# On generalization of Breuil-Schraen’s $\mathscr{L}$-invariants to
$\mathrm{GL}_{n}$

Zicheng Qian

###### Abstract.

Let $p$ be prime number and $K$ be a $p$-adic field. We systematically compute
the higher $\mathrm{Ext}$-groups between locally analytic generalized
Steinberg representations (LAGS for short) of $\mathrm{GL}_{n}(K)$ via a new
combinatorial treatment of some spectral sequences arising from the so-called
Tits complex. Such spectral sequences degenerate at the second page and each
$\mathrm{Ext}$-group admits a canonical filtration whose graded pieces are
terms in the second page of the corresponding spectral sequence. For each pair
of LAGS, we are particularly interested their $\mathrm{Ext}$-groups in the
bottom two non-vanishing degrees. We write down an explicit basis for each
graded piece (under the canonical filtration) of such an $\mathrm{Ext}$-group,
and then describe the cup product maps between such $\mathrm{Ext}$-groups
using these bases. As an application, we generalize Breuil’s
$\mathscr{L}$-invariants for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ and Schraen’s
higher $\mathscr{L}$-invariants for $\mathrm{GL}_{3}(\mathbb{Q}_{p})$ to
$\mathrm{GL}_{n}(K)$. Along the way, we also establish a generalization of
Bernstein–Zelevinsky geometric lemma to admissible locally analytic
representations constructed by Orlik–Strauch, generalizing a result in
Schraen’s thesis for $\mathrm{GL}_{3}(\mathbb{Q}_{p})$.

###### Contents

  1. 1 Introduction
    1. 1.1 Statement of the main results
    2. 1.2 Organization of the paper
    3. 1.3 Acknowledgements
    4. 1.4 Notation and preliminary
  2. 2 Cohomology of generalized Steinberg
    1. 2.1 Standard bases of group cohomology
    2. 2.2 Standard spectral sequences
    3. 2.3 $(I_{0},I_{1})$-standard and $(I_{0},I_{1})$-atomic tuples
    4. 2.4 A combinatorial subcomplex of $E_{1,I_{0},I_{1}}^{\bullet,k}$
    5. 2.5 Computations in bottom two degrees
    6. 2.6 Twisted $(I_{0},I_{1})$-atomic tuples
  3. 3 A result on decomposition of locally analytic distributions
  4. 4 Extensions between locally analytic generalized Steinberg
    1. 4.1 Locally analytic geometric lemma
    2. 4.2 Extensions between two complex
    3. 4.3 Explicit cup product
  5. 5 Breuil-Schraen $\mathscr{L}$-invariant
    1. 5.1 Definition of Breuil-Schraen $\mathscr{L}$-invariant
    2. 5.2 Breuil-Schraen $\mathscr{L}$-invariant and Galois representations

