We claim the following two facts

  1. (i)

If $d_{1}=d_{0}-1$, then we always have
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$, and
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\notin\Omega^{s_{0},d_{0},+}$
if and only if $\Theta^{\prime}\in\Omega^{s_{0},d_{0}-1}$.

  2. (ii)

If $d_{1}=d_{0}$, then we always have
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$,
and $p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\notin\Omega^{s_{0},d_{0},+}$
if and only if $\Theta^{\prime}\in\Omega^{s_{0},d_{0}}$.

We first treat item (i), namely the case when $d_{1}=d_{0}-1$. We note that
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$ (resp.
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$)
forces
$i_{\Theta^{\prime}}^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}-2}n^{s_{0},d_{0}}_{d^{\prime}}$
(resp.
$i_{\Theta^{\prime}}\leq\sum_{d^{\prime}=1}^{d_{0}-2}n^{s_{0},d_{0}}_{d^{\prime}}$),
therefore we always have

$\sum_{d^{\prime}=1}^{d_{0}-2}n^{\prime}_{d^{\prime}}=i_{\Theta^{\prime}}^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}-2}n^{s_{0},d_{0}}_{d^{\prime}}=\sum_{d^{\prime}=1}^{d_{0}-2}n^{s_{0},d_{0}-1}_{d^{\prime}}.$

So it follows from Lemma 2.31 (with $d_{0}$ replacing with $d_{0}-1$) that
$\Theta^{\prime}\in\Omega^{s_{0},d_{0}-1}$ if and only if

(2.19)
$\sum_{d^{\prime}=1}^{d_{0}-1}n^{\prime}_{d^{\prime}}=i_{\Theta^{\prime}}\geq\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}-1}_{d^{\prime}}.$

We have the following possibilities

  * •

If (2.19) holds, then we clearly deduce
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\notin\Omega^{s_{0},d_{0},+}$
from the definition of $\Omega^{s_{0},d_{0},+}$ (namely any
$\Theta^{\prime\prime}\in\Omega^{s_{0},d_{0},+}$ should satisfy
$\sum_{d^{\prime}=1}^{d_{0}-2}n^{\prime\prime}_{d^{\prime}}\leq\sum_{d^{\prime}=1}^{d_{0}-2}n^{s_{0},d_{0}}_{d^{\prime}}=\sum_{d^{\prime}=1}^{d_{0}-2}n^{s_{0},d_{0}-1}_{d^{\prime}}$).

  * •

If (2.19) is false, then we deduce from $i_{\Theta^{\prime}}\notin I_{0}$ that

(2.20)
$\sum_{d^{\prime}=1}^{d_{0}-1}n^{\prime}_{d^{\prime}}=i_{\Theta^{\prime}}\leq\sum_{d^{\prime}=1}^{d_{0}-2}n^{s_{0},d_{0}-1}_{d^{\prime}}=\sum_{d^{\prime}=1}^{d_{0}-2}n^{s_{0},d_{0}}_{d^{\prime}}.$

Assume on the contrary that
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\notin\Omega^{s_{0},d_{0},+}$,
then we must have
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$ by the
choice of $\Theta^{\prime}$, and thus there exists
$\Theta^{\prime\prime\prime}\in\Omega^{s_{0},d_{0}}$ such that
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})=p_{i_{\Theta^{\prime\prime\prime}}^{\prime}}^{+}(\Theta^{\prime\prime\prime})$.
But this together with (2.20) and Lemma 2.31 implies that
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}p_{i_{\Theta^{\prime}}}^{-}(\Theta^{\prime\prime\prime})\in\Omega^{s_{0},d_{0}}$
and

$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})=p_{i_{\Theta^{\prime\prime\prime}}^{\prime}}^{+}\left(p_{i_{\Theta^{\prime}}^{\prime}}^{+}p_{i_{\Theta^{\prime}}}^{-}(\Theta^{\prime\prime\prime})\right)\in\Omega^{s_{0},d_{0},+},$

a contradiction. Hence we deduce that
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$
and thus there exists
$\Theta^{\prime\prime\prime\prime}\in\Omega^{s_{0},d_{0}}$ such that
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})=p_{i_{\Theta^{\prime\prime\prime\prime}}^{\prime}}^{+}(\Theta^{\prime\prime\prime\prime})$
(with
$i_{\Theta^{\prime\prime\prime\prime}}^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}}_{d^{\prime}}$
and
$i_{\Theta^{\prime\prime\prime\prime}}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$).
We consider the unique tuple $\Theta^{\prime\prime\prime\prime\prime}$ that
satisfies
$\Theta^{\prime\prime\prime\prime}=p_{i_{\Theta^{\prime}}^{\prime}}^{+}p_{i_{\Theta^{\prime}}}^{-}(\Theta^{\prime\prime\prime\prime\prime})$,
which clearly satisfies
$i_{\Theta^{\prime\prime\prime\prime\prime}}^{\prime}=i_{\Theta^{\prime\prime\prime\prime}}^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}}_{d^{\prime}}$
and
$i_{\Theta^{\prime\prime\prime\prime\prime}}=i_{\Theta^{\prime\prime\prime\prime}}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$,
and thus $\Theta^{\prime\prime\prime\prime\prime}\in\Omega^{s_{0},d_{0}}$ by
Lemma 2.31. Consequently, we have
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})=p_{i_{\Theta^{\prime\prime\prime\prime\prime}}^{\prime}}^{+}(\Theta^{\prime\prime\prime\prime\prime})\in\Omega^{s_{0},d_{0},+}$.

Now we treat item (ii), namely the case when $d_{1}=d_{0}$. We note that
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$
(resp.
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$)
forces
$i_{\Theta^{\prime}}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$
(resp.
$i_{\Theta^{\prime}}^{\prime}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$),
therefore we always have

$\sum_{d^{\prime}=1}^{d_{0}}n^{\prime}_{d^{\prime}}=i_{\Theta^{\prime}}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}.$

So it follows from Lemma 2.31 that $\Theta^{\prime}\in\Omega^{s_{0},d_{0}}$ if
and only if

(2.21)
$\sum_{d^{\prime}=1}^{d_{0}-1}n^{\prime}_{d^{\prime}}=i_{\Theta^{\prime}}^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}}_{d^{\prime}}.$

We have the following possibilities

  * •

If (2.21) holds, then we clearly deduce
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\notin\Omega^{s_{0},d_{0},+}$
from the definition of $\Omega^{s_{0},d_{0},+}$ (namely any
$\Theta^{\prime\prime}\in\Omega^{s_{0},d_{0},+}$ should satisfy
$\sum_{d^{\prime}=1}^{d_{0}-1}n^{\prime\prime}_{d^{\prime}}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$).

  * •

If (2.21) is false, then we deduce from $i_{\Theta^{\prime}}^{\prime}\notin
I_{0}$ that

(2.22)
$\sum_{d^{\prime}=1}^{d_{0}-1}n^{\prime}_{d^{\prime}}=i_{\Theta^{\prime}}^{\prime}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}.$

Assume on the contrary that
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\notin\Omega^{s_{0},d_{0},+}$,
then we must have
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$
by the choice of $\Theta^{\prime}$, and thus there exists
$\Theta^{\prime\prime\prime}\in\Omega^{s_{0},d_{0}}$ such that
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})=p_{i_{\Theta^{\prime\prime\prime}}^{\prime}}^{+}(\Theta^{\prime\prime\prime})$
and
$i_{\Theta^{\prime\prime\prime}}^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$.
But this together with (2.22) and Lemma 2.31 implies that
$p_{i_{\Theta^{\prime}}}^{+}p_{i_{\Theta^{\prime}}^{\prime}}^{-}(\Theta^{\prime\prime\prime})\in\Omega^{s_{0},d_{0}}$
and

$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})=p_{i_{\Theta^{\prime\prime\prime}}^{\prime}}^{+}\left(p_{i_{\Theta^{\prime}}}^{+}p_{i_{\Theta^{\prime}}^{\prime}}^{-}(\Theta^{\prime\prime\prime})\right)\in\Omega^{s_{0},d_{0},+},$

a contradiction. Hence we deduce that
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$ and
thus there exists $\Theta^{\prime\prime\prime\prime}\in\Omega^{s_{0},d_{0}}$
such that
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})=p_{i_{\Theta^{\prime\prime\prime\prime}}^{\prime}}^{+}(\Theta^{\prime\prime\prime\prime})$
(with
$i_{\Theta^{\prime\prime\prime\prime}}^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}}_{d^{\prime}}$
and
$i_{\Theta^{\prime\prime\prime\prime}}=i_{\Theta^{\prime}}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$).
We consider the unique tuple $\Theta^{\prime\prime\prime\prime\prime}$ that
satisfies
$\Theta^{\prime\prime\prime\prime}=p_{i_{\Theta^{\prime}}}^{+}p_{i_{\Theta^{\prime}}^{\prime}}^{-}(\Theta^{\prime\prime\prime\prime\prime})$,
which clearly satisfies
$i_{\Theta^{\prime\prime\prime\prime\prime}}^{\prime}=i_{\Theta^{\prime\prime\prime\prime}}^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}}_{d^{\prime}}$
and
$i_{\Theta^{\prime\prime\prime\prime\prime}}=i_{\Theta^{\prime}}^{\prime}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$,
and thus $\Theta^{\prime\prime\prime\prime\prime}\in\Omega^{s_{0},d_{0}}$ by
Lemma 2.31. Consequently, we have
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})=p_{i_{\Theta^{\prime\prime\prime\prime\prime}}^{\prime}}^{+}(\Theta^{\prime\prime\prime\prime\prime})\in\Omega^{s_{0},d_{0},+}$.

Finally, we note that there exists a number
$\varepsilon(\Omega,s_{0},d_{0})\in\\{1,-1\\}$ depending only on $\Omega$,
$s_{0}$ and $d_{0}$ such that, for each $\Theta^{\prime}$ considered above, we
have

$\varepsilon(p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})}))=-\varepsilon(\Omega,s_{0},d_{0})\varepsilon(\Theta^{\prime})$

if $d_{1}=d_{0}-1$, and

$\varepsilon(p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})}))=\varepsilon(\Omega,s_{0},d_{0})\varepsilon(\Theta^{\prime})$

if $d_{1}=d_{0}$. This together with (2.17) and (2.18) finish the proof of
(2.14). ∎

##  3\. A result on decomposition of locally analytic distributions

In this section, we prove a technical result on (tensor) decomposition of
certain space of locally analytic distributions in Proposition 3.9, which is
essential for our computations of $N$-(co)homologies using Bruhat
stratifications in Section 4.1.

######  Lemma 3.1.

Let $I$ be a set and $(V_{i})_{i\in I}$ be a projective system of locally
convex $E$ vector space and $W$ another locally convex $E$ vector space. Then
we have a canonical topological isomorphism

$(\varprojlim_{i}V_{i})\widehat{\otimes}_{E,\pi}W\cong\varprojlim_{i}(V_{i}\widehat{\otimes}_{E,\pi}W).$

If there exists a locally convex $E$-vector space $A$ with separately
continuous algebra structure such that $W$ is a separately continuous left
$A$-module and $V_{i}$ is a separately continuous right $A$-module for each
$i\in I$, then we have a canonical isomorphism

$(\prod_{i}V_{i})\widehat{\otimes}_{A,\pi}W\cong\prod_{i}(V_{i}\widehat{\otimes}_{A,\pi}W).$

###### Proof.

This is standard (using for example Corollary 17.9 of [S02]). ∎

According to Proposition 17.6 of [S02], we can identify
$V_{1}\widehat{\otimes}_{E,\iota}V_{2}$ and
$V_{1}\widehat{\otimes}_{E,\pi}V_{2}$ for two Frechét spaces $V_{1}$, $V_{2}$
and sometimes write $V_{1}\widehat{\otimes}_{E}V_{2}$ for both of them. Note
that a countable projective limit of Fréchet spaces is still Fréchet, but a
countable inductive limit (in particular countable direct sum) of Fréchet
spaces is in general not Fréchet. Consequently, the notation
$V_{1}\widehat{\otimes}_{E,\iota}V_{2}$ would appear in the sequel typically
when $V_{2}$ is not Fréchet (say a countable direct sum of Fréchet spaces).

For each $p$-adic manifold $M$ (paracompact and finite dimensional), we can
consider the space of $E$-valued locally analytic functions
$C^{\rm{an}}(M,E)$, whose strong dual is the space of distributions $D(M,E)$.
For each closed subset $C\subseteq M$, we consider the closed subspace
$D(M,E)_{C}\subseteq D(M,E)$ consisting of distributions supported in $C$. If
$C$ is furthermore compact, we have an isomorphism
$D(M,E)_{C}\cong(\varinjlim_{U}C^{\rm{an}}(U,E))^{\prime}$ where $U$ runs
through open subset of $M$ containing $C$. In particular, for each open subset
$U\subseteq M$ containing $C$, we have a canonical isomorphism
$D(U,E)_{C}\cong D(M,E)_{C}$.

######  Lemma 3.2.

Let $M$ be a $p$-adic manifold and $C_{1}\subseteq C_{2}$ two compact subsets
of $M$. Then we have a topological isomorphism

$D(M,E)_{C_{2}}/D(M,E)_{C_{1}}\cong\varprojlim_{C}D(M,E)_{C}$

where $C$ runs through compact open subsets of $C_{2}\setminus C_{1}$.

###### Proof.

As $C$ is compact open in $C_{2}$, $C_{2}\cong C\sqcup(C_{2}\setminus C)$
(with disjoint union topology) and thus $D(M,E)_{C_{2}}\cong D(M,E)_{C}\oplus
D(M,E)_{C_{2}\setminus C}$ which induces a canonical projection
$D(M,E)_{C_{2}}\twoheadrightarrow D(M,E)_{C}$ with kernel
$D(M,E)_{C_{2}\setminus C}\subseteq D(M,E)_{C_{2}}$. For each pair of compact
open subsets $C\subseteq C^{\prime}$ of $C_{2}\setminus C_{1}$, we have a
canonical projection map $D(M,E)_{C^{\prime}}\cong D(M,E)_{C}\oplus
D(M,E)_{C^{\prime}\setminus C}\twoheadrightarrow D(M,E)_{C}$, and thus we
obtain a continuous map $D(M,E)_{C_{2}}\rightarrow\varprojlim_{C}D(M,E)_{C}$
with dense image and kernel $\bigcap_{C}D(M,E)_{C_{2}\setminus C}$. As
$C_{2}\setminus C_{1}$ is a union of its compact open subsets, the
intersection of $C_{2}\setminus C$ for all possible compact open $C\subseteq
C_{2}\setminus C_{1}$ is exactly $C_{1}$, which implies that
$\bigcap_{C}D(M,E)_{C_{2}\setminus C}=D(M,E)_{C_{1}}$. Now we prove the
surjectivity of $D(M,E)_{C_{2}}\rightarrow\varprojlim_{C}D(M,E)_{C}$ (which
implies that
$D(M,E)_{C_{2}}/D(M,E)_{C_{1}}\rightarrow\varprojlim_{C}D(M,E)_{C}$ is a
topological isomorphism). As $C_{2}$ is compact, it suffices to replace $M$
with a compact open neighborhood of $C_{2}$ and thus assume that $M$ is
compact, which implies that
$C^{\rm{an}}(M,E)=\varinjlim_{r}C^{\rm{an}}_{r}(M,E)$ where
$C^{\rm{an}}_{r}(M,E)\subseteq C^{\rm{an}}(M,E)$ is the subspace consisting of
locally analytic functions that are analytic on each open disc of radius $r$.
By taking dual, we have $D(M,E)=\varprojlim_{r}D_{r}(M,E)$ with
$D_{r}(M,E)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(C^{\rm{an}}_{r}(M,E))^{\prime}$.
For each compact subset $C\subseteq M$, we write $D_{r}(M,E)_{C}$ for the
image of $D(M,E)_{C}\subseteq D(M,E)$ in $D_{r}(M,E)$, then we have
$D(M,E)_{C}\cong\varprojlim_{r}D_{r}(M,E)_{C}$. If we write $C_{r}$ for the
union of all (finite number of) open disc of radius $r$ that intersect $C$
non-trivially, then $C_{r}$ is a compact open subset of $M$ that contains $C$
and satisfies $D_{r}(M,E)_{C}=D_{r}(M,E)_{C_{r}}$. Now let $C$ be a compact
open subset of $C_{2}\setminus C_{1}$. For each fixed $r$, $C_{r}$ stablizes
when $C$ is sufficiently large, and thus
$\varprojlim_{C}D_{r}(M,E)_{C}=D_{r}(M,E)_{C}$ for some sufficiently large
$C$. Hence, the canonical map
$D_{r}(M,E)_{C_{2}}\rightarrow\varprojlim_{C}D_{r}(M,E)_{C}$ is a continuous
surjection (of Banach spaces). For each $0<r^{\prime}<r<1$,
$D_{r^{\prime}}(M,E)_{C_{2}}\rightarrow D_{r}(M,E)_{C_{2}}$ and
$\varprojlim_{C}D_{r^{\prime}}(M,E)_{C}\rightarrow\varprojlim_{C}D_{r}(M,E)_{C}$
are injective, and thus
$\varprojlim_{r}D_{r}(M,E)_{C_{2}}\rightarrow\varprojlim_{r}\varprojlim_{C}D_{r}(M,E)_{C}$
is a continuous surjection. Note that we clearly have
$\varprojlim_{r}\varprojlim_{C}D_{r}(M,E)_{C}=\varprojlim_{C}\varprojlim_{r}D_{r}(M,E)_{C}=\varprojlim_{C}D(M,E)_{C}$,
and the proof is thus finished. ∎

######  Lemma 3.3.

Let $M_{1},M_{2}$ two $p$-adic manifolds with closed subsets $C_{1}\subseteq
M_{1}$ and $C_{2}\subseteq M_{2}$. Then we have a canonical isomorphism
$D(M_{1}\times M_{2},E)_{C_{1}\times C_{2}}\cong
D(M_{1},E)_{C_{1}}\widehat{\otimes}_{E,\iota}D(M_{2},E)_{C_{2}}$.

###### Proof.

This is Lemma 3.2.12 of [BD19]. ∎

Let $G_{/\mathbb{Q}_{p}}$ be a reductive group and $P_{0}\subseteq G$ be a
fixed minimal parabolic subgroup containing a fixed maximal split torus
$L_{0}$. Let $P_{1},P_{2}\subseteq G$ be two parabolic subgroups containing
$P_{0}$. We can associate with $P_{1},P_{2}$ the Bruhat stratification
$\\{P_{1}wP_{2}\\}_{W^{P_{1},P_{2}}}$ where $W^{P_{1},P_{2}}$ is the
corresponding set of minimal length representatives. We write
$P_{1}^{\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}w^{-1}P_{1}w$,
$Y\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}P_{1}^{\prime}P_{2}\subseteq
G$, $\overline{Y}$ for the (reduced) Zariski closure of $Y$ in $G$ and
$Z\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\overline{Y}\setminus Y$
with the reduced scheme structure. We write $N_{2}$ for the unipotent radical
of $P_{2}$ and $L_{2}\subseteq P_{2}$ its maximal Levi subgroup containing
$L_{0}$. Then we set
$L_{2}^{\flat}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}L_{2}\cap
P_{1}^{\prime}$ and
$N_{2}^{\flat}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}N_{2}\cap
P_{1}^{\prime}$. Note that

$Y\cong P_{1}^{\prime}\times(P_{1}^{\prime}\backslash Y)\cong
P_{1}^{\prime}\times(P_{1}^{\prime}\cap P_{2}\backslash P_{2})\cong
P_{1}^{\prime}\times(L_{2}^{\flat}\backslash
L_{2})\times(N_{2}^{\flat}\backslash N_{2})$

and the isomorphisms preserve the natural left $P_{1}^{\prime}$-action and
right $P_{2}$-action on each term. Note that $L_{2}^{\flat}$ is a parabolic
subgroup inside $L_{2}$, and thus $L_{2}^{\flat}\backslash L_{2}$ is proper.

We write
$D(G)_{\overline{Y}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}D(G(\mathbb{Q}_{p}),E)_{\overline{Y}(\mathbb{Q}_{p})}$
for short and similarly for others (in particular, we omit $E$ from all spaces
of distributions). As $G$, $\overline{Y}$ and $Z$ are left
$P_{1}^{\prime}$-stable and right $P_{2}$-stable, $D(G)_{\overline{Y}}$ and
$D(G)_{Z}$ are naturally left $D(P_{2})$-modules and right
$D(P_{1}^{\prime})$-modules. In the following, we will frequently abbreviate
$P_{1}^{\prime}(\mathbb{Q}_{p})$ as $P_{1}^{\prime}$ and the others are
similar.

