2. Gradings and Koszul duality

We first recall some well-known results about standard Koszul graded algebras and Koszul duality.

Let $\hat A = \bigoplus _{i \in \mathbb{Z}} A_i$ be a finite-dimensional graded algebra over a field $\Bbbk$ , with underlying (ungraded) algebra $A$ , and let $\mathcal{C}=A\text{-}\textrm{mod}$ and $\mathcal{C}_{\textrm{gr}} = \hat A\text{-}\textrm{grmod}$ be the categories of finite-dimensional left $A$ -modules and of finite-dimensional graded left $\hat A$ -modules, respectively. In the following, we simply refer to the objects of $\mathcal{C}$ as $A$ -modules and to the objects of $\mathcal{C}_{\textrm{gr}}$ as graded $\hat A$ -modules. A graded $\hat A$ -module $M = \bigoplus _i M_i$ is called pure of degree $d \in \mathbb{Z}$ if $M=M_d$ . For $m \in \mathbb{Z}$ , let us further denote by $\langle m \rangle \colon \mathcal{C}_{\textrm{gr}} \to \mathcal{C}_{\textrm{gr}}$ the $m$ -th grading shift functor, which sends a graded $\hat A$ -module $M = \bigoplus _i M_i$ to the graded $\hat A$ -module $M\langle m \rangle = M$ , with the grading defined by $M\langle m \rangle _i = M_{i-m}$ , and let $\textrm{f} \colon \mathcal{C}_{\textrm{gr}} \to \mathcal{C}$ be the functor that sends a graded $\hat A$ -module to the underlying (ungraded) $A$ -module. We call a graded $\hat A$ -module $\tilde M$ a graded lift of an $A$ -module $M$ if $M \cong \textrm{f}(\tilde M)$ . Now suppose additionally that $\hat A$ is positively graded, i.e. that $A_i = 0$ for all $i\lt 0$ , and that $A_0$ is a semisimple $\Bbbk$ -algebra. Observe that $A_{\gt i} = \bigoplus _{j\gt i} A_j$ is an ideal in $A$ for any $i \geq 0$ and that we can consider $A_0$ as an $A$ -module via the identification $A_0 \cong A/ A_{\gt 0}$ .

Definition 1.1. We say that $\hat A$ is Koszul if for all $i \in \mathbb{Z}_{\geq 0}$ and $j \in \mathbb{Z}$ with $i \neq j$ , we have

\begin{equation*} \textrm {Ext}_{\mathcal {C}_{\textrm {gr}}}^i\big ( A_0, A_0 \langle j \rangle \big ) = 0. \end{equation*}

In that case, we also say that $A$ admits a Koszul grading.

Consider the $\textrm{Ext}$ -algebra

\begin{equation*} E(\hat A)\, :\!= \,\bigoplus _{i \in \mathbb {Z}} \textrm {Ext}_{\mathcal {C}}^i(A_0,A_0), \end{equation*}

with the natural grading defined by $E(\hat A)_i = \textrm{Ext}_{\mathcal{C}}^i(A_0,A_0)$ for $i \in \mathbb{Z}$ . If $\hat A$ is Koszul then so is $E(\hat A)$ , and there is a canonical isomorphism of graded algebras $\hat A \cong E(E(\hat A))$ by Proposition 2.9.1 and Theorems 2.10.1 and 2.10.2 in [Reference Beilinson, Ginzburg and Soergel4]. When $\hat{A}$ is Koszul, we call the graded algebra $\hat A^! = E(\hat A)^{\textrm{op}}$ the Koszul dual of $\hat{A}$ ; it is Koszul because the opposite algebra of a Koszul graded algebra is Koszul by Proposition 2.2.1 in [Reference Beilinson, Ginzburg and Soergel4]. We also write $A^!$ for the underlying (ungraded) algebra of $\hat A^!$ .Footnote ¹ Now additionally suppose that $A$ has finite global dimension, so that $E(A)$ is finite-dimensional. Then according to Theorems 2.12.5 and 2.12.6 in [Reference Beilinson, Ginzburg and Soergel4], there is an equivalence of triangulated categories

\begin{equation*} K \colon D^b( \mathcal {C}_{\textrm {gr}} ) \longrightarrow D^b( \hat A^!\text {-}\textrm {grmod} ) \end{equation*}

such that $K( M \langle i \rangle ) \cong K(M) \langle -i \rangle [{-}i ]$ for every graded $\hat A$ -module $M$ .

Now suppose that $\mathcal{C}$ is a highest weight category (see Definition 3.1 in [Reference Cline, Parshall and Scott9]) with finite weight poset $(\Lambda,\leq )$ and with simple objects $L_\lambda$ , standard objects $\Delta _\lambda$ and costandard objects $\nabla _\lambda$ , for $\lambda \in \Lambda$ . Let us further write $I_\lambda$ and $P_\lambda$ for the injective hull and the projective cover of $L_\lambda$ , let $P = \bigoplus _{\lambda \in \Lambda } P_\lambda$ be a projective generator of $\mathcal{C}$ and set $A\, :\!= \,\textrm{End}_{\mathcal{C}}(P)^{\textrm{op}}$ , so that $\mathcal{C}$ is equivalent to $A\text{-}\textrm{mod}$ via the functor $\textrm{Hom}_{\mathcal{C}}(P,-)$ . In the following, we suppress the functor $\textrm{Hom}_{\mathcal{C}}(P,-)$ from the notation and simply treat objects of $\mathcal{C}$ as $A$ -modules. From now on, additionally suppose that $A$ admits a positive grading $\hat{A} = \bigoplus _{i \in \mathbb{Z}} A_i$ such that $A_0$ is a semisimple $\Bbbk$ -algebra. Then, up to an isomorphism of graded $\hat A$ -modules, any simple object $L_\lambda$ of highest weight $\lambda \in \Lambda$ in $\mathcal{C}$ admits a unique graded lift $\hat L_\lambda$ that is pure of degree $0$ , and the objects $\Delta _\lambda$ , $\nabla _\lambda$ , $P_\lambda$ , and $I_\lambda$ of $\mathcal{C}$ admit unique graded lifts $\hat \Delta _\lambda$ , $\hat \nabla _\lambda$ , $\hat P_\lambda$ and $\hat I_\lambda$ such that all non-zero homomorphisms $\Delta _\lambda \to L_\lambda$ , $L_\lambda \to \nabla _\lambda$ , $P_\lambda \to L_\lambda$ , and $L_\lambda \to I_\lambda$ are homomorphisms of graded $\hat A$ -modules. (See Corollary 4 and the introduction to Section 5 in [Reference Mazorchuk and Ovsienko21] for the existence and Lemma 2.5.1 in [Reference Beilinson, Ginzburg and Soergel4] for the uniqueness of these graded lifts.)

Definition 1.2. We say that $\hat A$ is standard Koszul if for all $\lambda,\mu \in \Lambda$ and all $i \in \mathbb{Z}_{\geq 0}$ and $j \in \mathbb{Z}$ with $i \neq j$ , we have

\begin{equation*} \textrm {Ext}_{\mathcal {C}_{\textrm {gr}}}^i\big ( \hat \Delta _\lambda, \hat L_\mu \langle j \rangle \big ) = 0 \qquad \text {and} \qquad \textrm {Ext}_{\mathcal {C}_{\textrm {gr}}}^i\big ( \hat L_\mu \langle -j \rangle, \hat \nabla _\mu \big ) = 0. \end{equation*}

In that case, we also say that $A$ admits a standard Koszul grading.