##  1\. Introduction

Let $p$ be a prime number and $E$ be a sufficiently large finite extension of
$\mathbb{Q}_{p}$. The theory of $\mathscr{L}$-invariant(s) has a long history,
and was first introduced by Mazur-Tate-Teitelbaum [MTT86] to describe the
derivative of the $p$-adic $L$-function (for a certain weight $2$ modular form
$f$) at its exceptional zero. If we write $\rho_{f}$ for the
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-representation attached to
$f$ via Langlands correspondence, then we are interested in the case when
$\rho_{f,p}$ is semi-stable non-crystalline, and this derivative, written
$\mathscr{L}(f)$, depends only on $\rho_{f,p}$ and can be read off explicitly
from the filtered $(\varphi,N)$-module attached to $\rho_{f,p}$ via Fontaine’s
theory (see [Fon94]). We usually use the term _Fontaine-Mazur
$\mathscr{L}$-invariant_ for invariants (of semi-stable $p$-adic Galois
representations) which are defined using an admissible Hodge filtration on a
$(\varphi,N)$-module. Among many different definitions of the
$\mathscr{L}$-invariant $\mathscr{L}(f)$ (which turn out to be equivalent, see
the book [Ast] for a comprehensive study), Breuil constructs in [Bre04] an
explicit finite length locally analytic representation $\Pi(\rho_{f,p})$
(written $\Sigma(2,\mathcal{L})$ in _loc.it._ with
$\mathcal{L}=\mathscr{L}(f)$) of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ whose
isomorphism class recovers $\mathscr{L}(f)$, and shows in [Bre10] that
$\Pi(\rho_{f,p})$ embeds into the $f$-isotypic component of completed
cohomology of modular curves. (Such results are usually called $p$-adic local-
global compatibility). Breuil’s construction is actually one of the first
instances of the $p$-adic Langlands correspondence, and the map
$\rho_{f,p}\mapsto\Pi(\rho_{f,p})$ can be improved to be a bijection between
the family of semi-stable non crystalline $p$-adic continuous representations
$\rho:\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})\rightarrow\mathrm{GL}_{2}(E)$
and an explicit family of locally analytic ($E$-)representations $\Pi(\rho)$
of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$. Motivated by the general philosophy of
$p$-adic Langlands correspondence, it is natural to seek for generalizations
of Breuil’s $\mathscr{L}$-invariants to $\mathrm{GL}_{n}(K)$ with
$[K:\mathbb{Q}_{p}]<\infty$, namely to recover all Fontaine-Mazur
$\mathscr{L}$-invariants of a semi-stable
$\rho:\mathrm{Gal}(\overline{K}/K)\rightarrow\mathrm{GL}_{n}(E)$ from a
certain locally analytic representation $\Pi(\rho)$ of $\mathrm{GL}_{n}(K)$.
When $\rho$ is crystalline with generic $\varphi$-eigenvalues, it is expected
that (for $\mathrm{GL}_{n}(K)\neq\mathrm{GL}_{2}(\mathbb{Q}_{p})$) $\Pi(\rho)$
has to involve some locally analytic representations which do not appear in
any parabolic induction. Hence, an explicit construction of $\Pi(\rho)$ (which
recovers all Fontaine-Mazur $\mathscr{L}$-invariants) is not yet available
beyond the case of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$. However, according to
Breuil’s $\mathrm{Ext}^{1}$ conjecture in [Bre19], we do expect the existence
of some explicit $\Pi(\rho)$ that recovers _all_ Fontaine-Mazur
$\mathscr{L}$-invariants of $\rho$, when $\rho$ is semi-stable with maximal
rank monodromy (namely $N^{n-1}\neq 0$).

We assume from now that $\rho$ is semi-stable with $N^{n-1}\neq 0$. We also
assume for simplicity in this introduction that $\rho$ is _ordinary_ , in
which case $\rho$ is up to a twist of the following form

(1.1) $\left(\begin{array}[]{ccc}\varepsilon^{n-1}&\cdots&\ast\\\
&\ddots&\vdots\\\ &&1\\\ \end{array}\right)$

with
$\varepsilon:\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})\rightarrow
E^{\times}$ being the $p$-adic cyclotomic character. In [Ding19], Ding has
constructed locally analytic representations of $\mathrm{GL}_{n}(K)$ that
recover those Fontaine-Mazur $\mathscr{L}$-invariants of $\rho$ coming from a
two dimensional subquotient of $\rho$, which are morally the extension
parameters at simple roots. The next objects of interest are therefore
generalizations of Breuil’s $\mathscr{L}$-invariants that conjecturally
corresponds to Fontaine-Mazur $\mathscr{L}$-invariants of $\rho$ _at non-
simple roots_ , the so-called _higher $\mathscr{L}$-invariants_. Much work has
been done towards higher $\mathscr{L}$-invariants and there are essentially
two different approaches which are closely related. Following Breuil’s
$\mathrm{Ext}^{1}$ conjecture in [Bre19], Breuil-Ding construct explicit
locally analytic representations $\Pi^{\rm{BD}}(\rho)$ of
$\mathrm{GL}_{3}(\mathbb{Q}_{p})$ (by which we mean either $\Pi(D)^{-}$ or
$\Pi(D)$ in (3.111) and (3.112) of _loc.it._ with $\lambda=0$), establish
their ($p$-adic) local-global compatibility (crucially using $p$-adic
Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$) in [BD20], and
then study their image under certain functor towards
$(\varphi,\Gamma)$-modules in [BD19]. In another direction, Schraen has done
extensive computations in his thesis [Schr11] and find higher
$\mathscr{L}$-invariants inside certain (higher) $\mathrm{Ext}$-groups between
locally analytic generalized Steinberg representations. Afterwards, part of
the author’s thesis has connected Breuil-Ding and Schraen’s approaches in an
explicit way, and in particular leading to a new family of locally analytic
representations $\Pi^{\rm{Q}}(\rho)$ (see (1.1) of [Qian21]) which contains
$\Pi^{\rm{BD}}(\rho)$. Breuil-Ding’s approach has the advantage of
constructing explicit true representations (instead of objects in certain
derived category), but its generalization to $\mathrm{GL}_{n}(K)$ is à priori
difficult due to complexity of $\Pi^{\rm{BD}}(\rho)$ and $\Pi^{\rm{Q}}(\rho)$.
Schraen’s approach is certainly not producing explicit representations, but
has the obvious advantage of involving only locally analytic generalized
Steinberg representations in its formulation. So it seems fair to first
generalize Schraen’s approach, which is the main focus of this paper.