We set
$\widehat{D}(G)_{Y}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\left(\varprojlim_{C}D(P_{1}^{\prime}\backslash
G)_{C}\right)\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime})$ where $C$ running
through all compact open subsets of $P_{1}^{\prime}\backslash
Y(\mathbb{Q}_{p})$.

######  Lemma 3.4.

The limit $\widehat{D}(G)_{Y}=\left(\varprojlim_{C}D(P_{1}^{\prime}\backslash
G)_{C}\right)\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime})$ is canonically a
left $D(P_{2})$-module and a right $D(P_{1}^{\prime})$-module.

###### Proof.

The right $D(P_{1}^{\prime})$-module structure is obvious from the definition
of $\widehat{D}(G)_{Y}$. Let $H_{2}\subseteq P_{2}(\mathbb{Q}_{p})$ be an
arbitrary compact open subgroup. Then for each compact open subset $C\subseteq
P_{1}^{\prime}\backslash Y(\mathbb{Q}_{p})$, $CH_{2}\subseteq
P_{1}^{\prime}\backslash Y(\mathbb{Q}_{p})$ is an open closed subset
(containing $C$) which is right $H_{2}$-stable. Consequently, right
$H_{2}$-stable compact open subsets of $P_{1}^{\prime}\backslash
Y(\mathbb{Q}_{p})$ are cofinal among all compact open subsets of
$P_{1}^{\prime}\backslash Y(\mathbb{Q}_{p})$, which implies that
$\varprojlim_{C}D(P_{1}^{\prime}\backslash G)_{C}$ is canonically a left
$D(H_{2})$-module. There clearly exist a discrete subset $J_{2}\subseteq
P_{2}(\mathbb{Q}_{p})$ such that $P_{2}(\mathbb{Q}_{p})=\bigsqcup_{g_{2}\in
J_{2}}H_{2}g_{2}$ as a $p$-adic manifold. For each compact open subset $C$ of
$P_{1}^{\prime}\backslash Y(\mathbb{Q}_{p})$ which is right $H_{2}$-stable, we
have canonical isomorphisms $\ell_{g_{2}^{-1}}:D(P_{1}^{\prime}\backslash
G)_{C}\rightarrow\delta_{g_{2}^{-1}}\ast D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}C}=D(P_{1}^{\prime}\backslash G)_{C\cdot g_{2}}$. If we take
inverse limit among all such $C$, we obtain the canonical left
$D(P_{2})$-action on $\varprojlim_{C}D(P_{1}^{\prime}\backslash G)_{C}$ and
thus on $\widehat{D}(G)_{Y}=\left(\varprojlim_{C}D(P_{1}^{\prime}\backslash
G)_{C}\right)\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime})$ as well. ∎

######  Proposition 3.5.

There exists a canonical isomorphisms
$D(G)_{\overline{Y}}/D(G)_{Z}\cong\widehat{D}(G)_{Y}$ which respects the left
$D(P_{2})$-action and right $D(P_{1}^{\prime})$-action.

###### Proof.

It follows from Lemma 3.3 that

$D(G)_{\overline{Y}}\cong D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash\overline{Y}}\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime})\mbox{
and }D(G)_{Z}\cong D(P_{1}^{\prime}\backslash G)_{P_{1}^{\prime}\backslash
Z}\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime}),$

and thus

$(D(G)_{\overline{Y}}/D(G)_{Z}\cong\left(D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash\overline{Y}}/D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash
Z})\right)\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime}).$

Consequently, it remains to construct a canonical $D(P_{2})$-equivariant
isomorphism

(3.1) $D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash\overline{Y}}/D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash Z}\cong\varprojlim_{C}D(P_{1}^{\prime}\backslash
G)_{C}$

according to the definition of $\widehat{D}(G)_{Y}$. This follows from Lemma
3.2 as well as the facts that both
$(P_{1}^{\prime}\backslash\overline{Y})(\mathbb{Q}_{p})$ and
$(P_{1}^{\prime}\backslash Z)(\mathbb{Q}_{p})$ are compact subsets of
$(P_{1}^{\prime}\backslash G)(\mathbb{Q}_{p})$. Now we check the left
$D(P_{2})$-equivariance. We borrow notation from the proof of Lemma 3.4. If
$C$ is a right $H_{2}$-stable compact open subset of $P_{1}^{\prime}\backslash
Y(\mathbb{Q}_{p})$, then the map $D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash\overline{Y}}/D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash Z}\rightarrow D(P_{1}^{\prime}\backslash G)_{C}$
is clearly left $D(H_{2})$-equivariant (as the $D(H_{2})$-actions on both
sides are compatible with that on $D(G)$). The extension to left
$D(P_{2})$-equivariance of (3.1) follows from the same argument as in the
proof of Lemma 3.4. ∎

Let $G_{0}\subseteq G(\mathbb{Q}_{p})$ be a compact open subgroup and choose
$H_{1}\subseteq G_{0}\cap P_{1}^{\prime}(\mathbb{Q}_{p})$ and $H_{2}\subseteq
G_{0}\cap P_{2}(\mathbb{Q}_{p})$. Then exactly the same proof as that of
(4.87) of [Schr11] shows that we have a canonical topological isomorphism
(with all the involved spaces being Fréchet)

(3.2) $D(G_{0})_{H_{1}H_{2}}\cong
D(H_{2})\widehat{\otimes}_{D(H_{2})_{H_{2}\cap H_{1}}}D(G_{0})_{H_{1}}.$

It is clearly harmless to replace $G_{0}$ with $G$ as it does not change the
corresponding space of distributions. So we drop $G_{0}$ from now on and
consider compact open subgroups $H_{1}\subseteq
P_{1}^{\prime}(\mathbb{Q}_{p})$ and $H_{2}\subseteq P_{2}(\mathbb{Q}_{p})$. We
fix discrete subset $J_{1}\subseteq P_{1}^{\prime}(\mathbb{Q}_{p})$ and
$J_{2}\subseteq P_{2}(\mathbb{Q}_{p})$ such that
$P_{1}^{\prime}(\mathbb{Q}_{p})=\bigsqcup_{g_{1}\in J_{1}}g_{1}H_{1}$ and
$P_{2}(\mathbb{Q}_{p})=\bigsqcup_{g_{2}\in J_{2}}H_{2}g_{2}$ as $p$-adic
manifolds. We set
$H_{2}^{\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}H_{2}\cap
L_{2}(\mathbb{Q}_{p})$ and
$H_{2}^{\prime\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}H_{2}\cap
N_{2}(\mathbb{Q}_{p})$.

######  Definition 3.6.

We say the pair $(H_{1},H_{2})$ is _good_ if

  1. (i)

$H_{2}=H_{2}^{\prime}H_{2}^{\prime\prime}$, $H_{2}\cap
P_{1}^{\prime}(\mathbb{Q}_{p})=H_{2}\cap H_{1}=(H_{2}^{\prime}\cap
H_{1})(H_{2}^{\prime\prime}\cap H_{1})$;

  2. (ii)

$(H_{2}^{\prime}\cap H_{1})\backslash H_{2}^{\prime}\cong
L_{2}^{\flat}(\mathbb{Q}_{p})\backslash L_{2}(\mathbb{Q}_{p})$; and

  3. (iii)

$P_{1}^{\prime}(\mathbb{Q}_{p})H_{2}=\bigsqcup_{g_{1}\in
J_{1}}g_{1}H_{1}H_{2}$.

Item (i) clearly implies topological isomorphisms (of Fréchet spaces)
$D(H_{2})\cong D(H_{2}^{\prime})\widehat{\otimes}_{E}D(H_{2}^{\prime\prime})$
and $D(H_{2})_{H_{2}\cap H_{1}}\cong D(H_{2}^{\prime})_{H_{2}^{\prime}\cap
H_{1}}\widehat{\otimes}_{E}D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
H_{1}}$ with the convolution having the form induced from semi-direct product.
These topological isomorphisms together with (3.2) implies that

$D(G)_{H_{1}H_{2}}\cong
D(H_{2}^{\prime})\widehat{\otimes}_{D(H_{2}^{\prime})_{H_{2}^{\prime}\cap
H_{1}}}(D(H_{2}^{\prime\prime})\widehat{\otimes}_{D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
H_{1}}}D(G)_{H_{1}}).$

By convolution with $\delta_{g_{1}^{-1}}$ from the right, we obtain

$D(G)_{g_{1}H_{1}H_{2}}\cong
D(H_{2}^{\prime})\widehat{\otimes}_{D(H_{2}^{\prime})_{H_{2}^{\prime}\cap
H_{1}}}(D(H_{2}^{\prime\prime})\widehat{\otimes}_{D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
H_{1}}}D(G)_{g_{1}H_{1}}).$

for each $g_{1}\in P_{1}^{\prime}(\mathbb{Q}_{p})$. As
$P_{1}^{\prime}(\mathbb{Q}_{p})$ is closed in $G(\mathbb{Q}_{p})$ and $H_{2}$
is compact, $P_{1}^{\prime}(\mathbb{Q}_{p})H_{2}$ is closed in
$G(\mathbb{Q}_{p})$. By taking direct sum over all $g_{1}\in J_{1}$ and using
item (iii) of Definition 3.6 (and the fact that $\widehat{\otimes}_{\iota}$
commutes with direct sum), we obtain (by abusing $P_{1}^{\prime}$ for
$P_{1}^{\prime}(\mathbb{Q}_{p})$)

$D(G)_{P_{1}^{\prime}H_{2}}\cong
D(H_{2}^{\prime})\widehat{\otimes}_{D(H_{2}^{\prime})_{H_{2}^{\prime}\cap
P_{1}^{\prime}}}(D(H_{2}^{\prime\prime})\widehat{\otimes}_{D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
P_{1}^{\prime}}}D(G)_{P_{1}^{\prime}}).$

According to item (ii) of Definition 3.6, we can replace
$D(H_{2}^{\prime})\widehat{\otimes}_{D(H_{2}^{\prime})_{H_{2}^{\prime}\cap
H_{1}}}$ with $D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}}}$ and
deduce

(3.3) $D(G)_{P_{1}^{\prime}H_{2}}\cong
D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}}}(D(H_{2}^{\prime\prime})\widehat{\otimes}_{D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
P_{1}^{\prime}}}D(G)_{P_{1}^{\prime}}).$

(Note that $L_{2}^{\flat}\backslash L_{2}$ is compact and thus we do not
distinguish between
$D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}},\iota}$ with
$D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}},\pi}$.) By factoring out
$\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime})$ from both sides of (3.3), we
obtain the following isomorphism (with $D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash P_{1}^{\prime}H_{2}}$, $D(H_{2}^{\prime\prime})$
and $D(P_{1}^{\prime}\backslash G)_{1}$ being Fréchet)

(3.4) $D(P_{1}^{\prime}\backslash G)_{P_{1}^{\prime}\backslash
P_{1}^{\prime}H_{2}}\cong
D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}}}(D(H_{2}^{\prime\prime})\widehat{\otimes}_{D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
P_{1}^{\prime}}}D(P_{1}^{\prime}\backslash G)_{1}).$

We have the following two observations

  * •

If $(H_{1},H_{2})$ is a good pair, then $(gH_{1}g^{-1},gH_{2}g^{-1})$ is a
good pair for any $g\in L_{0}(\mathbb{Q}_{p})$.

  * •

For each compact open subgroup $H\subseteq N_{2}(\mathbb{Q}_{p})$, we have
$N_{2}(\mathbb{Q}_{p})=\bigcup_{g\in L_{0}(\mathbb{Q}_{p})}gHg^{-1}$. In
particular $\\{gHg^{-1}\\}_{g\in L_{0}(\mathbb{Q}_{p})}$ is cofinal among all
compact open subgroups of $N_{2}(\mathbb{Q}_{p})$ under inclusion.

Consequently, if we take (countable) projective limit with respect all
possible $H_{2}$ that appears in a good pair $(H_{1},H_{2})$ (and then use
Lemma 3.1 for $D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}},\pi}$), we
conclude that

(3.5) $\varprojlim_{H_{2}}D(P_{1}^{\prime}\backslash
G)_{P_{1}^{\prime}\backslash P_{1}^{\prime}H_{2}}\cong
D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}}}\left(\varprojlim_{H_{2}^{\prime\prime}}D(H_{2}^{\prime\prime})\widehat{\otimes}_{D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
P_{1}^{\prime}}}D(P_{1}^{\prime}\backslash G)_{1}\right).$

We have $H_{2}^{\prime\prime}\cap P_{1}^{\prime}=H_{2}^{\prime\prime}\cap
N_{2}^{\flat}$, $(H_{2}^{\prime\prime}\cap N_{2}^{\flat})\backslash
H_{2}^{\prime\prime}\cong N_{2}^{\flat}\backslash
N_{2}^{\flat}H_{2}^{\prime\prime}$, as well as canonical topological
isomorphisms

$D(N_{2}^{\flat}H_{2}^{\prime\prime})\cong D(N_{2}^{\flat}\backslash
N_{2}^{\flat}H_{2}^{\prime\prime})\widehat{\otimes}_{E,\iota}D(N_{2}^{\flat})$

and

$D(N_{2}^{\flat}\backslash
N_{2}^{\flat}H_{2}^{\prime\prime})_{1}\widehat{\otimes}_{E,\iota}D(N_{2}^{\flat})\cong
D(N_{2}^{\flat}H_{2}^{\prime\prime})_{N_{2}^{\flat}}\cong
D(N_{2})_{N_{2}^{\flat}}\cong D(N_{2}^{\flat}\backslash
N_{2})_{1}\widehat{\otimes}_{E,\iota}D(N_{2}^{\flat}).$

Note that for each Fréchet space $V$ and $\ast\in\\{\pi,\iota\\}$,
$D(N_{2}^{\flat}H_{2}^{\prime\prime})\widehat{\otimes}_{D(N_{2})_{N_{2}^{\flat}},\ast}V$
is a quotient of $D(N_{2}^{\flat}\backslash
N_{2}^{\flat}H_{2}^{\prime\prime})\widehat{\otimes}_{E,\ast}V$ by a closed
subspace with $D(N_{2}^{\flat}\backslash N_{2}^{\flat}H_{2}^{\prime\prime})$
being Fréchet, so we do not distinguish between
$D(N_{2}^{\flat}H_{2}^{\prime\prime})\widehat{\otimes}_{D(N_{2})_{N_{2}^{\flat}},\pi}$
and
$D(N_{2}^{\flat}H_{2}^{\prime\prime})\widehat{\otimes}_{D(N_{2})_{N_{2}^{\flat}},\iota}$.
Consequently, we have a topological isomorphism

$D(H_{2}^{\prime\prime})\widehat{\otimes}_{D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
P_{1}^{\prime}}}D(P_{1}^{\prime}\backslash G)_{1}\cong
D(N_{2}^{\flat}H_{2}^{\prime\prime})\widehat{\otimes}_{D(N_{2})_{N_{2}^{\flat}}}D(P_{1}^{\prime}\backslash
G)_{1}$

for each $H_{2}$ that appears in a good pair $(H_{1},H_{2})$, which implies
that (using Lemma 3.1)

$\varprojlim_{H_{2}^{\prime\prime}}D(H_{2}^{\prime\prime})\widehat{\otimes}_{D(H_{2}^{\prime\prime})_{H_{2}^{\prime\prime}\cap
P_{1}^{\prime}}}D(P_{1}^{\prime}\backslash
G)_{1}\cong\varprojlim_{H_{2}^{\prime\prime}}D(N_{2}^{\flat}H_{2}^{\prime\prime})\widehat{\otimes}_{D(N_{2})_{N_{2}^{\flat}}}D(P_{1}^{\prime}\backslash
G)_{1}\cong\widehat{D}(N_{2})\widehat{\otimes}_{D(N_{2})_{N_{2}^{\flat}}}D(P_{1}^{\prime}\backslash
G)_{1}$

where
$\widehat{D}(N_{2})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\left(\varprojlim_{H_{2}^{\prime\prime}}D(N_{2}^{\flat}\backslash
N_{2}^{\flat}H_{2}^{\prime\prime})\right)\widehat{\otimes}_{E,\iota}D(N_{2}^{\flat})$.
This together with (3.5) implies that

(3.6) $\widehat{D}(G)_{Y}\cong
D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}}}(\widehat{D}(N_{2})\widehat{\otimes}_{D(N_{2})_{N_{2}^{\flat}}}D(P_{1}^{\prime}\backslash
G)_{1})\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime}).$

Now we consider the following condition

######  Condition 3.7.

There exists a $p$-adic field $K$, a split reductive group
$\widetilde{G}_{/L}$ together with its two parabolic subgroups
$\widetilde{P}_{1}^{\prime}$ and $\widetilde{P}_{2}$ such that
$G\cong\mathrm{Res}_{K/\mathbb{Q}_{p}}(\widetilde{G})$,
$P_{1}^{\prime}\cong\mathrm{Res}_{K/\mathbb{Q}_{p}}(\widetilde{P}_{1}^{\prime})$
and $P_{2}\cong\mathrm{Res}_{K/\mathbb{Q}_{p}}(\widetilde{P}_{2})$.

######  Lemma 3.8.

If Condition 3.7 holds, then a good pair $(H_{1},H_{2})$ exists.

###### Proof.

We fix $K$, $\widetilde{G}$, $\widetilde{P}_{1}^{\prime}$ and
$\widetilde{P}_{2}$ as in Condition 3.7. As $\widetilde{G}$ is split, we abuse
the same notation for the Chevalley group scheme over $\mathcal{O}_{K}$, where
$\mathcal{O}_{K}$ is the ring of integers of $K$. Similarly,
$\widetilde{P}_{1}^{\prime}$ and $\widetilde{P}_{2}$ can be made closed
subgroup schemes of $\widetilde{G}_{/\mathcal{O}_{K}}$. Let
$\widetilde{L}_{0}\subseteq\widetilde{P}_{1}^{\prime}\cap\widetilde{P}_{2}$ be
a (split) maximal torus which always exists. Upon replacing $L_{0}$ with
$\mathrm{Res}_{K/\mathbb{Q}_{p}}\widetilde{L}_{0}$, we may assume that
$L_{0}\cong\mathrm{Res}_{K/\mathbb{Q}_{p}}\widetilde{L}_{0}$ and moreover
$\widetilde{L}_{0}$ extends to a closed subgroup scheme of
$\widetilde{G}_{/\mathcal{O}_{K}}$. Given the pair
$(\widetilde{G},\widetilde{L}_{0})$, we consider the corresponding set of
roots $\Phi$, and $\widetilde{P}_{1}^{\prime}$ (resp. $\widetilde{P}_{2}$)
corresponds to a (closed) subset $\Phi_{1}$ (resp. $\Phi_{2}$) of $\Phi$. We
write $\widetilde{N}_{2}$ for the unipotent radical of $\widetilde{P}_{2}$ and
there exists a unique maximal Levi subgroup
$\widetilde{L}_{2}\subseteq\widetilde{P}_{2}$ containing $\widetilde{L}_{0}$,
such that $\widetilde{P}_{2}=\widetilde{L}_{2}\widetilde{N}_{2}$ is a semi
direct product. In the following, we understand all the group schemes
mentioned above to be over $\mathcal{O}_{K}$. The closed subgroup
$\widetilde{L}_{2}$ (resp. $\widetilde{N}_{2}$) of $\widetilde{P}_{2}$
corresponds to a closed subset $\Phi_{2}^{\prime}$ (resp.
$\Phi_{2}^{\prime\prime}$) of $\Phi_{2}$ such that
$\Phi_{2}=\Phi_{2}^{\prime}\sqcup\Phi_{2}^{\prime\prime}$. The closed subsets
$\Phi_{1}\cap\Phi_{2}$, $\Phi_{1}\cap\Phi_{2}^{\prime}$ and
$\Phi_{1}\cap\Phi_{2}^{\prime\prime}$ of $\Phi_{2}$ corresponds to closed
subgroup schemes $\widetilde{P}_{1}\cap\widetilde{P}_{2}$,
$\widetilde{P}_{1}\cap\widetilde{L}_{2}$ and
$\widetilde{P}_{1}\cap\widetilde{N}_{2}$ such that
$\widetilde{P}_{1}\cap\widetilde{P}_{2}=(\widetilde{P}_{1}\cap\widetilde{L}_{2})(\widetilde{P}_{1}\cap\widetilde{N}_{2})$
is a semi direct product. Now we claim that
$H_{1}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\widetilde{P}_{1}^{\prime}(\mathcal{O}_{K})$
and
$H_{2}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\widetilde{P}_{2}(\mathcal{O}_{K})$
gives a good pair as in Definition 3.6. As we clearly have
$H_{2}^{\prime}=\widetilde{L}_{2}(\mathcal{O}_{K})$ and
$H_{2}^{\prime\prime}=\widetilde{N}_{2}(\mathcal{O}_{K})$, item (i) follows
from the corresponding facts on group schemes, and the fact that