If $\hat{A}$ is standard Koszul then $\hat{A}$ is Koszul, $\hat{A}^!$ is standard Koszul and the category $\mathcal{C}^! = A^!\text{-}\textrm{mod}$ is a highest weight category with weight poset $(\Lambda,\leq ^{\textrm{op}})$ by Theorems 1, 2, and 3 in [Reference Ágoston, Dlab and Lukács1]. We denote the simple, standard, costandard, indecomposable projective, and indecomposable injective object of $\mathcal{C}^!$ of highest weight $\lambda \in \Lambda$ by

\begin{equation*} L_\lambda ^!, \qquad \Delta _\lambda ^!, \qquad \nabla _\lambda ^!, \qquad P_\lambda ^!, \qquad I_\lambda ^!, \end{equation*}

respectively, and we write

\begin{equation*} \hat L_\lambda ^!, \qquad \hat \Delta _\lambda ^!, \qquad \hat \nabla _\lambda ^!, \qquad \hat P_\lambda ^!, \qquad \hat I_\lambda ^! \end{equation*}

for their canonical graded lifts in the category $\mathcal{C}^!_{\textrm{gr}} =\hat A^!\text{-}\textrm{grmod}$ .

Now we are ready to explain how Koszul duality can be used to relate dimensions of $\textrm{Ext}$ -groups with graded composition multiplicities. For $\nu,\lambda \in \Lambda$ , consider the polynomials

\begin{equation*} p_{\nu,\lambda } = \sum _{i \geq 0} \dim \textrm {Ext}_{\mathcal {C}}^i( L_\lambda, \nabla _\nu ) \cdot v^i \qquad \text {and} \qquad q_{\nu,\lambda } = \sum _{i \in \mathbb {Z}} [ \hat \Delta _\nu\, :\, \hat L_\lambda \langle i \rangle ] \cdot v^i. \end{equation*}

Lemma 1.3. Suppose that $\hat A$ is standard Koszul. Then,

\begin{equation*} \sum _{\nu \in \Lambda } p_{\nu,\lambda }({-}v) \cdot q_{\nu,\mu } = \delta _{\lambda,\mu } \end{equation*}

for all $\lambda,\mu \in \Lambda$ .

Proof. The proof is completely analogous to the proof of part (iii) of Theorem 3.11.4 in [Reference Beilinson, Ginzburg and Soergel4].

Now for $\nu,\lambda \in \Lambda$ , let us additionally consider the polynomials

\begin{equation*} p_{\nu,\lambda }^! = \sum _{i \geq 0} \dim \textrm {Ext}_{\mathcal {C}^!}^i( L_\lambda ^!, \nabla _\nu ^! ) \cdot v^i \qquad \text {and} \qquad q_{\nu,\lambda }^! = \sum _{i \in \mathbb {Z}} [ \hat \Delta _\nu ^!\, :\, \hat L_\lambda ^! \langle i \rangle ] \cdot v^i. \end{equation*}

Lemma 1.4. Suppose that $\hat A$ is standard Koszul. Then,

\begin{equation*} q_{\nu,\lambda }^! = p_{\nu,\lambda } \qquad {and} \qquad p_{\nu,\lambda }^! = q_{\nu,\lambda } \end{equation*}

for all $\nu,\lambda \in \Lambda$ .

Proof. The proof is completely analogous to the proof of part (i) of Theorem 3.11.4 in [Reference Beilinson, Ginzburg and Soergel4], using the fact that the Koszul duality functor satisfies

\begin{equation*} K( \hat L_\lambda ) \cong \hat P_\lambda ^!, \qquad K( \hat \nabla _\lambda ) \cong \hat \Delta _\lambda ^!, \qquad K( \hat I_\lambda ) \cong \hat L_\lambda ^! \end{equation*}

for all $\lambda \in \Lambda$ by Proposition 2.7 in [Reference Ágoston, Dlab and Lukács1] and Theorem 2.12.5 in [Reference Beilinson, Ginzburg and Soergel4].

3. Complex semisimple Lie algebras

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with a Borel subalgebra $\mathfrak{b}$ and a Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{b}$ , and let $\Phi \subseteq \mathfrak{h}^*$ be the root system of $\mathfrak{g}$ with respect to $\mathfrak{h}$ . We write $\Phi ^+ \subseteq \Phi$ for the positive system corresponding to $\mathfrak{b}$ and let $\Pi \subseteq \Phi ^+$ be a base of $\Phi$ . The Weyl group $W = \langle s_\alpha \mid \alpha \in \Phi \rangle \subseteq \textrm{GL}(\mathfrak{h}^*)$ is a finite Coxeter group with simple reflections $S = \{ s_\alpha \mid \alpha \in \Pi \}$ , and we denote by $w_0$ the longest element of $W$ . We write $\alpha ^\vee \in \mathfrak{h}$ for the coroot of $\alpha \in \Phi$ and let

\begin{equation*} X = \{ \lambda \in \mathfrak {h}^* \mid \lambda (\alpha ^\vee ) \in \mathbb {Z} \text { for all } \alpha \in \Phi \} \qquad \text {and} \qquad X^+ = \{ \lambda \in X \mid \lambda (\alpha ^\vee ) \geq 0 \text { for all } \alpha \in \Phi ^+ \} \end{equation*}

be the sets of integral weights and dominant weights, respectively. Furthermore, we write $\rho = \frac 12 \cdot \sum _{\alpha \in \Phi ^+} \alpha$ for the half-sum of all positive roots and consider the dot action of $W$ on $\mathfrak{h}^*$ , which is defined by

\begin{equation*} w \boldsymbol {\cdot } \lambda = w( \lambda + \rho ) - \rho \end{equation*}

for $w \in W$ and $\lambda \in \mathfrak{h}^*$ .

We consider the category $\mathcal{O}$ of finitely generated $\mathfrak{g}$ -modules that are locally $\mathfrak{b}$ -finite and admit a weight space decomposition with respect to $\mathfrak{h}$ ; see for instance [Reference Humphreys10]. For all $\lambda \in \mathfrak{h}^*$ , the Verma module $\Delta _\lambda$ of highest weight $\lambda$ and its unique simple quotient $L_\lambda$ are objects of $\mathcal{O}$ , and so is the dual Verma module $\nabla _\lambda$ . For every weight $\lambda \in - \rho - X^+ = \{ - \rho - \lambda \mid \lambda \in X^+ \}$ , we denote by $\mathcal{O}_\lambda$ the block of $\mathcal{O}$ containing the simple $\mathfrak{g}$ -module $L_\lambda$ . The stabilizer of $\lambda$ with respect to the dot action of $W$ on $\mathfrak{h}^*$ is a parabolic subgroup of $W$ , and for $I \subseteq S$ such that $\textrm{Stab}_W(\lambda ) = W_I = \langle I \rangle$ , the isomorphism classes of simple $\mathfrak{g}$ -modules in $\mathcal{O}_\lambda$ are in bijection with the set $W^I$ of elements $w \in W$ that have minimal length in the coset $w W_I$ (see Appendix A), via $w \mapsto L_{w\boldsymbol{\cdot }\lambda }$ .