###  1.1. Statement of the main results

We only state our main result for $\mathrm{GL}_{n}(\mathbb{Q}_{p})$ for
simplicity. Let $B$ (resp. $B^{+}$, resp. $T$) be the lower triangular Borel
subgroup (resp. upper triangular Borel subgroup, resp. diagonal maximal torus)
of $\mathrm{GL}_{n/\mathbb{Z}}$ and $\Delta$ be the set of positive simple
roots with respect to the pair $B^{+}$, $T$. Then for each subset
$I\subseteq\Delta$, we can attach a parabolic subgroup $P_{I}\supseteq B$ and
a standard Levi subgroup $L_{I}\subseteq\mathrm{GL}_{n}$. Note that
$I^{\prime}\subseteq I$ if and only if $P_{I^{\prime}}\subseteq P_{I}$. For
each $I\subseteq\Delta$, we define

$i_{I}^{\rm{an}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\left(\mathrm{Ind}_{P_{I}(\mathbb{Q}_{p})}^{\mathrm{GL}_{n}(\mathbb{Q}_{p})}1_{L_{I}(\mathbb{Q}_{p})}\right)^{\rm{an}}$

and then

$v_{I}^{\rm{an}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}i_{I}^{\rm{an}}/\sum_{I^{\prime}\supsetneq
I}i_{I^{\prime}}^{\rm{an}},$

which is the so-called _locally analytic generalized Steinberg representation_
attached to $I$. Note that
$\mathrm{St}_{n}^{\rm{an}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}v_{\emptyset}^{\rm{an}}$
is called the _locally analytic Steinberg representation_ and
$v_{\Delta}^{\rm{an}}$ is the trivial representation $1_{n}$ of
$\mathrm{GL}_{n}(\mathbb{Q}_{p})$. Similarly, for each $I\subseteq\Delta$, we
have a _smooth generalized Steinberg representation_ $v_{I}^{\infty}$ (with
$\mathrm{St}_{n}^{\infty}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}v_{\emptyset}^{\infty}$)
which can be identified with the set of smooth vectors of $v_{I}^{\rm{an}}$.
Each $v_{I}^{\rm{an}}$ clearly factors as a locally analytic representation of
$\mathrm{PGL}_{n}(\mathbb{Q}_{p})$. We write
$\mathrm{Ext}_{\mathrm{PGL}_{n}(\mathbb{Q}_{p})}^{\bullet}(-,-)$ for locally
analytic $\mathrm{Ext}$-groups between admissible locally analytic
representations of $\mathrm{PGL}_{n}(\mathbb{Q}_{p})$(see Section 1.4 for
further details). For each pair of subsets $I^{\prime}\subseteq
I\subseteq\Delta$, we write

(1.2)
$\mathbf{E}_{I,I^{\prime}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ext}_{\mathrm{PGL}_{n}(\mathbb{Q}_{p})}^{\\#I-\\#I^{\prime}}(v_{I}^{\rm{an}},v_{I^{\prime}}^{\rm{an}}),$

and note that each triple $I^{\prime\prime}\subseteq I^{\prime}\subseteq I$
induces a cup product map

(1.3)
$\mathbf{E}_{I,I^{\prime}}\otimes\mathbf{E}_{I^{\prime},I^{\prime\prime}}\xrightarrow{\cup}\mathbf{E}_{I,I^{\prime\prime}}.$

The following is our main result

######  Theorem 1.1 (Theorem 4.22, Corollary 5.1).

The $\mathrm{Ext}$-groups (1.2) and cup product maps (1.3) satisfy the
following properties.