$H_{2}\cap
P_{1}^{\prime}(\mathbb{Q}_{p})=\widetilde{P}_{2}(\mathcal{O}_{K})\cap\widetilde{P}_{1}^{\prime}(L)=\widetilde{P}_{2}(\mathcal{O}_{K})\cap(\widetilde{P}_{2}\cap\widetilde{P}_{1}^{\prime})(L)=(\widetilde{P}_{2}\cap\widetilde{P}_{1}^{\prime})(\mathcal{O}_{K}),$

as $\widetilde{P}_{2}\cap\widetilde{P}_{1}^{\prime}$ is a closed subgroup
scheme of $\widetilde{P}_{2}$. Item (ii) follows from the fact that
$\widetilde{L}_{2}\cap\widetilde{P}_{1}\backslash\widetilde{L}_{2}$ is proper
over $\mathcal{O}_{K}$. For item (iii), we clearly have
$P_{1}^{\prime}(\mathbb{Q}_{p})H_{2}=\bigcup_{g_{1}\in J_{1}}g_{1}H_{1}H_{2}$.
Let $g_{1},g_{1}^{\prime}\in J_{1}$ be elements such that $g_{1}H_{1}H_{2}\cap
g_{1}^{\prime}H_{1}H_{2}\neq\emptyset$, namely there exists
$h_{1},h_{1}^{\prime}\in H_{1}$ and $h_{2},h_{2}^{\prime}\in H_{2}$ such that
$g_{1}h_{1}h_{2}=g_{1}^{\prime}h_{1}^{\prime}h_{2}^{\prime}$. As we have a
natural bijection $P_{1}^{\prime}(\mathbb{Q}_{p})\backslash
P_{1}^{\prime}(\mathbb{Q}_{p})H_{2}\cong(P_{1}^{\prime}(\mathbb{Q}_{p})\cap
H_{2})\backslash H_{2}=(H_{1}\cap H_{2})\backslash H_{2}$,
$P_{1}^{\prime}(\mathbb{Q}_{p})h_{2}=P_{1}^{\prime}(\mathbb{Q}_{p})h_{2}^{\prime}$
implies that $h_{2}^{\prime}h_{2}^{-1}\in H_{1}\cap H_{2}$, which together
with $g_{1}h_{1}h_{2}=g_{1}^{\prime}h_{1}^{\prime}h_{2}^{\prime}$ forces
$g_{1}H_{1}=g_{1}^{\prime}H_{1}$ and thus $g_{1}=g_{1}^{\prime}$ by the
definition of $J_{1}$. The proof is thus finished. ∎

######  Proposition 3.9.

If Condition 3.7 holds, then we have

$D(G)_{\overline{Y}}/D(G)_{Z}\cong\widehat{D}(G)_{Y}\cong
D(L_{2})\widehat{\otimes}_{D(L_{2})_{L_{2}^{\flat}}}(\widehat{D}(N_{2})\widehat{\otimes}_{D(N_{2})_{N_{2}^{\flat}}}D(P_{1}^{\prime}\backslash
G)_{1})\widehat{\otimes}_{E,\iota}D(P_{1}^{\prime}).$

###### Proof.

This follows directly from Proposition 3.5, (3.6) and Lemma 3.8. ∎

##  4\. Extensions between locally analytic generalized Steinberg

In this section, we compute various $\mathrm{Ext}$-groups between locally
analytic generalized Steinberg representations ( see Theorem 4.22 and
Corollary 4.23). Based on computations of the spectral sequence
$E_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}$ in Section 2 and the decomposition
in Section 3 (see Proposition 3.9), the main extra technical ingredients are
the computations of some $N$-(co)homologies (see Lemma 4.14 and Lemma 4.15)
which, via a standard spectral sequence, compute the $\mathrm{Ext}$-groups
between locally analytic principal series in Proposition 4.16.

###  4.1. Locally analytic geometric lemma

In this section, we generalize the classical Bernstein–Zelevinsky geometric
lemma in smooth representation theory (see Section 2.12 of [BZ77]) to Orlik-
Strauch representations in Proposition 4.7. As an application, we compute the
$\mathrm{Ext}$-groups between various $i_{n,I}^{\rm{an}}(\lambda)$ in
Proposition 4.16. We write $N_{n,I}$ for the unipotent radical of $P_{n,I}$,
for each $I\subseteq\Delta_{n}$.

Let $I,I^{\prime}\subseteq\Delta_{n}$ be two subsets, and we write
$W_{n}^{I^{\prime},I}\subseteq W(G_{n})$ be a set of (minimal length)
representative so that $\\{P_{n,I^{\prime}}wP_{n,I}\\}_{w\in
W_{n}^{I^{\prime},I}}$ is a stratification on the group scheme $G_{n}$. For
each $w\in W_{n}^{I^{\prime},I}$, we write
$C_{w}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}P_{n,I^{\prime}}wP_{n,I}$
(which is a locally closed subscheme of $G_{n}$) and $\overline{C_{w}}$ for
its Zariski closure in $G_{n}$. We will abuse $C_{w}$ and $\overline{C_{w}}$
for their $K$-points from now on. As in Section 3, we use the shorted notation
$D(G_{n})_{\overline{C_{w}}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}D(G_{n},E)_{\overline{C_{w}}}$
and similarly for other spaces of distributions. Then
$\\{D(G_{n})_{\overline{C_{w}}}\\}_{w\in W_{n}^{I^{\prime},I}}$ is an
increasing $W_{n}^{I^{\prime},I}$-filtration on $D(G_{n})$ with respect to the
Bruhat order on $W_{n}^{I^{\prime},I}$. As $\overline{C_{w}}\setminus
C_{w}=\bigcup_{w^{\prime}\in
W_{n}^{I^{\prime},I},w^{\prime}<w}\overline{C_{w^{\prime}}}$, we clearly have

$D(G_{n})_{\overline{C_{w}}\setminus C_{w}}=\sum_{w^{\prime}\in
W_{n}^{I^{\prime},I},w^{\prime}<w}D(G_{n})_{\overline{C_{w^{\prime}}}},$

and thus $D(G_{n})_{\overline{C_{w}}}/D(G_{n})_{\overline{C_{w}}\setminus
C_{w}}$ is the $w$-graded piece associated with the filtration
$\\{D(G_{n})_{\overline{C_{w}}}\\}_{w\in W_{n}^{I^{\prime},I}}$. For technical
convenience, we set
$P_{n,I^{\prime}}^{w}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}w^{-1}P_{n,I^{\prime}}w$,
$L_{n,I,w}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}L_{n,I}\cap
P_{n,I^{\prime}}^{w}$,
$N_{n,I,w}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}N_{n,I}\cap
P_{n,I^{\prime}}^{w}$,
$Y_{w}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}w^{-1}C_{w}$ whose
closure in $G_{n}$ is
$\overline{Y_{w}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}w^{-1}\overline{C_{w}}$.
According to Proposition 3.5, there exists a canonical left
$D(P_{n,I})$-equivariant and right $D(P_{n,I^{\prime}}^{w})$-equivariant
isomorphism

$\widehat{D}(G_{n})_{Y_{w}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\varprojlim_{C}D(P_{n,I^{\prime}}^{w}\backslash
G_{n})_{C}\widehat{\otimes}_{E,\iota}D(P_{n,I^{\prime}}^{w})\cong
D(G_{n})_{\overline{Y_{w}}}/D(G_{n})_{\overline{Y_{w}}\setminus Y_{w}}$

where $C$ runs through compact open subsets of $P_{n,I^{\prime}}^{w}\backslash
P_{n,I^{\prime}}^{w}P_{n,I}$.

Given a coadmissible $D(L_{n,I^{\prime}})$-module $\mathcal{M}_{I^{\prime}}$
and an integer $k\geq 0$, it is a fundamental question to compute

$H^{k}(N_{n,I},\mathcal{M})$

with
$\mathcal{M}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}D(G_{n})\widehat{\otimes}_{D(P_{n,I^{\prime}}),\iota}\mathcal{M}_{I^{\prime}}$.
With the help of the filtration $\\{D(G_{n})_{\overline{C_{w}}}\\}_{w\in
W_{n}^{I^{\prime},I}}$, we usually first compute

(4.1)
$H^{k}(N_{n,I},(D(G_{n})_{\overline{C_{w}}}/D(G_{n})_{\overline{C_{w}}\setminus
C_{w}})\widehat{\otimes}_{D(P_{n,I^{\prime}}),\iota}\mathcal{M}_{I^{\prime}})$

for each $w\in W_{n}^{I^{\prime},I}$. We clearly have a canonical isomorphism

(4.2)
$\mathrm{gr}_{w}(\mathcal{M})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(D(G_{n})_{\overline{C_{w}}}/D(G_{n})_{\overline{C_{w}}\setminus
C_{w}})\widehat{\otimes}_{D(P_{n,I^{\prime}}),\iota}\mathcal{M}_{I^{\prime}}\cong(D(G_{n})_{\overline{Y_{w}}}/D(G_{n})_{\overline{Y_{w}}\setminus
Y_{w}})\widehat{\otimes}_{D(P_{n,I^{\prime}}^{w}),\iota}\mathcal{M}_{I^{\prime}}^{w}$

with $\mathcal{M}_{I^{\prime}}^{w}$ being the $w$-conjugate of
$\mathcal{M}_{I^{\prime}}$. Combine with Proposition 3.9, (4.1) can be reduced
to

(4.3)
$H^{k}(N_{n,I},D(L_{n,I})\widehat{\otimes}_{D(L_{n,I})_{L_{n,I,w}}}(\widehat{D}(N_{n,I})\widehat{\otimes}_{D(N_{n,I})_{N_{n,I,w}}}D(P_{n,I^{\prime}}^{w}\backslash
G_{n})_{1}\widehat{\otimes}_{E,\iota}\mathcal{M}_{I^{\prime}}^{w}))$

for each $w\in W_{n}^{I^{\prime},I}$, where
$\widehat{D}(N_{n,I})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\left(\varprojlim_{C}D(C)\right)\widehat{\otimes}_{E,\iota}D(N_{n,I,w})$
and $C$ runs through compact open subsets of $N_{n,I,w}\backslash N_{n,I}$.
See for example the discussion after Lemme 5.1.1 of [Bre19] that (4.3) admits
a natural $D(L_{n,I})$-module structure (and in particular a
$D(Z_{n,I})$-module structure).

We assume from now on that
$\mathcal{M}_{I^{\prime}}=L^{I^{\prime}}(\mu)\otimes_{E}(\pi^{\infty})^{\prime}$
with $\mu\in X(T_{n,E})$ dominant with respect to $B_{n,E}\cap
L_{n,I^{\prime},E}$ and $\pi^{\infty}$ an admissible smooth representation of
$L_{n,I^{\prime}}$. Now we set

$\mathcal{M}^{1}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}D(P_{n,I^{\prime}}\backslash
G_{n})_{1}\widehat{\otimes}_{E}L^{I^{\prime}}(\mu)\cong
D(G_{n})_{1}\widehat{\otimes}_{D(P_{n,I^{\prime}})_{1}}L^{I^{\prime}}(\mu)$
$\mathrm{gr}_{w}^{1}(\mathcal{M})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(\mathcal{M}^{1})^{w}\widehat{\otimes}_{E}(\pi^{\infty,w})^{\prime}\cong
D(G_{n})_{1}\widehat{\otimes}_{D(P_{n,I^{\prime}}^{w})_{1}}(L^{I^{\prime}}(\mu)^{w}\otimes_{E}(\pi^{\infty,w})^{\prime}),$
$\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\widehat{D}(N_{n,I})\widehat{\otimes}_{D(N_{n,I})_{N_{n,I,w}}}\mathrm{gr}_{w}^{1}(\mathcal{M})$

and note that

$\mathrm{gr}_{w}(\mathcal{M})\cong
D(L_{n,I})\widehat{\otimes}_{D(L_{n,I})_{L_{n,I,w}}}\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M})$

for each $w\in W_{n}^{I^{\prime},I}$.

Let $\mathfrak{p}$ be a (finite dimensional) Lie algebra over $\mathbb{Q}_{p}$
with an ideal $\mathfrak{n}$ and a commutative sub Lie algebra $\mathfrak{t}$
such that $\mathfrak{t}\subseteq\mathfrak{p}$ induces an embedding
$\mathfrak{t}\hookrightarrow\mathfrak{p}/\mathfrak{n}$. We set
$\mathfrak{p}_{E}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathfrak{p}\otimes_{\mathbb{Q}_{p}}E$
and similarly for others. We write $X(\mathfrak{t}_{E})$ for the set of all
eigencharacters of $\mathfrak{t}_{E}$ (which is countable).

######  Definition 4.1.

Let $V$ be a Fréchet unclear space equipped with a
$U(\mathfrak{p}_{E})$-action. We say the topological
$U(\mathfrak{p}_{E})$-module $V$ is _nice_ if there exists a dense sub
$U(\mathfrak{p}_{E})$-module $V^{\flat}\subseteq V$ such that the
$\mathfrak{t}_{E}$-eigenspace $V^{\flat}_{\eta}$ is finite dimensional for
each $\eta\in X(\mathfrak{t}_{E})$ and $V^{\flat}=\bigoplus_{\eta\in
X(\mathfrak{t}_{E})}V^{\flat}_{\eta}$. Here $V^{\flat}_{\eta}$ for the
$\eta$-eigenspace of $V^{\flat}$ under the $\mathfrak{t}_{E}$-action.

We endow $V^{\flat}$ with the subspace topology induced from $V$, and set
$V^{\flat}_{X}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\bigoplus_{\eta\in
X}V^{\flat}_{\eta}$ for each subset $X\subseteq X(\mathfrak{t}_{E})$. Now we
consider the Chevalley-Eilenberg complex
$\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{E},V)$ that computes the Lie
algebra cohomology $H^{\bullet}(\mathfrak{n}_{E},V)$. Equipped with its
natural topology induced from $V$,
$\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{E},V)$ contains a (dense)
subcomplex $\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{E},V^{\flat})$ with
each term endowed with the induced subspace topology. Then there exists a
finite set $\Phi_{\mathfrak{p}}\subseteq X(\mathfrak{t}_{E})$ depending only
on our choice of $\mathfrak{p}$, such that the image of
$\mathrm{Hom}_{E}(\wedge^{k}\mathfrak{n}_{E},V^{\flat}_{X})$ under

(4.4)
$\varphi^{k}:~{}\mathrm{Hom}_{E}(\wedge^{k}\mathfrak{n}_{E},V^{\flat})\rightarrow\mathrm{Hom}_{E}(\wedge^{k+1}\mathfrak{n}_{E},V^{\flat})$

is contained in
$\mathrm{Hom}_{E}(\wedge^{k+1}\mathfrak{n}_{E},V^{\flat}_{X+\Phi_{\mathfrak{p}}})$
for each finite subset $X\subseteq X(\mathfrak{t}_{E})$ (with
$X+\Phi_{\mathfrak{p}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{\eta+\eta^{\prime}\mid\eta\in
X,\eta^{\prime}\in\Phi_{\mathfrak{p}}\\}$).

######  Lemma 4.2.

Let $V$ be a nice topological $U(\mathfrak{p}_{E})$-module as in Definition
4.1. Each map of the complex
$\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{E},V^{\flat})$ is strict with
closed image, with respect to the topology induced from
$\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{E},V)$.

###### Proof.

Let $k\geq 0$ and consider the map (4.4). Let

$X_{0}\subseteq X_{1}\subseteq\cdots$

be a sequence of subsets of $X(\mathfrak{t}_{E})$ satisfying
$X_{i}+\Phi_{\mathfrak{p}}\subseteq X_{i+1}$ for each $i\geq 0$ and
$X(\mathfrak{t}_{E})=\bigcup_{i=0}^{\infty}X_{i}$. As
$V^{\flat}=\bigoplus_{\eta\in X(\mathfrak{t}_{E})}V^{\flat}_{\eta}$, we have
$V^{\flat}=\varinjlim_{i}V^{\flat}_{X_{i}}$. By the definition of
$\Phi_{\mathfrak{p}}$, the map $\varphi^{k}$ has the form
$\varphi^{k}=\varinjlim_{i}\varphi^{k}_{i}$ where
$\varphi^{k}_{i}:\mathrm{Hom}_{E}(\wedge^{k}\mathfrak{n}_{E},V^{\flat}_{X_{i}})\rightarrow\mathrm{Hom}_{E}(\wedge^{k+1}\mathfrak{n}_{E},V^{\flat}_{X_{i+1}})$
is the restriction of $\varphi^{k}$. As $\varphi^{k}_{i}$ is a map between
finite dimensional $E$-vector spaces for each $i\geq 0$, we deduce that
$\varphi^{k}$ is strict with closed image. ∎

######  Lemma 4.3.

Let $V$ be a nice topological $U(\mathfrak{p}_{E})$-module as in Definition
4.1. For each $k\geq 0$, the space $H^{k}(\mathfrak{n}_{E},V)$ is a Fréchet
nuclear space and the map $H^{k}(\mathfrak{n}_{E},V^{\flat})\rightarrow
H^{k}(\mathfrak{n}_{E},V)$ is an injection with dense image.

###### Proof.

This follows from Lemma 4.2 and Theorem 7.4 of [Koh11]. ∎

For each $p$-adic manifold $M$ (paracompact and finite dimensional), we write
$C^{\infty}(M)=C^{\infty}(M,E)$ for the space of locally constant $E$-valued
functions on $M$ and set
$D^{\infty}(M)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}C^{\infty}(M)^{\prime}$.
The closed embedding $C^{\infty}(M)\hookrightarrow C^{\rm{an}}(M)$ induces a
canonical surjection

(4.5) $D(M)\twoheadrightarrow D^{\infty}(M).$

For each closed subset $C\subseteq M$, we write $D^{\infty}(M)_{C}\subseteq
D^{\infty}(M)$ for the closed subspace consists of locally constant
distributions that are supported on $C$, and note that $D^{\infty}(M)_{C}$ is
simply the image of $D(M)_{C}$ under (4.5). If $C$ is a closed submanifold of
$M$, then we actually have a canonical isomorphism

$D^{\infty}(M)_{C}\cong D^{\infty}(C):\delta\mapsto\delta|_{C}$

which makes $D^{\infty}(M)_{C}$ a much simpler object than $D(M)_{C}$. Lemma
3.2 and Lemma 3.3 have their analogue for $D^{\infty}(-)$, and $D^{\infty}(M)$
is Fréchet nuclear if $M$ is compact. In particular, we have a topological
isomorphism

$D^{\infty}(N_{n,I})\cong D^{\infty}(N_{n,I,w}\backslash
N_{n,I})\widehat{\otimes}_{E,\iota}D^{\infty}(N_{n,I,w}).$

We set
$\widehat{D}^{\infty}(N_{n,I})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(\varprojlim_{C}D^{\infty}(C))\widehat{\otimes}_{E,\iota}D^{\infty}(N_{n,I,w})$
where $C$ runs through compact open subsets of $N_{n,I,w}\backslash N_{n,I}$.
Note that there exists a canonical $D(N_{n,I})$-equivariant surjection

$\widehat{D}(N_{n,I})\twoheadrightarrow\widehat{D}^{\infty}(N_{n,I})$

with the $D(N_{n,I})$-action on the target factoring through
$D^{\infty}(N_{n,I})$.