Now let $J \subseteq S$ and denote by $\mathfrak{p}_J$ the parabolic subalgebra of $\mathfrak{g}$ that is generated by $\mathfrak{b}$ together with the root spaces $\mathfrak{g}_{-\alpha }$ for $\alpha \in \Pi$ with $s_\alpha \in J$ . We write $\mathcal{O}^{\mathfrak{p}_J}$ for the full subcategory of $\mathcal{O}$ whose objects are the locally $\mathfrak{p}_J$ -finite $\mathfrak{g}$ -modules in $\mathcal{O}$ , and for $\lambda \in -\rho - X^+$ , let $\mathcal{O}_\lambda ^{\mathfrak{p}_J}$ be the full subcategory of $\mathcal{O}_\lambda$ whose objects are the locally $\mathfrak{p}_J$ -finite $\mathfrak{g}$ -modules in $\mathcal{O}_\lambda$ . For $I \subseteq S$ such that $\textrm{Stab}_W(\lambda ) = W_I$ and for $w \in W^I$ , the simple $\mathfrak{g}$ -module $L_{w\boldsymbol{\cdot }\lambda }$ belongs to $\mathcal{O}_\lambda ^{\mathfrak{p}_J}$ if and only if $(w\boldsymbol{\cdot }\lambda )(\alpha ^\vee ) \geq 0$ for all $\alpha \in J$ , or equivalently, if $w \in w_J{}^{J}{W} \cap W^I = w_J{}^{J}{W}^I_{\textrm{reg}}$ . Here, we write $w_J$ for the longest element of $W_J$ and ${}^{J}{W}$ for the set of elements $w \in W$ that have minimal length in the coset $W_J w$ , and we denote by ${}^{J}{W}^I_{\textrm{reg}}$ the set of minimal length representatives for the regular double cosets in $W_J \backslash W/ W_I$ (see Appendix A). In particular, the isomorphism classes of simple $\mathfrak{g}$ -modules in $\mathcal{O}_\lambda ^{\mathfrak{p}_J}$ are in bijection with the set $w_J{}^{J}{W}^I_{\textrm{reg}}$ via $w \mapsto L_{w\boldsymbol{\cdot }\lambda }$ . For $\mu \in \mathfrak{h}^*$ , let us write $\Delta ^{\mathfrak{p}_J}_\mu$ for the largest quotient of $\Delta _\mu$ that belongs to $\mathcal{O}^{\mathfrak{p}_J}$ and $\nabla ^{\mathfrak{p}_J}_\mu$ for the largest submodule of $\nabla _\mu$ that belongs to $\mathcal{O}^{\mathfrak{p}_J}$ and call these $\mathfrak{g}$ -modules the generalized or parabolic Verma module and dual Verma module, respectively, of highest weight $\mu$ with respect to $\mathfrak{p}_J$ . Then, $\mathcal{O}_\lambda ^{\mathfrak{p}_J}$ is a highest weight category with weight poset $(w_J{}^{J}{W}^I_{\textrm{reg}},\leq )$ , where $\leq$ denotes the Bruhat order, with simple objects $L_{w\boldsymbol{\cdot }\lambda }$ , standard objects $\Delta ^{\mathfrak{p}_J}_{w\boldsymbol{\cdot }\lambda }$ , and costandard objects $\nabla ^{\mathfrak{p}_J}_{w\boldsymbol{\cdot }\lambda }$ for $w \in w_J{}^{J}{W}^I_{\textrm{reg}}$ . Let us further write $P^{\mathfrak{p}_J}_{w\boldsymbol{\cdot }\lambda }$ for the projective cover of $L_{w\boldsymbol{\cdot }\lambda }$ in $\mathcal{O}^{\mathfrak{p}_J}_\lambda$ , for $w \in w_J{}^{J}{W}^I_{\textrm{reg}}$ . We define

\begin{equation*} \textbf {P}^{\mathfrak {p}_J}_\lambda\, :\!= \,\bigoplus _{ w \in w_J {}^{J}{W}^I_{\textrm {reg}} } P^{\mathfrak {p}_J}_{w \boldsymbol {\cdot } \lambda } \qquad \text {and} \qquad A_\lambda ^{\mathfrak {p}_J}\, :\!= \,\textrm {End}_{\mathcal {O}^{\mathfrak {p}_J}}\big ( \textbf {P}^{\mathfrak {p}_J}_\lambda \big )^{\textrm {op}}, \end{equation*}

so that $\mathcal{O}^{\mathfrak{p}_J}_\lambda$ is equivalent to the category $A^{\mathfrak{p}_J}_\lambda \text{-}\textrm{mod}$ . The following theorem combines the results of E. Backelin and V. Mazorchuk; it will be crucial for our determination of the dimensions of $\textrm{Ext}$ -groups between simple objects and costandard objects in $\mathcal{O}^{\mathfrak{p}_J}_\lambda$ .

Proof. By Theorem 1.1 in [Reference Backelin2], the algebra $A_\lambda ^{\mathfrak{p}_J}$ admits a Koszul grading and its Koszul dual $(A_\lambda ^{\mathfrak{p}_J})^!$ is isomorphic to the algebra $A_{-w_0\mu }^{\mathfrak{p}_I}$ . The standard Koszulity of the Koszul graded version of $A_\lambda ^{\mathfrak{p}_J}$ is established in Theorem 5.1 of [Reference Mazorchuk20].

We keep the notation and assumptions of Theorem 2.1 and define

\begin{equation*} \textbf {L}^{\mathfrak {p}_J}_\lambda\, :\!= \,\bigoplus _{ w \in w_J {}^{J}{W}^I_{\textrm {reg}} } L_{w \boldsymbol {\cdot } \lambda }, \qquad \text {so that} \qquad ( A_\lambda ^{\mathfrak {p}_J} )^! \cong \Big ( \bigoplus _{i \geq 0} \textrm {Ext}_{\mathcal {O}^{\mathfrak {p}_J}}^i\big ( \textbf {L}^{\mathfrak {p}_J}_\lambda, \textbf {L}^{\mathfrak {p}_J}_\lambda \big ) \Big )^{\textrm {op}}. \end{equation*}

For $w \in w_J{}^{J}{W}^I_{\textrm{reg}}$ , we further write $e^{\mathfrak{p}_J}_{\lambda,w}$ for the idempotent in $A^{\mathfrak{p}_J}_\lambda$ corresponding to the canonical projection $\textrm{P}^{\mathfrak{p}_J}_\lambda \to P^{\mathfrak{p}_J}_{w\boldsymbol{\cdot }\lambda }$ and $f^{\mathfrak{p}_J}_{\lambda,w}$ for the idempotent in the Koszul dual $( A^{\mathfrak{p}_J}_\lambda )^!$ corresponding to the canonical projection $\textrm{L}^{\mathfrak{p}_J}_\lambda \to L_{w\boldsymbol{\cdot }\lambda }$ . Then, according to Remark 3.8 (and Proposition 3.1) in [Reference Backelin2], the isomorphism

\begin{equation*} \varphi \colon ( A^{\mathfrak {p}_J}_\lambda )^! \longrightarrow A^{\mathfrak {p}_I}_{-w_0\mu } \end{equation*}

from Theorem 2.1 can be chosen in such a way that

\begin{equation*} \varphi \big ( f^{\mathfrak {p}_J}_{\lambda,w} \big ) = e^{\mathfrak {p}_I}_{-w_0\mu,w^{-1} w_0} \end{equation*}

for all $w \in w_J{}^{J}{W}^I_{\textrm{reg}}$ .Footnote ² In particular, if we write $L^{\mathfrak{p}_J,!}_{\lambda,w}$ for the simple $( A^{\mathfrak{p}_J}_\lambda )^!$ -module corresponding to an element $w \in w_J{}^{J}{W}^I_{\textrm{reg}}$ (as described in Section 1), then the functor

\begin{equation*} ( \mathcal {O}^{\mathfrak {p}_J}_\lambda )^!\, :\!= \,( A^{\mathfrak {p}_J}_\lambda )^!\text {-}\textrm {mod} \longrightarrow A^{\mathfrak {p}_J}_{-w_0\mu }\text {-}\textrm {mod} \cong \mathcal {O}^{\mathfrak {p}_J}_{-w_0\mu } \end{equation*}

induced by $\varphi ^{-1}$ sends the simple $( A^{\mathfrak{p}_J}_\lambda )^!$ -module $L^{\mathfrak{p}_J,!}_{\lambda,w}$ to the simple $\mathfrak{g}$ -module $L_{w^{-1} w_0\boldsymbol{\cdot }({-}w_0\mu )}$ .