  1. (i)

for each $\alpha\in\Delta$, we have a canonical isomorphism
$\mathbf{E}_{\\{\alpha\\},\emptyset}\cong\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)$;

  2. (ii)

for each $I^{\prime}\subseteq I$, we have a canonical isomorphism
$\mathbf{E}_{I,I^{\prime}}\cong\mathbf{E}_{I\setminus I^{\prime},\emptyset}$;

  3. (iii)

the map (1.3) is injective, and is an isomorphism if $\alpha+\beta$ is not a
root for any $\alpha\in I\setminus I^{\prime}$ and $\beta\in
I^{\prime}\setminus I^{\prime\prime}$;

  4. (iv)

the map (1.3) together with item (ii) fits into the following commutative
diagram

$\displaystyle{\mathbf{E}_{I,I^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\displaystyle{\otimes}$$\displaystyle{\mathbf{E}_{I^{\prime},I^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\cong}$$\scriptstyle{\cup}$$\displaystyle{\mathbf{E}_{I,I^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\displaystyle{\mathbf{E}_{I^{*},I^{\prime\prime}}}$$\displaystyle{\otimes}$$\displaystyle{\mathbf{E}_{I,I^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\displaystyle{\mathbf{E}_{I,I^{\prime\prime}}}$

with $I^{*}=I^{\prime\prime}\cup(I\setminus I^{\prime})$ and
$\varepsilon=(-1)^{(\\#I\setminus I^{\prime})(\\#I^{\prime}\setminus
I^{\prime\prime})}$;

  5. (v)

The map

$\bigoplus_{\emptyset\neq I^{\prime}\subsetneq
I}\mathbf{E}_{I,I^{\prime}}\otimes\mathbf{E}_{I^{\prime},\emptyset}\rightarrow\mathbf{E}_{I,\emptyset}$

has a codimension one image if $\sum_{\alpha\in I}\alpha$ is a root, and is
surjective otherwise.

######  Remark 1.2.

  1. (1)

We have

$\mathrm{Ext}_{\mathrm{PGL}_{n}(\mathbb{Q}_{p})}^{k}(v_{I}^{\rm{an}},v_{I^{\prime}}^{\rm{an}})=0$

for each $k<\\#I-\\#I^{\prime}$, and thus $\mathbf{E}_{I,I^{\prime}}$ is the
bottom degree non-vanishing $\mathrm{Ext}$-group for each $I^{\prime}\subseteq
I\subseteq\Delta$.

  2. (2)

The space $\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)$ admits a basis
$\\{\mathrm{val},\log\\}$ where
$\mathrm{val}:\mathbb{Q}_{p}^{\times}\rightarrow\mathbb{Z}\hookrightarrow E$
is $p$-adic valuation and $\log$ is any branch of the $p$-adic logarithm. In
particular, $\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)$ contains a
canonical line $E\mathrm{val}$ spanned by $\mathrm{val}$.

  3. (3)

In Corollary 5.1, we also study
$\mathbf{E}_{I,I^{\prime}}^{\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ext}_{\mathrm{PGL}_{n}(\mathbb{Q}_{p})}^{\\#I-\\#I^{\prime}+1}(v_{I}^{\rm{an}},v_{I^{\prime}}^{\rm{an}})$
as well as cup product map of the form

$\mathbf{E}_{I,I^{\prime}}\otimes\mathbf{E}_{I^{\prime},I^{\prime\prime}}^{\prime}\xrightarrow{\cup}\mathbf{E}_{I,I^{\prime\prime}}^{\prime},$

and Theorem 1.1 has a natural variant in this setting.

  4. (4)

When $n=2$, Theorem 1.1 goes back to [Bre04]. The $n=3$ case is proven by
Schraen in [Schr11] as a key ingredient in his definition of (higher)
$\mathscr{L}$-invariants for $\mathrm{GL}_{3}(\mathbb{Q}_{p})$.

