) u \ \ (\nu \in P) \,\}$ be the $\mu $ -weight space of V. For $u\in V_\mu \setminus \{0\}$ , we call u a weight vector with weight $\mu $ and put $\mathrm {wt}(u)=\mu $ . Denote by $\mathrm {wt}(V) =\{ \mu \in P \, | \, V_\mu \ne 0 \}$ the set of weights of V. For $\Lambda \in P$ , let $V_{\mathfrak a}(\Lambda )$ be the irreducible highest weight ${\mathcal U}(\mathfrak {a})$ -module generated by a highest weight vector $v^{\mathfrak a}_\Lambda $ or simply $v_\Lambda $ of weight $\Lambda $ .

We also remark that the representation theory of ${\mathcal U}(\mathfrak {g})$ is equivalent to that of the quantum superalgebra associated to $\mathfrak {g}$ [Reference Yamane26] (see [Reference Jang, Kwon and Uruno13, Section 2.2] for more details).

2.3 Root vectors

Let $\mathbf {f}_\beta $ ( $\beta \in \Phi ^+$ ) be a root vector in ${\mathcal U}(\mathfrak {g})^-$ , which is defined in [Reference Jang, Kwon and Uruno13, Section 3.1] following [Reference Clark, Hill and Wang3, Reference Leclerc24]. Let $\prec $ be the linear order on $\Phi ^+$ induced from that on the set of good Lyndon words $\mathcal {GL}(\mathfrak {g})$ under the bijection with $\Phi ^+$ . Then we have a PBW type basis B of ${\mathcal U}(\mathfrak {g})^-$ with respect to $\prec $ . Note that $\alpha \prec \beta $ for $\alpha \in \Phi ^+(\mathfrak {u})$ and $\beta \in \Phi ^+(\mathfrak {l})$ . For $\beta \in \Phi ^+(\mathfrak {u})$ , we simply write

$$ \begin{align*} \beta= \begin{cases} (i,i) & \text{if }\beta = -2\delta_i/r_{\mathfrak{g}}\text{ for }1\le i\le m+n,\\ (i,j) & \text{if }\beta = -\delta_i-\delta_j\text{ for }1\le i<j\le m+n. \end{cases} \end{align*} $$

Then

(2.2) $$ \begin{align} \Phi^+(\mathfrak{u})= \begin{cases} \{\,(i,i)\,|\,1\le i\le m+n\,\} \sqcup \{\,(i,j)\,|\,1\le i<j\le m+n\,\} & \text{if }\mathfrak{g}=\mathfrak{b}_{m|n},\\ \{\,(i,i)\,|\,1\le i\le m\,\} \sqcup \{\,(i,j)\,|\,1\le i<j\le m+n\,\} & \text{if }\mathfrak{g}=\mathfrak{c}_{m|n},\\ \{\,(i,i)\,|\,m+1\le i\le m+n\,\} \sqcup \{\,(i,j)\,|\,1\le i<j\le m+n\,\} & \text{if }\mathfrak{g}=\mathfrak{d}_{m|n}, \end{cases} \end{align} $$

and for $(i_1,j_1), (i_2,j_2)\in \Phi ^+(\mathfrak {u})$ ,

$$ \begin{align*} (i_1,j_1) \prec (i_2,j_2) \ \Longleftrightarrow \ \begin{cases} i_1<i_2 \text{ or } (i_1=i_2 \text{ and } j_1<j_2) & \text{if }\mathfrak{g}=\mathfrak{b}_{m|n}, \\ j_1<j_2 \text{ or } (j_1=j_2 \text{ and } i_1<i_2) & \text{if }\mathfrak{g}=\mathfrak{c}_{m|n}, \mathfrak{d}_{m|n}. \end{cases} \end{align*} $$

Let ${\mathcal U}(\mathfrak {u}^-)$ be the $\Bbbk $ -space spanned by the vectors in B where the product is over $\beta \in \Phi ^+(\mathfrak {u})$ . It is a $\Bbbk $ -subalgebra of ${\mathcal U}(\mathfrak {g})^-$ generated by $\mathbf {f}_\beta $ ( $\beta \in \Phi ^+(\mathfrak {u})$ ), which we denote by $\mathcal {N}$ . As a $\Bbbk $ -space

$$ \begin{align*} {\mathcal U}(\mathfrak{g})^-\cong {\mathcal U}(\mathfrak{u}^-)\otimes {\mathcal U}(\mathfrak{l})^-, \end{align*} $$

where $\otimes =\otimes _\Bbbk $ .

2.4 Irreducible oscillator modules $V(\lambda ,\ell )$

Let us recall a semisimple tensor category of ${\mathcal U}(\mathfrak {g})$ -modules studied in [Reference Kwon20]. Let

• • $\mathtt {P}=\sum _{a\in \mathbb {I}}\mathbb {Z}\delta _a \subset P$ ,

• • $\mathtt {P}^+=\{\,\lambda =\sum _{i\in \mathbb {I}}\lambda _i\delta _i \in \mathtt {P}\ |\ \lambda _{1}\geq \ldots \geq \lambda _{m},\ \ \lambda _{m+1}\geq \ldots \geq \lambda _{m+n}\,\}$ ,

• • $\lambda _{+}=\sum _{i\in \mathbb {I}_0}\lambda _{i}\delta _i$ , $\lambda _{-}=\sum _{j\in \mathbb {I}_1}\lambda _{j}\delta _j$ for $\lambda \in \mathtt { P}$ ,

• • $\mathtt {P}_{\geq 0}=\sum _{a\in \mathbb {I}}\mathbb {Z}_{\ge 0}\delta _a$ .

Let $\mathcal {O}(\mathfrak l)_{\geq 0}$ be the category of ${\mathcal U}(\mathfrak {l})$ -modules V such that $V=\bigoplus _{\mu \in \mathtt {P}_{\ge 0}}V_\mu $ with $\dim V_\mu < \infty $ . Let $\mathscr {P}$ be the set of all partitions $\lambda =(\lambda _i)_{i\ge 1}$ , and let $\mathscr {P}_{m|n}$ be the set of $\lambda \in \mathscr {P}$ such that $\lambda _{m+1}\leq n$ . For $\lambda \in \mathscr {P}_{m|n}$ , let

(2.3) $$ \begin{align} \Lambda_\lambda = \lambda_1\delta_1 +\cdots +\lambda_m\delta_m + \mu_1\delta_{m+1}+\cdots+\mu_n\delta_{m+n} \in \mathtt{P}_{\geq 0}, \end{align} $$

where $(\mu _1,\ldots ,\mu _n)$ is the conjugate of the partition $(\lambda _{m+1},\lambda _{m+2},\ldots )$ . We often identify $\lambda \in \mathscr {P}_{m|n}$ with $\Lambda _\lambda $ if there is no confusion. We simply write $V_{\mathfrak {l}}(\lambda )=V_{\mathfrak {l}}(\Lambda _{\lambda })$ and $v_\lambda =v_{\Lambda _\lambda }$ . Then any irreducible ${\mathcal U}(\mathfrak {l})$ -module in $\mathcal {O}(\mathfrak l)_{\geq 0}$ is isomorphic to $V_{\mathfrak {l}}(\lambda )$ for some $\lambda \in \mathscr {P}_{m|n}$ [Reference Benkart, Kang and Kashiwara1, Propositions 3.4 and 4.5] (see also [Reference Jang, Kwon and Uruno13, Remark 4.2]). Moreover, $\mathcal {O}(\mathfrak l)_{\geq 0}$ is semisimple [Reference Jang, Kwon and Uruno13, Proposition 4.3].

We may understand that $V\in \mathcal {O}(\mathfrak {g})_{\ge 0}$ belongs to $\mathcal {O}(\mathfrak l)_{\ge 0}$ as a ${\mathcal U}(\mathfrak {l})$ -module if we ignore $\Lambda _0$ , and hence is a direct sum of $V_{\mathfrak l}(\mu )$ for $\mu \in \mathscr {P}_{m|n}$ with finite multiplicity. Let $\mathscr {P}(\mathfrak {g})$ be given by

$$ \begin{align*} \begin{aligned} \mathscr{P}(\mathfrak{b}_{m|n})&=\{(\lambda,\ell)\in \mathscr{P}_{m|n}\times\mathbb{Z}_{> 0}\,|\,\ell-2\lambda_1\in \mathbb{Z}_{\geq 0}\},\\ \mathscr{P}(\mathfrak{c}_{m|n})&=\{(\lambda,\ell)\in \mathscr{P}_{m|n}\times\mathbb{Z}_{> 0}\,|\,\ell-\lambda_1\in \mathbb{Z}_{\geq 0}\},\\ \mathscr{P}(\mathfrak{d}_{m|n})&=\{(\lambda,\ell)\in \mathscr{P}_{m|n}\times\mathbb{Z}_{> 0}\,|\,\ell-\lambda_1-\lambda_2\in \mathbb{Z}_{\geq 0}\,\}, \end{aligned} \end{align*} $$

and for $(\lambda ,\ell )\in \mathscr {P}(\mathfrak {g})$ , let

$$ \begin{align*} \Lambda(\lambda,\ell) = \ell\Lambda_0 + \Lambda_\lambda. \end{align*} $$

For simplicity, let us write $V(\lambda ,\ell )=V(\Lambda (\lambda ,\ell ))$ and $v_{(\lambda ,\ell )}=v_{\Lambda (\lambda ,\ell )}$ . For $V\in \mathcal {O}(\mathfrak {g})_{\ge 0}$ , we have

$$ \begin{align*} V \cong \bigoplus_{(\lambda,\ell)\in \mathscr{P}(\mathfrak{g})}V(\lambda,\ell)^{\oplus m(\lambda,\ell)}, \end{align*} $$

for some $m(\lambda ,\ell )\in \mathbb {Z}_{\ge 0}$ .

2.5 Parabolic Verma modules

Let ${\mathcal U}(\mathfrak {p})$ be the subalgebra of ${\mathcal U}(\mathfrak {g})$ generated by ${\mathcal U}(\mathfrak {l})$ and ${\mathcal U}(\mathfrak {g})^{+}$ . For $\lambda \in \mathscr {P}_{m|n}$ , let

$$ \begin{align*} P(\lambda)={\mathcal U}(\mathfrak{g})\otimes_{{\mathcal U}(\mathfrak{p})}V_{\mathfrak{l}}(\lambda), \end{align*} $$

where we regard $V_{\mathfrak {l}}(\lambda )$ as a ${\mathcal U}(\mathfrak {p})$ -module with $e_0 v_\lambda =0$ . We have an isomorphism of $\Bbbk $ -spaces

(2.4) $$ \begin{align} \mathcal{N}\otimes V_{\mathfrak l}(\lambda)\cong P(\lambda), \end{align} $$

sending $u\otimes v$ to itself since ${\mathcal U}(\mathfrak {g})\cong {\mathcal U}(\mathfrak {u}^-)\otimes {\mathcal U}(\mathfrak {l})^-\otimes {\mathcal U}(\mathfrak {g})^{\ge 0}\cong \mathcal {N} \otimes {\mathcal U}(\mathfrak {p})$ , where ${\mathcal U}(\mathfrak {g})^{\ge 0} = {\mathcal U}(\mathfrak {g})^+{\mathcal U}(\mathfrak {g})^0$ . It is a highest weight ${\mathcal U}(\mathfrak {g})$ -module with highest weight $\Lambda _\lambda $ (2.3).

Let ${\mathcal U}(\mathfrak {q})$ be the subalgebra of ${\mathcal U}(\mathfrak {g})$ generated by ${\mathcal U}(\mathfrak {l}_{\overline 0})$ and ${\mathcal U}(\mathfrak {g})^{+}$ . For $\lambda \in \mathtt {P}^+$ , let

(2.5) $$ \begin{align} Q(\lambda)={\mathcal U}(\mathfrak{g})\otimes_{{\mathcal U}(\mathfrak{q})}V_{\mathfrak{l}_{\overline 0}}(\lambda), \end{align} $$

where we regard $V_{\mathfrak {l}_{\overline 0}}(\lambda )$ as a ${\mathcal U}(\mathfrak {q})$ -module with $e_0 v_{\lambda }=e_m v_{\lambda }=0$ .

Finally, let ${\mathcal U}(\mathfrak {r})$ be the subalgebra of ${\mathcal U}(\mathfrak {l})$ generated by ${\mathcal U}(\mathfrak {l}_{0|n})$ and ${\mathcal U}(\mathfrak {g})^{+}$ . For $\lambda \in \mathtt {P}$ with $\lambda _-\in \mathtt {P}^+$ , let

(2.6) $$ \begin{align} R(\lambda)={\mathcal U}(\mathfrak{g})\otimes_{{\mathcal U}(\mathfrak{r})}V_{\mathfrak{l}_{0|n}}(\lambda), \end{align} $$

where we regard $V_{\mathfrak {l}_{0|n}}(\lambda )$ as a ${\mathcal U}(\mathfrak {r})$ -module with $e_i v_{\lambda }=0$ for $i=0,\dots ,m$ .

For $(\lambda ,\ell )\in \mathscr {P}(\mathfrak {g})$ , we have natural surjective ${\mathcal U}(\mathfrak {g})^-$ -linear maps

where each map sends a highest weight vector to a highest weight vector.