Now for a subset $I \subseteq S$ and $x,y \in{}^{I}{W}$ , let us denote by $m^I_{x,y}$ , $m_I^{x,y}$ , $n^I_{x,y}$ , and $n_I^{x,y}$ the corresponding spherical and anti-spherical (inverse) parabolic Kazhdan-Lusztig polynomials, as defined in Appendix A. With all of the above notation in place, we can prove the first main result.

Theorem 2.2. Let $\lambda,\mu \in - \rho - X^+$ and let $I,J \subseteq S$ such that $\textrm{Stab}_W(\lambda ) = W_I$ and $\textrm{Stab}_W(\mu ) = W_J$ . For $x,z \in w_J{}^{J}{}{W}^I_{\textrm{reg}}$ , we have

\begin{equation*} \sum _{i \geq 0} \dim \textrm {Ext}_{\mathcal {O}^{\mathfrak {p}_J}}^i\big ( L_{x\boldsymbol {\cdot }\lambda }, \nabla ^{\mathfrak {p}_J}_{z\boldsymbol {\cdot }\lambda } \big ) \cdot v^i = n^I_{z^{-1},x^{-1}}. \end{equation*}

Proof. For $w \in w_J{}^{J}{W}^I_{\textrm{reg}}$ , let us write $\hat L^{\mathfrak{p}_J}_{w\boldsymbol{\cdot }\lambda }$ and $\hat \Delta ^{\mathfrak{p}_J}_{w\boldsymbol{\cdot }\lambda }$ for the canonical graded lifts of $L_{w\boldsymbol{\cdot }\lambda }$ and $\Delta ^{\mathfrak{p}_J}_{w\boldsymbol{\cdot }\lambda }$ , respectively, in the category of graded modules over a Koszul graded version of $A^{\mathfrak{p}_J}_\lambda$ . Then, according to Lemma 1.4 and the above discussion of Koszul duality for $\mathcal{O}^{\mathfrak{p}_J}_\lambda$ , the polynomials

\begin{equation*} p^{\mathfrak {p}_J}_{\lambda,z,x} = \sum _{i \geq 0} \dim \textrm {Ext}_{\mathcal {O}^{\mathfrak {p}_J}}^i( L_{x\boldsymbol {\cdot }\lambda }, \nabla ^{\mathfrak {p}_J}_{z\boldsymbol {\cdot } \lambda } ) \cdot v^i \qquad \text {and} \qquad q^{\mathfrak {p}_J}_{\lambda,z,y} = \sum _{i \geq 0} [ \hat \Delta ^{\mathfrak {p}_J}_{z\boldsymbol {\cdot } \lambda }\, :\, \hat L^{\mathfrak {p}_J}_{y\boldsymbol {\cdot }\lambda } \langle i \rangle ] \cdot v^i \end{equation*}

with $x,y,z \in w_J{}^{J}{W}^I_{\textrm{reg}}$ are related via the equations

(3.1) \begin{equation} p^{\mathfrak{p}_J}_{\lambda,z,x} = q^{\mathfrak{p}_I}_{-w_0\mu,z^{-1}w_0,x^{-1}w_0} \qquad \text{and} \qquad q^{\mathfrak{p}_J}_{\lambda,z,y} = p^{\mathfrak{p}_I}_{-w_0\mu,z^{-1}w_0,y^{-1}w_0}. \end{equation}

We first consider a special case of the formula in the theorem, which we will then use to prove the general statement: For $\lambda ^\prime \in - \rho - X^+$ with $\textrm{Stab}_W(\lambda ^\prime ) = \{ e \}$ , we have

\begin{equation*} q^{\mathfrak {p}_J}_{\lambda ^\prime,z,y} = p^{\mathfrak {b}}_{-w_0\mu,z^{-1}w_0,y^{-1}w_0} = n^J_{zw_0,yw_0} \end{equation*}

for all $y,z \in w_J{}^{J}{W}$ by parts (i) and (iv) of Theorem 3.11.4 in [Reference Beilinson, Ginzburg and Soergel4]. Now let $\hat A^{\mathfrak{p}_J}_\lambda$ and $\hat A^{\mathfrak{p}_J}_{\lambda ^\prime }$ be Koszul graded algebras whose underlying ungraded algebras are $A^{\mathfrak{p}_J}_\lambda$ and $A^{\mathfrak{p}_J}_{\lambda ^\prime }$ , respectively, and let

\begin{equation*} \mathcal {O}^{\mathfrak {p}_J}_{\lambda,\textrm {gr}} = \hat A^{\mathfrak {p}_J}_\lambda \text {-} \textrm {mod} \qquad \text {and} \qquad \mathcal {O}^{\mathfrak {p}_J}_{\lambda ^\prime,\textrm {gr}} = \hat A^{\mathfrak {p}_J}_{\lambda ^\prime } \text {-} \textrm {mod} \end{equation*}

be the corresponding categories of graded modules. For $J=\varnothing$ , we omit the superscript $\mathfrak{p}_J$ and write

\begin{equation*} \mathcal {O}_{\lambda,\textrm {gr}} = \hat A_\lambda \text {-} \textrm {mod} \qquad \text {and} \qquad \mathcal {O}_{\lambda ^\prime,\textrm {gr}} = \hat A_{\lambda ^\prime } \text {-} \textrm {mod}. \end{equation*}

Then, by the proof of Proposition 3.2 in [Reference Backelin2], the graded algebra $\hat A^{\mathfrak{p}_J}_\lambda$ is the quotient of $\hat A_\lambda$ by some homogeneous ideal, whence we can consider $\mathcal{O}^{\mathfrak{p}_J}_{\lambda,\textrm{gr}}$ as a full subcategory of $\mathcal{O}_{\lambda,\textrm{gr}}$ , and similarly for $\mathcal{O}^{\mathfrak{p}_J}_{\lambda ^\prime,\textrm{gr}}$ and $\mathcal{O}_{\lambda ^\prime,\textrm{gr}}$ . (See also Lemma 2.2 in [Reference Shan, Varagnolo and Vasserot25] and Theorem 3.5.3 in [Reference Beilinson, Ginzburg and Soergel4].) Furthermore, as shown in part (d) of the proof of Lemma 3.3 in [Reference Backelin2], there is an exact functor

\begin{equation*} \hat T_{\lambda ^\prime }^\lambda \colon \mathcal {O}_{\lambda ^\prime,\textrm {gr}} \longrightarrow \mathcal {O}_{\lambda,\textrm {gr}} \end{equation*}

which lifts the usual translation functor $T_{\lambda ^\prime }^\lambda \colon \mathcal{O}_{\lambda ^\prime } \to \mathcal{O}_\lambda$ and sends any pure object in $\mathcal{O}_{\lambda ^\prime,\textrm{gr}}$ to a pure object of the same degree in $\mathcal{O}_{\lambda,\textrm{gr}}$ . Using well-known properties of translation functors (see Section 7.9 in [Reference Humphreys10]), it follows that

\begin{equation*} \hat T_{\lambda ^\prime }^\lambda \hat L_{y\boldsymbol {\cdot }\lambda ^\prime } \cong \begin {cases} \hat L_{y\boldsymbol {\cdot }\lambda } & \text {if } y \in W^I, \\ 0 & \text {otherwise} \end {cases} \end{equation*}

for all $y \in W$ and that $\hat T_{\lambda ^\prime }^\lambda$ restricts to a functor from $\mathcal{O}^{\mathfrak{p}_J}_{\lambda ^\prime,\textrm{gr}}$ to $\mathcal{O}^{\mathfrak{p}_J}_{\lambda,\textrm{gr}}$ with

\begin{equation*} \hat T_{\lambda ^\prime }^\lambda \hat \Delta ^{\mathfrak {p}_J}_{z\boldsymbol {\cdot }\lambda ^\prime } \cong \hat \Delta ^{\mathfrak {p}_J}_{z\boldsymbol {\cdot }\lambda } \end{equation*}