When $n=2$, we have a natural exact sequence

$v_{\Delta}^{\rm{an}}\hookrightarrow i_{\emptyset}^{\rm{an}}\twoheadrightarrow
v_{\emptyset}^{\rm{an}}$

which induces canonical isomorphisms

(1.4)
$\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)\cong\mathrm{Ext}_{\mathbb{Q}_{p}^{\times}}^{1}(1,1)\cong\mathrm{Ext}_{\mathrm{PGL}_{2}(\mathbb{Q}_{p})}^{1}(i_{\emptyset}^{\rm{an}},i_{\emptyset}^{\rm{an}})\cong\mathrm{Ext}_{\mathrm{PGL}_{2}(\mathbb{Q}_{p})}^{1}(v_{\Delta}^{\rm{an}},v_{\emptyset}^{\rm{an}}).$

Consequently, given a $E$-line
$W\subseteq\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)$, Breuil
attaches a representation $V_{W}$ that fits into

(1.5) $v_{\emptyset}^{\rm{an}}\hookrightarrow V_{W}\twoheadrightarrow
v_{\Delta}^{\rm{an}}.$

Note that $\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)$ contains a
canonical $E$-line $E\mathrm{val}$, and $W\neq E\mathrm{val}$ if and only if
$V_{W}$ is uniserial if and only if the set of smooth vectors
$V_{W}^{\infty}\subseteq V_{W}$ is $\mathrm{St}_{2}^{\infty}$ if and only if
the corresponding
$\rho:\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})\rightarrow\mathrm{GL}_{2}(E)$
is semi-stable non-crystalline (namely $N\neq 0$).

Now we consider the case $n=3$ which is first studied in [Schr11]. Similar to
(1.4), we have canonical isomorphisms

(1.6)
$\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)\cong\mathrm{Ext}_{\mathrm{PGL}_{3}(\mathbb{Q}_{p})}^{1}(v_{\Delta}^{\rm{an}},v_{\Delta\setminus\\{\alpha\\}}^{\rm{an}})\cong\mathrm{Ext}_{\mathrm{PGL}_{3}(\mathbb{Q}_{p})}^{1}(v_{\\{\alpha\\}}^{\rm{an}},v_{\emptyset}^{\rm{an}}),$

and we write $\mathrm{val}_{\alpha}$ for the element corresponding to
$\mathrm{val}$ under (1.6) in the second and third space, for each
$\alpha\in\Delta$. Consequently, each $E$-line
$W_{\alpha}\subseteq\mathrm{Ext}_{\mathrm{PGL}_{3}(\mathbb{Q}_{p})}^{1}(v_{\\{\alpha\\}}^{\rm{an}},v_{\emptyset}^{\rm{an}})$
determines a representation $V_{W_{\alpha}}$ that fits into

$v_{\emptyset}^{\rm{an}}\hookrightarrow V_{W_{\alpha}}\twoheadrightarrow
v_{\\{\alpha\\}}^{\rm{an}}.$

We assume that

(1.7) $W_{\alpha}\neq E\mathrm{val}_{\alpha}$

for each $\alpha\in\Delta$. Then Schraen computes that

(1.8)
$\mathrm{Ext}_{\mathrm{PGL}_{3}(\mathbb{Q}_{p})}^{2}(v_{\Delta}^{\rm{an}},V_{W_{\alpha_{1}},W_{\alpha_{2}}})\cong\mathbf{E}_{\Delta,\emptyset}/\left(\sum_{i=1}^{2}\mathbf{E}_{\Delta,\\{\alpha_{i}\\}}\cup
W_{\alpha_{i}}\right)$

is two-dimensional, where $V_{W_{\alpha_{1}},W_{\alpha_{2}}}$ is the
amalgamate sum of $V_{W_{\alpha_{1}}}$ and $V_{W_{\alpha_{2}}}$ over
$v_{\emptyset}^{\rm{an}}$ with $\Delta=\\{\alpha_{1},\alpha_{2}\\}$. (Note
that (1.7) for each $\alpha\in\Delta$ if and only if the set of smooth vectors
$V_{W_{\alpha_{1}},W_{\alpha_{2}}}^{\infty}\subseteq
V_{W_{\alpha_{1}},W_{\alpha_{2}}}$ is $\mathrm{St}_{3}^{\infty}$.) We abuse
$\mathrm{val}_{\alpha_{1}}\cup\mathrm{val}_{\alpha_{2}}$ for its image in
(1.8). Hence, he can choose a third $E$-line $W_{\alpha_{3}}$ (with
$\alpha_{3}=\alpha_{1}+\alpha_{2}$ being the unique non-simple positive root)
inside (1.8) satisfying