3.1 Crystal bases of $V(\lambda ,\ell )$ and $P(\lambda )$

Let us briefly recall the results on crystal bases of $V(\lambda ,\ell )$ [Reference Kwon20, Reference Kwon21] and $P(\lambda )$ [Reference Jang, Kwon and Uruno13].

3.1.1 Crystal base of $V(\lambda ,\ell )$

Let V be a ${\mathcal U}(\mathfrak {g})$ -module in $\mathcal {O}(\mathfrak {g})_{\ge 0}$ . For a weight vector $u\in V$ and $i\in I$ , we define $\tilde {e}_i u$ and $\tilde {f}_i u$ as follows:

1. (1) Suppose that $i\neq m$ . We have $u=\sum _{k \geq 0} f_i^{(k)}u_k$ , where $f_i^{(k)}=f_i^k/[k]_i!$ and $e_iu_k=0$ for $k\geq 0$ . We define

   $$ \begin{align*}\begin{aligned} \tilde{e}_iu&=\sum_{k\geq1}f_i^{(k-1)}u_k,\quad \tilde{f}_iu=\sum_{k\geq0}f_i^{(k+1)}u_k \quad (i<m),\\\tilde{e}_iu&=\sum_{k\geq1}q_i^{-l_k+2k-1}f_i^{(k-1)}u_k,\quad \tilde{f}_iu=\sum_{k\geq 0}q_i^{l_k-2k-1}f_i^{(k+1)}u_k\quad (i>m), \end{aligned}\end{align*} $$
   where $l_k= - (\mathrm {wt}(u_k)|\alpha _i)/r_{\mathfrak {g}}$ . Here we follow the convention in [Reference Jang, Kwon and Uruno12, Section 4.2].

2. (2) Suppose that $i=m$ . We define

   $$ \begin{align*} \tilde{e}_m u =\eta(f_m) u =q_m^{-1}k_me_m u,\quad \tilde{f}_m u=f_m u, \end{align*} $$
   where $\eta $ is the anti-involution on ${\mathcal U}(\mathfrak {g})$ defined by $\eta (k_\mu )=k_\mu $ , $\eta (e_i)=q_if_ik^{-1}_i$ , and $\eta (f_i)=q^{-1}_ik_ie_i$ for $\mu \in P$ and $i\in I$ .

Let $A_0$ be the subring of rational functions $f(q)\in \Bbbk $ regular at $q=0$ . We define a crystal base of V to be a pair $(L,B)$ in the usual way with respect to $\tilde {e}_i$ and $\tilde {f}_i$ ( $i\in I$ ), where L is an $A_0$ -lattice of V, and B is a signed basis of $L/qL$ , that is, $B=\mathbf {B}\cup -\mathbf {B}$ , where $\mathbf {B}$ is a $\mathbb {Q}$ -basis of $L/qL$ . We call the set $B/\{\pm 1\}$ , which has a colored oriented graph structure, a ${\mathcal U}(\mathfrak {g})$ -crystal of V or simply a crystal of V. We identify B with $B/\{\pm 1\}$ when we consider B as a crystal, if there is no confusion. The following is obtained in [Reference Kwon20, Theorem 8.8] for $\mathfrak {g}=\mathfrak {b}_{m|n}, \mathfrak {c}_{m|n}$ and [Reference Kwon21, Theorem 5.7] for $\mathfrak {g}=\mathfrak {d}_{m|n}$ .

Theorem 3.1. For $(\lambda ,\ell )\in \mathscr {P}(\mathfrak {g})$ , $V(\lambda ,\ell )$ has a crystal base $(\mathscr {L}(V(\lambda ,\ell )), \mathscr {B}(V(\lambda ,\ell )))$ , where the crystal $\mathscr {B}(V(\lambda ,\ell ))$ is connected.

Since $\mathscr {B}(V(\lambda ,\ell ))$ is connected, we have a unique crystal base of $V(\lambda ,\ell )$ up to scalar multiplication, that is,

(3.1) $$ \begin{align} \begin{aligned} \mathscr{L}(V(\lambda,\ell))&=\sum_{r \ge 0, i_1,\dots,i_r \in I}A_0\widetilde{x}_{i_1}\dots\widetilde{x}_{i_r}v_{(\lambda,\ell)} \\ \mathscr{B}(V(\lambda,\ell))&=\left\{\,\pm\widetilde{x}_{i_1}\dots\widetilde{x}_{i_r}v_{(\lambda,\ell)} \!\!\!\pmod{q\mathscr{L}(V(\lambda,\ell))}\,|\,r\ge 0, i_1,\dots,i_r\in I\,\right\}\setminus\{0\}, \end{aligned} \end{align} $$

where $x = e,f$ for each $i_k$ . Note that (3.1) holds in general for a highest weight module that has a crystal base (with respect to $\tilde {e}_i$ and $\tilde {f}_i$ defined suitably) such that the associated crystal is connected. A combinatorial model for $\mathscr {B}(V(\lambda ,\ell ))$ is also given in [Reference Kwon20, Reference Kwon21], which will be reviewed in Section 6.2.

Regarding $V\in \mathcal {O}(\mathfrak {g})_{\ge 0}$ as a ${\mathcal U}(\mathfrak {l})$ -module, we may consider a crystal base of $V\in \mathcal {O}(\mathfrak l)_{\ge 0}$ with respect to $\tilde {e}_i$ and $\tilde {f}_i$ ( $i\in I\setminus \{0\}$ ). For $\lambda \in \mathscr {P}_{m|n}$ , $V_{\mathfrak {l}}(\lambda )$ has a crystal base $(\mathscr {L}(V_{\mathfrak l}(\lambda )), \mathscr {B}(V_{\mathfrak l}(\lambda )))$ [Reference Benkart, Kang and Kashiwara1], where the connected crystal $\mathscr {B}(V_{\mathfrak l}(\lambda ))$ can be realized as the set $SST_{m|n}(\lambda )$ of $(m|n)$ -hook semistandard tableaux of shape $\lambda $ . Hence $(\mathscr {L}(V(\lambda ,\ell )), \mathscr {B}(V(\lambda ,\ell )))$ is a direct sum of crystal bases of $V_{\mathfrak {l}}(\mu )$ ’s for $\mu \in \mathscr {P}_{m|n}$ as a ${\mathcal U}(\mathfrak {l})$ -module.

3.1.2 Crystal base of $P(\lambda )$

Let $\lambda \in \mathscr {P}_{m|n}$ be given. For a weight vector $u\in P(\lambda )$ , we define

(3.2) $$ \begin{align} \tilde{e}_0 u = \sum_{k\ge 1}f_0^{(k-1)}u_k,\quad \tilde{f}_0 u = \sum_{k\ge 0}f_0^{(k+1)}u_k, \end{align} $$

where $u=\sum _{k\ge 0}f_0^{(k)}u_k$ for $e^{\prime }_0 ( u_k ) = 0$ . Then we define a crystal base of $P(\lambda )$ in the same way as in Section 3.1.1, where $\tilde {e}_0$ and $\tilde {f}_0$ are replaced by (3.2). We may define $\tilde {e}_0$ and $\tilde {f}_0$ on ${\mathcal U}(\mathfrak {g})^-$ in the same way, and we have $(\widetilde {x}_0P)(1\otimes v_\lambda ) = \widetilde {x}_0(P(1\otimes v_\lambda ))$ for $P\in {\mathcal U}(\mathfrak {g})^-$ (cf. [Reference Jang, Kwon and Uruno13, Section 4.4]).

Theorem 3.2 [Reference Jang, Kwon and Uruno13, Section 4.4].

For $\lambda \in \mathscr {P}_{m|n}$ , $P(\lambda )$ has a crystal base $(\mathscr {L}(P(\lambda )), \mathscr {B}(P(\lambda )))$ , where the crystal $\mathscr {B}(P(\lambda ))$ is connected.

Let $\mathrm {ad}_q: {\mathcal U}(\mathfrak {g}) \longrightarrow \mathrm {End}_{\Bbbk }({\mathcal U}(\mathfrak {g}))$ be the adjoint representation of ${\mathcal U}(\mathfrak {g})$ with respect to $\Delta $ , where for $\mu \in P$ , $i \in I$ , and $u \in {\mathcal U}(\mathfrak {g})$ ,

(3.3) $$ \begin{align} \begin{aligned} & \mathrm{ad}_q(k_\mu)(u) = k_\mu u k_\mu^{-1}, \ \ \mathrm{ad}_q(e_i)(u) = \left( e_i u - u e_i \right) k_i, \ \ \mathrm{ad}_q(f_i)(u) = f_i u - k_i u k_i^{-1} f_i. \end{aligned} \end{align} $$

Let us write $\mathrm {ad}_q(x)(u)=x\cdot u$ for simplicity. Then the subalgebra $\mathcal {N}$ is a ${\mathcal U}(\mathfrak {l})$ -submodule of ${\mathcal U}(\mathfrak {g})$ with respect to $\mathrm {ad}_q$ such that $x\cdot m_{\mathcal N}(u_1\otimes u_2) = m_{\mathcal N}(\Delta (x)\cdot (u_1\otimes u_2))$ for $x\in {\mathcal U}(\mathfrak {l})$ and $u_1\otimes u_2\in \mathcal {N}\otimes \mathcal {N}$ , where $m_{\mathcal N}$ denotes the multiplication on $\mathcal {N}$ [Reference Jang, Kwon and Uruno13, Proposition 3.16]. We have $\mathcal {N}\in \mathcal {O}(\mathfrak l)_{\ge 0}$ as a ${\mathcal U}(\mathfrak {l})$ -module, and the map (2.4) is an isomorphism of ${\mathcal U}(\mathfrak {l})$ -modules.

For $\beta \in \Phi ^+(\mathfrak {u})$ and $k\in \mathbb {Z}_{\ge 0}$ , let

$$ \begin{align*} \mathbf{F}_\beta^{(k)} = \begin{cases} \mathbf{f}_\beta^{(k)} & \text{if }(\beta|\beta)\ge 0,\\ q^{-\frac{k(k-1)}{2}r_{\mathfrak g}}\mathbf{f}_\beta^{(k)} & \text{if }(\beta|\beta)< 0\text{ and }\beta\ne(i,i),\\ q^{-\frac{k(k-1)}{2}}\mathbf{f}_\beta^{(k)} & \text{if }\mathfrak{g}=\mathfrak{b}_{m|n}, (\beta|\beta)< 0, \text{and }\beta=(i,i),\\ q^{-k^2}\mathbf{f}_\beta^{(k)} & \text{if }\mathfrak{g}=\mathfrak{d}_{m|n}, (\beta|\beta)< 0, \text{and} \beta=(i,i), \end{cases} \end{align*} $$

(see [Reference Jang, Kwon and Uruno13, Section 3.1] for the definition of $\mathbf {f}_\beta ^{(k)}$ ). Let

$$ \begin{align*} M(\mathfrak{u}) = \left\{\, (c_\beta)_{\beta\in \Phi^+(\mathfrak{u})}\in \mathbb{Z}_{\ge 0}^{N} \, | \, c_{\beta}\in\mathbb{Z}_{\ge 0}\ (\beta\in \Phi^+_{\overline 0}\cup \Phi^+_{\mathrm{{n\text{-}iso}}}),\quad c_{\beta}=0,1\ (\beta\in \Phi^+_{\mathrm{ iso}}) \,\right\}, \end{align*} $$

where $N=|\Phi ^+(\mathfrak {u})|$ and $\Phi ^+_{\text {n-iso}}$ is the set of nonisotropic roots in $\Phi ^+_{\overline 1}$ . For $\mathbf {c}=(c_\beta )_{\beta \in \Phi ^+(\mathfrak {u})}\in M(\mathfrak {u})$ , let

$$ \begin{align*} \mathbf{F}^{(\mathbf c)} = \prod^{\rightarrow}_{\beta\in\Phi^+(\mathfrak{u})}\mathbf{F}_\beta^{(c_\beta)}, \end{align*} $$

where the product is taken in increasing order with respect to $\prec $ .

Theorem 3.3 [Reference Jang, Kwon and Uruno13, Theorem 4.15].

Let

$$ \begin{align*} \begin{aligned} \mathscr{L}\left(\mathcal{N}\right) &= \bigoplus_{\mathbf{c}\in M(\mathfrak{u})} A_0 \mathbf{F}^{(\mathbf c)},\ \ \mathscr{B}\left(\mathcal{N}\right) = \left\{\,\pm \mathbf{F}^{(\mathbf c)}\!\!\!\pmod{q\mathscr{L}\left(\mathcal{N}\right)}\,|\,\mathbf{c}\in M(\mathfrak{u})\,\right\}. \end{aligned} \end{align*} $$

Then $(\mathscr {L}\left (\mathcal {N}\right ),\mathscr {B}\left (\mathcal {N}\right ))$ is a crystal base of $\mathcal {N}$ as a ${\mathcal U}(\mathfrak {l})$ -module.