for all $z \in w_J{}^{J}{W}^I_{\textrm{reg}}$ .Footnote ³ Now the exactness of $\hat T_{\lambda ^\prime }^\lambda$ implies that

\begin{equation*} [ \hat \Delta ^{\mathfrak {p}_J}_{z\boldsymbol {\cdot }\lambda }\, :\, \hat L^{\mathfrak {p}_J}_{y\boldsymbol {\cdot }\lambda } \langle i \rangle ] = [ \hat \Delta ^{\mathfrak {p}_J}_{z\boldsymbol {\cdot }\lambda ^\prime }\, :\, \hat L^{\mathfrak {p}_J}_{y\boldsymbol {\cdot }\lambda ^\prime } \langle i \rangle ] \end{equation*}

for all $y,z \in w_J{}^{J}{W}^I_{\textrm{reg}}$ and all $i \in \mathbb{Z}$ . In particular, the special case at the beginning of the proof implies that $q^{\mathfrak{p}_J}_{\lambda,z,y} = q^{\mathfrak{p}_J}_{\lambda ^\prime,z,y} = n^J_{zw_0,yw_0}$ for all $y,z \in w_J{}^{J}{W}^I_{\textrm{reg}}$ , and using (2.1), we obtain

\begin{equation*} p^{\mathfrak {p}_J}_{\lambda,z,x} = q^{\mathfrak {p}_I}_{-w_0 \mu,z^{-1}w_0,x^{-1}w_0} = n^I_{z^{-1},x^{-1}} \end{equation*}

for all $x,z \in w_J{}^{J}{W}^I_{\textrm{reg}}$ , as required.

4. Affine Kac-Moody algebras

Let us keep the notation from the previous section and additionally assume that $\mathfrak{g}$ is a complex simple Lie algebra. Recall that we fix a Borel subalgebra $\mathfrak{b}$ and a Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{b}$ of $\mathfrak{g}$ . We consider the affine Kac-Moody Lie algebra

\begin{equation*} \mathfrak {\tilde g} = ( \mathbb {C}[t^{\pm 1}] \otimes _{\mathbb {C}} \mathfrak {g} ) \oplus \mathbb {C} c \oplus \mathbb {C} d \end{equation*}

as in [Reference Tanisaki26, Section 6] and its subalgebras

\begin{equation*} \mathfrak {\tilde b} = \mathfrak {b} \oplus ( t \mathbb {C}[t] \otimes \mathfrak {g} ) \oplus \mathbb {C} c \oplus \mathbb {C} d \qquad \text {and} \qquad \mathfrak {\tilde h} = \mathfrak {h} \oplus \mathbb {C} c \oplus \mathbb {C} d, \end{equation*}

and we view $\mathfrak{h}^*$ as a subspace of $\mathfrak{\tilde h}^*$ , with the convention that $\lambda (c) = \lambda (d) = 0$ for all $\lambda \in \mathfrak{h}^*$ . Let $\tilde \Phi$ be the root system of $\mathfrak{\tilde g}$ and let $\tilde \Phi ^+$ be the set of positive roots corresponding to $\mathfrak{\tilde b}$ , again as in [Reference Tanisaki26, Section 6]. Further, let $\tilde \Pi = \Pi \sqcup \{ \alpha _0 \}$ be the set of simple roots in $\tilde \Phi$ and let

\begin{equation*}\tilde W = \langle s_\alpha \mid \alpha \in \tilde \Pi \rangle \subseteq \textrm {GL}(\mathfrak {\tilde h}^*)\end{equation*}

be the (affine) Weyl group of $\mathfrak{\tilde g}$ . It is a Coxeter group with set of simple reflections $\tilde S = \{ s_\alpha \mid \alpha \in \tilde \Pi \}$ , and it is canonically isomorphic to the semidirect product $W \rtimes \mathbb{Z}\Phi ^\vee$ , where $W$ denotes the Weyl group of $\mathfrak{g}$ and $\mathbb{Z}\Phi ^\vee$ is the coroot lattice of $\mathfrak{g}$ (see Proposition 13.1.7 in [Reference Kumar17]). Let us write $\alpha ^\vee \in \mathfrak{\tilde h}$ for the coroot corresponding to $\alpha \in \tilde \Pi$ and define $\tilde \rho \in \mathfrak{\tilde h}^*$ by $\tilde \rho (\alpha ^\vee ) = 1$ for all $\alpha \in \tilde \Pi$ and $\tilde \rho (d) = 0$ . The dot action of $\tilde W$ on $\mathfrak{\tilde h}^*$ is given by

\begin{equation*} w \boldsymbol {\cdot } \lambda = w( \lambda + \tilde \rho ) - \tilde \rho \end{equation*}

for all $w \in \tilde W$ and $\lambda \in \mathfrak{\tilde h}^*$ . We say that $\lambda \in \mathfrak{\tilde h}^*$ has level $e \in \mathbb{C}$ if $e = ( \lambda + \tilde \rho )(c)$ , and we set

\begin{equation*} \lambda ^\prime\, :\!= \,- \lambda - 2 \tilde \rho, \end{equation*}

so that $\lambda ^\prime$ has level $-e$ when $\lambda$ has level $e$ and $(w \boldsymbol{\cdot } \lambda )^\prime = w\boldsymbol{\cdot }\lambda ^\prime$ for all $w \in \tilde W$ . The sets of integral and dominant weights are defined by

\begin{equation*} \tilde X = \{ \lambda \in \mathfrak {\tilde h}^* \mid \lambda (\alpha ^\vee ) \in \mathbb {Z} \text { for all } \alpha \in \tilde \Pi \} \qquad \text {and} \qquad \tilde X^+ = \{ \lambda \in \tilde X \mid \lambda (\alpha ^\vee ) \geq 0 \text { for all } \alpha \in \tilde \Pi \}, \end{equation*}

respectively, and we note that for every integral weight $\mu \in \tilde X$ of positive (or negative) level, there is a unique weight $\nu \in \tilde X^+ \setminus \{0\}$ such that $\mu$ belongs to the $\tilde W$ -orbit of $\nu - \tilde \rho$ (respectively $-\nu - \tilde \rho$ ) with respect to the dot action; see for instance Section 6.3 in [Reference Brundan and Stroppel8].

Now let us consider the category $\mathcal{\tilde O}$ of $\mathfrak{\tilde g}$ -modules that admit a weight space decomposition with finite-dimensional weight spaces with respect to $\mathfrak{\tilde h}$ and whose set of weights is contained in the union of finitely many sets of the form $\{ \mu \in \mathfrak{\tilde h}^* \mid \mu \leq \lambda \}$ for $\lambda \in \mathfrak{\tilde h}^*$ , where we write $\leq$ for the partial order on $\mathfrak{\tilde h}^*$ that is defined by

\begin{equation*} \mu \leq \lambda \qquad \text {if and only if} \qquad \lambda - \mu \in \sum _{\alpha \in \tilde \Pi } \mathbb {Z}_{\geq 0} \cdot \alpha. \end{equation*}