######  Lemma 4.4.

For each $k\geq 0$ and each $w\in W_{n}^{I^{\prime},I}$, the cohomology groups

$H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1})^{w}),H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M})),~{}H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))$

are Fréchet nuclear spaces. Moreover, we have canonical
$D(L_{n,I})_{L_{n,I,w}}$-equivariant and $D^{\infty}(N_{n,I})$-equivariant
topological isomorphisms

$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))\cong\widehat{D}^{\infty}(N_{n,I})\widehat{\otimes}_{D^{\infty}(N_{n,I,w})}H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M}))\\\
\cong\left(\widehat{D}^{\infty}(N_{n,I})\widehat{\otimes}_{D^{\infty}(N_{n,I,w})}(\pi^{\infty,w})^{\prime}\right)\widehat{\otimes}_{E}H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1})^{w}).$

###### Proof.

We fix an element $w\in W_{n}^{I^{\prime},I}$ throughout. As
$(\mathcal{M}^{1})^{w}\cong
D(G_{n})_{1}\widehat{\otimes}_{D(P_{n,I^{\prime}}^{w})_{1}}L^{I^{\prime}}(\mu)^{w}$,
we can consider the dense subspace $(\mathcal{M}^{1,\flat})^{w}$ with

$\mathcal{M}^{1,\flat}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}U(\mathfrak{g}_{n,E})\otimes_{U(\mathfrak{p}_{n,I^{\prime},E})}L^{I^{\prime}}(\mu)$

and observe that $(\mathcal{M}^{1})^{w}$ is a nice
$U(\mathfrak{p}_{n,I,E})$-module (see Definition 4.1). Therefore it follows
from Lemma 4.2 (with $\mathfrak{p}=\mathfrak{p}_{n,I}$,
$\mathfrak{n}=\mathfrak{n}_{n,I}$ and $\mathfrak{t}=\mathfrak{t}_{n}$) that
each map of the complex
$\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{n,I,E},(\mathcal{M}^{1})^{w})$
is strict with closed image and
$H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1})^{w})$ is a Fréchet nuclear space
for each $k\geq 0$. As $\pi^{\infty}$ is admissible smooth, it is a countable
direct limit of finite dimensional subspaces, making
$(\pi^{\infty,w})^{\prime}$ a countable inverse limit of finite dimensional
$E$-vector spaces. In particular, $(\pi^{\infty,w})^{\prime}$ is Fréchet
nuclear and for each strict exact sequence of Fréchet nuclear spaces
$V_{1}\hookrightarrow V_{2}\twoheadrightarrow V_{3}$, we have a strict exact
sequence of Fréchet nuclear spaces

$V_{1}\widehat{\otimes}_{E}(\pi^{\infty,w})^{\prime}\hookrightarrow
V_{2}\widehat{\otimes}_{E}(\pi^{\infty,w})^{\prime}\twoheadrightarrow
V_{3}\widehat{\otimes}_{E}(\pi^{\infty,w})^{\prime}.$

Consequently, each map of the complex

$\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M}))\cong\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{n,I,E},(\mathcal{M}^{1})^{w})\widehat{\otimes}_{E}(\pi^{\infty,w})^{\prime}$

is strict with closed image, making
$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M}))$ a Fréchet
nuclear space for each $k\geq 0$.

Now we consider $D(N_{n,I})$-equivariant topological isomorphism of complexes

(4.6)
$\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))\cong\widehat{D}(N_{n,I})\widehat{\otimes}_{D(N_{n,I})_{N_{n,I,w}}}\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M})).$

For each Fréchet nuclear space $V$ we have the following topological
isomorphism of Fréchet nuclear spaces

(4.7)
$\widehat{D}(N_{n,I})\widehat{\otimes}_{D(N_{n,I})_{N_{n,I,w}}}V\cong\varprojlim_{C,r}D(N_{n,I,w}C)_{r}\widehat{\otimes}_{D(N_{n,I,w}C)_{N_{n,I,w},r}}V$

with the radius $\frac{1}{p}<r<1$ and $C$ running through compact open
subgroups of $N_{n,I}$. Following the argument of Lemme 4.27 of [Schr11], we
observe that $D(N_{n,I,w}C)_{r}$ is finite free over
$D(N_{n,I,w}C)_{N_{n,I,w},r}$ and that the system

$\\{D(N_{n,I,w}C)_{r}\widehat{\otimes}_{D(N_{n,I,w}C)_{N_{n,I,w},r}}V\\}_{C,r}$

is Mittag-Leffler. Consequently, for each strict exact sequence of Fréchet
nuclear spaces $V_{1}\hookrightarrow V_{2}\twoheadrightarrow V_{3}$, we have a
strict exact sequence of Fréchet nuclear spaces

$\widehat{D}(N_{n,I})\widehat{\otimes}_{D(N_{n,I})_{N_{n,I,w}}}V_{1}\hookrightarrow\widehat{D}(N_{n,I})\widehat{\otimes}_{D(N_{n,I})_{N_{n,I,w}}}V_{2}\twoheadrightarrow\widehat{D}(N_{n,I})\widehat{\otimes}_{D(N_{n,I})_{N_{n,I,w}}}V_{3}.$

In particular, we deduces that each map of the complex (4.6) is strict with
closed image, and
$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))$ is a
Fréchet nuclear space for each $k\geq 0$. The above argument clearly produces
a $D(N_{n,I})$-equivariant isomorphism

(4.8)
$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))\cong\widehat{D}(N_{n,I})\widehat{\otimes}_{D(N_{n,I})_{N_{n,I,w}}}H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M}))$

for each $k\geq 0$. As $\mathfrak{n}_{n,I,E}$ acts trivially on
$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M}))$, the
$D(N_{n,I})_{N_{n,I,w}}$-action on
$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M}))$ factors through
$D^{\infty}(N_{n,I})_{N_{n,I,w}}\cong D^{\infty}(N_{n,I,w})$, and thus the
$D(N_{n,I})$-equivariant isomorphism (4.8) factors through a
$D^{\infty}(N_{n,I})$-equivariant isomorphism

(4.9)
$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))\cong\widehat{D}^{\infty}(N_{n,I})\widehat{\otimes}_{D^{\infty}(N_{n,I,w})}H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M}))$

Then we observe that we have a $D^{\infty}(N_{n,I,w})$-equivariant isomorphism

(4.10) $H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{1}(\mathcal{M}))\cong
H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1})^{w})\widehat{\otimes}_{E}(\pi^{\infty,w})^{\prime}$

for each $k\geq 0$. The $D(L_{n,I})_{L_{n,I,w}}$-equivariance of both (4.9)
and (4.10) are clear. ∎

Note that we have a canonical isomorphism

$\widehat{D}^{\infty}(N_{n,I})\widehat{\otimes}_{D^{\infty}(N_{n,I,w})}(\pi^{\infty,w})^{\prime}\cong\left((\rm{c}\textit{-}\mathrm{Ind}_{N_{n,I,w}}^{N_{n,I}}\pi^{\infty,w})^{\infty}\right)^{\prime}$

by the definition of the smooth compact induction functor
$(\rm{c}\textit{-}\mathrm{Ind}_{N_{n,I,w}}^{N_{n,I}}-)^{\infty}$ and that of
$\widehat{D}^{\infty}(N_{n,I})$.

######  Lemma 4.5.

For each $k\geq 0$ and each $w\in W_{n}^{I^{\prime},I}$, the cohomology group
$H^{k}(N_{n,I},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))$ is a Fréchet nuclear
space, and we have a canonical $D(L_{n,I})_{L_{n,I,w}}$-equivariant
topological isomorphism

$H^{k}(N_{n,I},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))\cong
J_{n,I,I^{\prime},w}(\pi^{\infty})^{\prime}\widehat{\otimes}_{E}H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1})^{w}).$

with
$J_{n,I,I^{\prime},w}(\pi^{\infty})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\left((\rm{c}\textit{-}\mathrm{Ind}_{N_{n,I,w}}^{N_{n,I}}\pi^{\infty,w})^{\infty}\right)_{N_{n,I}}$
an admissible smooth $E$-representation of the Levi quotient of $L_{n,I,w}$.

###### Proof.

This follows directly from Lemma 4.4 and Theorem 7.1 of [Koh11], namely the
$D(L_{n,I})_{L_{n,I,w}}$-equivariant isomorphism

$H^{k}(N_{n,I},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))^{\prime}\cong\left(H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))^{\prime}\right)_{N_{n,I}}.$

∎

######  Lemma 4.6.

For each $k\geq 0$ and each $w\in W_{n}^{I^{\prime},I}$, the cohomology groups
$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}(\mathcal{M}))$ and
$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))$ are Fréchet nuclear spaces.
Moreover, we have a canonical $D(L_{n,I})$-equivariant topological
isomorphisms

$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))\cong
D(L_{n,I})\widehat{\otimes}_{D(L_{n,I})_{L_{n,I,w}}}H^{k}(N_{n,I},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M})).$

###### Proof.

The same argument as Lemme 4.27 of [Schr11] shows that
$D(L_{n,I})\widehat{\otimes}_{D(L_{n,I})_{L_{n,I,w}}}$ sends a strict complex
of Fréchet spaces to a strict complex of Fréchet spaces, and in particular

$H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}(\mathcal{M}))\cong
D(L_{n,I})\widehat{\otimes}_{D(L_{n,I})_{L_{n,I,w}}}H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))$

for each $k\geq 0$, using the isomorphism of complexes

$\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}(\mathcal{M}))\cong
D(L_{n,I})\widehat{\otimes}_{D(L_{n,I})_{L_{n,I,w}}}\mathrm{Hom}_{E}(\wedge^{\bullet}\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M})).$

The desired isomorphism thus follows from Theorem 7.1 of [Koh11], namely the
$D(L_{n,I})_{L_{n,I,w}}$-equivariant isomorphism

$H^{k}(N_{n,I},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))^{\prime}\cong\left(H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M}))^{\prime}\right)_{N_{n,I}}$

and the $D(L_{n,I})$-equivariant isomorphism

$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))^{\prime}\cong\left(H^{k}(\mathfrak{n}_{n,I,E},\mathrm{gr}_{w}(\mathcal{M}))^{\prime}\right)_{N_{n,I}}.$

∎

Given a split reductive group $G$ over $K$ and a parabolic subgroup $P$ with
Levi quotient $L$, we recall from [OS15] the so-called Orlik–Strauch functor
$\mathcal{F}_{P}^{G}(\cdot,\cdot)$ which sends a pair $(V,\pi^{\infty})$ to an
admissible locally analytic representation
$\mathcal{F}_{P}^{G}(V,\pi^{\infty})$ of $G(K)$. Here $V$ is a locally
$\mathfrak{p}$-finite object in category $\mathcal{O}$ and $\pi^{\infty}$ is
an admissible smooth $E$-representation of $L(K)$. The main properties of
$\mathcal{F}_{P}^{G}(\cdot,\cdot)$ are summarized in the main theorem of
_loc.it._ and in particular $\mathcal{F}_{P}^{G}(\cdot,\cdot)$ is exact in
both arguments.

Combining all the results above, we arrive at the following locally analytic
geometric lemma (with the terminology borrowed from the classical geometric
lemma in smooth representation theory).

######  Proposition 4.7.

For each $k\geq 0$ and each $w\in W_{n}^{I^{\prime},I}$, we have a
$D(L_{n,I})$-equivariant isomorphism of Fréchet nuclear spaces

$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))\cong\mathcal{F}_{L_{n,I,w}}^{L_{n,I}}(H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1,\flat})^{w}),J_{n,I,I^{\prime},w}(\pi^{\infty}))^{\prime}.$

###### Proof.

As $\mathcal{M}^{1,\flat}$ is $\mathfrak{t}_{n,E}$-semisimple and locally
$\mathfrak{p}_{n,I^{\prime},E}$-finite, $(\mathcal{M}^{1,\flat})^{w}$ is
$\mathfrak{t}_{n,E}$-semisimple and locally
$\mathfrak{p}_{n,I^{\prime},E}^{w}$-finite, making
$H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1,\flat})^{w})$
$\mathfrak{t}_{n,E}$-semisimple and locally
$\mathfrak{l}_{n,I,w}=\mathfrak{l}_{n,I,E}\cap\mathfrak{p}_{n,I^{\prime},E}^{w}$-finite,
and thus the Orlik-Strauch representation
$\mathcal{F}_{L_{n,I,w}}^{L_{n,I}}(H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1,\flat})^{w}),J_{n,I,I^{\prime},w}(\pi^{\infty}))$
is well-defined. We conclude by Lemma 4.5, Lemma 4.6 and the following
$D(L_{n,I})$-equivariant isomorphism

$\mathcal{F}_{L_{n,I,w}}^{L_{n,I}}(H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1,\flat})^{w}),J_{n,I,I^{\prime},w}(\pi^{\infty}))^{\prime}\cong
D(L_{n,I})\widehat{\otimes}_{D(L_{n,I})_{L_{n,I,w}}}\left(J_{n,I,I^{\prime},w}(\pi^{\infty})^{\prime}\widehat{\otimes}_{E}H^{k}(\mathfrak{n}_{n,I,E},(\mathcal{M}^{1})^{w})\right).$

∎

For each locally algebraic character $\chi$ of $Z_{n,I}$ and each
$D(Z_{n,I})$-module $V$, we write $V_{\chi^{\prime}}$ for its maximal quotient
on which $D(Z_{n,I})$ acts by $\chi^{\prime}$. There exists a finite set
$\Sigma_{w,k}(\mathcal{M})$ of locally algebraic characters of $Z_{n,I}$ such
that the $\chi^{\prime}$-isotypic component
$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))_{\chi^{\prime}}$ is non-zero for
some locally algebraic character $\chi$ of $Z_{n,I}$ if and only if
$\chi\in\Sigma_{w,k}(\mathcal{M})$, and moreover

$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))\cong\bigoplus_{\chi\in\Sigma_{w,k}(\mathcal{M})}H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))_{\chi^{\prime}}.$

Similarly, we write $\Sigma_{w}^{\infty}(\pi^{\infty})$ for the set of smooth
characters of $Z_{n,I}$ such that
$J_{n,I,I^{\prime},w}(\pi^{\infty})_{\chi^{\infty}}\neq 0$ if and only if
$\chi^{\infty}\in\Sigma_{w}^{\infty}(\pi^{\infty})$. There clearly exists a
natural surjection
$\Sigma_{w,k}(\mathcal{M})\twoheadrightarrow\Sigma_{w}^{\infty}(\pi^{\infty})$
sending each $\chi\in\Sigma_{w,k}$ to its smooth part $\chi^{\infty}$.

######  Definition 4.8.

Let $I,I^{\prime}\subseteq\Delta_{n}$ be a pair of subsets and $\pi^{\infty}$
an admissible smooth $E$-representation of $L_{n,I^{\prime}}$. We say that
$\pi^{\infty}$ is _$(I,I^{\prime})$ -regular_ if
$\Sigma_{w}\cap\Sigma_{w^{\prime}}=\emptyset$ for each $w\neq w^{\prime}\in
W_{n}^{I^{\prime},I}$.

We have the following corollary of Proposition 4.7

######  Corollary 4.9.

Let $I,I^{\prime}\subseteq\Delta_{n}$ be a pair of subsets and $\pi^{\infty}$
a $(I,I^{\prime})$-regular admissible smooth $E$-representation of
$L_{n,I^{\prime}}$. For each $k\geq 0$, the cohomology
$H^{k}(N_{n,I},\mathcal{M})$ is a Fréchet nuclear space and there exists a
$D(L_{n,I})$-equivariant isomorphism

$H^{k}(N_{n,I},\mathcal{M})\cong\bigoplus_{w\in
W_{n}^{I^{\prime},I}}H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))$

###### Proof.

Note that $\mathcal{M}$ admits a canonical filtration
$\\{\mathrm{Fil}_{w}(\mathcal{M})\\}_{w\in W_{n}^{I^{\prime},I}}$. Our
assumption implies that, for each smooth character $\chi^{\infty}$ of
$Z_{n,I}$, there exists at most one $w\in W_{n}^{I^{\prime},I}$ such that

$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))_{\chi^{\prime}}\neq 0$

for some locally algebraic character $\chi$ of $Z_{n,I}$ whose smooth part is
$\chi^{\infty}$.

We prove by induction on $w\in W_{n}^{I^{\prime},I}$ with respect to Bruhat
order. For each $w\in W_{n}^{I^{\prime},I}$, we set
$\mathrm{Fil}_{<w}(\mathcal{M})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{w^{\prime}<w}\mathrm{Fil}_{w^{\prime}}(\mathcal{M})$
and consider the following long exact sequence

$\cdots\rightarrow H^{k}(N_{n,I},\mathrm{Fil}_{<w}(\mathcal{M}))\rightarrow
H^{k}(N_{n,I},\mathrm{Fil}_{w}(\mathcal{M}))\\\ \rightarrow
H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))\rightarrow
H^{k+1}(N_{n,I},\mathrm{Fil}_{<w}(\mathcal{M}))\rightarrow\cdots$

induced from
$\mathrm{Fil}_{<w}(\mathcal{M})\hookrightarrow\mathrm{Fil}_{w}(\mathcal{M})\twoheadrightarrow\mathrm{gr}_{w}(\mathcal{M})$.
Our inductive assumption says that

$H^{k}(N_{n,I},\mathrm{Fil}_{w^{\prime}}(\mathcal{M}))\cong\bigoplus_{w^{\prime\prime}\in
W_{n}^{I^{\prime},I},w^{\prime\prime}\leq
w^{\prime}}H^{k}(N_{n,I},\mathrm{gr}_{w^{\prime\prime}}(\mathcal{M}))$

for each $w^{\prime}<w$ and $k\geq 0$, which implies that

$H^{k}(N_{n,I},\mathrm{Fil}_{<w}(\mathcal{M}))\cong\bigoplus_{w^{\prime\prime}\in
W_{n}^{I^{\prime},I},w^{\prime\prime}<w}H^{k}(N_{n,I},\mathrm{gr}_{w^{\prime\prime}}(\mathcal{M})).$

This together with the previous long exact sequence clearly finishes our
induction step. The proof is thus finished. ∎

When $\pi^{\infty}$ is $(I,I^{\prime})$-regular, we can define
$H_{k}(N_{n,I},\mathcal{M}^{\prime})$ (for each $k\geq 0$) as the unique (up
to isomorphism) admissible locally analytic representation such that

(4.11) $H_{k}(N_{n,I},\mathcal{M}^{\prime})^{\prime}\cong
H^{k}(N_{n,I},\mathcal{M}),$

which exists by Proposition 4.7 and Corollary 4.9. For each locally algebraic
character $\chi$ of $Z_{n,I}$, we write
$H_{k}(N_{n,I},\mathcal{M}^{\prime})_{\chi}$ for the $\chi$-isotypic component
of $H_{k}(N_{n,I},\mathcal{M}^{\prime})$, namely the unique (up to
isomorphism) direct summand of $H_{k}(N_{n,I},\mathcal{M}^{\prime})$ so that
(4.11) induces an isomorphism

$\left(H_{k}(N_{n,I},\mathcal{M}^{\prime})_{\chi}\right)^{\prime}\cong
H^{k}(N_{n,I},\mathcal{M})_{\chi^{\prime}}.$

For each $w\in W(G_{n})$, we set
$\chi_{w}^{\infty}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}|\cdot|\circ\delta_{w\cdot
0}$ where $\delta_{w\cdot 0}:T_{n}\rightarrow E^{\times}$ is the algebraic
character associated with the weight $w\cdot 0\in X(T_{n})$.

######  Lemma 4.10.

Let $I,I^{\prime}\subseteq\Delta_{n}$ be a pair of subsets and $\pi^{\infty}$
an admissible smooth $E$-representation of $L_{n,I^{\prime}}$. Assume that
$\pi^{\infty}$ is the subquotient of
$i_{n,\emptyset,I^{\prime}}^{\infty}(\chi_{w}^{\infty})$ for some $w\in
W(G_{n})$, then $\pi^{\infty}$ is $(I,I^{\prime})$-regular. In particular, the
trivial representation $1_{L_{n,I^{\prime}}}$ is $(I,I^{\prime})$-regular.