By [Reference Jang, Kwon and Uruno13, Theorems 4.22 and 4.24], we have

(3.4) $$ \begin{align} \mathscr{L}(P(\lambda))=\mathscr{L}(\mathcal{N})\otimes \mathscr{L}(V_{\mathfrak{l}}(\lambda)),\quad \mathscr{B}(P(\lambda))=\mathscr{B}(\mathcal{N})\otimes \mathscr{B}(V_{\mathfrak{l}}(\lambda)). \end{align} $$

The crystal operators $\tilde {e}_i, \tilde {f}_i$ for $i\in I\setminus \{0\}$ act on $\mathscr {B}(\mathcal {N})\otimes \mathscr {B}(V_{\mathfrak {l}}(\lambda ))$ by the tensor product rule [Reference Benkart, Kang and Kashiwara1, Proposition 2.8], while $\tilde {e}_0, \tilde {f}_0$ act by decreasing and increasing the multiplicity $c_{\alpha _0}$ in $\mathbf {F}^{(\mathbf c)}$ by $1$ , respectively.

3.2 Compatibility of crystal bases of $V(\lambda ,\ell )$ and $P(\lambda )$

Let $(\lambda ,\ell )\in \mathscr {P}(\mathfrak {g})$ be given. We show that the crystal base of $P(\lambda )$ is compatible with that of $V(\lambda ,\ell )$ under $\pi _{V(\lambda ,\ell )}$ . This can be done by following the same idea as in [Reference Kashiwara14, Section 4].

Let $\ell '>\ell \ge 0$ be given. Note that $(\lambda ,\ell ')\in \mathscr {P}(\mathfrak {g})$ . For $t=\ell '-\ell $ , we have a ${\mathcal U}(\mathfrak {g})$ -linear embedding

and a ${\mathcal U}(\mathfrak {g})^-$ -linear map

for $v'$ such that $\mathrm {wt}(v') \ne \Lambda (0,t)=t\Lambda _0$ . Since $\mathcal {O}(\mathfrak {g})_{\ge 0}$ is a semisimple tensor category, $\Phi $ is a well-defined ${\mathcal U}(\mathfrak {g})$ -linear map, and in particular it commutes with $\tilde {e}_i$ and $\tilde {f}_i$ for $i\in I$ .

Proof. Since $\Phi $ commutes with $\tilde {e}_i$ and $\tilde {f}_i$ for $i\in I$ , it follows from the connectedness of $\mathscr {B}(V(\lambda ,\ell '))$ that $\Phi (\mathscr {L}(V(\lambda ,\ell ')))\subset \mathscr {L}(V(\lambda ,\ell ))\otimes \mathscr {L}(V(0,t))$ . Hence $S\circ \Phi (\mathscr {L}(V(\lambda ,\ell ')))\subset \mathscr {L}(V(\lambda ,\ell ))$ since $S(\mathscr {L}(V(\lambda ,\ell ))\otimes \mathscr {L}(V(0,t)))=\mathscr {L}(V(\lambda ,\ell ))$ by definition of S.

Let $b \in \mathscr {B}(V(\lambda ,\ell ))$ be given, where $b=\widetilde {x}_{i_k}\dots \widetilde {x}_{i_1} v_{(\lambda ,\ell )}$ for some $i_1, \dots , i_k \in I $ , where $x=e,f$ for each $i_j$ . By the tensor product rule (cf. [Reference Benkart, Kang and Kashiwara1, Proposition 2.8]), we have

$$ \begin{align*} \tilde{x}_{i_k}\dots \tilde{x}_{i_1} (v_{(\lambda,\ell)}\otimes v_{(0,t)}) \equiv \pm (\tilde{x}_{i_k}\dots \tilde{x}_{i_1} v_{(\lambda,\ell)})\otimes v_{(0,t)} \ (\mathrm{mod} \ q\mathscr{L}(V(\lambda,\ell))\otimes \mathscr{L}(V(0,t))), \end{align*} $$

since $\tilde {f}_i v_{(0,t)}=0$ for all $i>m$ . Hence $\overline {S\circ \Phi }(\tilde {x}_{i_k}\dots \tilde {x}_{i_1} v_{(\lambda ,\ell ')})=b$ . In particular, $\overline {S\circ \Phi }$ is surjective and hence $S\circ \Phi (\mathscr {L}(V(\lambda ,\ell '))) =\mathscr {L}(V(\lambda ,\ell ))$ . This proves (1).

To prove the injectivity of $\overline {S\circ \Phi }$ , suppose $b, b' \in \mathscr {B}(V(\lambda ,\ell '))$ satisfy $\overline {S\circ \Phi }(b) = \overline {S\circ \Phi }(b') \neq 0$ . Write $b=\widetilde {x}_{i_k}\dots \widetilde {x}_{i_1} v_{(\lambda ,\ell ')}$ , $b'=\widetilde {x}_{j_l}\dots \widetilde {x}_{j_1} v_{(\lambda ,\ell ')}$ for some $i_1, \dots , i_k, j_1, \dots j_l \in I $ , where each x stands for e or f. Then by assumption, we have $\widetilde {x}_{i_k}\dots \widetilde {x}_{i_1} v_{(\lambda ,\ell )} = \widetilde {x}_{j_l}\dots \widetilde {x}_{j_1} v_{(\lambda ,\ell )}$ . On the other hand,

$$ \begin{align*}\overline{\Phi}(b)=\tilde{x}_{i_k} \dots \tilde{x}_{i_1} (v_{(\lambda,\ell) }\otimes v_{(0,t)}) = (\tilde{x}_{i_k} \dots \tilde{x}_{i_1} v_{(\lambda,\ell)}) \otimes v_{(0,t)} = \overline{\Phi}(b').\end{align*} $$

Since $\overline {\Phi }$ is an embedding of crystals, it follows that $b=b'$ .

Let $\Phi _{\ell ', \ell }=S\circ \Phi $ , which is ${\mathcal U}(\mathfrak {g})^-$ -linear, and let $\Psi _{\ell }:{\mathcal U}(\mathfrak {g})^- \longrightarrow V(\lambda ,\ell )$ be the ${\mathcal U}(\mathfrak {g})^-$ -linear projection sending $1$ to $v_{(\lambda ,\ell )}$ . Since $\Phi _{\ell ',\ell } \circ \Psi _{\ell '} = \Psi _{\ell }$ , we have $\mathrm {Ker} \, \Psi _{\ell '} \subset \mathrm {Ker} \, \Psi _{\ell }$ , especially $(\mathrm {Ker} \, \Psi _{\ell '})_{\xi } \subset (\mathrm {Ker} \, \Psi _{\ell })_{\xi } \subset {\mathcal U}(\mathfrak {g})^-_\xi $ for $\xi \in Q^-$ . Hence for a given $\xi \in Q^-$ there is $N>0$ such that $(\mathrm {Ker} \, \Psi _{\ell +t})_{\xi } = (\mathrm {Ker} \, \Psi _{\ell +N})_{\xi }$ for all $t\ge N$ .

This implies that the restriction $\Phi _{\ell +t,\ell +N} : V(\lambda ,\ell +t)_{\Lambda (\lambda ,\ell +t)+\xi } \longrightarrow V(\lambda ,\ell +N)_{\Lambda (\lambda ,\ell +N)+\xi }$ is a $\Bbbk $ -linear isomorphism, and

(3.5) $$ \begin{align} \Phi_{\ell+t,\ell+N} : \mathscr{L}(V(\lambda,\ell+t))_{\Lambda(\lambda,\ell+t)+\xi} \longrightarrow \mathscr{L}(V(\lambda,\ell+N))_{\Lambda(\lambda,\ell+N)+\xi} \end{align} $$

is an isomorphism of $A_0$ -modules for all $t\ge N$ .

Lemma 3.5 (cf. [Reference Kashiwara14, Lemma 4.4.1]).

Let $\xi \in Q^-$ be given. For $P \in {\mathcal U}(\mathfrak {g})^-_{\xi }$ , there exists $N>0$ such that for all $t>N$ ,

$$ \begin{align*} \begin{aligned} (\widetilde{x}_0 P) v_{(\lambda,\ell+t)} & \equiv \widetilde{x}_0(P v_{(\lambda,\ell+t)}) \quad (\mathrm{mod} \ q\mathscr{L}(V(\lambda,\ell+t))) \quad (x=e,f), \end{aligned} \end{align*} $$

where $\widetilde {x}_0$ on ${\mathcal U}(\mathfrak {g})^-$ is defined in the same way as in (3.2).

Proof. The argument is essentially the same as in [Reference Kashiwara14, Lemma 4.4.1] after a minor modification. Assume that $P=f_0^{(k)}Q$ with $e^{\prime }_0 Q=0$ . Then $\tilde {e}_0 P = f_0^{(k-1)}Q$ and $\tilde {f}_0 P = f_0^{(k+1)}Q$ . We will show that

$$ \begin{align*} \begin{aligned} \tilde{e}_0(P v_{(\lambda,\ell+t)}) & \equiv f_0^{(k-1)}Q v_{(\lambda,\ell+t)},\ \ \tilde{f}_0(P v_{(\lambda,\ell+t)}) \equiv f_0^{(k+1)}Q v_{(\lambda,\ell+t)} \ \ (\mathrm{mod} \ q\mathscr{L}(V(\lambda,\ell+t))), \end{aligned} \end{align*} $$

for all sufficiently large t.

We remark that [Reference Kashiwara14, Corollary 3.4.6, Lemma 4.3.9] still hold for ${\mathcal U}(\mathfrak {g})^-$ and $V(\lambda ,\ell )$ , where $(L(\lambda ),B(\lambda ))$ and u are replaced by $(\mathscr {L}(V(\lambda ,\ell )),\mathscr {B}(V(\lambda ,\ell )))$ and $Qv_{(\lambda ,\ell +t)}$ respectively. Thus by [Reference Kashiwara14, Lemma 4.3.9], it is enough to show that

$$ \begin{align*} k_0^\nu e_0^{(\nu)} Q v_{(\lambda,\ell+t)} \in q_0^{\nu(\nu+k+1)}q \mathscr{L}(V(\lambda,\ell+t)) \quad \text{for all } 1 \leq \nu \leq k+1, \end{align*} $$

where we have by [Reference Kashiwara14, Corollary 3.4.6],

$$ \begin{align*} \begin{aligned} k_0^\nu e_0^{(\nu)} Q v_{(\lambda,\ell+t)} & = \frac{\mathbf{q}(\alpha_0, \Lambda_\lambda+(\ell+t)\Lambda_0+\xi-k\alpha_0)^{2\nu} q_0 ^{3\nu^2+\nu} }{(q_0 - q_0^{-1})^\nu} ({e^{\prime\prime}_0}^{\, \nu} Q) v_{(\lambda,\ell+t)}, \\ & = \frac{\mathbf{q}(\alpha_0, \Lambda_\lambda+\ell \Lambda_0 +\xi-k\alpha_0)^{2\nu} q_0 ^{3\nu^2+\nu +t\nu} }{(q_0 - q_0^{-1})^\nu} ({e^{\prime\prime}_0}^{\, \nu} Q) v_{(\lambda,\ell+t)}. \end{aligned} \end{align*} $$

Let N be as in (3.5). Then we have $P v_{(\lambda ,\ell +t)} {\kern-1pt}\in{\kern-1pt} \mathscr {L}(V(\lambda ,\ell {\kern-1pt}+{\kern-1pt}t))$ if and only if ${P v_{(\lambda ,\ell {\kern-1pt}+{\kern-1pt}N)} {\kern-1pt}\in{\kern-1pt} \mathscr {L}(V(\lambda ,\ell +N))}$ for all $t\ge N$ . On the other hand, there exists $M_\nu \in \mathbb {Z}$ such that $q^{M_\nu } ({e^{\prime \prime }_0}^{\, \nu } Q) v_{(\lambda ,\ell )} \in \mathscr {L}(V(\lambda ,\ell +t))$ for all $t \geq N$ . Then one can choose $N'\geq N$ such that

$$ \begin{align*} \frac{\mathbf{q}(\alpha_0, \Lambda_\lambda+\ell \Lambda_0 +\xi-k\alpha_0)^{2\nu} q_0 ^{3\nu^2+\nu +t\nu} }{(q_0 - q_0^{-1})^\nu} \in q_0^{\nu(\nu+k+1)+M_\nu} q A_0 \end{align*} $$

for all $t\geq N'$ and $1 \leq \nu \leq k+1$ . This completes the proof.

The following lemma shows that $P(\lambda )$ can be viewed as the limit of $V(\lambda ,\ell )$ as $\ell \to \infty $ .

Proof. Since $P(\lambda )$ and $V(\lambda ,\ell )$ are semisimple ${\mathcal U}(\mathfrak {l})$ -modules, every ${\mathcal U}(\mathfrak {l})$ -irreducible component L of $P(\lambda )$ is isomorphic to its image under $\pi _{V(\lambda ,\ell )}$ if it is not zero. Thus $\pi _{V(\lambda ,\ell )}$ commutes with $\tilde {e}_i$ and $\tilde {f}_i$ if $i\in I\setminus \{ 0 \}$ .

Now we claim that

(3.6) $$ \begin{align} \pi_{V(\lambda,\ell)} (\tilde{x}_{i_k} \dots \tilde{x}_{i_1} (1 \otimes v_\lambda) ) \equiv \pm \tilde{x}_{i_k} \dots \tilde{x}_{i_1} v_{(\lambda,\ell)} \quad (\mathrm{mod} \ q \mathscr{L}(V(\lambda,\ell))), \end{align} $$

for all $(\lambda ,\ell ) \in \mathscr {P}(\mathfrak {g})$ and $i_i,\dots ,i_k\in I$ . We use induction on k. It is clearly true when $k=0$ . Suppose that (3.6) is true for $k-1$ .