For all $\lambda \in \mathfrak{\tilde h}^*$ , the Verma module $\Delta _\lambda$ of highest weight $\lambda$ , the dual Verma module $\nabla _\lambda$ , and the unique simple quotient $L_\lambda$ of $\Delta _\lambda$ (which is also the unique simple submodule of $\nabla _\lambda$ ) are objects of $\mathcal{\tilde O}$ . Even though a $\mathfrak{\tilde g}$ -module $M$ in $\mathcal{\tilde O}$ does not need to admit a finite composition series, the composition multiplicities $[ M\, :\, L_\mu ]$ are still (well-defined and) finite for $\mu \in \mathfrak{\tilde h}^*$ (see Lemma 2.1.9 in [Reference Kumar17]). Now let us fix

\begin{equation*} \lambda \in ( {-} \tilde \rho + \tilde X^+ ) \cup ( {-} \tilde \rho - \tilde X^+ ) \qquad \text {with} \qquad \lambda \neq - \tilde \rho \end{equation*}

and write $\mathcal{\tilde O}_\lambda$ for the full subcategory of $\mathcal{\tilde O}$ whose objects are the $\mathfrak{\tilde g}$ -modules $M$ in $\mathcal{\tilde O}$ such that $[ M\, :\, L_\mu ] = 0$ for all $\mu \in \mathfrak{\tilde h}^* \setminus \tilde W \boldsymbol{\cdot } \lambda$ . Furthermore, for $J \subseteq \tilde S$ , we write $\mathfrak{p}_J$ for the Levi subalgebra of $\mathfrak{\tilde g}$ corresponding to $J$ and we let $\mathcal{\tilde O}^{\mathfrak{p}_J}$ (or $\mathcal{\tilde O}^{\mathfrak{p}_J}_\lambda$ ) be the full subcategory of $\mathcal{\tilde O}$ (or $\mathcal{\tilde O}_\lambda$ ) whose objects are the locally $\mathfrak{p}_J$ -finite $\mathfrak{\tilde g}$ -modules in $\mathcal{\tilde O}$ (or $\mathcal{\tilde O}_\lambda$ ). As in the previous section, we write $\Delta ^{\mathfrak{p}_J}_\mu$ for the largest locally $\mathfrak{p}_J$ -finite quotient of the Verma module $\Delta _\mu$ of highest weight $\mu \in \mathfrak{\tilde h}^*$ and $\nabla ^{\mathfrak{p}_J}_\mu$ for the largest locally $\mathfrak{p}_J$ -finite submodule of $\nabla _\mu$ , and we call these $\mathfrak{\tilde g}$ -modules the generalized Verma module and the dual generalized Verma module of highest weight $\mu$ with respect to $\mathfrak{p}_J$ .

Now let $I \subseteq \tilde S$ such that $\textrm{Stab}_{\tilde W}(\lambda ) = \tilde W_I$ . If $\lambda$ has positive (or negative) level then we parameterize the set of isomorphism classes of simple $\mathfrak{\tilde g}$ -modules in $\mathcal{\tilde O}_\lambda$ by $X_\lambda\, :\!= \,\tilde W^I w_I$ (respectively $X_\lambda\, :\!= \,\tilde W^I$ ) via the map $w \mapsto L_{w\boldsymbol{\cdot }\lambda }$ , and this parametrization restricts to a bijection between the set of isomorphism classes of simple $\mathfrak{\tilde g}$ -modules in $\mathcal{\tilde O}_\lambda ^{\mathfrak{p}_J}$ and the set $X_\lambda ^J \,:\!=\, {}^J \tilde W^I_{\textrm{reg}} w_I$ (respectively $X_\lambda ^J\, :\!= \,w_J{}^J \tilde W^I_{\textrm{reg}}$ ) of representatives for the regular double cosets in $\tilde W_J \backslash \tilde W/ \tilde W_I$ (as defined in Appendix A). In order to apply the considerations from Section 1, we consider truncations of the category $\mathcal{\tilde O}_\lambda ^{\mathfrak{p}_J}$ as follows: If $\lambda$ has negative level then for $w \in X_\lambda ^J$ , we write $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda,\leq w}$ for the Serre subcategory of $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda }$ generated by the simple $\mathfrak{\tilde g}$ -modules $L_{x\boldsymbol{\cdot }\lambda }$ with $x \in X_\lambda ^{J, \leq w}$ , where we set

\begin{equation*} X_\lambda ^{J, \leq w}\, :\!= \,\{ y \in X_\lambda ^J \mid y \leq w \}. \end{equation*}

Then, $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda,\leq w}$ is a highest weight category with weight poset $( X_\lambda ^{J, \leq w}, \leq )$ by Lemma 3.7 in [Reference Shan, Varagnolo and Vasserot25], where $\leq$ denotes the Bruhat order and the standard objects and costandard objects are given by $\Delta ^{\mathfrak{p}_J}_{x\boldsymbol{\cdot }\lambda }$ and $\nabla ^{\mathfrak{p}_J}_{x\boldsymbol{\cdot }\lambda }$ , respectively, for $x \in X_\lambda ^{J, \leq w}$ . If $\lambda$ has positive level then for $w \in X_\lambda ^J$ , we write $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda,\leq w}$ for the Serre quotient category of $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda }$ by the Serre subcategory generated by the simple $\mathfrak{\tilde g}$ -modules $L_{x\boldsymbol{\cdot }\lambda }$ for $x \in X_\lambda ^J$ with $x \nleq w$ , so that the simple objects of $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda,\leq w}$ are naturally parameterized by the set

\begin{equation*} X_\lambda ^{J,\leq w}\, :\!= \,\{ y \in X_\lambda ^J \mid y \leq w \}. \end{equation*}

Then, $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda,\leq w}$ is a highest weight category with weight poset $( X_\lambda ^{J, \leq w}, \geq\! )$ by Proposition 3.9 in [Reference Shan, Varagnolo and Vasserot25], where $\geq$ denotes the opposite Bruhat order and the standard objects and costandard objects are given by $\Delta ^{\mathfrak{p}_J}_{x\boldsymbol{\cdot }\lambda }$ and $\nabla ^{\mathfrak{p}_J}_{x\boldsymbol{\cdot }\lambda }$ , respectively, for $x \in X_\lambda ^{J, \leq w}$ . In either case (i.e. for $\lambda$ of positive or negative level), we write $P^{\mathfrak{p}_J,\leq w}_{x\boldsymbol{\cdot }\lambda }$ for the projective cover of $L_{x\boldsymbol{\cdot }\lambda }$ in $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda,\leq w}$ , for $x \in X_\lambda ^{J, \leq w}$ , and we set

\begin{equation*} \textrm {P}_{\lambda,\leq w}^{\mathfrak {p}_J}\, :\!= \,\bigoplus _{x \in X_\lambda ^{\mathfrak {p}_J,\leq w}} P_{x\boldsymbol {\cdot }\lambda }^{\mathfrak {p}_J,\leq w} \qquad \text {and} \qquad A_{\lambda,\leq w}^{\mathfrak {p}_J} = \textrm {End}_{\mathcal {\tilde O}^{\mathfrak {p}_J}_{\lambda,\leq w}}\big ( \textrm {P}_{\lambda,\leq w}^{\mathfrak {p}_J} \big )^{\textrm {op}}, \end{equation*}

so that $\mathcal{\tilde O}_{\lambda,\leq w}^{\mathfrak{p}_J}$ is equivalent to $A_{\lambda,\leq w}^{\mathfrak{p}_J}\text{-}\textrm{mod}$ .

As in the preceding section, we want to use Koszul duality as a tool to compute the dimensions of $\textrm{Ext}$ -groups in $\mathcal{\tilde O}_{\lambda,\leq w}^{\mathfrak{p}_J}$ . This is possible by the following result; see Theorem 3.12 in [Reference Shan, Varagnolo and Vasserot25].