###### Proof.

According to Definition 4.8, it is harmless to assume that
$\pi^{\infty}=i_{n,\emptyset,I^{\prime}}^{\infty}(\chi_{w}^{\infty})$.
According to the classical geometric lemma in representation theory (cf.
Section VI.5.1 of [Ren]), we have

(4.12)
$\left(i_{n,\emptyset}^{\infty}(\chi_{w}^{\infty})\right)_{N_{n,\emptyset}}\cong\bigoplus_{w^{\prime}\in
W(G_{n})}\chi_{w^{\prime}}^{\infty}$

and

(4.13)
$\left(i_{n,\emptyset}^{\infty}(\chi_{w}^{\infty})\right)_{N_{n,I}}\cong\left(i_{n,I^{\prime}}^{\infty}i_{n,\emptyset,I^{\prime}}^{\infty}(\chi_{w}^{\infty})\right)_{N_{n,I}}\cong\bigoplus_{w^{\prime}\in
W_{n}^{I^{\prime},I}}(\mathrm{Ind}_{L_{n,I,w}}^{L_{n,I}}J_{n,I,I^{\prime},w}(\pi^{\infty}))^{\infty}.$

Compare (4.13) with (4.12), we observe that the set
$\Sigma_{w^{\prime}}^{\infty}(\pi^{\infty})$ consists of elements of the form
$\chi_{w^{\prime\prime}}^{\infty}|_{Z_{n,I}}$ with $w^{\prime\prime}\in
wW(L_{n,I^{\prime}})w^{\prime}W(L_{n,I})$. Given two elements
$w^{\prime\prime},w^{\prime\prime\prime}\in W(G_{n})$, we have
$\chi_{w^{\prime\prime}}^{\infty}|_{Z_{n,I}}=\chi_{w^{\prime\prime\prime}}^{\infty}|_{Z_{n,I}}$
if and only if $w^{\prime\prime}W(L_{n,I})=w^{\prime\prime\prime}W(L_{n,I})$,
and thus

$\Sigma_{w^{\prime}}^{\infty}(\pi^{\infty})=\\{\chi_{w^{\prime\prime}}^{\infty}|_{Z_{n,I}}\mid
w^{\prime\prime}\in wW(L_{n,I^{\prime}})w^{\prime}W(L_{n,I})\\}$

are disjoint for different $w^{\prime}\in W_{n}^{I^{\prime},I}$. The last
claim follows from the fact that $1_{L_{n,I^{\prime}}}$ is a subrepresentation
of $i_{n,\emptyset,I^{\prime}}^{\infty}(\chi_{1}^{\infty})$. ∎

######  Remark 4.11.

Concerning the results up to Proposition 4.7 and Corollary 4.9, it is actually
possible to treat $\mathcal{M}$ of the form
$\mathcal{F}_{P_{n,I^{\prime}}}^{G_{n}}(V,\pi^{\infty})^{\prime}$, namely the
dual of an arbitrary Orlik-Strauch representation, without assuming that $V$
has the form $M^{\Delta_{n}}_{I^{\prime}}(\mu)$ for some $\mu\in X(T_{n,E})$
dominant for $L_{n,I^{\prime},E}\cap B_{n,E}$. In fact, each
$V\in\mathcal{O}_{n,\Delta_{n}}^{I^{\prime}}$ naturally determines a
$D(G_{n})_{1}$-module $\mathcal{M}^{1}$ which contains $V$ as the (dense)
subset of $\mathfrak{p}_{n,I^{\prime}}$-finite vectors and satisfies

$\mathcal{M}\cong
D(G_{n})\widehat{\otimes}_{D(G_{n})_{P_{n,I^{\prime}}}}(\mathcal{M}^{1}\widehat{\otimes}_{E}(\pi^{\infty,w})^{\prime}).$

We can thus define $\mathrm{gr}_{w}^{1}(\mathcal{M})$,
$\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M})$ and $\mathrm{gr}_{w}(\mathcal{M})$ by
exactly the same formula as before. Note that the computation of such
$H^{k}(N_{n,I},\mathcal{M})$ largely reduces to that of
$H^{k}(\mathfrak{n}_{n,I,E},V^{w})$ for
$V\in\mathcal{O}_{n,\Delta_{n}}^{I^{\prime}}$ and $w\in W_{n}^{I^{\prime},I}$,
which is complicated in general.

######  Remark 4.12.

Proposition 4.7 is actually a true generalization of the classical geometric
lemma in smooth representation theory. According to Remark 4.11, we can simply
take $\mathcal{M}^{1}$ to be the trivial $D(G_{n})_{1}$-module, in which case
we have

$\mathrm{gr}_{w}^{1}(\mathcal{M})=(\pi^{\infty,w})^{\prime},$
$\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M})=\widehat{D}^{\infty}(N_{n,I})\widehat{\otimes}_{D^{\infty}(N_{n,I,w})}(\pi^{\infty,w})^{\prime}\cong\left(\rm{c}\textit{-}\mathrm{Ind}_{N_{n,I,w}}^{N_{n,I}}\pi^{\infty,w}\right)^{\prime}$

and

$\mathrm{gr}_{w}(\mathcal{M})\cong
D^{\infty}(L_{n,I})\widehat{\otimes}_{D^{\infty}(L_{n,I,w})}\mathrm{gr}_{w}^{\rm{u}}(\mathcal{M})\cong\left(\rm{c}\textit{-}\mathrm{Ind}_{P_{n,I^{\prime}}^{w}}^{P_{n,I^{\prime}}^{w}P_{n,I}}\pi^{\infty,w}\right)^{\prime}.$

For each $I\subseteq\Delta_{n}$, we write
$\mathcal{O}_{n,I}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathcal{O}_{\rm{alg}}^{\mathfrak{l}_{n,I,E}\cap\mathfrak{b}_{n,E}}$
for the BGG category $\mathcal{O}$ attached to $\mathfrak{l}_{n,I,E}$ and
$\mathfrak{l}_{n,I,E}\cap\mathfrak{b}_{n,E}$. For each $\mu\in X(T_{n,E})$
which is dominant for $L_{n,I,E}\cap B_{n,E}$, we write $L^{I}(\mu)$ for the
unique simple object in $\mathcal{O}_{n,I}$ with highest weight $\mu$,
$M^{\Delta_{n}}_{I}(\mu)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}U(\mathfrak{g}_{n,E})\otimes_{U(\mathfrak{p}_{n,I,E})}L^{I}(\mu)$
for the generalized Verma module, and $\mathcal{O}_{n,I,\mu}$ for the maximal
subcategory of $\mathcal{O}_{n,I}$ so that $U(\mathfrak{z}_{n,I,E})$ acts on
each object of $\mathcal{O}_{n,I,\mu}$ by the same character as that of
$M^{\Delta_{n}}_{I}(\mu)$. More generally, we write
$\mathcal{O}_{n,I}^{I^{\prime}}\subseteq\mathcal{O}_{n,I}$ for the parabolic
BGG category corresponding to
$\mathfrak{l}_{n,I,E}\cap\mathfrak{p}_{n,I^{\prime},E}\supseteq\mathfrak{l}_{n,I,E}\cap\mathfrak{b}_{n,E}$
and
$\mathcal{O}_{n,I,\mu}^{I^{\prime}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathcal{O}_{n,I}^{I^{\prime}}\cap\mathcal{O}_{n,I,\mu}$.
For a $U(\mathfrak{z}_{n,I,E})$-module $V$, we write $V_{n,I,\mu}$ for its
maximal quotient on which $U(\mathfrak{z}_{n,I,E})$ acts by the same character
as that of $M^{\Delta_{n}}_{I}(\mu)$, and in particular $V\mapsto V_{n,I,\mu}$
gives a projection functor
$\mathcal{O}_{n,I}\rightarrow\mathcal{O}_{n,I,\mu}$. We assume from now that
$\mu=-\lambda$ where $\lambda\in X(T_{n,E})$ is dominant with respect to
$B_{n,E}^{+}$.

######  Lemma 4.13.

Let $I,I^{\prime}\subseteq\Delta_{n}$ be two subsets. Then we have

  1. (i)

$H^{k}(\mathfrak{n}_{n,I,E},V)_{n,I,\mu}=0$ for each
$V\in\mathcal{O}_{n,\Delta_{n},\mu}$ and each $k\geq 1$;

  2. (ii)

$H^{0}(\mathfrak{n}_{n,I,E},-)_{n,I,\mu}$ induces an exact functor
$\mathcal{O}_{n,\Delta_{n},\mu}\rightarrow\mathcal{O}_{n,I,\mu}$; and

  3. (iii)

$H^{0}(\mathfrak{n}_{n,I,E},M^{\Delta_{n}}_{I^{\prime}}(\mu))_{n,I,\mu}\cong
F_{n,I}(\lambda)^{\prime}$ for each $I\subseteq
I^{\prime}\subseteq\Delta_{n}$.

###### Proof.

Let $V$ be an object in $\mathcal{O}_{n,\Delta_{n},\mu}$. We can compute
$H^{k}(\mathfrak{n}_{n,I,E},V)$ as the cohomology of the Chevalley–Eilenberg
complex with the $k$-th term given by
$\mathrm{Hom}_{E}(\wedge^{k}\mathfrak{n}_{n,I,E},V)$. Note that
$\mathrm{Hom}_{E}(\wedge^{k}\mathfrak{n}_{n,I,E},V)$ is semisimple as a
$U(\mathfrak{z}_{n,I,E})$-module and contains $H^{k}(\mathfrak{n}_{n,I,E},V)$
as a $U(\mathfrak{z}_{n,I,E})$-submodule. Then we conclude by the observation
that the $U(\mathfrak{z}_{n,I,E})$-module
$\mathrm{Hom}_{E}(\wedge^{k}\mathfrak{n}_{n,I,E},V)\cong
V\otimes_{E}\wedge^{k}\mathfrak{n}_{n,I,E}^{+}$ (with
$\mathfrak{n}_{n,I,E}^{+}$ the Lie algebra of the unipotent radical of the
parabolic subgroup opposite to $P_{n,I,E}\subseteq G_{n,E}$) satisfies
$(V\otimes_{E}\wedge^{k}\mathfrak{n}_{n,I,E}^{+})_{n,I,\mu}=0$ for each $k\geq
1$. The vanishing of $H^{1}(\mathfrak{n}_{n,I,E},-)_{n,I,\mu}$ implies the
exactness of $H^{0}(\mathfrak{n}_{n,I,E},-)_{n,I,\mu}$. It is not difficult to
see that, for each simple object
$L^{\Delta_{n}}(\mu^{\prime})\in\mathcal{O}_{n,\Delta_{n},\mu}$, we have

$H^{0}(\mathfrak{n}_{n,I,E},L^{\Delta_{n}}(\mu^{\prime}))\cong
L^{I}(\mu^{\prime})$

and thus

$H^{0}(\mathfrak{n}_{n,I,E},L^{\Delta_{n}}(\mu^{\prime}))_{n,I,\mu}\neq 0$

if and only if $\mu^{\prime}=w\cdot\mu$ for some element $w\in W(L_{n,I,E})$
(in the Weyl group of $L_{n,I,E}$). Now let $I\subseteq
I^{\prime}\subseteq\Delta_{n}$ be two subsets, then the surjection
$M^{\Delta_{n}}_{I^{\prime}}(\mu)\twoheadrightarrow L^{\Delta_{n}}(\mu)$
induces a surjection

$H^{0}(\mathfrak{n}_{n,I,E},M^{\Delta_{n}}_{I^{\prime}}(\mu))_{n,I,\mu}\twoheadrightarrow
H^{0}(\mathfrak{n}_{n,I,E},L^{\Delta_{n}}(\mu))_{n,I,\mu}\cong L^{I}(\mu).$

If this surjection is not an isomorphism, then there exists $1\neq w\in
W(L_{n,I,E})$ such that $w\cdot\mu$ is dominant with respect to $B_{n,E}\cap
L_{n,I^{\prime},E}$, but this is impossible as $I\subseteq I^{\prime}$. ∎

We assume from now on that $\pi^{\infty}=1_{L_{n,I^{\prime}}}$, which implies
that $\mathrm{gr}_{w}^{1}(\mathcal{M})=(\mathcal{M}^{1})^{w}$ for each $w\in
W_{n}^{I^{\prime},I}$. We continue to assume that $\mu=-\lambda$ where
$\lambda\in X(T_{n,E})$ is dominant with respect to $B_{n,E}^{+}$. Note that
$Z_{n,I}$ acts on $F_{n,I}(\lambda)$ by an algebraic character
$\delta_{n,I,\lambda}$.

######  Lemma 4.14.

If $w=1$ and thus
$C_{1}=\overline{C_{1}}=Y_{1}=\overline{Y_{1}}=P_{n,I^{\prime}}P_{n,I}$ is the
closed cell, then we have

  1. (i)

$H^{k}(N_{n,I},\mathrm{gr}_{1}(\mathcal{M}))_{\delta_{n,I,\lambda}^{\prime}}=0$
if $k\geq 1$;

  2. (ii)

$H^{0}(N_{n,I},\mathrm{gr}_{1}(\mathcal{M}))_{\delta_{n,I,\lambda}^{\prime}}=F_{n,I}(\lambda)$
if $I\subseteq I^{\prime}$; and

  3. (iii)

$H^{0}(N_{n,I},\mathrm{gr}_{1}(\mathcal{M}))_{\delta_{n,I,\lambda}^{\prime}}=\mathcal{F}_{L_{n,I}\cap
P_{n,I\cap I^{\prime}}}^{L_{n,I}}(V,1_{L_{n,I\cap I^{\prime}}})$ for some
$V\in\mathcal{O}_{n,I,\mu}^{I\cap I^{\prime}}$ if $I\not\subseteq I^{\prime}$.

###### Proof.

When $w=1$, we have $N_{n,I}=N_{n,I,w}$, $L_{n,I,1}=L_{n,I}\cap
P_{n,I^{\prime}}$ and
$\mathrm{gr}_{1}^{\rm{u}}(\mathcal{M})=\mathrm{gr}_{1}^{1}(\mathcal{M})=\mathcal{M}^{1}$.
The key observation is Proposition 4.7 induces an isomorphism

$H^{k}(N_{n,I},\mathrm{gr}_{1}(\mathcal{M}))_{\delta_{n,I,\lambda}^{\prime}}\cong\mathcal{F}_{L_{n,I,1}}^{L_{n,I}}(H^{k}(\mathfrak{n}_{n,I,E},\mathcal{M}^{1,\flat})_{n,I,\mu},(\pi^{\infty})_{N_{n,I,1}})^{\prime}$

with $N_{n,I,1}$ being the unipotent radical of $L_{n,I,1}$. As
$\mathcal{M}^{1,\flat}=M^{\Delta_{n}}_{I^{\prime}}(\mu)$, the desired results
follow directly from Lemma 4.13. ∎

######  Lemma 4.15.

If $1\neq w\in W_{n}^{I^{\prime},I}$, then we have

$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))_{\delta_{n,I,\lambda}^{\prime}}=0$

for all $k\geq 0$.

###### Proof.

This follows directly from Proposition 4.7, Lemma 4.10 and Lemma 4.14. In
fact, Lemma 4.10 already implies that, if
$H^{k}(N_{n,I},\mathrm{gr}_{1}(\mathcal{M}))_{\delta_{n,I,\lambda}^{\prime}}\neq
0$ for certain $k\geq 0$, then
$H^{k}(N_{n,I},\mathrm{gr}_{w}(\mathcal{M}))_{\delta_{n,I,\lambda}^{\prime}}\neq
0$ for all $k\geq 0$ and $1\neq w\in W_{n}^{I^{\prime},I}$. ∎

######  Proposition 4.16.

Let $I,I^{\prime}\subseteq\Delta_{n}$ be two subsets. Then we have

  1. (i)

$\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\cong
H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$ for each $k\geq 0$, if
$I\subseteq I^{\prime}$; and

  2. (ii)

$\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))=0$
for each $k\geq 0$, if $I\not\subseteq I^{\prime}$.

###### Proof.

Recall that we write $\mathrm{Ext}^{\bullet}_{D(G_{n}),\lambda}$ (resp.
$\mathrm{Ext}^{\bullet}_{D(L_{n,I}),\lambda}$) for the $\mathrm{Ext}$-groups
between abstract $D(G_{n})$-modules (resp. $D(L_{n,I})$-modules) fixing
$Z_{n}$-character (which equals that of $F_{n,I}(\lambda)^{\prime}$). There
exists a standard spectral sequence (see Lemma 1.8)

$\mathrm{Ext}^{k_{1}}_{D(L_{n,I}),\lambda}(F_{n,I}(\lambda)^{\prime},H^{k_{2}}(N_{n,I},i_{n,I^{\prime}}^{\rm{an}}(\lambda)^{\prime}))\Rightarrow\mathrm{Ext}^{k_{1}+k_{2}}_{D(G_{n}),\lambda}(i_{n,I}^{\rm{an}}(\lambda)^{\prime},i_{n,I^{\prime}}^{\rm{an}}(\lambda)^{\prime})=\mathrm{Ext}^{k_{1}+k_{2}}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda)).$

It follows from Lemma 4.14 and Lemma 4.15 that

$\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\cong\mathrm{Ext}^{k}_{L_{n,I},\lambda}(H_{0}(N_{n,I},i_{n,I^{\prime}}^{\rm{an}}(\lambda))_{\delta_{n,I,\lambda}},F_{n,I}(\lambda)).$

If $I\subseteq I^{\prime}$, we have

$\mathrm{Ext}^{k}_{L_{n,I},\lambda}(H_{0}(N_{n,I},i_{n,I^{\prime}}^{\rm{an}}(\lambda))_{\delta_{n,I,\lambda}},F_{n,I}(\lambda))\cong\mathrm{Ext}^{k}_{L_{n,I},\lambda}(F_{n,I}(\lambda),F_{n,I}(\lambda))\cong
H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$

where the second isomorphism follows from for example the translation functor
in [JL21]. If $I\not\subseteq I^{\prime}$, then it suffices to show that

$\mathrm{Ext}^{k}_{L_{n,I},\lambda}(\mathcal{F}_{L_{n,I}\cap P_{n,I\cap
I^{\prime}}}^{L_{n,I}}(V,1_{L_{n,I\cap I^{\prime}}}),F_{n,I}(\lambda))=0$

for each $V\in\mathcal{O}_{n,I,\mu}^{I\cap I^{\prime}}$ and $k\geq 0$. As each
$V\in\mathcal{O}_{n,I,\mu}^{I\cap I^{\prime}}$ admits BGG resolution by
generalized Verma modules in $\mathcal{O}_{n,I,\mu}^{I\cap I^{\prime}}$, we
reduce to the case when $V$ has the form $M^{I}_{I\cap
I^{\prime}}(\omega)=U(\mathfrak{l}_{n,I})\otimes_{U(\mathfrak{l}_{n,I}\cap\mathfrak{p}_{n,I\cap
I^{\prime}})}F_{n,I\cap I^{\prime}}(\omega)\in\mathcal{O}_{n,I,\lambda}^{I\cap
I^{\prime}}$ where $\omega\in X(T_{n})$ is a weight dominant for
$\mathfrak{b}_{n}\cap\mathfrak{l}_{n,I\cap I^{\prime}}$. Then it follows from
Proposition 6.5 of [ST05] that

$\mathrm{Ext}^{k}_{L_{n,I},\lambda}(\mathcal{F}_{L_{n,I}\cap P_{n,I\cap
I^{\prime}}}^{L_{n,I}}(V,1_{L_{n,I\cap
I^{\prime}}}),F_{n,I}(\lambda))\cong\mathrm{Ext}^{k}_{L_{n,I},\lambda}(i_{n,I,I\cap
I^{\prime}}^{\rm{an}}(F_{n,I\cap I^{\prime}}(\omega)),F_{n,I}(\lambda))\\\
\cong\mathrm{Ext}^{\ell}_{L_{n,I},-\lambda}(F_{n,I}(\lambda)^{\vee},i_{n,I,I\cap
I^{\prime}}^{\rm{an}}(F_{n,I\cap
I^{\prime}}(\omega)^{\vee}\otimes_{E}\mathfrak{d}_{L_{n,I}\cap P_{n,I\cap
I^{\prime}}}))$

for each $k,\ell\geq 0$ satisfying $\ell=k+\mathrm{dim}L_{n,I}\cap P_{n,I\cap
I^{\prime}}-\mathrm{dim}L_{n,I}$. Here $\mathfrak{d}_{L_{n,I}\cap P_{n,I\cap
I^{\prime}}}$ is locally algebraic character of $L_{n,I}\cap P_{n,I\cap
I^{\prime}}$ with non-trivial smooth part as $I\not\subseteq I^{\prime}$.
Hence, $F_{n,I\cap
I^{\prime}}(\omega)^{\vee}\otimes_{E}\mathfrak{d}_{L_{n,I}\cap P_{n,I\cap
I^{\prime}}}$ and $H_{\ell_{2}}(N_{n,I\cap
I^{\prime}},F_{n,I}(\lambda)^{\vee})$ never share the same $Z_{n,I\cap
I^{\prime}}$-character for any $\ell_{2}\geq 0$, which together with the
spectral sequence

$\mathrm{Ext}^{\ell_{1}}_{L_{n,I},-\lambda}(H_{\ell_{2}}(N_{n,I\cap
I^{\prime}},F_{n,I}(\lambda)^{\vee}),F_{n,I\cap
I^{\prime}}(\omega)^{\vee}\otimes_{E}\mathfrak{d}_{L_{n,I}\cap P_{n,I\cap
I^{\prime}}})\\\
\Rightarrow\mathrm{Ext}^{\ell_{1}+\ell_{2}}_{L_{n,I},-\lambda}(F_{n,I}(\lambda)^{\vee},i_{n,I,I\cap
I^{\prime}}^{\rm{an}}(F_{n,I\cap
I^{\prime}}(\omega)^{\vee}\otimes_{E}\mathfrak{d}_{L_{n,I}\cap P_{n,I\cap
I^{\prime}}}))$

implies that

$\mathrm{Ext}^{\ell}_{L_{n,I},-\lambda}(F_{n,I}(\lambda)^{\vee},i_{n,I,I\cap
I^{\prime}}^{\rm{an}}(F_{n,I\cap
I^{\prime}}(\omega)^{\vee}\otimes_{E}\mathfrak{d}_{L_{n,I}\cap P_{n,I\cap
I^{\prime}}}))=0$

for each $\ell\geq 0$. ∎

######  Lemma 4.17.

Let $I\subseteq I^{\prime}\subseteq I^{\prime\prime}\subseteq\Delta_{n}$ be
three subsets and $k,k^{\prime}\geq 0$ be two integers. Then the following
commutative diagram is commutative.