If $i_k \ne 0$ , then (3.6) is true since $\tilde {x}_i$ commutes with $\pi _{V(\lambda ,\ell )}$ . So we assume $i_k=0$ . Let $P\in {\mathcal U}(\mathfrak {g})^-$ such that $\tilde {x}_{i_{k-1}}\dots \tilde {x}_{i_1}(1 \otimes v_\lambda )=P(1\otimes v_\lambda )$ . By the induction hypothesis,

(3.7) $$ \begin{align} P v_{(\lambda,\ell)} \equiv \pm \tilde{x}_{i_{k-1}} \dots \tilde{x}_{i_1} v_{(\lambda,\ell)} \ (\mathrm{mod} \ q \mathscr{L}(V(\lambda,\ell))) \end{align} $$

for all $(\lambda ,\ell ) \in \mathscr {P}(\mathfrak {g})$ . By Lemma 3.5 and (3.7), there exists N such that for all $t \geq N$

(3.8) $$ \begin{align} (\tilde{x}_0 P) v_{(\lambda,\ell+t)} \equiv \tilde{x}_0 (P v_{(\lambda,\ell+t)}) \equiv \pm \tilde{x}_0 \tilde{x}_{i_{k-1}} \dots \tilde{x}_{i_1} v_{(\lambda,\ell+t)} \ (\mathrm{mod} \ q \mathscr{L}(V(\lambda,\ell+t))). \end{align} $$

Applying $\Phi _{\ell +t,\ell }$ to (3.8) and using Lemma 3.4 together with the induction hypothesis, we obtain

(3.9) $$ \begin{align} (\tilde{x}_0 P) v_{(\lambda,\ell)} \equiv \pm \tilde{x}_0 \tilde{x}_{i_{k-1}} \dots \tilde{x}_{i_1} v_{(\lambda, \ell)} \equiv \pm \tilde{x}_0 (P v_{(\lambda,\ell)}) \ (\mathrm{mod} \ q \mathscr{L}(V(\lambda,\ell))). \end{align} $$

Finally, since $\tilde {x}_0 (P(1\otimes v_\lambda )) = (\tilde {x}_0 P)(1 \otimes v_\lambda )$ by definition, we have

$$ \begin{align*} \begin{aligned} \pi_{V(\lambda,\ell)} (\tilde{x}_{i_k} \dots \tilde{x}_{i_1} (1 \otimes v_\lambda) ) & = \pi_{V(\lambda,\ell)} (\tilde{x}_{0} (P (1 \otimes v_\lambda)) ) \\ & = \pi_{V(\lambda,\ell)} ((\tilde{x}_0 P) (1 \otimes v_\lambda) ) \\ & = (\tilde{x}_0 P) v_{(\lambda,\ell)} \\ & \equiv \pm \tilde{x}_0 (P v_{(\lambda,\ell)}) \quad{(\mathrm{mod} \ q\mathscr{L}(V(\lambda,\ell)))} \\ & \equiv \pm \tilde{x}_0 \tilde{x}_{i_{k-1}} \dots \tilde{x}_{i_1} v_{(\lambda,\ell)} \quad{(\mathrm{mod} \ q\mathscr{L}(V(\lambda,\ell)))}, \end{aligned} \end{align*} $$

where the second-to-last equivalence follows from (3.9). This completes the proof of (3.6). Then (1) and (2) follow from (3.6) and the connectedness of $\mathscr {B}(P(\lambda ))$ and $\mathscr {B}(V(\lambda ,\ell ))$ . We now prove (3). Suppose there exist $b,b' \in \mathscr {B}(P(\lambda ))_{\Lambda (\lambda ,0)+\xi }$ for some $\xi \in Q^-$ with $b \neq b'$ and ${\overline {\pi }_{V(\lambda ,\ell )}(b)=\overline {\pi }_{V(\lambda ,\ell )}(b') \neq 0}$ . By Lemma 3.6, for a sufficiently large integer t we have an isomorphism $\mathscr {B}(P(\lambda ))_{\Lambda (\lambda ,0)+\xi } \cong \mathscr {B}(V(\lambda ,\ell +t))_{\Lambda (\lambda ,t)+\xi }$ , and therefore $\overline {\pi }_{V(\lambda ,\ell +t)}(b) \neq \overline {\pi }_{V(\lambda ,\ell +t)}(b')$ . Combining this with Lemma 3.4 (2) and $\pi _{V(\lambda ,\ell )} = \Phi _{\ell +t,\ell +N} \circ \pi _{V(\lambda ,\ell +t)}$ yields $\overline {\pi }_{V(\lambda ,\ell )}(b) \neq \overline {\pi }_{V(\lambda ,\ell )}(b')$ , a contradiction. Together with (3.6), the proof of (3) is complete.

4.1 Crystal base of a Kac module

Let us first recall the crystal bases of q-deformed Kac modules [Reference Kwon19, Section 4]. Let $\Phi ^+(\mathfrak {l})_{\kappa }=\Phi ^+(\mathfrak {l})\cap \Phi ^+_{\kappa }$ ( $\kappa ={\overline 0}, {\overline 1}$ ). Let $\textbf {f}^{\prime }_\beta $ be the root vector for $\beta \in \Phi ^+(\mathfrak {l})$ with a partial order $\prec '$ on $\Phi ^+(\mathfrak {l})$ defined as in [Reference Jang, Kwon and Uruno12, Section 2.5], where $\alpha \prec '\beta $ for $\alpha \in \Phi ^+(\mathfrak {l})_{\overline 1}$ and $\beta \in \Phi ^+(\mathfrak {l})_{\overline 0}$ . Note that $\textbf {f}^{\prime }_\beta $ is different from $\textbf {f}_\beta $ in Section 2.3.

Let $\mathcal {K}$ be the $\Bbbk $ -span of $\mathbf {f}^{\prime }_S$ for $S\subset \Phi ^+(\mathfrak {l})_{\overline 1}$ , where $\mathbf {f}^{\prime }_S$ is the product of $\textbf {f}^{\prime }_\alpha $ ( $\alpha \in S$ ) with respect to $\prec '$ . With respect to the associated PBW type basis, we have

(4.1) $$ \begin{align} {\mathcal U}(\mathfrak{l})^- \cong \mathcal{K} \otimes {\mathcal U}(\mathfrak{l}_{\overline 0})^- \cong \mathcal{K} \otimes {\mathcal U}(\mathfrak{l}_{m|0})^-\otimes {\mathcal U}(\mathfrak{l}_{0|n})^-. \end{align} $$

Moreover $\mathcal {K}$ is equal to the subalgebra generated by $\textbf {f}^{\prime }_\alpha $ for $\alpha \in \Phi ^+(\mathfrak {l})_{\overline 1}$ , and it is a ${\mathcal U}(\mathfrak {l}_{\overline 0})$ -module with respect to (3.3), which is a direct sum of $V_{\mathfrak {l}_{\overline 0}}(\lambda )$ for some $\mathfrak {l}_{\overline 0}$ -dominant weights $\lambda \in \texttt {P}^+$ . The pair $(\mathscr {L}({\mathcal K}),\mathscr {B}({\mathcal K}))$ with

$$ \begin{align*} \mathscr{L}({\mathcal K})=\bigoplus_{S\subset \Phi^+(\mathfrak{l})_{\overline 1}}A_0\mathbf{f}^{\prime}_S,\quad \mathscr{B}({\mathcal K})=\{\,\pm \mathbf{f}^{\prime}_S\ \pmod{q\mathscr{L}({\mathcal K})}\,|\,S\subset \Phi^+(\mathfrak{l})_{\overline 1}\,\} \end{align*} $$

is a crystal base of $\mathcal {K}$ as a ${\mathcal U}(\mathfrak {l}_{\overline 0})$ -module.

For $\lambda \in \mathtt {P}^+$ , let

$$ \begin{align*} K(\lambda) = {\mathcal U}(\mathfrak{l})\otimes_{\mathcal{Q}}V_{\mathfrak{l}_{\overline 0}}(\lambda), \end{align*} $$

where $\mathcal {Q}$ is the subalgebra of ${\mathcal U}(\mathfrak {l})$ generated by ${\mathcal U}(\mathfrak {l}_{\overline 0})$ and ${\mathcal U}(\mathfrak {l})^+$ , and $V_{\mathfrak {l}_{\overline 0}}(\lambda )$ is extended to a $\mathcal {Q}$ -module trivially. It is a highest weight ${\mathcal U}(\mathfrak {l})$ -module generated by $1\otimes v_{\lambda }$ with highest weight $\lambda $ . As a ${\mathcal U}(\mathfrak {l}_{\overline 0})$ -module, we have

(4.2) $$ \begin{align} K(\lambda) \cong \mathcal{K}\otimes V_{\mathfrak{l}_{\overline 0}}(\lambda). \end{align} $$

We define a crystal base of $K(\lambda )$ as in Section 3.1.1, where $\tilde {e}_m, \tilde {f}_m$ are replaced by

(4.3) $$ \begin{align} \tilde{e}_m u = e^{\prime}_m(u),\quad \tilde{f}_m u = f_m u. \end{align} $$

Then $K(\lambda )$ has a unique crystal base $\left (\mathscr {L}({K(\lambda )}),\mathscr {B}(K(\lambda ))\right )$ , where $\mathscr {B}(K(\lambda ))$ is connected. Let $(\mathscr {L}(V_{\mathfrak {l}_{\overline 0}}(\lambda )), \mathscr {B}(V_{\mathfrak {l}_{\overline 0}}(\lambda )))$ be the crystal base of $V_{\mathfrak {l}_{\overline 0}}(\lambda )$ generated by $v_{\lambda }$ . Then we have

$$ \begin{align*} (\mathscr{L}({K(\lambda)}),\mathscr{B}(K(\lambda))) \cong (\mathscr{L}(\mathcal{K})\otimes \mathscr{L}(V_{\mathfrak{l}_{\overline 0}}(\lambda)),\mathscr{B}(\mathcal{K}) \otimes \mathscr{B}(V_{\mathfrak{l}_{\overline 0}}(\lambda))), \end{align*} $$

under (4.2), where $\cong $ means the isomorphism of crystal bases.

For $\lambda \in \mathscr {P}_{m|n}$ , let

(4.4)

be the ${\mathcal U}(\mathfrak {l})$ -module homomorphism such that $\pi _{V_{\mathfrak {l}}(\lambda )}(1\otimes v_\lambda )=v_\lambda $ . Then $\pi _{V_{\mathfrak {l}}(\lambda )}$ preserves the crystal bases in the sense of Theorem 3.7.

For $\lambda ,\mu \in \mathtt {P}^+$ , we define $\lambda <\mu $ if and only if $\mu -\lambda \in \mathtt {P}^+$ . It is shown in [Reference Jang, Kwon and Uruno12] that $\{\,\mathscr {B}(K(\lambda ))\,|\,\lambda \in \mathtt {P}^+\,\}$ forms an inductive system of (abstract) ${\mathcal U}({\mathfrak l}_{\overline 0})$ -crystals (not ${\mathcal U}(\mathfrak l)$ -crystal), with respect to the injective morphism of ${\mathcal U}({\mathfrak l}_{\overline 0})$ -crystals;

for $\lambda < \mu $ . Then its limit, say $\mathscr {B}(K(\infty ))$ , becomes an (abstract) ${\mathcal U}(\mathfrak l)$ -crystal (see [Reference Jang, Kwon and Uruno12, Section 5.3] for more details), where for $b\in \mathscr {B}(K(\infty ))$

(4.5) $$ \begin{align} \varepsilon_i(b)=\max\{\,k\,|\,\tilde{e}_i^kb\neq \mathbf{0}\,\},\quad \varphi_i(b) = \begin{cases} (-1)^{\epsilon_i}(\mathrm{wt}(b),\alpha_i ) + \varepsilon_i(b) & \text{for }i\neq m,\\ \max\{\,k\,|\,\tilde{f}_i^kb\neq \mathbf{0}\,\} & \text{for }i= m, \end{cases} \end{align} $$

where $\mathbf {0}$ denotes the formal zero (when we consider an abstract crystal).

4.2 Crystal base of ${\mathcal U}(\mathfrak {l})^-$

Let us recall a crystal base of ${\mathcal U}(\mathfrak {l})^-$ [Reference Jang, Kwon and Uruno12, Section 6]. It is given in such a way that the associated crystal is $\mathscr {B}(K(\infty ))$ .