Theorem 3.2. Let $I,J \subsetneq \tilde S$ and let $\lambda \in - \tilde \rho - \tilde X^+$ and $\mu \in - \tilde \rho + \tilde X^+$ such that $\textrm{Stab}_{\tilde W}(\lambda ) = \tilde W_I$ and $\textrm{Stab}_{\tilde W}(\mu ) = \tilde W_J$ . Then, for $w \in X_\lambda ^J$ and $u \in X_\mu ^I$ , the algebras $A_{\lambda,\leq w}^{\mathfrak{p}_J}$ and $A_{\mu,\leq u}^{\mathfrak{p}_I}$ are standard Koszul, and their Koszul duals are given by

\begin{equation*} ( A_{\lambda,\leq w}^{\mathfrak {p}_J} )^! \cong A_{\mu,\leq w^{-1}}^{\mathfrak {p}_I} \qquad \text {and} \qquad ( A_{\mu, \leq u}^{\mathfrak {p}_I} )^! \cong A_{\lambda,\leq u^{-1}}^{\mathfrak {p}_J}. \end{equation*}

Now let us return to a weight $\lambda \in ( {-} \tilde \rho + \tilde X^+ ) \cup ( {-} \tilde \rho - \tilde X^+ )$ with $\lambda \neq - \rho$ and subsets $I,J \subsetneq \tilde S$ such that $\textrm{Stab}_{\tilde W}(\lambda ) = \tilde W_I$ . Let us further fix a weight $\mu \in ( {-} \tilde \rho + \tilde X^+ ) \cup ( {-} \tilde \rho - \tilde X^+ )$ with $\textrm{Stab}_{\tilde W}(\mu ) = \tilde W_J$ such that $\mu$ has negative (or positive) level if $\lambda$ has positive (or negative) level. For $w \in X_\lambda ^J$ , we set

\begin{equation*} \textbf {L}^{\mathfrak {p}_J}_{\lambda,\leq w}\, :\!= \,\bigoplus _{ x \in X_\lambda ^{J,\leq w} } L_{x \boldsymbol {\cdot } \lambda }, \qquad \text {so that} \qquad ( A_{\lambda,\leq w}^{\mathfrak {p}_J} )^! \cong \Big ( \bigoplus _{i \geq 0} \textrm {Ext}_{\mathcal {\tilde O}_{\lambda,\leq w}^{\mathfrak {p}_J}}^i\big ( \textbf {L}^{\mathfrak {p}_J}_{\lambda,\leq w}, \textbf {L}^{\mathfrak {p}_J}_{\lambda,\leq w} \big ) \Big )^{\textrm {op}}. \end{equation*}

For $x \in X_\lambda ^{J,\leq w}$ , we further write $e^{\mathfrak{p}_J,\leq w}_{\lambda,x}$ for the idempotent in $A^{\mathfrak{p}_J}_{\lambda,\leq w}$ corresponding to the canonical projection $\textrm{P}^{\mathfrak{p}_J}_{\lambda,\leq w} \to P^{\mathfrak{p}_J,\leq w}_{x\boldsymbol{\cdot }\lambda }$ and $f^{\mathfrak{p}_J,\leq w}_{\lambda,x}$ for the idempotent in the Koszul dual $( A^{\mathfrak{p}_J}_{\lambda,\leq w} )^!$ corresponding to the canonical projection $\textrm{L}^{\mathfrak{p}_J}_{\lambda,\leq w} \to L_{x\boldsymbol{\cdot }\lambda }$ . Then, again by Theorem 3.12 in [Reference Shan, Varagnolo and Vasserot25], the isomorphism

\begin{equation*} A_{\mu,\leq w^{-1}}^{\mathfrak {p}_I} \mathop{\longrightarrow}^{\sim} ( A_{\lambda,\leq w}^{\mathfrak {p}_J} )^! \end{equation*}

from Theorem 3.2 can be chosen in such a way that it maps $e^{\mathfrak{p}_I,\leq w^{-1}}_{\mu,x^{-1}}$ to $f^{\mathfrak{p}_J,\leq w}_{\lambda,x}$ for all $x \in X_\lambda ^J$ . In particular, if we write $L^{\mathfrak{p}_J,\leq w,!}_{\lambda,x}$ for the simple $( A^{\mathfrak{p}_J}_{\lambda,\leq w} )^!$ -module corresponding to $x \in X_\lambda ^{J,\leq w}$ (as described in Section 1) then the functor

\begin{equation*} ( \mathcal {\tilde O}^{\mathfrak {p}_J}_{\lambda,\leq w} )^!\, :\!= \,( A^{\mathfrak {p}_J}_{\lambda,\leq w} )^!\text {-}\textrm {mod} \longrightarrow A^{\mathfrak {p}_I}_{\mu,\leq w^{-1}} \text {-} \textrm {mod} \simeq \mathcal {\tilde O}^{\mathfrak {p}_I}_{\mu,\leq w^{-1}} \end{equation*}

induced by the aforementioned isomorphism sends the simple $( A^{\mathfrak{p}_J}_{\lambda,\leq w} )^!$ -module $L^{\mathfrak{p}_J,\leq w,!}_{\lambda,x}$ to the simple $\mathfrak{\tilde g}$ -module $L_{x^{-1} \boldsymbol{\cdot } \mu }$ . We can now prove the main result of this section; it is an analogue of Theorem 2.2 in the setting of affine Kac-Moody algebras.

Theorem 3.3. Let $\lambda \in ( {-} \tilde \rho + \tilde X^+ ) \cup ( {-} \tilde \rho - \tilde X^+ )$ and $I,J \subsetneq \tilde S$ such that $\textrm{Stab}_{\tilde W}(\lambda ) = \tilde W_I$ . If $\lambda$ has negative level then we have

\begin{equation*} \sum _{i \geq 0} \dim \textrm {Ext}_{\mathcal {\tilde O}^{\mathfrak {p}_J}}^i( L_{y\boldsymbol {\cdot }\lambda }, \nabla ^{\mathfrak {p}_J}_{x\boldsymbol {\cdot }\lambda } ) \cdot v^i = n^I_{x^{-1},y^{-1}} \end{equation*}

for all $x,y \in X_\lambda ^J$ , and if $\lambda$ has positive level then we have

\begin{equation*} \sum _{i \geq 0} \dim \textrm {Ext}_{\mathcal {\tilde O}^{\mathfrak {p}_J}}^i( L_{y\boldsymbol {\cdot }\lambda }, \nabla ^{\mathfrak {p}_J}_{x\boldsymbol {\cdot }\lambda } ) \cdot v^i = m_I^{w_Ix^{-1},w_Iy^{-1}} \end{equation*}

for all $x,y \in X_\lambda ^J$ .

Proof. Let us start by observing that by Theorems 3.42 and 3.59 in [Reference Brundan and Stroppel8], we have

\begin{equation*} \textrm {Ext}_{\mathcal {\tilde O}^{\mathfrak {p}_J}}^i( L_{y\boldsymbol {\cdot }\lambda }, \nabla ^{\mathfrak {p}_J}_{x\boldsymbol {\cdot }\lambda } ) \cong \textrm {Ext}_{\mathcal {\tilde O}_{\lambda,\leq w}^{\mathfrak {p}_J}}^i( L_{y\boldsymbol {\cdot }\lambda }, \nabla ^{\mathfrak {p}_J}_{x\boldsymbol {\cdot }\lambda } ) \end{equation*}

for all $i \geq 0$ and $w \in X_\lambda ^J$ such that $x,y \leq w$ ; hence, it suffices to prove the formulas in the theorem for the highest weight category $\mathcal{\tilde O}_{\lambda,\leq w}^{\mathfrak{p}_J}$ .