$\displaystyle{\mathrm{Ext}^{k^{\prime}}_{G_{n},\lambda}(i_{n,I^{\prime\prime}}^{\rm{an}}(\lambda),i_{n,I^{\prime}}^{\rm{an}}(\lambda))\times\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\sim}$$\displaystyle{H^{k^{\prime}}(\overline{L}_{n,I^{\prime}},1_{\overline{L}_{n,I^{\prime}}})\times
H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{H^{k^{\prime}}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})\times
H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\displaystyle{\mathrm{Ext}^{k+k^{\prime}}_{G_{n},\lambda}(i_{n,I^{\prime\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\displaystyle{H^{k+k^{\prime}}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})}$

###### Proof.

It is clear that
$\mathrm{Hom}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))$
is one dimensional and spanned by the natural embedding
$\iota:i_{n,I^{\prime}}^{\rm{an}}(\lambda)\hookrightarrow
i_{n,I}^{\rm{an}}(\lambda)$. Moreover, Proposition 4.16 also implies that the
cup product map

$\mathrm{Hom}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\times\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\xrightarrow{\cup}\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))$

induces an isomorphism

$\iota\cup\bullet:\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\xrightarrow{\sim}\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))$

which corresponds to identity map of
$H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$. For each
$x\in\mathrm{Ext}^{k^{\prime}}_{G_{n},\lambda}(i_{n,I^{\prime\prime}}^{\rm{an}}(\lambda),i_{n,I^{\prime}}^{\rm{an}}(\lambda))$
and
$y\in\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))$,
there exists
$y^{\prime}\in\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))$
such that $y=\iota\cup y^{\prime}$, and thus $x\cup y=(x\cup\iota)\cup
y^{\prime}$. We finish the proof by noting that the map

$\bullet\cup\iota:\mathrm{Ext}^{k^{\prime}}_{G_{n},\lambda}(i_{n,I^{\prime\prime}}^{\rm{an}}(\lambda),i_{n,I^{\prime}}^{\rm{an}}(\lambda))\rightarrow\mathrm{Ext}^{k^{\prime}}_{G_{n},\lambda}(i_{n,I^{\prime\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))$

corresponds to the restriction map
$H^{k^{\prime}}(\overline{L}_{n,I^{\prime}},1_{\overline{L}_{n,I^{\prime}}})\rightarrow
H^{k^{\prime}}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$ under Proposition
4.16. ∎

We use the shortened notation
$H_{I}^{\bullet}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}H^{\bullet}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$,
$H_{I^{\prime}}^{\bullet}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}H^{\bullet}(\overline{L}_{n,I^{\prime}},1_{\overline{L}_{n,I^{\prime}}})$
and write $\mathrm{Res}_{n,I,I^{\prime}}^{\bullet}:H_{I}^{\bullet}\rightarrow
H_{I^{\prime}}^{\bullet}$ for the restriction map, for each
$I^{\prime}\subseteq I\subseteq\Delta_{n}$. We consider two pair of subsets
$I_{0}\subseteq I_{1}\subseteq\Delta_{n}$ and $I_{2}\subseteq
I_{3}\subseteq\Delta_{n}$ satisfying $I_{3}\subseteq I_{1}$ and
$I_{1}\setminus I_{0}=I_{3}\setminus I_{2}=\\{i\\}$ for some $i\in\Delta_{n}$.
Let $\theta,\theta^{\prime}\in E^{\times}$ be two scalars. We define a complex
$\mathbf{C}_{\theta}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}[i_{n,I_{1}}^{\rm{an}}(\lambda)\hookrightarrow
i_{n,I_{0}}^{\rm{an}}(\lambda)]$ (resp.
$\mathbf{C}_{\theta^{\prime}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}[i_{n,I_{3}}^{\rm{an}}(\lambda)\hookrightarrow
i_{n,I_{2}}^{\rm{an}}(\lambda)]$) supported in degree $[-1,0]$ with the
injection given by $\theta\kappa_{I_{1},I_{0}}^{\rm{an}}(\lambda)$ (resp.
$\theta^{\prime}\kappa_{I_{3},I_{2}}^{\rm{an}}(\lambda)$). We have canonical
maps $p_{\theta}:i_{n,I_{0}}^{\rm{an}}(\lambda)\rightarrow\mathbf{C}_{\theta}$
and $q_{\theta}:\mathbf{C}_{\theta}\rightarrow
i_{n,I_{1}}^{\rm{an}}(\lambda)[-1]$ and similarly for $p_{\theta^{\prime}}$
and $q_{\theta^{\prime}}$. In the following, we will use the isomorphisms in
item (i) Proposition 4.16 without further explanation. It follows directly
from Proposition 4.16 that

  * •

$\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(i_{n,I_{0}}^{\rm{an}}(\lambda),i_{n,I_{3}}^{\rm{an}}(\lambda))=0$;

  * •

$\kappa_{I_{1},I_{0}}^{\rm{an}}(\lambda)$ induces an isomorphism
$\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(i_{n,I_{0}}^{\rm{an}}(\lambda),i_{n,I_{2}}^{\rm{an}}(\lambda))\xrightarrow{\sim}\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda),i_{n,I_{2}}^{\rm{an}}(\lambda))$
which corresponds to identity map on $H_{I_{2}}^{\bullet}$;

  * •

$\kappa_{I_{3},I_{2}}^{\rm{an}}(\lambda)$ induces a map
$\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda),i_{n,I_{3}}^{\rm{an}}(\lambda))\rightarrow\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda),i_{n,I_{2}}^{\rm{an}}(\lambda))$
which corresponds to
$\mathrm{Res}_{n,I_{3},I_{2}}^{\bullet}:H_{I_{3}}^{\bullet}\rightarrow
H_{I_{2}}^{\bullet}$.

Consequently, for each $\theta,\theta^{\prime}\in E^{\times}$, we have

  * •

$\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(\mathbf{C}_{\theta},i_{n,I_{2}}^{\rm{an}}(\lambda))=0$
as we can compute it using a spectral sequence whose second page is
identically zero;

  * •

$q_{\theta^{\prime}}$ and $q_{\theta}$ induce isomorphisms
$\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(\mathbf{C}_{\theta},\mathbf{C}_{\theta^{\prime}})\xrightarrow{q_{\theta^{\prime},\ast}}\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(\mathbf{C}_{\theta},i_{n,I_{3}}^{\rm{an}}(\lambda)[-1])\xleftarrow{q_{\theta}^{\ast}}H_{I_{3}}^{\bullet}$;

  * •

$p_{\theta^{\prime}}$ induces a isomorphism
$H_{I_{2}}^{\bullet}\xrightarrow{p_{\theta^{\prime},\ast}}\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(i_{n,I_{0}}^{\rm{an}}(\lambda),\mathbf{C}_{\theta^{\prime}})$.

The following is our main goal.

######  Lemma 4.18.

The composition

$H_{I_{3}}^{\bullet}\xrightarrow{(q_{\theta^{\prime},\ast})^{-1}q_{\theta}^{\ast}}\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(\mathbf{C}_{\theta},\mathbf{C}_{\theta^{\prime}})\xrightarrow{p_{\theta}^{\ast}}\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(i_{n,I_{0}}^{\rm{an}}(\lambda),\mathbf{C}_{\theta^{\prime}})\xrightarrow{p_{\theta^{\prime},\ast}^{-1}}H_{I_{2}}^{\bullet}$

is given by
$\theta^{\prime}\theta^{-1}\mathrm{Res}_{n,I_{3},I_{2}}^{\bullet}$.

###### Proof.

Let $k\geq 0$ be an integer. It follows from item (i) of Lemma 1.7 that we can
compute
$\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda)[-1],\mathbf{C}_{\theta^{\prime}})$
via a spectral sequence whose degenerate at its second page due to degree
reason. Therefore we have a canonical isomorphism in the derived category of
$E$-vector spaces

$\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda)[-1],\mathbf{C}_{\theta^{\prime}})\cong
H_{\theta^{\prime}}^{k}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}[H_{I_{3}}^{k}\rightarrow
H_{I_{2}}^{k}]$

with RHS supported in degree $[0,1]$ and the map given by
$\theta^{\prime}\mathrm{Res}_{n,I_{3},I_{2}}^{k}$. The distinguished triangle

(4.14)
$i_{n,I_{0}}^{\rm{an}}(\lambda)\xrightarrow{p_{\theta}}\mathbf{C}_{\theta}\xrightarrow{q_{\theta}}i_{n,I_{1}}^{\rm{an}}(\lambda)[-1]\rightarrow$

determines the distinguish triangle

(4.15)
$\mathbf{C}_{\theta}\xrightarrow{q_{\theta}}i_{n,I_{1}}^{\rm{an}}(\lambda)[-1]\xrightarrow{\theta\kappa_{I_{1},I_{0}}^{\rm{an}}(\lambda)[-1]}i_{n,I_{0}}^{\rm{an}}(\lambda)[-1]\xrightarrow{p_{\theta}[-1]}.$

This induces a distinguish triangle

(4.16)
$\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{0}}^{\rm{an}}(\lambda)[-1],\mathbf{C}_{\theta^{\prime}})\xrightarrow{\theta\kappa_{I_{1},I_{0}}^{\rm{an}}(\lambda)[-1]^{\ast}}\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda)[-1],\mathbf{C}_{\theta^{\prime}})\xrightarrow{q_{\theta}^{\ast}}\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{\theta},\mathbf{C}_{\theta^{\prime}})\xrightarrow{p_{\theta}^{\ast}}.$

The commutative diagram

$\displaystyle{\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{0}}^{\rm{an}}(\lambda)[-1],\mathbf{C}_{\theta^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta\kappa_{I_{1},I_{0}}^{\rm{an}}(\lambda)[-1]^{\ast}}$$\displaystyle{\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda)[-1],\mathbf{C}_{\theta^{\prime}})}$$\displaystyle{\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{0}}^{\rm{an}}(\lambda)[-1],i_{n,I_{2}}^{\rm{an}}(\lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\scriptstyle{\theta\kappa_{I_{1},I_{0}}^{\rm{an}}(\lambda)[-1]^{\ast}}$$\displaystyle{\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda)[-1],i_{n,I_{2}}^{\rm{an}}(\lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$

implies that $\theta\kappa_{I_{1},I_{0}}^{\rm{an}}(\lambda)[-1]^{\ast}$ in
(4.16) corresponds to the composition
$H_{I_{2}}^{k}[1]\xrightarrow{\theta}H_{I_{2}}^{k}[1]\xrightarrow{f_{\theta^{\prime}}}H_{\theta^{\prime}}^{k}$
with the second map $f_{\theta^{\prime}}$ being the standard truncation map.
On the other hand, using the commutative diagram

$\displaystyle{\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda)[-1],\mathbf{C}_{\theta^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{\theta}^{\ast}}$$\displaystyle{\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{\theta},\mathbf{C}_{\theta^{\prime}})}$$\displaystyle{\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda)[-1],i_{n,I_{3}}^{\rm{an}}(\lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\scriptstyle{q_{\theta}^{\ast}}$$\displaystyle{\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{\theta},i_{n,I_{3}}^{\rm{an}}(\lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$

we know that the map $q_{\theta}^{\ast}$ in (4.16) corresponds to the
truncation map $g_{\theta^{\prime}}:H_{\theta^{\prime}}^{k}\rightarrow
H_{I_{3}}^{k}$. Consequently, we deduce from (4.16) the following distinguish
triangle

$H_{I_{2}}^{k}[1]\xrightarrow{\theta
f_{\theta^{\prime}}}H_{\theta^{\prime}}^{k}\xrightarrow{g_{\theta^{\prime}}}H_{I_{3}}^{k}\xrightarrow{(q_{\theta^{\prime},\ast})^{-1}q_{\theta}^{\ast}p_{\theta}^{\ast}p_{\theta^{\prime},\ast}^{-1}}.$

If we compare it with the obvious distinguish triangle

$H_{I_{2}}^{k}[1]\xrightarrow{f_{\theta^{\prime}}}H_{\theta^{\prime}}^{k}\xrightarrow{g_{\theta^{\prime}}}H_{I_{3}}^{k}\xrightarrow{\theta^{\prime}\mathrm{Res}_{n,I_{3},I_{2}}^{k}}$

from the definition of $H_{\theta^{\prime}}^{k}$, we conclude that
$(q_{\theta^{\prime},\ast})^{-1}q_{\theta}^{\ast}p_{\theta}^{\ast}p_{\theta^{\prime},\ast}^{-1}=\theta^{-1}\theta^{\prime}\mathrm{Res}_{n,I_{3},I_{2}}^{k}$,
which finishes the proof. ∎

###  4.2. Extensions between two complex

In this section, we compute the $\mathrm{Ext}$-groups between certain
complexes (with each term given by direct sum of various
$i_{n,I}^{\rm{an}}(\lambda)$) using the computation of
$E_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}$ in Section 2 and Proposition 4.16.

We first introduce here some general complexes that will be essential in
Section 4.3. Note that $\mathbf{C}_{I_{0},I_{1}}(\lambda)$ (for various
$I_{0},I_{1}\subseteq\Delta_{n}$) are special examples of the more general
complexes we consider here.

We consider a tuple
$\mathcal{P}=(\underline{I},\underline{\ell}^{+},\underline{\ell}^{-})$ that
satisfies the following conditions

  * •

$\underline{I}=(I_{r^{\prime}})_{1\leq r^{\prime}\leq r}$ with
$\Delta_{n}=\bigsqcup_{r^{\prime}=1}^{r}I_{r^{\prime}}$ a partition;

  * •

$\underline{\ell}^{+}=(\ell_{r^{\prime}}^{+})_{1\leq r^{\prime}\leq r}$ and
$\underline{\ell}^{-}=(\ell_{r^{\prime}}^{-})_{1\leq r^{\prime}\leq r}$ with
$0\leq\ell_{r^{\prime}}^{-}\leq\ell_{r^{\prime}}^{+}\leq\\#I_{r^{\prime}}$ for
each $1\leq r^{\prime}\leq r$.

Given a tuple $\mathcal{P}$ as above, we define a complex
$\mathbf{C}_{\mathcal{P}}(\lambda)$ whose degree $-\ell$ term is the direct
sum of all $i_{n,I^{\prime}}^{\rm{an}}(\lambda)$ satisfying $\\#I^{\prime}\cap
I_{r^{\prime}}\in[\ell_{r^{\prime}}^{-},\ell_{r^{\prime}}^{+}]$ for each
$1\leq r^{\prime}\leq r$.

Given an extra $I\subseteq\Delta_{n}$, we can compute
$\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(\mathbf{C}_{\mathcal{P}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))$
by a spectral sequence ${}_{\mathcal{P}}E_{\bullet,I}^{\bullet,\bullet}$ (see
item (ii) of Lemma 1.7) with ${}_{\mathcal{P}}E_{1,I}^{-\ell,k}$ isomorphic to
the direct sum of

$\mathrm{Ext}^{k}_{G_{n},\lambda}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\cong
H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$

for $I\subseteq I^{\prime}\subseteq\Delta_{n}$ satisfying $\\#I^{\prime}=\ell$
and $\\#I^{\prime}\cap
I_{r^{\prime}}\in[\ell_{r^{\prime}}^{-},\ell_{r^{\prime}}^{+}]$ for each
$1\leq r^{\prime}\leq r$. For three integers
$\ell^{\prime}\leq\ell^{\prime}\leq\ell$, we use the shorted notation
$c(\ell,\ell^{\prime},\ell^{\prime\prime})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\frac{(\ell-\ell^{\prime\prime})!}{(\ell-\ell^{\prime})!(\ell^{\prime}-\ell^{\prime\prime})!}$.
We fix a $1\leq r^{\prime}\leq r$ for now and set
$\ell_{r^{\prime},I}^{-}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\max\\{\ell_{r^{\prime}}^{-},\\#I\cap
I_{r^{\prime}}\\}$. We set
$d_{r^{\prime}}^{+}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}c(\\#I_{r^{\prime}},\ell_{r^{\prime}}^{+},\\#I\cap
I_{r^{\prime}})$ if $\ell_{r^{\prime}}^{+}=\ell_{r^{\prime},I}^{-}$,
$d_{r^{\prime}}^{+}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}0$ if
$\\#I_{r^{\prime}}=\ell_{r^{\prime}}^{+}>\ell_{r^{\prime},I}^{-}$, and
$d_{r^{\prime}}^{+}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}c(\\#I_{r^{\prime}},\ell_{r^{\prime}}^{+},1+\\#I\cap
I_{r^{\prime}})$ if
$\\#I_{r^{\prime}}>\ell_{r^{\prime}}^{+}>\ell_{r^{\prime},I}^{-}$. We also set
$d_{I,r^{\prime}}^{-}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}0$ if
$\ell_{r^{\prime}}^{+}>\ell_{r^{\prime},I}^{-}=\\#I\cap I_{r^{\prime}}$, and
$d_{I,r^{\prime}}^{-}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}c(\\#I_{r^{\prime}}-1,\ell_{r^{\prime},I}^{-},\\#I\cap
I_{r^{\prime}})$ if $\ell_{r^{\prime}}^{+}>\ell_{r^{\prime},I}^{-}>\\#I\cap
I_{r^{\prime}}$.

Then we consider the complex $\mathbf{C}_{\mathcal{P},I,r^{\prime}}$ of
$E$-vector spaces (having support in degree
$[\ell_{r^{\prime},I}^{-},\ell_{r^{\prime}}^{+}]$) with the dimension of the
degree $\ell_{r^{\prime}}$ term of $\mathbf{C}_{\mathcal{P},I,r^{\prime}}$
counting the number of sets $I^{\prime}_{r^{\prime}}$ satisfying $I\cap
I_{r^{\prime}}\subseteq I^{\prime}_{r^{\prime}}\subseteq I_{r^{\prime}}$ and
$\\#I^{\prime}_{r^{\prime}}=\ell_{r^{\prime}}\in[\ell_{r^{\prime},I}^{-},\ell_{r^{\prime}}^{+}]$.
The differential of $\mathbf{C}_{\mathcal{P},I,r^{\prime}}$ is naturally
induced from inclusion between various $I^{\prime}_{r^{\prime}}$. We write
$H^{\bullet}(\mathbf{C}_{\mathcal{P},I,r^{\prime}})$ for its cohomology
(understood to be a complex with zero differentials) and observe that
$H^{\bullet}(\mathbf{C}_{\mathcal{P},I,r^{\prime}})=0$ if
$\ell_{r^{\prime},I}^{-}>\ell_{r^{\prime}}^{+}$,
$H^{\bullet}(\mathbf{C}_{\mathcal{P},I,r^{\prime}})=E^{\oplus
d_{r^{\prime}}^{+}}[\ell_{r^{\prime}}^{+}]$ if
$\ell_{r^{\prime},I}^{-}=\ell_{r^{\prime}}^{+}$, and
$H^{\bullet}(\mathbf{C}_{\mathcal{P},I,r^{\prime}})=E^{\oplus
d_{I,r^{\prime}}^{-}}[\ell_{r^{\prime},I}^{-}]\oplus E^{\oplus
d_{r^{\prime}}^{+}}[\ell_{r^{\prime}}^{+}]$ if
$\ell_{r^{\prime},I}^{-}<\ell_{r^{\prime}}^{+}$.

######  Lemma 4.19.

We have an isomorphism of complexes

(4.17) $_{\mathcal{P}}E_{1,I}^{\bullet,k}\cong
H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})\otimes\bigotimes_{r^{\prime}=1}^{r}\mathbf{C}_{\mathcal{P},I,r^{\prime}}$

which induces an isomorphism of cohomologies

${}_{\mathcal{P}}E_{2,I}^{\bullet,k}\cong
H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})\otimes\bigotimes_{r^{\prime}=1}^{r}H^{\bullet}(\mathbf{C}_{\mathcal{P},I,r^{\prime}}),$

and the spectral sequence ${}_{\mathcal{P}}E_{\bullet,I}^{\bullet,\bullet}$
degenerates at the second page. In particular, if
$\ell_{r^{\prime},I}^{-}=\ell_{r^{\prime}}^{+}$ for each $1\leq r^{\prime}\leq
r$, then ${}_{\mathcal{P}}E_{2,I}^{\bullet,k}\cong
H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})^{\oplus
d_{\mathcal{P}}^{+}}[\ell_{\mathcal{P}}^{+}]$ with
$d_{\mathcal{P}}^{+}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{r^{\prime}=1}^{r}d_{r^{\prime}}^{+}$
and
$\ell_{\mathcal{P}}^{+}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{r^{\prime}=1}^{r}\ell_{r^{\prime}}^{+}$.