We first need a parabolic Verma module and its crystal base, which interpolates those of $K(\lambda )$ and ${\mathcal U}(\mathfrak {l})^-$ [Reference Jang, Kwon and Uruno12, Section 6.2]. For $\lambda \in \mathtt {P}$ with $\lambda _-\in \mathtt {P}^+$ , let

$$ \begin{align*} X(\lambda) = {\mathcal U}(\mathfrak{l})\otimes_{\mathcal{R}}V_{\mathfrak{l}_{0|n}}(\lambda), \end{align*} $$

where $\mathcal {R}$ is the subalgebra of ${\mathcal U}(\mathfrak {l})$ generated by ${\mathcal U}(\mathfrak {l}_{0|n})$ and ${\mathcal U}(\mathfrak {l})^+$ , and $V_{\mathfrak {l}_{0|n}}(\lambda )$ is extended to an $\mathcal {R}$ -module trivially. It is also a highest weight ${\mathcal U}(\mathfrak {l})$ -module generated by $1\otimes v_{\lambda }$ with highest weight $\lambda $ . As a $\Bbbk $ -space, we have

(4.6) $$ \begin{align} X(\lambda) \cong {\mathcal K}\otimes {\mathcal U}(\mathfrak{l}_{m|0})^-\otimes V_{\mathfrak{l}_{0|n}}(\lambda). \end{align} $$

We define a crystal base of $X(\lambda )$ as in $K(\lambda )$ , where $\tilde {e}_i, \tilde {f}_i$ for $i<m$ are replaced by those defined as in (3.2) with respect to $e^{\prime }_i$ . Then $X(\lambda )$ also has a unique crystal base $\left (\mathscr {L}({X(\lambda )}),\mathscr {B}(X(\lambda ))\right )$ , where $\mathscr {B}(X(\lambda ))$ is connected.

Let $(\mathscr {L}({\mathcal U}(\mathfrak {l}_{m|0})^-),\mathscr {B}({\mathcal U}(\mathfrak {l}_{m|0})^-))$ and $(\mathscr {L}(V_{\mathfrak {l}_{0|n}}(\lambda )),\mathscr {B}(V_{\mathfrak {l}_{0|n}}(\lambda )))$ be the crystal bases of ${\mathcal U}(\mathfrak {l}_{m|0})^-$ and $V_{\mathfrak {l}_{0|n}}(\lambda )$ , respectively. We have

$$ \begin{align*} \begin{aligned} \mathscr{L}(X(\lambda)) &\cong \left(\mathscr{L}({\mathcal K})\cdot \mathscr{L}({\mathcal U}(\mathfrak{l}_{m|0})^-)\right)\otimes \mathscr{L}(V_{\mathfrak{l}_{0|n}}(\lambda)),\\ \mathscr{B}(X(\lambda)) &\cong \left(\mathscr{B}({\mathcal K}) \cdot \mathscr{B}({\mathcal U}(\mathfrak{l}_{m|0})^-)\right) \otimes \mathscr{B}(V_{\mathfrak{l}_{0|n}}(\lambda)), \end{aligned} \end{align*} $$

under (4.6).

For $\lambda \in \mathtt {P}^+$ , let

be the ${\mathcal U}(\mathfrak {l})$ -module homomorphism such that $\pi _{K(\lambda )}(1\otimes v_{\lambda })=1\otimes v_{\lambda }$ . Then $\pi _{K(\lambda )}$ preserves the crystal bases in the sense of Theorem 3.7.

Let $u\in {\mathcal U}(\mathfrak {l})^-$ be given with $u=u_1 u_2 u_3$ for $u_1\in \mathcal {K}$ , $u_2\in {\mathcal U}(\mathfrak {l}_{m|0})^-$ and $u_3\in {\mathcal U}(\mathfrak {l}_{0|n})^-$ by (4.1). For $i\in I$ with $i>m$ , we define

(4.7) $$ \begin{align} \tilde{e}_i u = u_1u_2(\tilde{e}_iu_3),\quad \tilde{f}_i u = u_1u_2(\tilde{f}_iu_3), \end{align} $$

where $\tilde {e}_i$ and $\tilde {f}_i$ are those on ${\mathcal U}(\mathfrak {l}_{0|n})^-$ . Then we define a crystal base of ${\mathcal U}(\mathfrak {l})^-$ as in $X(\lambda )$ , where $\tilde {e}_i, \tilde {f}_i$ for $i>m$ are replaced by (4.7).

Theorem 4.1 [Reference Jang, Kwon and Uruno13, Theorem 6.1].

Let

$$ \begin{align*} \begin{aligned} \mathscr{L}({\mathcal U}(\mathfrak{l})^-) & = \mathscr{L}({\mathcal K})\cdot \mathscr{L}({\mathcal U}(\mathfrak{l}_{m|0})^-)\cdot \mathscr{L}({\mathcal U}(\mathfrak{l}_{0|n})^-), \\ \mathscr{B}({\mathcal U}(\mathfrak{l})^-)&=\mathscr{B}({\mathcal K}) \cdot \mathscr{B}({\mathcal U}(\mathfrak{l}_{m|0})^-) \cdot \mathscr{B}({\mathcal U}(\mathfrak{l}_{0|n})^-), \end{aligned} \end{align*} $$

where $\cdot $ denotes the multiplication in ${\mathcal U}^-$ and the induced one in $\mathscr {L}({\mathcal U}(\mathfrak {l})^-)/q\mathscr {L}({\mathcal U}(\mathfrak {l})^-)$ respectively. Then $(\mathscr {L}({\mathcal U}(\mathfrak {l})^-),\mathscr {B}({\mathcal U}(\mathfrak {l})^-))$ is a crystal base of ${\mathcal U}(\mathfrak {l})^-$ , where $\mathscr {B}({\mathcal U}(\mathfrak {l})^-)$ is isomorphic to $\mathscr {B}(K(\infty ))$ .

Let $\lambda \in \mathtt {P}$ with $\lambda _-\in \mathtt {P}^+$ . Since $\left (\mathscr {L}({X(\lambda )}),\mathscr {B}(X(\lambda ))\right )$ is an upper crystal base of $X(\lambda )$ as a ${\mathcal U}(\mathfrak {l}_{0|n})$ -module, it is not compatible with that of ${\mathcal U}(\mathfrak l)^-$ under the canonical projection from ${\mathcal U}(\mathfrak {l})^-$ to $X(\lambda )$ .

Let $(\mathscr {L}({\mathcal U}(\mathfrak {l}_{0|n})^-),\mathscr {B}({\mathcal U}(\mathfrak {l}_{0|n})^-))$ be the crystal base of ${\mathcal U}(\mathfrak {l}_{0|n})^-$ (see [Reference Jang, Kwon and Uruno12, (3.8)] for more details). For $\lambda \in \mathtt {P}$ with $\lambda _-\in \mathtt {P}^+$ , let $\pi _{\lambda }: {\mathcal U}(\mathfrak {l}_{0|n})^- \longrightarrow V_{\mathfrak {l}_{0|n}}(\lambda )$ be the canonical projection, and let

be the dual of $\pi _{\lambda }$ induced from nondegenerate bilinear forms on ${\mathcal U}(\mathfrak {l}_{0|n})^-$ and $V_{\mathfrak {l}_{0|n}}(\lambda )$ , which is injective. Then we have

(4.8) $$ \begin{align} \pi_\lambda^\vee(\mathscr{L}(V_{\mathfrak{l}_{0|n}}(\lambda)))\subset \mathscr{L}({\mathcal U}(\mathfrak{l}_{0|n})^-), \quad \overline{\pi}_\lambda^\vee(\mathscr{B}(V_{\mathfrak{l}_{0|n}}(\lambda)))\subset \mathscr{B}({\mathcal U}(\mathfrak{l}_{0|n})^-), \end{align} $$

where $\overline {\pi }_\lambda ^\vee $ is the induced map at $q=0$ and $\overline {\pi }_\lambda ^\vee (\tilde {e}_i b)= \tilde {e}_i \overline {\pi }_\lambda ^\vee (b)$ for $i>m$ and $b\in \mathscr {B}(V_{\mathfrak {l}_{0|n}}(\lambda ))$ (cf. [Reference Jang, Kwon and Uruno12, Section 3.2.2] and references therein).

For $\lambda \in \mathtt {P}$ with $\lambda _-\in \mathtt {P}^+$ , let $\pi _{X(\lambda )}: {\mathcal U}(\mathfrak {l})^- \longrightarrow X(\lambda )$ be the ${\mathcal U}(\mathfrak {l})^-$ -linear map such that $\pi _{X(\lambda )}(1)=1\otimes v_{\lambda }$ . By (4.1), (4.6), and (4.8), we immediately have

(4.9)

given by $\pi _{X(\lambda )}^\vee =1_{\mathcal {K}\otimes {\mathcal U}(\mathfrak {l}_{m|0})^-}\otimes \pi ^\vee _\lambda $ . Then we have

(4.10) $$ \begin{align} \pi_{X(\lambda)}^\vee(\mathscr{L}(X(\lambda)))\subset \mathscr{L}({\mathcal U}(\mathfrak{l})^-),\quad \overline{\pi}_{X(\lambda)}^\vee(\mathscr{B}(X(\lambda)))\subset \mathscr{B}({\mathcal U}(\mathfrak{l})^-), \end{align} $$

where $ \overline {\pi }_{X(\lambda )}^\vee $ is the induced map at $q=0$ . Moreover, $\overline {\pi }^\vee _{X(\lambda )}$ restricts to a bijection

(4.11) $$ \begin{align} \overline{\pi}^\vee_{X(\lambda)} : \mathscr{B}(X(\lambda)) \longrightarrow \{\,b=b_1b_2b_3\in \mathscr{B}({\mathcal U}(\mathfrak{l})^-)\,|\ b_3 \otimes t_{\lambda} \in \mathscr{B}(V_{\mathfrak{l}_{0|n}}(\lambda)) \,\}, \end{align} $$

where $b=b_1b_2b_3{\kern-1pt}\in{\kern-1pt} \mathscr {B}({\mathcal U}(\mathfrak {l})^-)$ is as in Theorem 4.1 and $\{t_\lambda \}$ is the (abstract) crystal with ${\tilde {e}_i t_\lambda {\kern-1pt}={\kern-1pt}\tilde {f}_i t_\lambda =0}$ , $\varepsilon _i(t_\lambda )=\varphi _i(t_\lambda )=-\infty $ for all $i\in I$ , and $\mathrm { wt}(t_\lambda )=\Lambda _{0,\lambda }$ . Here $\overline {\pi }^\vee _{X(\lambda )}$ commutes with $\tilde {e}_i$ , $\tilde {f}_i$ for $i<m$ , while it commutes with $\tilde {e}_i$ , $\tilde {f}_i$ for $i\ge m$ and $b'=\overline {\pi }^\vee _{X(\lambda )}(b)$ with $\lambda $ sufficiently large (see [Reference Jang, Kwon and Uruno12, Remark 6.6]).

Let $\lambda \in \mathscr {P}_{m|n}$ be given. Summarizing, the crystal bases mentioned in this subsection are compatible with the following maps:

(4.12)

The maps in (4.12) induce injective maps at $q=0$ :

(4.13)

A combinatorial description of $\iota _{V_{\mathfrak {l}}(\lambda )}$ can be seen in [Reference Kwon19], and $\iota _{K(\lambda )}$ is essentially the one from $\mathscr {B}(V_{\mathfrak {l}_{\overline 0}}(\lambda ^+))$ into $\mathscr {B}({\mathcal U}(\mathfrak {l}_{m|0})^-)$ for $\lambda =\lambda ^+\in \texttt {P}^+$ given by counting the number of entries in each row of a semistandard Young tableau to give multiplicities of root vectors (and hence Lusztig data for a PBW type crystal), and the description of $\iota _{X(\lambda )}$ is similar.

We remark that the crystal $\mathscr {B}({\mathcal U}(\mathfrak {l})^-)$ is not connected for $n\ge 2$ unlike $\mathscr {B}(X(\lambda ))$ , $\mathscr {B}(K(\lambda ))$ , and $\mathscr {B}(V_{\mathfrak l}(\lambda ))$ . More precisely, it is not connected if and only if $n\ge 2$ , and

$$ \begin{align*} \mathscr{B}({\mathcal U}(\mathfrak{l})^-) \cong \left(\mathscr{B}({\mathcal U}(\mathfrak{l}_{m|1})^-)\times \mathscr{B}({\mathcal U}(\mathfrak{l}_{0|n})^-)\right)^{\oplus 2^{m(n-1)}} \end{align*} $$

(see [Reference Jang, Kwon and Uruno12, Theorem 6.8] for more details).

5.1 Crystal base of $Q(\lambda )$

Let $B_q(\mathfrak {l}_{1|1})$ be the q-boson algebra associated with $\mathfrak {l}_{1|1}$ , that is, an associative $\Bbbk $ -algebra generated by $e'$ and f subject to the relations

$$ \begin{align*} e^{\prime2}=f^2=0, \quad e'f+fe'=1. \end{align*} $$

As these relations have no q-dependence, $B_q(\mathfrak {l}_{1|1})$ is simply isomorphic to the rank 2 Clifford algebra with f (resp. $e'$ ) representing a single fermionic particle creation (resp. annihilation) operator. Note that $\langle e^{\prime }_m, f_m \rangle \cong B_q(\mathfrak {l}_{1|1})$ , where $e^{\prime }_m$ acts as in (2.1), and $f_m$ acts by left multiplication on ${\mathcal U}(\mathfrak {g})^-$ .