Next, for $x \in X_\lambda ^{J,\leq w}$ , let us write $\hat L^{\mathfrak{p}_J,\leq w}_{x\boldsymbol{\cdot }\lambda }$ and $\hat \Delta ^{\mathfrak{p}_J,\leq w}_{x\boldsymbol{\cdot }\lambda }$ for the canonical graded lifts of $L_{x\boldsymbol{\cdot }\lambda }$ and $\Delta ^{\mathfrak{p}_J}_{x\boldsymbol{\cdot }\lambda }$ , respectively, in the category $\mathcal{\tilde O}_{\lambda,\leq w,\textrm{gr}}^{\mathfrak{p}_J}$ of graded modules over a Koszul graded version $\hat A^{\mathfrak{p}_J}_{\lambda,\leq w}$ of $A^{\mathfrak{p}_J}_{\lambda,\leq w}$ . Then, according to Lemma 1.3, the polynomials

\begin{equation*} p^{\mathfrak {p}_J,\leq w}_{\lambda,z,x} = \sum _{i \geq 0} \dim \textrm {Ext}_{\mathcal {\tilde O}^{\mathfrak {p}_J}}^i( L_{x\boldsymbol {\cdot }\lambda }, \nabla ^{\mathfrak {p}_J}_{z\boldsymbol {\cdot } \lambda } ) \cdot v^i \qquad \text {and} \qquad q^{\mathfrak {p}_J,\leq w}_{\lambda,z,y} = \sum _{i \geq 0} [ \hat \Delta ^{\mathfrak {p}_J,\leq w}_{z\boldsymbol {\cdot } \lambda }\, :\, \hat L^{\mathfrak {p}_J,\leq w}_{y\boldsymbol {\cdot }\lambda } \langle i \rangle ] \cdot v^i \end{equation*}

with $x,y,z \in X_\lambda ^{J,\leq w}$ satisfy the inversion formula

(3.1) \begin{equation} \sum _{z \in X_\lambda ^{J,\leq w}} p^{\mathfrak{p}_J, \leq w}_{\lambda,z,x}({-}v) \cdot q^{\mathfrak{p}_J,\leq w}_{\lambda,z,y} = \delta _{x,y}, \end{equation}

and for $\mu \in ( {-} \tilde \rho + \tilde X^+ ) \cup ( {-} \tilde \rho - \tilde X^+ )$ with $\textrm{Stab}_{\tilde W}(\mu ) = \tilde W_J$ and such that $\mu$ has negative (or positive) level if $\lambda$ has positive (or negative) level, we have

(3.2) \begin{equation} p^{\mathfrak{p}_J,\leq w}_{\lambda,z,x} = q^{\mathfrak{p}_I,\leq w^{-1}}_{\mu,z^{-1},x^{-1}} \qquad \text{and} \qquad q^{\mathfrak{p}_J,\leq w}_{\lambda,z,y} = p^{\mathfrak{p}_I, \leq w^{-1}}_{\mu,z^{-1},y^{-1}} \end{equation}

by Lemma 1.4 and the above discussion of Koszul duality for $\mathcal{\tilde O}^{\mathfrak{p}_J}_{\lambda,\leq w}$ .

As in the proof of Theorem 2.2, we first consider a special case of the formulas in the theorem, which we will then use to prove the general statement: Let $\nu \in - \tilde \rho - \tilde X^+$ with $\nu \neq - \tilde \rho$ and $\textrm{Stab}_{\tilde W}(\nu ) = \{ e \}$ , so that $L_{w_J\boldsymbol{\cdot }\nu } = \Delta ^{\mathfrak{p}_J}_{w_J\boldsymbol{\cdot }\nu }$ and

\begin{equation*} \sum _{i \geq 0} \dim \textrm {Ext}_{ \mathcal {\tilde O}^{\mathfrak {p}_J} }^i( L_{w_J\boldsymbol {\cdot }\nu }, \nabla ^{\mathfrak {p}_J}_{x\boldsymbol {\cdot }\nu } ) \cdot v^i = \delta _{x,w_J} = m^J_{w_Jx,e} \end{equation*}

for all $x \in X_\nu ^J$ by Theorem 3.56 in [Reference Brundan and Stroppel8].

Now one can show by induction on the length of $y \in X_\nu ^J$ that the validity of the Kazhdan-Lusztig character formula in $\mathcal{\tilde O}_\nu$ (as conjectured in this case by G. Lusztig [Reference Lusztig19] and proven by M. Kashiwara and T. Tanisaki [Reference Kashiwara and Tanisaki15]) and of its parabolic analogue in $\mathcal{\tilde O}_\nu ^{\mathfrak{p}_J}$ (see Section 3.3 in [Reference Lanini, Ram and Sobaje18]) implies that

\begin{equation*} \sum _{i \geq 0} \dim \textrm {Ext}_{\mathcal {\tilde O}}^i( L_{y\boldsymbol {\cdot }\nu }, \nabla ^{\mathfrak {p}_J}_{x\boldsymbol {\cdot }\nu } ) \cdot v^i = m^J_{w_Jx,w_Jy} \end{equation*}

for all $x,y \in X_\nu ^J$ , exactly as in the proof of Proposition C.2 in [Reference Jantzen12].Footnote ⁴ This implies that

\begin{equation*} p^{\mathfrak {p}_J,\leq w}_{\nu,z,x} = m^J_{w_Jz,w_Jx} \end{equation*}

for all $w \in X_\nu ^J$ and $x,y \in X_\nu ^{J,\leq w}$ , and by equations (3.1) and (A1), we further have

\begin{equation*} q^{\mathfrak {p}_J,\leq w}_{\nu,x,y} = m_J^{w_Jx,w_Jy}. \end{equation*}

Now suppose that $\lambda$ has negative level. By Lemma 5.10 in [Reference Shan, Varagnolo and Vasserot25] (see also Remark 4.2 in [Reference Ko14]), the ordinary translation functor

\begin{equation*} T_\nu ^\lambda \colon \mathcal {\tilde O}_{\nu,\leq w} \longrightarrow \mathcal {\tilde O}_{\lambda,\leq w} \end{equation*}

admits a graded lift

\begin{equation*} \hat T_\nu ^\lambda \colon \mathcal {\tilde O}_{\nu,\leq w,\textrm {gr}} \longrightarrow \mathcal {\tilde O}_{\lambda,\leq w,\textrm {gr}} \end{equation*}

for all $w \in X_\lambda$ , and the latter restricts to a functor from $\mathcal{\tilde O}_{\nu,\leq w,\textrm{gr}}^{\mathfrak{p}_J}$ to $\mathcal{\tilde O}_{\lambda,\leq w,\textrm{gr}}^{\mathfrak{p}_J}$ for $w \in X_\lambda ^J$ , as in the proof of Theorem 2.2. Still as in the proof of Theorem 2.2, we conclude that

\begin{equation*} q^{\mathfrak {p}_J,\leq w}_{\lambda,z,y} = q^{\mathfrak {p}_J,\leq w}_{\nu,z,y} = m_J^{w_Jz,w_Jy} \end{equation*}

for all $w \in X_\lambda ^J$ and $y,z \in X_\lambda ^{J,\leq w}$ , and using the inversion formulas from equation (3.1) and Theorem A.1, it follows that

\begin{equation*} p^{\mathfrak {p}_J,\leq w}_{\lambda,z,x} = n^I_{z^{-1},x^{-1}} \end{equation*}

for all $w \in X_\lambda ^J$ and $x,z \in X_\lambda ^{J,\leq w}$ , matching the first formula in the theorem. Furthermore, for a weight $\mu \in - \tilde \rho + \tilde X^+$ with $\mu \neq - \tilde \rho$ and $\textrm{Stab}_{\tilde W}(\mu ) = \tilde W_J$ , equation (3.2) yields

\begin{equation*} q^{\mathfrak {p}_I,\leq w}_{\mu,z,x} = p^{\mathfrak {p}_J,\leq w^{-1}}_{\lambda,z^{-1},x^{-1}} = n^I_{z,x} \qquad \text {and} \qquad p^{\mathfrak {p}_I,\leq w}_{\mu,z,y} = q^{\mathfrak {p}_J,\leq w^{-1}}_{\lambda,z^{-1},y^{-1}} = m_J^{w_Jz^{-1},w_Jy^{-1}} \end{equation*}

for all $w \in X_\mu ^I$ and $x,y,z \leq X_\mu ^{I,\leq w}$ , and by relabeling, we obtain the second equation in the theorem.