###### Proof.

We notice that the complex ${}_{\mathcal{P}}E_{1,I}^{\bullet,k}$ is isomorphic
to the tensor of $H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$ with a
complex $\mathbf{C}_{\mathcal{P},I}$ of vector spaces, whose degree $\ell$
term has dimension counting the number of $I\subseteq
I^{\prime}\subseteq\Delta_{n}$ satisfying $\\#I^{\prime}=\ell$ and
$\\#I^{\prime}\cap
I_{r^{\prime}}\in[\ell_{r^{\prime}}^{-},\ell_{r^{\prime}}^{+}]$ for each
$1\leq r^{\prime}\leq r$. As each such $I^{\prime}$ admits a partition
$I^{\prime}=\bigsqcup_{r^{\prime}=1}^{r}I^{\prime}\cap I_{r^{\prime}}$, the
complex $\mathbf{C}_{\mathcal{P},I}$ is isomorphic to the tensor product of
complexes $\mathbf{C}_{\mathcal{P},I,r^{\prime}}$ for $1\leq r^{\prime}\leq r$
with $\mathbf{C}_{\mathcal{P},I,r^{\prime}}$ defined before this lemma. We
also note that $H^{\bullet}(\mathbf{C}_{\mathcal{P},I})$ is clearly isomorphic
to the tensor of various $H^{\bullet}(\mathbf{C}_{\mathcal{P},I,r^{\prime}})$.
The decomposition (4.17) actually indicates that
${}_{\mathcal{P}}E_{\bullet,I}^{\bullet,\bullet}$ is actually the tensor
product of two spectral sequences, one supported inside the $0$-th row and the
other supported inside the $0$-th column with both of them degenerate at the
second page, which forces ${}_{\mathcal{P}}E_{\bullet,I}^{\bullet,\bullet}$ to
degenerate at the second page as well. ∎

The following is a special case of Lemma 4.19 when we take $r=3$ and consider
the partition $\Delta_{n}=I_{0}\sqcup(I_{1}\setminus
I_{0})\sqcup(\Delta_{n}\setminus I_{1})$ with
$\ell_{1}^{+}=\ell_{1}^{-}=\\#I_{0}$, $\ell_{2}^{+}=\\#I_{1}\setminus I_{0}$,
$\ell_{2}^{-}=0$ and $\ell_{3}^{+}=\ell_{3}^{-}=0$. Note that we have
$\mathbf{C}_{\mathcal{P}}(\lambda)\cong\mathbf{C}_{I_{0},I_{1}}(\lambda)$ in
this case.

######  Lemma 4.20.

For each $k\in\mathbb{Z}$, we have

$\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\xleftarrow{\sim}\mathrm{Ext}^{k-\\#I_{1}}_{G_{n},\lambda}(i_{n,I_{1}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\cong
H^{k-\\#I_{1}}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$

if $I_{0}\cup I=I_{1}$ and
$\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))=0$
otherwise.

Note that $I_{0}\cup I=I_{1}$ if and only if $I_{1}\setminus I_{0}\subseteq
I\subseteq I_{1}$. Now we consider a second pair of subsets $I_{2}\subseteq
I_{3}\subseteq\Delta_{n}$. Using Lemma 4.20, we can compute
$\mathrm{Ext}^{\bullet}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))$
by a spectral sequence $\\{_{I_{0},I_{1}}E_{r,I_{2},I_{3}}^{-\ell,k}\\}_{r\geq
1,\\#I_{2}\leq\ell\leq\\#I_{3},k\geq 0}$ whose first page has $(-\ell,k)$ term
given by

$\bigoplus_{I_{2}\subseteq I\subseteq
I_{3},\\#I=\ell}\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\\\
\cong\bigoplus_{I_{2}\subseteq I\subseteq I_{3},I_{0}\cup
I=I_{1},\\#I=\ell}H^{k-\\#I_{1}}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})\cong\bigoplus_{I_{2}\cup(I_{1}\setminus
I_{0})\subseteq I\subseteq I_{1}\cap
I_{3},\\#I=\ell}H^{k-\\#I_{1}}(\overline{L}_{n,I},1_{\overline{L}_{n,I}}),$

which is precisely $E_{1,I_{2}\cup(I_{1}\setminus I_{0}),I_{1}\cap
I_{3}}^{-\ell,k-\\#I_{1}}$ as defined at the beginning of Section 2.2. The
following cases are particularly simple.

######  Lemma 4.21.

  1. (i)

If $I_{2}\not\subseteq I_{1}$ or $I_{1}\not\subseteq I_{0}\cup I_{3}$, then
$\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))=0$
for each $k\in\mathbb{Z}$.

  2. (ii)

If $I_{0}\subseteq I_{2}\subseteq I_{1}\subseteq I_{3}$, then
$\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))\cong
H^{k}(\overline{L}_{n,I_{1}},1_{\overline{L}_{n,I_{1}}})$ for each
$k\in\mathbb{Z}$. In particular,
$\mathrm{Hom}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))$
is spanned by the composition of two truncation morphisms
$\mathbf{C}_{I_{0},I_{1}}(\lambda)\rightarrow\mathbf{C}_{I_{2},I_{1}}(\lambda)\rightarrow\mathbf{C}_{I_{2},I_{3}}(\lambda)$.

  3. (iii)

The truncation map $\mathbf{C}_{I_{2},I_{1}\cap
I_{3}}(\lambda)\rightarrow\mathbf{C}_{I_{2},I_{3}}(\lambda)$ induces an
isomorphism of spectral sequences

${}_{I_{0},I_{1}}E_{\bullet,I_{2},I_{1}\cap
I_{3}}^{\bullet,\bullet}\xrightarrow{\sim}_{I_{0},I_{1}}E_{\bullet,I_{2},I_{3}}^{\bullet,\bullet}.$

Here we understand $\mathbf{C}_{I_{2},I_{1}\cap I_{3}}(\lambda)$ to be zero if
$I_{2}\not\subseteq I_{1}$.

  4. (iv)

If $I_{2}\subseteq I_{0}\subseteq I_{1}=I_{3}$, then we have a canonical
isomorphism of spectral sequences

${}_{I_{0},I_{1}}E_{\bullet,I_{2},I_{1}}^{\bullet,\bullet}\cong_{I_{0}\setminus
I,I_{1}}E_{\bullet,I_{2}\setminus I,I_{1}}^{\bullet,\bullet}$

for each $I\subseteq I_{2}$.

###### Proof.

The first part follows from the observation that $I_{2}\cup(I_{1}\setminus
I_{0})\subseteq I_{1}\cap I_{3}$ is equivalent to $I_{2}\subseteq
I_{1}\subseteq I_{0}\cup I_{3}$. For the second part, it suffices to note that
$I_{0}\subseteq I_{2}\subseteq I_{1}\subseteq I_{3}$ implies that
$I_{2}\cup(I_{1}\setminus I_{0})=I_{1}=I_{1}\cap I_{3}$, and thus

$\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))\cong_{I_{0},I_{1}}E_{1,I_{2},I_{3}}^{-\\#I_{1},k+\\#I_{1}}\cong
E_{1,I_{2}\cup(I_{1}\setminus I_{0}),I_{1}\cap I_{3}}^{-\\#I_{1},k}\cong
H^{k}(\overline{L}_{n,I_{1}},1_{\overline{L}_{n,I_{1}}}).$

In particular,
$\mathrm{Hom}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))$
is one dimensional and thus spanned by
$\mathbf{C}_{I_{0},I_{1}}(\lambda)\rightarrow\mathbf{C}_{I_{2},I_{1}}(\lambda)\rightarrow\mathbf{C}_{I_{2},I_{3}}(\lambda)$.
The third part follows from the following isomorphism of first page of
spectral sequences

${}_{I_{0},I_{1}}E_{1,I_{2},I_{3}}^{-\ell,k}\cong
E_{1,I_{2}\cup(I_{1}\setminus I_{0}),I_{1}\cap
I_{3}}^{-\ell,k-\\#I_{1}}\cong_{I_{0},I_{1}}E_{1,I_{2},I_{1}\cap
I_{3}}^{-\ell,k}.$

The fourth part follows from the isomorphism of first page of spectral
sequences

${}_{I_{0},I_{1}}E_{\bullet,I_{2},I_{1}}^{\bullet,\bullet}\cong
E_{1,I_{2}\cup(I_{1}\setminus
I_{0}),I_{1}}^{\bullet,\bullet-\\#I_{1}}\cong_{I_{0}\setminus
I,I_{1}}E_{1,I_{2}\setminus I,I_{1}}^{\bullet,\bullet}$

based on the equality $I_{2}\cup(I_{1}\setminus I_{0})=(I_{2}\setminus
I)\cup(I_{1}\setminus(I_{0}\setminus I))$. ∎

Now we apply computations from Section 2.2 to
${}_{I_{0},I_{1}}E_{r,I_{2},I_{3}}^{-\ell,k}\cong
E_{1,I_{2}\cup(I_{1}\setminus I_{0}),I_{1}\cap I_{3}}^{-\ell,k-\\#I_{1}}$.
Lemma 2.27 implies that
$\\{_{I_{0},I_{1}}E_{r,I_{2},I_{3}}^{-\ell,k}\\}_{r\geq
1,\\#I_{2}\leq\ell\leq\\#I_{3},k\geq 0}$ degenerates at the second page.
Hence, we deduce from Lemma 2.16 and Proposition 2.26 that

######  Theorem 4.22.

Let $I_{0}\subseteq I_{1}\subseteq\Delta_{n}$ and $I_{2}\subseteq
I_{3}\subseteq\Delta_{n}$ be two pairs of subsets with
$I_{0}^{\sharp}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}I_{2}\cup(I_{1}\setminus
I_{0})$ and
$I_{1}^{\sharp}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}I_{1}\cap
I_{3}$.

  * •

If
$\mathrm{Ext}_{G_{n},\lambda}^{h}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))\neq
0$, then we have $I_{2}\subseteq I_{1}\subseteq I_{0}\cup I_{3}$ and
$\\#I_{1}^{\sharp}-2\\#I_{0}^{\sharp}+\\#I_{1}\leq h\leq n^{2}-n+\\#I_{1}$.

  * •

For each
$h\in\\{\\#I_{1}^{\sharp}-2\\#I_{0}^{\sharp}+\\#I_{1},\\#I_{1}^{\sharp}-2\\#I_{0}^{\sharp}+\\#I_{1}+1\\}$,
the space

$M_{I_{0},I_{1},I_{2},I_{3}}^{h}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ext}_{G_{n},\lambda}^{\\#I_{1}^{\sharp}-2\\#I_{0}^{\sharp}+\\#I_{1}}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))$

admits a canonical decreasing filtration

$0=\mathrm{Fil}^{-\\#I_{0}^{\sharp}+1}(M_{I_{0},I_{1},I_{2},I_{3}}^{h})\subseteq\mathrm{Fil}^{-\\#I_{0}^{\sharp}}(M_{I_{0},I_{1},I_{2},I_{3}}^{h})\subseteq\cdots\subseteq\mathrm{Fil}^{-\\#I_{1}^{\sharp}}(M_{I_{0},I_{1},I_{2},I_{3}}^{h})=M_{I_{0},I_{1},I_{2},I_{3}}^{h}$

such that
$\mathrm{Fil}^{-\ell}(M_{I_{0},I_{1},I_{2},I_{3}}^{h})/\mathrm{Fil}^{-\ell+1}(M_{I_{0},I_{1},I_{2},I_{3}}^{h})\cong
E_{2,I_{0}^{\sharp},I_{1}^{\sharp}}^{-\ell,\ell+h-\\#I_{1}}$ admits a basis
indexed by $\Psi_{I_{0}^{\sharp},I_{1}^{\sharp}}^{-\ell,\ell+h-\\#I_{1}}$ for
each $\\#I_{0}^{\sharp}\leq\ell\leq\\#I_{1}^{\sharp}$.

Recall that $\mathbf{C}_{I,\Delta_{n}}(\lambda)\cong
V_{n,I}^{\rm{an}}(\lambda)[-\\#I]$ for each $I\subseteq\Delta_{n}$. If we take
$I_{1}=I_{3}=\Delta_{n}$, we have
$I_{0}^{\sharp}=I_{2}\cup(\Delta_{n}\setminus
I_{0})\subseteq\Delta_{n}=I_{1}^{\sharp}$. We set

$h_{I_{0},I_{2}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(\\#I_{1}^{\sharp}-2\\#I_{0}^{\sharp}+\\#I_{1})+\\#I_{2}-\\#I_{0}=2n-2-2\\#I_{2}\cup(\Delta_{n}\setminus
I_{0})+\\#I_{2}-\\#I_{0}\\\ =2n-2+2\\#I_{2}\cap(\Delta_{n}\setminus
I_{0})-2\\#I_{2}-2\\#\Delta_{n}\setminus
I_{0}+\\#I_{2}-\\#I_{0}=2\\#I_{2}\setminus I_{0}+\\#I_{0}-\\#I_{2}\\\
=\\#I_{0}+\\#I_{2}-2\\#I_{0}\cap I_{2}=\\#I_{0}\setminus
I_{2}+\\#I_{2}\setminus I_{0}.$

We observe that $I_{2}\subseteq I_{0}$ if and only if
$h_{I_{0},I_{2}}=\\#I_{0}-\\#I_{2}$, and $I_{0}\subseteq I_{2}$ if and only if
$h_{I_{0},I_{2}}=\\#I_{2}-\\#I_{0}$. As a consequence of Theorem 4.22, we
obtain

######  Corollary 4.23.

Let $I_{0},I_{2}\subseteq\Delta_{n}$ be two subsets.

  * •

If
$\mathrm{Ext}_{G_{n},\lambda}^{h}(V_{n,I_{0}}^{\rm{an}}(\lambda),V_{n,I_{2}}^{\rm{an}}(\lambda))\neq
0$, then $h_{I_{0},I_{2}}\leq h\leq n^{2}-1+\\#I_{2}-\\#I_{0}$.

  * •

The space
$\mathbf{E}_{I_{0},I_{2}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ext}_{G_{n},\lambda}^{h_{I_{0},I_{2}}}(V_{n,I_{0}}^{\rm{an}}(\lambda),V_{n,I_{2}}^{\rm{an}}(\lambda))$
admits a canonical decreasing filtration

$0=\mathrm{Fil}^{-\\#I_{0}^{\sharp}+1}(\mathbf{E}_{I_{0},I_{2}})\subseteq\mathrm{Fil}^{-\\#I_{0}^{\sharp}}(\mathbf{E}_{I_{0},I_{2}})\subseteq\cdots\subseteq\mathrm{Fil}^{-n+1}(\mathbf{E}_{I_{0},I_{2}})=\mathbf{E}_{I_{0},I_{2}}$

such that
$\mathrm{Fil}^{-\ell}(\mathbf{E}_{I_{0},I_{2}})/\mathrm{Fil}^{-\ell+1}(\mathbf{E}_{I_{0},I_{2}})\cong
E_{2,I_{0}^{\sharp},\Delta_{n}}^{-\ell,\ell+n-1-2\\#I_{0}^{\sharp}}$ admits a
basis indexed by
$\Psi_{I_{0}^{\sharp},\Delta_{n}}^{-\ell,\ell+n-1-2\\#I_{0}^{\sharp}}$ for
each $\\#I_{0}^{\sharp}\leq\ell\leq n-1$.

  * •

The space
$\mathbf{E}_{I_{0},I_{2}}^{\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ext}_{G_{n},\lambda}^{h_{I_{0},I_{2}}+1}(V_{n,I_{0}}^{\rm{an}}(\lambda),V_{n,I_{2}}^{\rm{an}}(\lambda))$
admits a canonical decreasing filtration

$0=\mathrm{Fil}^{-\\#I_{0}^{\sharp}+1}(\mathbf{E}_{I_{0},I_{2}}^{\prime})\subseteq\mathrm{Fil}^{-\\#I_{0}^{\sharp}}(\mathbf{E}_{I_{0},I_{2}}^{\prime})\subseteq\cdots\subseteq\mathrm{Fil}^{-n+1}(\mathbf{E}_{I_{0},I_{2}}^{\prime})=\mathbf{E}_{I_{0},I_{2}}^{\prime}$

such that
$\mathrm{Fil}^{-\ell}(\mathbf{E}_{I_{0},I_{2}}^{\prime})/\mathrm{Fil}^{-\ell+1}(\mathbf{E}_{I_{0},I_{2}}^{\prime})\cong
E_{2,I_{0}^{\sharp},\Delta_{n}}^{-\ell,\ell+n-2\\#I_{0}^{\sharp}}$ admits a
basis indexed by
$\Psi_{I_{0}^{\sharp},\Delta_{n}}^{-\ell,\ell+n-\\#I_{0}^{\sharp}}$ for each
$\\#I_{0}^{\sharp}\leq\ell\leq n-1$.

######  Remark 4.24.

Recall that
$i_{n,I}^{\infty}=i_{n,I,\Delta_{n}}^{\infty}(1_{\overline{L}_{n,I}})$ and
$V_{n,I}^{\infty}=i_{n,I}^{\infty}/\sum_{I\subsetneq
I^{\prime}\subseteq\Delta_{n}}i_{n,I^{\prime}}^{\infty}$ for each
$I\subseteq\Delta_{n}$. Then for each pair $I_{0}\subseteq
I_{1}\subseteq\Delta_{n}$, we can define a complex
$\mathbf{C}_{I_{0},I_{1}}^{\infty}$ supported in degree
$[-\\#I_{1},-\\#I_{0}]$ whose degree $-\ell$ term is
$\bigoplus_{I_{0}\subseteq I\subseteq I_{1},\\#I=\ell}i_{n,I}^{\infty}$. For
each $\lambda\in X(T_{n,E})$ which is dominant with respect to $B_{n,E}$, we
have
$i_{n,I}^{\rm{alg}}(\lambda)=F_{n,\Delta_{n}}(\lambda)\otimes_{E}i_{n,I}^{\infty}$,
$V_{n,I}^{\rm{alg}}(\lambda)=F_{n,\Delta_{n}}(\lambda)\otimes_{E}V_{n,I}^{\infty}$
and
$\mathbf{C}_{I_{0},I_{1}}^{\rm{alg}}(\lambda)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}F_{n,\Delta_{n}}(\lambda)\otimes_{E}\mathbf{C}_{I_{0},I_{1}}^{\infty}$.
It follows from Theorem 7.1 of [Koh11], Lemma 4.14 and Lemma 4.15 that the
natural embedding $i_{n,I^{\prime}}^{\rm{alg}}(\lambda)\hookrightarrow
i_{n,I^{\prime}}^{\rm{an}}(\lambda)$ induces isomorphisms

$H^{k}(N_{n,I},i_{n,I^{\prime}}^{\rm{an}}(\lambda)^{\prime})_{\delta_{n,I,\lambda}^{\prime}}\cong
H^{k}(N_{n,I},i_{n,I^{\prime}}^{\rm{alg}}(\lambda)^{\prime})_{\delta_{n,I,\lambda}^{\prime}}$

for each $k\geq 0$, which together with standard spectral sequences implies
the isomorphism

$\mathrm{Ext}_{G_{n},\lambda}^{k}(i_{n,I^{\prime}}^{\rm{an}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\xrightarrow{\sim}\mathrm{Ext}_{G_{n},\lambda}^{k}(i_{n,I^{\prime}}^{\rm{alg}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))$

for each $I,I^{\prime}\subseteq\Delta_{n}$ and $k\geq 0$. Consequently, if
$I_{2}\subseteq I_{2}\subseteq\Delta_{n}$ is a second pair of subsets, then
$\mathrm{Ext}_{G_{n},\lambda}^{\bullet}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{2}}(\lambda))$
and
$\mathrm{Ext}_{G_{n},\lambda}^{\bullet}(\mathbf{C}_{I_{0},I_{1}}^{\rm{alg}}(\lambda),\mathbf{C}_{I_{2},I_{2}}(\lambda))$
can be computed by exactly the same spectral sequence, and thus isomorphic as
filtered spaces (equipped with the canonical filtrations coming from the same
spectral sequence). As a special case, we conclude that the natural embedding
$V_{n,I_{0}}^{\rm{alg}}(\lambda)\hookrightarrow
V_{n,I_{0}}^{\rm{an}}(\lambda)$ induces an isomorphism

$\mathrm{Ext}_{G_{n},\lambda}^{h}(V_{n,I_{0}}^{\rm{an}}(\lambda),V_{n,I_{2}}^{\rm{an}}(\lambda))\xrightarrow{\sim}\mathrm{Ext}_{G_{n},\lambda}^{h}(V_{n,I_{0}}^{\rm{alg}}(\lambda),V_{n,I_{2}}^{\rm{an}}(\lambda))$

for each $I_{0},I_{2}\subseteq\Delta_{n}$ and $h\geq 0$.