It is easy to see that ${\mathcal U}(\mathfrak {l}_{1|1})^-=\Bbbk 1 \oplus \Bbbk f$ is the unique irreducible $B_q(\mathfrak {l}_{1|1})$ -module up to isomorphism, where ${\mathcal U}(\mathfrak {l}_{1|1})=\langle e,f,k^{\pm 1} \rangle $ . For a $B_q(\mathfrak {l}_{1|1})$ -module, we define crystal operators $\widetilde {e}$ and $\widetilde {f}$ as in (4.3). Then ${\mathcal U}(\mathfrak {l}_{1|1})^-$ has a unique crystal base $(L,B)$ up to scalar multiplication, where $L=A_0 1 \oplus A_0 f$ and $B=\{1, f \ (\mathrm {mod} \ qL) \}$ with $\tilde {f} 1= f (\mathrm {mod} \ qL)$ . Furthermore, the natural basis $\{1, f\}$ of this unique simple module actually serves as a global crystal basis, which coincides with the canonical basis introduced in [Reference Clark4] for the $m=1$ case.

For $\lambda =a \delta _1 + b \delta _2$ with $a, b\in \mathbb {Z}_{\ge 0}$ , let $V_{\mathfrak {l}_{1|1}}(\lambda )$ be the irreducible highest weight ${\mathcal U}(\mathfrak {l}_{1|1})$ -module with highest weight $\lambda $ . Then $V_{\mathfrak {l}_{1|1}}(\lambda )\otimes {\mathcal U}(\mathfrak {l}_{1|1})^-$ is a $B_q(\mathfrak {l}_{1|1})$ -module under the $\Bbbk $ -algebra homomorphism $\Delta _{B_q(\mathfrak {l}_{1|1})} : B_q(\mathfrak {l}_{1|1}) \longrightarrow {\mathcal U}(\mathfrak {l}_{1|1})\otimes B_q(\mathfrak {l}_{1|1})$ , which is defined by

$$ \begin{align*} \begin{aligned} \Delta_{B_q(\mathfrak{l}_{1|1})}(e') &= (q^{-1}-q) k e \otimes 1 + k \otimes e', \\ \Delta_{B_q(\mathfrak{l}_{1|1})}(f) &= f \otimes 1 + k \otimes f. \end{aligned} \end{align*} $$

Lemma 5.1. Let $V_1=V_{\mathfrak {l}_{1|1}}(\lambda )$ and $V_2={\mathcal U}(\mathfrak {l}_{1|1})^-$ with crystal bases $(L_1,B_1)$ and $({L}_2,{B}_2)$ respectively. Then $(L_1 \otimes L_2,B_1 \otimes B_2)$ is a crystal base of $V_1\otimes V_2$ , where $\tilde {e}$ and $\tilde {f}$ act on $B_1 \otimes B_2$ by

$$ \begin{align*} \begin{aligned} \tilde{e}(b_1\otimes b_2)&= \begin{cases} \tilde{e} b_1\otimes b_2, & \text{if } a+b>0, \\ b_1\otimes \tilde{e} b_2, & \text{if } a=b=0, \end{cases} \\\tilde{f}(b_1\otimes b_2)&= \begin{cases} \tilde{f} b_1\otimes b_2, & \text{if } a+b >0, \\ b_1\otimes \tilde{f} b_2, & \text{if } a=b=0. \end{cases} \end{aligned} \end{align*} $$

Proof. The proof for the case when $a=b=0$ is clear. Suppose that $a+b>0$ . Then we have $V_1=\Bbbk v_1 \oplus \Bbbk f v_1$ and it has a unique crystal base up to scalar multiplication. So we may assume that $L_1= A_0 v_1 \oplus A_0 f v_1$ and $L_2=A_0 v_2 \oplus A_0 fv_2$ with $v_2=1\in {\mathcal U}(\mathfrak {l}_{1|1})^-$ . We have

$$ \begin{align*}\begin{aligned} e' \cdot (v_1 \otimes v_2)=0,\quad f \cdot (v_1 \otimes v_2) = f v_1 \otimes v_2 + q^{a+b} v_1 \otimes f v_2 \equiv f v_1 \otimes v_2 \ (\mathrm{mod} \ qL_1\otimes L_2),\\e' \cdot X=0,\quad f \cdot X = fv_1 \otimes fv_2, \qquad\qquad\qquad\qquad\qquad\qquad\end{aligned}\end{align*} $$

where $X=(1-q^{2(a+b)}) v_1 \otimes f v_2 - q^{a+b} f v_1 \otimes v_2 \equiv v_1 \otimes f v_2 \ (\mathrm {mod} \ qL_1\otimes L_2)$ .

For $\lambda \in \mathtt {P}^+$ , consider $Q(\lambda )$ in (2.5). As a $\Bbbk $ -space, we have

(5.1) $$ \begin{align} Q(\lambda) \cong \mathcal{N}\otimes K(\lambda) \cong \mathcal{N}\otimes \mathcal{K}\otimes V_{\mathfrak{l}_{\overline 0}}(\lambda), \end{align} $$

where the second isomorphism is given by sending $u_1\otimes u_2 \otimes v$ to $u_1 \otimes u_2 v$ for $u_1\in \mathcal {N}$ , $u_2\in \mathcal {K}$ and $v\in V_{\mathfrak {l}_{\overline 0}}(\lambda )$ from the right to the left.

We define a crystal base of $Q(\lambda )$ as in Section 3.1.2 for $P(\lambda )$ , where $\tilde {e}_m, \tilde {f}_m$ are replaced by the ones in (4.3).

Theorem 5.3. For $\lambda \in \mathtt {P}^+$ , let

$$ \begin{align*} \begin{aligned} \mathscr{L}(Q(\lambda))&= \left(\mathscr{L}(\mathcal{N})\cdot \mathscr{L}(\mathcal{K})\right)\otimes \mathscr{L}(V_{\mathfrak{l}_{\overline 0}}(\lambda))\subset Q(\lambda),\\ \mathscr{B}(Q(\lambda))&=\left(\mathscr{B}(\mathcal{N})\cdot \mathscr{B}(\mathcal{K}) \right)\otimes \mathscr{B}(V_{\mathfrak{l}_{\overline 0}}(\lambda))\subset \mathscr{L}(Q(\lambda))/q\mathscr{L}(Q(\lambda)). \end{aligned} \end{align*} $$

where $\cdot $ denotes the multiplication in ${\mathcal U}(\mathfrak {g})^-$ and the induced one at $q=0$ , respectively. Then $(\mathscr {L}(Q(\lambda )), \mathscr {B}(Q(\lambda )))$ is a crystal base of $Q(\lambda )$ , and the crystal $\mathscr {B}(Q(\lambda ))$ is connected.

Proof. Put $\mathscr {L}=\mathscr {L}(Q(\lambda ))$ and $\mathscr {B}=\mathscr {B}(Q(\lambda ))$ . First we show that $(\mathscr {L},\mathscr {B})$ is a crystal base. It suffices to show that $\widetilde {x}_i \mathscr {L}\subset \mathscr {L}$ and $\widetilde {x}_i \mathscr {B}\subset \mathscr {B}\cup \{0\}$ for $i\in I$ and $x=e,f$ , which follows from arguments similar to those in [Reference Jang, Kwon and Uruno13, Theorem 4.22] when $i=0$ , and Lemma 5.2 when $i\in I\setminus \{m\}$ . If $i=m$ , it follows from Lemma 5.1, since $\mathcal {N}$ is a semisimple ${\mathcal U}(\mathfrak {l}_{1|1})$ -module and $\mathcal {K}$ is a semisimple $B_q(\mathfrak {l}_{1|1})$ -module.

Next, we show that $\mathscr {B}$ is connected. As a set, we have

(5.2) $$ \begin{align} \mathscr{B}\cong \mathscr{B}(\mathcal{N})\times \mathscr{B}(\mathcal{K}) \times \mathscr{B}(V_{\mathfrak{l}_{\overline 0}}(\lambda))=\mathscr{B}(\mathcal{N})\times \mathscr{B}(K(\lambda)). \end{align} $$

Note that the crystal graph $\mathscr {B}(K(\lambda ))$ is equal to that of $\mathscr {B}(K(\lambda +\ell \delta _+))$ (up to shifting weights by $\ell \delta _+$ ), where $\delta _+=\delta _1+\cdots +\delta _m$ for all $\ell \ge 0$ . Hence we may identify $\mathscr {B}$ with $\mathscr {B}(\mathcal {N})\times \mathscr {B}(K(\lambda +\ell \delta _+))$ (up to shifting weights by $\ell \delta _+$ ).

On the other hand, if $\ell $ is large enough, then $K(\lambda +\ell \delta _+)=V_{\mathfrak l}(\lambda +\ell \delta _+)$ , and $\mathscr {B}(K(\lambda +\ell \delta _+))=\mathscr {B}(V_{\mathfrak l}(\lambda +\ell \delta _+))$ . Since $\mathscr {B}(P(\lambda +\ell \delta _+))=\mathscr {B}(\mathcal {N})\times \mathscr {B}(V_{\mathfrak {l}}(\lambda +\ell \delta _+))$ is connected by Theorem 3.2, $\mathscr {B}$ is connected (see [Reference Kwon19, Section 6.4] and references therein).

The crystal structure on $\mathscr {B}(Q(\lambda ))$ can be described explicitly from (5.2). As an $\mathfrak {l}_{\overline 0}$ -crystal, we have

$$ \begin{align*} \mathscr{B}(Q(\lambda))\cong \mathscr{B}(\mathcal{N})\otimes \mathscr{B}(K(\lambda)), \end{align*} $$

while $\tilde {e}_0, \tilde {f}_0$ act only on $\mathscr {B}(\mathcal {N})$ , and the actions of $\tilde {e}_m, \tilde {f}_m$ are given by Lemma 5.1. A combinatorial description of $\mathscr {B}(\mathcal {N})$ and $\mathscr {B}(K(\lambda ))$ can be found in [Reference Jang, Kwon and Uruno13, Section 4.2] and [Reference Kwon19, Section 5.1], respectively.

5.2 Crystal base of $R(\lambda )$

Let $B_q=B_q({\mathfrak l}_{m|0})$ be the q-boson algebra associated with $\mathfrak {l}_{m|0}$ generated by $e^{\prime }_i$ and $f_i$ ( $i=1,\dots ,m-1$ ) [Reference Kashiwara14, Section 3.3]. Then ${\mathcal U}(\mathfrak {l}_{m|0})^-$ is the unique irreducible $B_q$ -module, where $e^{\prime }_i$ acts as in (2.1) and $f_i$ acts as left multiplication.

For $\lambda \in \mathtt {P}^+$ with $\lambda =\lambda _+$ , let $V_{\mathfrak {l}_{m|0}}(\lambda )$ be the irreducible highest weight ${\mathcal U}(\mathfrak {l}_{m|0})$ -module with highest weight $\lambda $ . Then $V_{\mathfrak {l}_{m|0}}(\lambda )\otimes {\mathcal U}(\mathfrak {l}_{m|0})^-$ is a $B_q$ -module under the $\Bbbk $ -algebra homomorphism $\Delta _{B_q} : B_q \longrightarrow {\mathcal U}(\mathfrak {l}_{m|0})\otimes B_q$ given by

$$ \begin{align*} \Delta_{B_q}(e_i') = (q_i^{-1}-q_i) k_ie_i \otimes 1 + k_i \otimes e_i', \quad \Delta_{B_q}(f_i) = f_i \otimes 1 + k_i \otimes f_i. \end{align*} $$

Let $V_1=V_{\mathfrak {l}_{m|0}}(\lambda )$ and $V_2={\mathcal U}(\mathfrak {l}_{0|n})^-$ with crystal bases $(L_1,B_1)$ and $({L}_2,{B}_2)$ , respectively. Then $(L_1 \otimes L_2,B_1 \otimes B_2)$ is a crystal base of $V_1\otimes V_2$ , where $\tilde {e}_i$ and $\tilde {f}_i\ (i \in I)$ act on $B_1 \otimes B_2$ by the same formula as in the case of a tensor product of crystal bases of ${\mathcal U}(\mathfrak {l}_{m|0})$ -modules [Reference Kashiwara14, Remark 3.5.1].

For $\lambda \in \mathtt {P}$ with $\lambda _-\in \mathtt {P}^+$ , consider $R(\lambda )$ in (2.6). As a $\Bbbk $ -space, we have

(5.3) $$ \begin{align} R(\lambda) \cong \mathcal{N}\otimes {\mathcal K}\otimes {\mathcal U}(\mathfrak{l}_{m|0})^-\otimes V_{\mathfrak{l}_{0|n}}(\lambda), \end{align} $$

where the isomorphism is given in a standard way (cf. (5.1)).

We define a crystal base of $R(\lambda )$ as in $Q(\lambda )$ , where $\tilde {e}_i, \tilde {f}_i$ for $i=1,\dots ,m-1$ are replaced by those defined as in (3.2) with respect to $e^{\prime }_i$ .