###  4.3. Explicit cup product

In this section, we explicitly compute certain cup product maps (see (4.18))
using canonical filtration on each$\mathrm{Ext}$-group as well as the basis of
each graded piece established in Section 2 and Section 4.2. The key new
ingredients are the commutative diagrams (4.20) and (4.23) which carefully
record the effect on $\mathrm{Ext}$-groups under various truncations of
complexes.

Recall from Theorem 4.22 that we use the shortened notation
$M^{k}_{I_{0},I_{1},I_{2},I_{3}}$ for
$\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{I_{2},I_{3}}(\lambda))$,
and we will use similar notation for other $\mathrm{Ext}$-groups (for example
$M^{k}_{\mathcal{P},\mathcal{P}^{\prime}}$ for
$\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{\mathcal{P}}(\lambda),\mathbf{C}_{\mathcal{P}^{\prime}}(\lambda))$
and $M^{k}_{I_{0},I_{1},\mathcal{P}^{\prime}}$ for
$\mathrm{Ext}^{k}_{G_{n},\lambda}(\mathbf{C}_{I_{0},I_{1}}(\lambda),\mathbf{C}_{\mathcal{P}^{\prime}}(\lambda))$).
We also set
$H^{k}_{S}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\bigoplus_{I\in
S}H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$ for each set $S$ of
subsets of $\Delta_{n}$ and $k\geq 0$. Given two integers
$k_{0},k_{1}\in\mathbb{Z}$ and three tuples
$\mathcal{P},\mathcal{P}^{\prime},\mathcal{P}^{\prime\prime}$ as introduced at
the beginning of Section 4.2, we have a canonical cup product map

$M^{k_{0}}_{\mathcal{P},\mathcal{P}^{\prime}}\otimes
M^{k_{1}}_{\mathcal{P}^{\prime},\mathcal{P}^{\prime\prime}}\xrightarrow{\cup}M^{k_{0}+k_{1}}_{\mathcal{P},\mathcal{P}^{\prime\prime}}.$

In particular, given two integers $k_{0},k_{1}\in\mathbb{Z}$ and three pairs
of subsets of $\Delta_{n}$ $I_{0}\subseteq I_{1}$, $I_{2}\subseteq I_{3}$ and
$I_{4}\subseteq I_{5}$, we have the following cup product map

(4.18) $M^{k_{0}}_{I_{0},I_{1},I_{2},I_{3}}\otimes
M^{k_{1}}_{I_{2},I_{3},I_{4},I_{5}}\xrightarrow{\cup}M^{k_{0}+k_{1}}_{I_{0},I_{1},I_{4},I_{5}}.$

We assume from now the following simplifying condition which is already
sufficient for our later application.

######  Condition 4.25.

We have $I_{1}=I_{3}=I_{5}$ and $I_{4}\subseteq I_{2}\subseteq I_{0}$.

Let $J\subseteq[0,\\#I_{2}\setminus I_{4}]$ and
$J^{\prime}\subseteq[0,\\#I_{0}\setminus I_{2}]$ be two subintervals. Under
Condition 4.25, we consider a list of tuples $\mathcal{P}_{J,J^{\prime}}$ with
a common (ordered) partition of set

$\Delta_{n}=I_{4}\sqcup(I_{2}\setminus I_{4})\sqcup(I_{0}\setminus
I_{2})\sqcup(I_{1}\setminus I_{0})\sqcup(\Delta_{n}\setminus I_{1})$

that satisfies
$[\ell_{J,J^{\prime},1}^{-},\ell_{J,J^{\prime},1}^{+}]=\\{\\#I_{4}\\}$,
$[\ell_{J,J^{\prime},2}^{-},\ell_{J,J^{\prime},2}^{+}]=J$,
$[\ell_{J,J^{\prime},3}^{-},\ell_{J,J^{\prime},3}^{+}]=J^{\prime}$,
$[\ell_{J,J^{\prime},4}^{-},\ell_{J,J^{\prime},4}^{+}]=\\{\\#I_{1}\setminus
I_{0}\\}$ and $[\ell_{J,J^{\prime},5}^{-},\ell_{J,J^{\prime},5}^{+}]=\\{0\\}$.
Given two intervals $J_{i}=[\ell_{i}^{-},\ell_{i}^{+}]$ for $i=1,2$, we write
$J_{1}\leq J_{2}$ if $\ell_{1}^{-}\leq\ell_{2}^{-}$ and
$\ell_{1}^{+}\leq\ell_{2}^{+}$. Given
$J_{1},J_{2}\subseteq[0,\\#I_{2}\setminus I_{4}]$ and
$J_{1}^{\prime},J_{2}^{\prime}\subseteq[0,\\#I_{0}\setminus I_{2}]$ satisfying
$J_{1}\leq J_{2}$ and $J_{1}^{\prime}\leq J_{2}^{\prime}$, then there exists a
canonical morphism

(4.19)
$\mathbf{C}_{\mathcal{P}_{J_{1},J_{1}^{\prime}}}(\lambda)\rightarrow\mathbf{C}_{\mathcal{P}_{J_{2},J_{2}^{\prime}}}(\lambda).$

We write
$t_{0}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\#I_{0}\setminus
I_{2}$,
$t_{1}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\#I_{2}\setminus
I_{4}$ and let $s_{0}\in[0,t_{0}]$ and $s_{1}\in[0,t_{1}]$ be two integers. We
define the following three set of subsets of $\Delta_{n}$.

  * •

$S_{0}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{I\subseteq\Delta_{n}\mid
I_{2}\cup(I_{1}\setminus I_{0})\subseteq I\subseteq
I_{1},~{}\\#I\cap(I_{0}\setminus I_{2})=s_{0}\\}$.

  * •

$S_{1}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{I\subseteq\Delta_{n}\mid
I_{4}\cup(I_{1}\setminus I_{2})\subseteq I\subseteq
I_{1},~{}\\#I\cap(I_{2}\setminus I_{4})=s_{1}\\}$.

  * •

$S_{2}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{I\subseteq\Delta_{n}\mid
I_{4}\cup(I_{1}\setminus I_{0})\subseteq I\subseteq
I_{1},~{}\\#I\cap(I_{2}\setminus I_{4})=s_{1},~{}\\#I\cap(I_{0}\setminus
I_{2})=s_{0}\\}$.

For each $I\in S_{0}$ (resp. $I\in S_{1}$, resp. $I\in S_{2}$), we have
$\\#I=\ell_{0}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\#I_{2}+\\#I_{1}\setminus
I_{0}+s_{0}$ (resp.
$\\#I=\ell_{1}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\#I_{4}+\\#I_{1}\setminus
I_{2}+s_{1}$, resp.
$\\#I=\ell_{2}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\#I_{4}+\\#I_{1}\setminus
I_{0}+s_{0}+s_{1}$). Note that we have
$\ell_{0}+\ell_{1}-\ell_{2}=\\#I_{2}+\\#I_{1}\setminus I_{2}=\\#I_{1}$ and the
map $S_{0}\times S_{1}\rightarrow S_{2}:~{}(I,I^{\prime})\mapsto I\cap
I^{\prime}$ is a bijection.

Assuming Condition 4.25 and using maps of the form (4.19), we can extend
(4.18) to the following commutative diagram

(4.20)
$\displaystyle{M^{k_{0}}_{I_{0},I_{1},I_{2},I_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\otimes}$$\displaystyle{M^{k_{1}}_{I_{2},I_{1},I_{4},I_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\displaystyle{M^{k_{0}+k_{1}}_{I_{0},I_{1},I_{4},I_{1}}}$$\displaystyle{M^{k_{0}}_{I_{0},I_{1},I_{2},I_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\otimes}$$\displaystyle{M^{k_{1}}_{I_{2},I_{1},\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{p_{0}}$$\scriptstyle{p_{2}}$$\displaystyle{M^{k_{0}+k_{1}}_{I_{0},I_{1},\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{3}}$$\displaystyle{M^{k_{0}}_{I_{0},I_{1},I_{2},I_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\otimes}$$\displaystyle{M^{k_{1}}_{I_{2},I_{1},\mathcal{P}_{\\{s_{1}\\},[0,t_{0}]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{p_{4}}$$\displaystyle{M^{k_{0}+k_{1}}_{I_{0},I_{1},\mathcal{P}_{\\{s_{1}\\},[0,t_{0}]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{5}}$$\displaystyle{M^{k_{0}}_{I_{0},I_{1},I_{2},I_{1}}}$$\displaystyle{\otimes}$$\displaystyle{M^{k_{1}}_{I_{2},I_{1},\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{p_{7}}$$\displaystyle{M^{k_{0}+k_{1}}_{I_{0},I_{1},\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}}$$\displaystyle{M^{k_{0}}_{I_{0},I_{1},\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{6}}$$\scriptstyle{p_{9}}$$\displaystyle{\otimes}$$\displaystyle{M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\displaystyle{M^{k_{0}+k_{1}}_{I_{0},I_{1},\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{8}}$$\displaystyle{M^{k_{0}}_{I_{0},I_{1},\mathcal{P}_{\\{t_{1}\\},\\{s_{0}\\}}}}$$\displaystyle{\otimes}$$\displaystyle{M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},\\{s_{0}\\}},\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{p_{10}}$$\displaystyle{M^{k_{0}+k_{1}}_{I_{0},I_{1},\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}}$$\displaystyle{H^{k_{0}+\ell_{0}-\\#I_{1}}_{S_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\scriptstyle{p_{11}}$$\displaystyle{\otimes}$$\displaystyle{H^{k_{1}+\ell_{1}-\\#I_{1}}_{S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\displaystyle{H^{k_{0}+k_{1}+\ell_{2}-\\#I_{1}}_{S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\displaystyle{H^{k_{0}+\ell_{0}-\\#I_{1}}_{S_{2}}}$$\displaystyle{\otimes}$$\displaystyle{H^{k_{1}+\ell_{1}-\\#I_{1}}_{S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\displaystyle{H^{k_{0}+k_{1}+\ell_{2}-\\#I_{1}}_{S_{2}}}$

The commutativity of (4.20) (including the definition of each map inside) is
clear except that between the fourth and the fifth rows. In fact, we have an
isomorphism between spectral sequences
${}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]}}E_{\bullet,\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}^{\bullet,\bullet}\xrightarrow{\sim}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]}}E_{\bullet,\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}^{\bullet,\bullet}$
from the isomorphism on their first pages. Consequently, there exists a
canonical isomorphism
$M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}\xrightarrow{\sim}M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}$
and we can define $p_{7}$ as the composition

(4.21)
$M^{k_{1}}_{I_{2},I_{1},\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}\rightarrow
M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}\cong
M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}.$

The commutativity between the fourth and the fifth row of (4.20) thus follows
from the fact that the composition

$M^{k_{0}}_{I_{0},I_{1},\mathcal{P}_{\\{t_{1}\\},\\{s_{0}\\}}}\otimes
M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}\xrightarrow{\sim}M^{k_{0}}_{I_{0},I_{1},\mathcal{P}_{\\{t_{1}\\},\\{s_{0}\\}}}\otimes
M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}\xrightarrow{\cup}M^{k_{0}+k_{1}}_{I_{0},I_{1},\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}$

clearly factors through $p_{8}$. The isomorphism between the sixth and seventh
rows is clear from the description of
${}_{I_{0},I_{1}}E_{1,\mathcal{P}_{\\{t_{1}\\},\\{s_{0}\\}}}^{\bullet,\bullet}$
(resp.
${}_{\mathcal{P}_{\\{t_{1}\\},\\{s_{0}\\}}}E_{1,\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}^{\bullet,\bullet}$,
resp.
${}_{I_{0},I_{1}}E_{1,\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}^{\bullet,\bullet}$)
which concentrate in bidegrees $(-\ell_{0},\bullet)$ (resp.
$(-\ell_{1},\bullet)$, resp. $(-\ell_{2},\bullet)$).

By the definition of the canonical filtration on each term of the first row of
(4.20), we observe that
$\mathrm{im}(p_{4})=\mathrm{Fil}^{-\ell_{0}}(M^{k_{0}}_{I_{0},I_{1},I_{2},I_{1}})$
and
$\mathrm{im}(p_{0})=\mathrm{Fil}^{-\ell_{1}}(M^{k_{1}}_{I_{2},I_{1},I_{4},I_{1}})$.
As $I_{4}\cup(I_{1}\setminus I_{0})\subseteq I\subseteq I_{1}$ satisfying
$\\#I\cap(I_{2}\setminus I_{4})\leq s_{1}$ and $\\#I\cap(I_{0}\setminus
I_{2})\leq s_{0}$ must also satisfy $\\#I\leq\ell_{2}$, we deduce that the
image of
$M^{k_{0}+k_{1}}_{I_{0},I_{1},\mathcal{P}_{[0,s_{1}],[0,s_{0}]}}\rightarrow
M^{k_{0}+k_{1}}_{I_{0},I_{1},I_{4},I_{1}}$ sits inside
$\mathrm{Fil}^{-\ell_{2}}(M^{k_{0}+k_{1}}_{I_{0},I_{1},I_{4},I_{1}})$. We
consider the following composition of canonical maps

$M^{k_{0}}_{I_{0},I_{1},\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]}}\otimes
M^{k_{1}}_{I_{2},I_{1},\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}\rightarrow
M^{k_{0}}_{I_{0},I_{1},\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]}}\otimes
M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}\\\
\cong M^{k_{0}}_{I_{0},I_{1},\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]}}\otimes
M^{k_{1}}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]},\mathcal{P}_{[0,s_{1}],[0,s_{0}]}}\xrightarrow{\cup}M^{k_{0}+k_{1}}_{I_{0},I_{1},\mathcal{P}_{[0,s_{1}],[0,s_{0}]}},$

which implies that the first row of (4.20) restricts to a canonical map

(4.22)
$\mathrm{Fil}^{-\ell_{0}}(M^{k_{0}}_{I_{0},I_{1},I_{2},I_{1}})\otimes\mathrm{Fil}^{-\ell_{1}}(M^{k_{1}}_{I_{2},I_{1},I_{4},I_{1}})\xrightarrow{\cup}\mathrm{Fil}^{-\ell_{2}}(M^{k_{0}+k_{1}}_{I_{0},I_{1},I_{4},I_{1}})$

for each $s_{0}\in[0,t_{0}]$ and $s_{1}\in[0,t_{1}]$.

We write
$s_{0}^{\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}t_{0}-s_{0}$
for short. Each map in (4.20) actually arises from a map between the
corresponding spectral sequences. In particular, we obtain the following
commutative diagram by considering specific terms of their second pages. Note
that each term in the following diagram is naturally a graded piece of the
canonical filtration on its corresponding term in (4.20).

(4.23)
$\displaystyle{{}_{I_{0},I_{1}}E_{2,I_{2},I_{1}}^{-\ell_{0},k_{0}+\ell_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\otimes}$$\displaystyle{{}_{I_{2},I_{1}}E_{2,I_{4},I_{1}}^{-\ell_{1},k_{1}+\ell_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,I_{4},I_{1}}^{-\ell_{2},k_{0}+k_{1}+\ell_{2}}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,I_{2},I_{1}}^{-\ell_{0},k_{0}+\ell_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\otimes}$$\displaystyle{{}_{I_{2},I_{1}}E_{2,\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{0}}$$\scriptstyle{q_{2}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}^{-\ell_{2},k_{0}+k_{1}+\ell_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{1}}$$\scriptstyle{q_{3}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,I_{2},I_{1}}^{-\ell_{0},k_{0}+\ell_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\otimes}$$\displaystyle{{}_{I_{2},I_{1}}E_{2,\mathcal{P}_{\\{s_{1}\\},[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{4}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,\mathcal{P}_{\\{s_{1}\\},[0,t_{0}]}}^{-\ell_{2},k_{0}+k_{1}+\ell_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{5}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,I_{2},I_{1}}^{-\ell_{0},k_{0}+\ell_{0}}}$$\displaystyle{\otimes}$$\displaystyle{{}_{I_{2},I_{1}}E_{2,\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{7}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,\mathcal{P}_{\\{s_{1}\\},[s_{0},t_{0}]}}^{-\ell_{2},k_{0}+k_{1}+\ell_{2}}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]}}^{-\ell_{0},k_{0}+\ell_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{6}}$$\scriptstyle{q_{9}}$$\displaystyle{\otimes}$$\displaystyle{{}_{\mathcal{P}_{\\{t_{1}\\},[0,s_{0}]}}E_{2,\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}^{-\ell_{1}+s_{0}^{\prime},k_{1}+\ell_{1}-s_{0}^{\prime}}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}^{-\ell_{2},k_{0}+k_{1}+\ell_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{8}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,\mathcal{P}_{\\{t_{1}\\},\\{s_{0}\\}}}^{-\ell_{0},k_{0}+\ell_{0}}}$$\displaystyle{\otimes}$$\displaystyle{{}_{\mathcal{P}_{\\{t_{1}\\},\\{s_{0}\\}}}E_{2,\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}^{-\ell_{1}+s_{0}^{\prime},k_{1}+\ell_{1}-s_{0}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{10}}$$\displaystyle{{}_{I_{0},I_{1}}E_{2,\mathcal{P}_{\\{s_{1}\\},\\{s_{0}\\}}}^{-\ell_{2},k_{0}+k_{1}+\ell_{2}}}$$\displaystyle{H^{k_{0}+\ell_{0}-\\#I_{1}}_{S_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\scriptstyle{p_{11}}$$\displaystyle{\otimes}$$\displaystyle{H^{k_{1}+\ell_{1}-\\#I_{1}}_{S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\displaystyle{H^{k_{0}+k_{1}+\ell_{2}-\\#I_{1}}_{S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\displaystyle{H^{k_{0}+\ell_{0}-\\#I_{1}}_{S_{2}}}$$\displaystyle{\otimes}$$\displaystyle{H^{k_{1}+\ell_{1}-\\#I_{1}}_{S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\displaystyle{H^{k_{0}+k_{1}+\ell_{2}-\\#I_{1}}_{S_{2}}}$

The main properties of maps in (4.23) (except $q_{7}$) are summarized as
follows.

  * •

For technical simplicity, we only keep track of the horizontal maps for the
first and last rows induced from (4.20), although there should be a well-
defined horizontal map for each row that makes the diagram commutative. Note
that the horizontal map in the first row of (4.23) exists thanks to (4.22) for
each $s_{0}\in[0,t_{0}]$ and $s_{1}\in[0,t_{1}]$.

  * •

We have the following factorization for the map $q_{0}$

(4.24)
$_{I_{2},I_{1}}E_{2,\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}}\cong\mathrm{ker}(_{I_{2},I_{1}}d_{1,\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}})\hookrightarrow\mathrm{ker}(_{I_{2},I_{1}}d_{1,I_{4},I_{1}}^{-\ell_{1},k_{1}+\ell_{1}})\twoheadrightarrow_{I_{2},I_{1}}E_{2,I_{4},I_{1}}^{-\ell_{1},k_{1}+\ell_{1}}$

and similarly for $q_{1}$, $q_{6}$ and $q_{8}$.

  * •

The map $q_{2}$ is injective with the following factorization

(4.25)
$_{I_{2},I_{1}}E_{2,\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}}\cong\mathrm{ker}(_{I_{2},I_{1}}d_{1,\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}})\hookrightarrow_{I_{2},I_{1}}E_{1,\mathcal{P}_{[0,s_{1}],[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}}\cong_{I_{2},I_{1}}E_{1,\mathcal{P}_{\\{s_{1}\\},[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}}\cong_{I_{2},I_{1}}E_{2,\mathcal{P}_{\\{s_{1}\\},[0,t_{0}]}}^{-\ell_{1},k_{1}+\ell_{1}}$

and so are $q_{3}$, $q_{4}$, $q_{5}$ and $q_{9}$. Here we use the following
fact for (4.25) of $q_{2}$: if $I_{4}\cup(I_{1}\setminus I_{2})\subseteq
I\subseteq I_{1}$ satisfies $\\#I=\ell_{1}$ and $\\#I\cap(I_{2}\setminus
I_{4})\leq s_{1}$, then we must have $\\#I\cap(I_{2}\setminus I_{4})=s_{1}$
(namely $I\in S_{1}$). It is easy to check that $q_{4}$ is actually an
isomorphism.