Theorem 5.6. For $\lambda \in \mathtt {P}$ with $\lambda _-\in \mathtt {P}^+$ , let

$$ \begin{align*} \begin{aligned} \mathscr{L}(R(\lambda))&= \left(\mathscr{L}(\mathcal{N})\cdot \mathscr{L}({\mathcal K})\cdot \mathscr{L}({\mathcal U}(\mathfrak{l}_{m|0})^-)\right)\otimes \mathscr{L}(V_{\mathfrak{l}_{0|n}}(\lambda))\subset R(\lambda),\\ \mathscr{B}(R(\lambda))&=\left(\mathscr{B}(\mathcal{N})\cdot \mathscr{B}({\mathcal K}) \cdot \mathscr{B}({\mathcal U}(\mathfrak{l}_{m|0})^-)\right) \otimes \mathscr{B}(V_{\mathfrak{l}_{0|n}}(\lambda))\subset \mathscr{L}(R(\lambda))/q\mathscr{L}(R(\lambda)). \end{aligned} \end{align*} $$

Then $(\mathscr {L}(R(\lambda )), \mathscr {B}(R(\lambda )))$ is a crystal base of $R(\lambda )$ , where the crystal $\mathscr {B}(R(\lambda ))$ is connected.

As a set, we have

(5.4) $$ \begin{align} \mathscr{B}(R(\lambda))\cong \mathscr{B}(\mathcal{N})\times \mathscr{B}({\mathcal K}) \times \mathscr{B}({\mathcal U}(\mathfrak{l}_{m|0})^-) \times \mathscr{B}(V_{\mathfrak{l}_{0|n}}(\lambda)). \end{align} $$

Recall that $\tilde {e}_0, \tilde {f}_0$ act only on $\mathscr {B}(\mathcal {N})$ . The actions of $\tilde {e}_i, \tilde {f}_i$ for $i\in I\setminus \{0\}$ can be described by applying them to

(5.5) $$ \begin{align} \begin{cases} \mathscr{B}(\mathcal{N})\otimes \mathscr{B}({\mathcal K}) \otimes \mathscr{B}(V_{\mathfrak{l}_{0|n}}(\lambda)) & \text{using}\ [14,\ \text{Theorem 1] for }i>m,\\ \mathscr{B}(\mathcal{N})\otimes \mathscr{B}({\mathcal K}) \otimes \mathscr{B}({\mathcal U}(\mathfrak{l}_{m|0})^-) & \text{using}\ [15,\ \text{Remark 3.5.1] for }i<m,\\ \mathscr{B}(\mathcal{N})\otimes \mathscr{B}({\mathcal K}) & \text{using Lemma}\ 5.1\ \text{for }i=m. \end{cases} \end{align} $$

5.3 Crystal base of ${\mathcal U}(\mathfrak {g})^-$

We define a crystal base of ${\mathcal U}(\mathfrak {g})^-$ as in $R(\lambda )$ except for $\tilde {e}_i$ and $ \tilde {f}_i$ for $i>m$ , which are defined by

(5.6) $$ \begin{align} \tilde{e}_i u = u_1(\tilde{e}_iu_2),\quad \tilde{f}_i u = u_1(\tilde{f}_iu_2), \end{align} $$

for $u=u_1u_2\in {\mathcal U}(\mathfrak {g})^-$ with $u_1\in \mathcal {N}$ and $u_2\in {\mathcal U}(\mathfrak l)^-$ , where $\tilde {e}_iu_2$ and $\tilde {f}_iu_2$ are given in (4.7).

As a set, let

$$ \begin{align*} \mathscr{B}(\mathcal{N}) \cdot \mathscr{B}({\mathcal U}(\mathfrak{l})^-)=\mathscr{B}(\mathcal{N}) \times \mathscr{B}({\mathcal U}(\mathfrak{l})^-). \end{align*} $$

We define a ${\mathcal U}(\mathfrak {g})$ -crystal structure on this set, which is intended to be the crystal of ${\mathcal U}(\mathfrak {g})^-$ , as follows: Let $b=(S,b_0)$ be given. We let $\mathrm {wt}(b) = \mathrm {wt}(S) + \mathrm {wt}(b_0)$ , and let $\varepsilon _i(b)$ and $\varphi _i(b)$ be as in (4.5). For $i\in I$ and $x=e,f$ , we define

$$ \begin{align*} \begin{aligned} \tilde{x}_i b &= \begin{cases} (\tilde{x}_0S,b_0) & \text{if }i=0,\\ (S',b^{\prime}_0) & \text{if }1\le i\le m\text{ and }\tilde{x}_i (S\otimes b_0)= S'\otimes b^{\prime}_0,\\ (S,\tilde{x}_ib_0) & \text{if }i>m,\\ \end{cases} \end{aligned} \end{align*} $$

where $\tilde {x}_i (S\otimes b_0)$ is defined by using [Reference Kashiwara14, Remark 3.5.1] for $1\le i<m$ and using Lemma 5.1 for $i=m$ . Here we assume that $\tilde {x}_i b=\mathbf {0}$ if any of its components on the right-hand side is $\mathbf {0}$ .

Theorem 5.8. Let

$$ \begin{align*} \begin{aligned} \mathscr{L}(\infty) & = \mathscr{L}(\mathcal{N})\cdot \mathscr{L}({\mathcal U}(\mathfrak l)^-), \ \ \mathscr{B}(\infty) = \mathscr{B}(\mathcal{N})\cdot \mathscr{B}({\mathcal U}(\mathfrak l)^-). \end{aligned} \end{align*} $$

Then $(\mathscr {L}(\infty ),\mathscr {B}(\infty ))$ is a crystal base of ${\mathcal U}(\mathfrak {g})^-$ .

Example 5.9. Let $\mathfrak {g}=\mathfrak {d}_{2|2}$ . Then, we have

$$ \begin{align*} \mathscr{B}(\infty)\cong \mathscr{B}(\mathcal{N}) \times \mathscr{B}({\mathcal U}(\mathfrak{l})^-) \cong (\mathscr{B}(\mathcal{N}) \times \mathscr{B}({\mathcal K}) \times \mathscr{B}({\mathcal U}(\mathfrak{l}_{2|0})^-)) \times \mathscr{B}({\mathcal U}(\mathfrak{l}_{0|2})^-) \end{align*} $$

as a ( $\mathfrak {g}_{2|1}$ , $\mathfrak {g}_{0|2}$ )-bicrystal. For $\tilde {x}=\tilde {e}$ or $\tilde {f}$ , it follows from (5.6) that $\tilde {x}_3$ acts exclusively on $\mathscr {B}({\mathcal U}(\mathfrak {l}_{0|2})^-)$ and commutes with $\tilde {x}_i$ ’s for $0 \leq i \leq 2$ . Since $\mathscr {B}({\mathcal U}(\mathfrak {l}_{0|2})^-) $ is isomorphic to $ \mathscr {B}(U_q^-(\mathfrak {gl}_2))$ , the crystal graph of $\mathscr {B}({\mathcal U}(\mathfrak {l}_{0|2})^-)$ is given as follows, where each arrow denotes $\tilde {f}_3$ :

On the other hand, the operators $\tilde {x}_i$ for $0 \leq i \leq 2$ act on $\mathscr {B}(\mathcal {N}) \otimes \mathscr {B}({\mathcal K}) \otimes \mathscr {B}({\mathcal U}(\mathfrak {l}_{2|0})^-)$ according to the two cases for $i\leq m$ given in (5.5), and this crystal is disconnected (as is the case for general $n \geq 2$ ). The figure below illustrates the connected component containing $1 \otimes 1 \otimes 1 \in \mathscr {B}(\mathcal {N}) \otimes \mathscr {B}({\mathcal K}) \otimes \mathscr {B}({\mathcal U}(\mathfrak {l}_{2|0})^-)$ (with $\tilde {x}_3$ omitted). Here, solid, dashed, and dotted arrows represent $\tilde {f}_0$ , $\tilde {f}_1$ , and $\tilde {f}_2$ , respectively:

For $\lambda \in \mathtt {P}$ with $\lambda _-\in \mathtt {P}^+$ , let

given by $\mathrm {id}_{\mathcal {N}}\otimes \pi _{X(\lambda )}^\vee $ (see (4.9) for $\pi _{X(\lambda )}^\vee $ ).

Summarizing, the crystal bases constructed in Section 5 are compatible with the following maps (partly compatible in the case of $\pi ^\vee _{R(\lambda )}$ ):

This induces the following commuting diagram at $q=0$ :

Since the maps in the bottom line are obtained from (4.13) by tensoring with $\mathrm {id}_{\mathscr {B}(\mathcal {N})}$ , it is enough to describe

(5.7) $$ \begin{align} \iota_{V(\lambda,\ell)}: \mathscr{B}(V(\lambda,\ell)) \longrightarrow \mathscr{B}(P(\lambda)). \end{align} $$

6.1 Crystal $\mathscr {B}(V_{\mathfrak l}(\lambda ))$

Let us recall the results on the crystal $\mathscr {B}(V_{\mathfrak l}(\lambda ))$ for $\lambda \in \mathscr {P}_{m|n}$ [Reference Benkart, Kang and Kashiwara1].

Let $\mathcal {A}$ be a $\mathbb {Z}_2$ -graded linearly ordered set. For a skew Young diagram $\lambda /\mu $ , we denote by $SST_{\mathcal {A}}(\lambda /\mu )$ the set of all $\mathcal {A}$ -semistandard tableaux of shape $\lambda /\mu $ with entries in $\mathcal {A}$ . We write $SST_{\mathcal {A}}(\lambda /\mu )$ as $SST_{m|n}(\lambda /\mu )$ when $\mathcal {A} = \mathbb {I}$ . For $T \in SST_{\mathcal {A}}(\lambda /\mu )$ , let $\mathrm {sh}(T)=\lambda /\mu $ denote the shape of T, and let $w(T)$ be the word of T, which is the word with letters in $\mathcal {A}$ obtained from T by reading its entries column by column from the right to the left, and from top to bottom in each column.

We may identify $\mathbb {I} =SST_{m|n}((1))$ , a crystal of $V_{\mathfrak l}((1))$ . Then $SST_{m|n}(\lambda /\mu )$ has a structure of an (abstract) ${\mathcal U}(\mathfrak {l})$ -crystal (or simply $\mathfrak l$ -crystal) under the identification of $T\in SST_{m|n}(\lambda /\mu )$ with $w(T)$ , where we regard $w(T)$ as an element in $SST_{m|n}((1))^{\otimes |\lambda /\mu |}$ ( $|\lambda /\mu |$ is the number of boxes in $\lambda /\mu $ ) and apply the tensor product rule. Then we have $SST_{m|n}(\lambda )\cong \mathscr {B}(V_{\mathfrak l}(\lambda ))$ for $\lambda \in \mathscr {P}_{m|n}$ .

For a skew Young diagram $\lambda /\mu $ , let $(\lambda /\mu )^\pi $ be the skew Young diagram obtained by rotating $\lambda /\mu $ by $180^\circ $ . We also write $\lambda ^\pi /\mu ^\pi =(\lambda /\mu )^\pi $ . For $T\in SST_{m|n}(\delta ^\pi )$ with $\delta \in \mathscr {P}_{m|n}$ and $i\in \mathbb {I}$ , we denote by $(T\leftarrow i)$ the tableau obtained by applying Schensted’s column insertion of i into T in a reverse way starting from the rightmost column of T so that $(T\leftarrow i)\in SST_{m|n}(\gamma ^\pi )$ for some $\gamma \supset \delta $ , which is given by adding a box in a corner of $\delta $ .

6.2 Crystal $\mathscr {B}(V(\lambda ,\ell ))$

Let us recall the results on the realization of the crystal $\mathscr {B}(V(\lambda ,\ell ))$ in [Reference Kwon20, Reference Kwon21]. Here we follow the convention in [Reference Kwon23, Section 3], which is slightly different from that in [Reference Kwon20, Reference Kwon21].

First we assume the following notations:

• • $\mathbf {T}^{\mathfrak {g}}(a)$ for $a\in \mathbb {Z}_{\ge 0}$ ,

• • $\mathbf {T}^{\texttt {sp}}$ for $\mathfrak {g} = \mathfrak {b}_{m|n}$ and $\mathfrak {d}_{m|n}$ ,

• • $\overline {\mathbf {T}}(0)$ , $\mathbf {T}^{\texttt {sp}+}$ , $\mathbf {T}^{\texttt {sp}-}$ for $\mathfrak {g}=\mathfrak {d}_{m|n}$ .

For precise definitions, we refer the reader to [Reference Kwon20, Definition 6.5] for the types $\mathfrak {b}_{m|n}$ and $\mathfrak {c}_{m|n}$ , and to [Reference Kwon21, Definition 3.1] for the type $\mathfrak {d}_{m|n}$ , where the only difference is that $\mathbb {N}$ -semistandardness is replaced by $\mathbb {I}$ -semistandardness. For example, $\mathbf {T}^{\mathfrak {c}_{m|n}}(a)=\bigsqcup _{c\ge 0}SST_{m|n}\left (\lambda (a,0,c)\right )$ , where $\lambda (a, b, c) = (2^{b+c}, 1^a) / (1^b)$ , while $\mathbf {T}^{\mathfrak {b}_{m|n}}(a)$ and $\mathbf {T}^{\mathfrak {d}_{m|n}}(a)$ consist of $T\in SST_{m|n}\left (\lambda (a,b,c)\right )$ ( $a,b\in \mathbb {Z}_{\ge 0}$ ) satisfying certain mild constraints. When drawing tableaux, we often assume that a horizontal line L is given such that any box in a tableau T is either below or above L, and denote by $T^{\texttt {body}}$ and $T^{\texttt {tail}}$ the subtableaux of T placed above and below L, respectively (see Example 6.2).

Let $\mathbf {B}$ be one of the sets above, and we regard it as an (abstract) $\mathfrak {g}$ -crystal such that the $\mathfrak {l}$ -crystal structure is given as in Section 6.1, and $\tilde {e}_0$ , $\tilde {f}_0$ are defined as in [Reference Kwon23, Section 3.1], where for $T \in \mathbf {B}$ ,

$$ \begin{align*} \begin{aligned} \mathrm{wt}(T) &= \begin{cases} 2\Lambda_0 + \mathrm{wt}_{\mathfrak{l}}(T) & \text{if }(\mathbf{B} = \mathbf{T}^{\mathfrak{g}}(a)\text{ with }\mathfrak{g} = \mathfrak{b}_{m|n}) \\ & \text{or }(\mathbf{B} = \mathbf{T}^{\mathfrak{g}}(a),\, \overline{\mathbf{T}}(0)\text{ with }\mathfrak{g} = \mathfrak{d}_{m|n}), \\ \Lambda_0 + \mathrm{wt}_{\mathfrak{l}}(T) & \text{otherwise,} \end{cases} \\ \varepsilon_i(T) &= \max\left\{ k \, | \, \tilde{e}_i^k T \neq \mathbf{0} \right\}, \qquad \varphi_i(T) = \max\left\{ k \, | \, \tilde{f}_i^k T \neq \mathbf{0} \right\}, \end{aligned} \end{align*} $$

and $\mathrm {wt}_{\mathfrak {l}}(T)$ denotes the $\mathfrak {l}$ -weight of $T \in \mathbf {B}$ . Indeed, they are crystals of $V(\lambda ,\ell )$ as follows:

• • when $\mathfrak {g} = \mathfrak {c}_{m|n}$ , $ \mathbf {T}^{\mathfrak {c}_{m|n}}(a) \cong \mathscr {B}\left ( V((1^a),1) \right ), $

• • when $\mathfrak {g} = \mathfrak {b}_{m|n}$ , $ \mathbf {T}^{\mathfrak {b}_{m|n}}(a) \cong \mathscr {B}\left ( V((1^a),2) \right ) $ and $ \mathbf {T}^{\texttt {sp}} \cong \mathscr {B}\left ( V((0), 1) \right ), $

• • when $\mathfrak {g} = \mathfrak {d}_{m|n}$ ,

  $$ \begin{align*} \begin{cases} \mathbf{T}^{\mathfrak{d}_{m|n}}(a) \cong \mathscr{B}\left( V((1^a),2) \right), \\ \overline{\mathbf{T}}(0) \cong \mathscr{B}\left( V((2), 2) \right), \end{cases} \quad \begin{cases} \mathbf{T}^{\texttt{sp}+} \cong \mathscr{B}\left( V((0),1) \right), \\ \mathbf{T}^{\texttt{sp}-} \cong \mathscr{B}\left( V((1),1) \right). \end{cases} \end{align*} $$

Here $(0)$ denotes the empty partition (that is, $(0,0,\dots )$ ) and $(1^0)$ is understood as $(0)$ .

For a partition $\lambda =(\lambda _i)_{i \ge 1}$ , let $\lambda '=(\lambda ^{\prime }_i)_{i \ge 1}$ be its conjugate. For $\mathfrak {g}=\mathfrak {d}_{m|n}$ and given $(\lambda , \ell ) \in \mathscr {P}(\mathfrak {d}_{m|n})$ , let $\overline {\lambda } = (\overline {\lambda }_i)_{i \ge 1}$ be given by $\overline {\lambda }_1 = \ell - \lambda _1$ and $\overline {\lambda }_i = \lambda _i$ for $i \ge 2$ . Let $\mu = \lambda '$ , $\overline {\mu } = \left ( \overline {\lambda } \right )'$ , $M_+ = \lambda _1$ , $M_- = \overline {\lambda }_1$ , and let $q_{\pm }$ and $r_{\pm }\in \left \{ 0, 1 \right \}$ be such that

$$ \begin{align*} \begin{cases} \ell - 2\lambda_1 = 2q_+ + r_+ & \text{if }\ell - 2\lambda_1 \ge 0, \\ 2\lambda_1 - \ell = 2q_- + r_- & \text{if }\ell - 2\lambda_1 < 0, \end{cases} \end{align*} $$

We define $\widehat {\mathbf {T}}^{\mathfrak {g}}(\lambda , \ell )$ as follows:

• • when $\mathfrak {g} = \mathfrak {b}_{m|n}$ or $\mathfrak {c}_{m|n}$ ,

  $$ \begin{align*} \widehat{\mathbf{T}}^{\mathfrak{g}}(\lambda, \ell) = \begin{cases} \mathbf{T}^{\mathfrak{g}}(\lambda^{\prime}_1) \times \dots \times \mathbf{T}^{\mathfrak{g}}(\lambda^{\prime}_\ell) & \text{if }\mathfrak{g} = \mathfrak{c}_{m|n}, \\ \mathbf{T}^{\mathfrak{g}}(\lambda^{\prime}_1) \times \dots \times \mathbf{T}^{\mathfrak{g}}(\lambda^{\prime}_{\ell/2}) & \text{if }\mathfrak{g} = \mathfrak{b}_{m|n}\text{ and }\ell - 2\lambda_1\text{ is even,} \\ \mathbf{T}^{\mathfrak{g}}(\lambda^{\prime}_1) \times \dots \times \mathbf{T}^{\mathfrak{g}}(\lambda^{\prime}_{(\ell-1)/2}) \times \mathbf{T}^{\texttt{sp}} & \text{if }\mathfrak{g} = \mathfrak{b}_{m|n}\text{ and }\ell - 2\lambda_1\text{ is odd,} \\ \end{cases} \end{align*} $$

• • when $\mathfrak {g} = \mathfrak {d}_{m|n}$ ,

  $$ \begin{align*} \widehat{\mathbf{T}}^{\mathfrak{g}}(\lambda, \ell) = \begin{cases} \mathbf{T}^{\mathfrak{g}}(\mu_1) \times \dots \times \mathbf{T}^{\mathfrak{g}}(\mu_{M_+}) \times \left( \mathbf{T}(0) \right)^{q_+} \times \left( \mathbf{ T}^{\texttt{sp}+} \right)^{r_+} & \text{if }\ell - 2\lambda_1 \ge 0, \\ \mathbf{T}^{\mathfrak{g}}(\overline{\mu}_1) \times \dots \times \mathbf{T}^{\mathfrak{g}}(\overline{\mu}_{M_-}) \times \left( \overline{\mathbf{T}}(0) \right)^{q_-} \times \left( \mathbf{T}^{\texttt{sp}-} \right)^{r_-} & \text{if }\ell - 2\lambda_1 < 0, \end{cases} \end{align*} $$

where one may define a $\mathfrak {g}$ -crystal structure on $\widehat {\mathbf {T}}^{\mathfrak {g}}(\lambda , \ell )$ by identifying $(\dots , T_2, T_1)$ with $T_1 \otimes T_2 \otimes \dots $ . Let

$$ \begin{align*} \mathbf{T}^{\mathfrak{g}}(\lambda, \ell) = \left\{ \left. (\dots, T_2, T_1) \in \widehat{\mathbf{T}}^{\mathfrak{g}}(\lambda, \ell) \, \right| \, T_{i+1} \prec T_i \text{ for }i \ge 1 \right\}, \end{align*} $$

(see [Reference Kwon23, Definition 3.2] for the definition of $T_{i+1} \prec T_i$ ). Then $\mathbf {T}^{\mathfrak {g}}(\lambda , \ell )$ is a $\mathfrak {g}$ -subcrystal of $\widehat {\mathbf {T}}^{\mathfrak {g}}(\lambda , \ell )$ , and it is shown in [Reference Kwon20, Reference Kwon21] that

$$ \begin{align*} \mathbf{T}^{\mathfrak{g}}(\lambda, \ell) \cong \mathscr{B}(V(\lambda,\ell)). \end{align*} $$

Example 6.2. (1) Let $\mathfrak {g}=\mathfrak {c}_{2|5}$ and take $\lambda = (3, 2, 1, 1) \in \mathscr {P}_{2|5}$ and $\ell = 3$ . Since $\ell - \lambda _1= 0$ , we know that $(\lambda , \ell ) \in \mathscr {P}(\mathfrak {c}_{2|5})$ . Let $\mathbf {T}=(T_3,T_2,T_1) \in \mathbf {T}^{\mathfrak {g}}(4) \times \mathbf {T}^{\mathfrak {g}}(2) \times \mathbf {T}^{\mathfrak {g}}(1)$ be given by

where the tableaux $T_1,\cdots ,T_3$ are placed on the plane with a horizontal line L drawn as the dotted line. In fact, the line L is determined by $\lambda '$ , in the sense that the length of $T_i^{\mathrm {tail}}$ is $\lambda _i'$ . Let us check that $T_3 \prec T_2$ . We have

where $T_i^{\mathtt {L}}$ and $T_i^{\mathtt {R}}$ denote the left and right columns of $T_i$ , respectively, and ${}^{\mathtt {L}} T_i$ (resp. ${}^{\mathtt {R}} T_i$ ) is the left (resp. right) column of the tableau obtained from $T_i$ by applying $\lambda _i'$ jeu de taquin slidings into the position below the bottom of $T_i^{\mathtt {L}}$ (resp. $T_i^{\mathtt {R}}$ ) (see [Reference Kwon23, Section 3.2] for more details). Then $({}^{\mathtt {R}}T_3,T^{\mathtt {L}}_2)$ (in blue) and $(T^{\mathtt {R}}_3,{}^{\mathtt {L}}T_2)$ (in red) form semistandard tableaux as shown below:

This precisely corresponds to the definition of $T_3 \prec T_2$ for type $\mathfrak {c}_{m|n}$ (and similarly for type $\mathfrak {b}_{m|n}$ ). In a similar way, we can also check that $T_2 \prec T_1$ , and hence $\mathbf {T} \in \mathbf {T}^{\mathfrak {g}}(\lambda ,\ell )$ .

(2) Let $\mathfrak {g} = \mathfrak {d}_{4|4}$ . Take $\lambda = (4, 4, 2) \in \mathscr {P}_{4|4} $ and $\ell = 8$ so that $\ell - 2\lambda _1 = 0$ and $\mu = \lambda ' = (3, 3, 2, 2)$ . Since $\ell - \lambda _1 - \lambda _2 = 0$ and $\ell - 2\lambda _1 = 0$ , we know that $(\lambda , \ell ) \in \mathscr {P}(\mathfrak {d}_{4|4})$ and $q_+ = r_+ = 0$ . Let $\mathbf {T} = (T_4, T_3, T_2, T_1) \in \mathbf {T}^{\mathfrak {g}}(3) \times \mathbf {T}^{\mathfrak {g}}(3) \times \mathbf {T}^{\mathfrak {g}}(2) \times \mathbf {T}^{\mathfrak {g}}(2)$ be given by

One can check that $T_4 \prec T_3 \prec T_2 \prec T_1$ , and hence $\mathbf {T} \in \mathbf {T}^{\mathfrak {g}}(\lambda ,\ell )$ . We remark that the definition of admissibility for type $\mathfrak {d}_{m|n}$ is more involved than for other types. See [Reference Kwon23, Section 3.2] for more details. Note that $T_4 \ntriangleleft T_3 \triangleleft T_2 \triangleleft T_1$ , where the definition of $\triangleleft $ is given in [Reference Jang and Kwon9, Definition 3.5].

6.3 Crystal $\mathscr {B}(P(\lambda ))$

Let us give some comments on the combinatorial realization of $\mathscr {B}(P(\lambda ))$ . By (3.4), it suffices to describe $\mathscr {B}(\mathcal {N})$ . We may regard $\mathscr {B}(\mathcal {N})= M(\mathfrak {u})$ by Theorem 3.3, and also identify it with the set $\mathbf {M}^{\mathfrak {g}}$ of upper-triangular matrices $\mathbf {c}=\left (c_{ij}\right )_{i, j \in \mathbb {I}, i \le j}$ such that

$$ \begin{align*} c_{ij} \in \begin{cases} \mathbb{Z}_{\ge 0} & \text{if }\epsilon_i = \epsilon_j, \\ \left\{ 0, 1 \right\} & \text{if }\epsilon_i \neq \epsilon_j, \end{cases} \quad \text{ and } \quad c_{ii} = \begin{cases} 0 & \text{if }\mathfrak{g} = \mathfrak{c}_{m|n}\text{ and }i \in \mathbb{I}_{\overline{1}}, \\ 0 & \text{if }\mathfrak{g} = \mathfrak{d}_{m|n}\text{ and }i \in \mathbb{I}_{\overline{0}}, \end{cases} \end{align*} $$

where $c_{ij}$ is equal to the multiplicity $c_\beta =c_{(i,j)}$ of $\mathbf {F}_\beta $ with $\beta $ corresponding to $(i,j)$ following (2.2). For example, when $\mathfrak {g} = \mathfrak {b}_{2|3}$ ,