(1.9) $W_{\alpha_{3}}\neq
E\mathrm{val}_{\alpha_{1}}\cup\mathrm{val}_{\alpha_{2}}$

and then define a derived object which (morally) fits into the following
distinguished triangle (see (1.14) of [Schr11])

(1.10) $V_{W_{\alpha_{1}},W_{\alpha_{2}}}\rightarrow
V_{W_{\alpha_{1}},W_{\alpha_{2}},W_{\alpha_{3}}}\rightarrow
v_{\Delta}^{\rm{an}}[-1]\rightarrow.$

Then he uses (1.10) to pin down a filtered $(\varphi,N)$-module from the de
Rham complex of the Drinfeld upper half space. Note that the filtered
$(\varphi,N)$-module in [Schr11] depends on some non-canonical choices of
normalization of parameters, and it is à priori unclear which choice gives the
correct $\rho$ that “corresponds to”
$V_{W_{\alpha_{1}},W_{\alpha_{2}},W_{\alpha_{3}}}$ under $p$-adic Langlands
correspondence. We slightly reformulate Schraen’s result by replacing the
three $E$-lines $W_{\alpha_{1}}$, $W_{\alpha_{2}}$ and $W_{\alpha_{3}}$ by a
single hyperplane $W\subseteq\mathbf{E}_{\Delta,\emptyset}$ which is the
preimage of $W_{\alpha_{3}}$ under

(1.11)
$\mathbf{E}_{\Delta,\emptyset}\twoheadrightarrow\mathbf{E}_{\Delta,\emptyset}/\left(\sum_{i=1}^{2}\mathbf{E}_{\Delta,\\{\alpha_{i}\\}}\cup
W_{\alpha_{i}}\right)\cong\mathrm{Ext}_{\mathrm{PGL}_{3}(\mathbb{Q}_{p})}^{2}(v_{\Delta}^{\rm{an}},V_{W_{\alpha_{1}},W_{\alpha_{2}}}).$

It is clear that the hyperplane $W\subseteq\mathbf{E}_{\Delta,\emptyset}$ is
determined by $W_{\alpha_{1}}$, $W_{\alpha_{2}}$ and $W_{\alpha_{3}}$.
Conversely, $W_{\alpha_{3}}$ is the image of $W$ under (1.11), and
$W_{\alpha_{i}}$ is uniquely characterized by the equality

(1.12) $E\mathrm{val}_{\alpha_{3-i}}\cup
W_{\alpha_{i}}=W\cap(E\mathrm{val}_{\alpha_{3-i}}\cup\mathbf{E}_{\\{\alpha_{i}\\},\emptyset})$

for each $i=1,2$. In other words, we simplify Schraen’s definition of
$\mathscr{L}$-invariants by gluing $W_{\alpha_{1}}$, $W_{\alpha_{2}}$ and
$W_{\alpha_{3}}$ together to a single hyperplane
$W\subseteq\mathbf{E}_{\Delta,\emptyset}$.

In general, as
$\mathbf{E}_{\\{\alpha\\},\emptyset}\cong\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)$
contains a canonical $E$-line $E\mathrm{val}_{\alpha}$ for each $\alpha\in I$,
we obtain a canonical $E$-line
$\mathbf{E}_{I,I^{\prime}}^{\infty}\cong\mathbf{E}_{I\setminus
I^{\prime},\emptyset}^{\infty}$ by cup product of $E\mathrm{val}_{\alpha}$
coming from each $\alpha\in I\setminus I^{\prime}$. Hence, we can define
$\widehat{\mathbf{E}}_{I,I^{\prime}}\subseteq\mathbf{E}_{\Delta,\emptyset}$ as
the image of

(1.13)
$\mathbf{E}_{\Delta,I}^{\infty}\otimes\mathbf{E}_{I,I^{\prime}}\otimes\mathbf{E}_{I^{\prime},\emptyset}^{\infty}\xrightarrow{\cup}\mathbf{E}_{\Delta,\emptyset}.$

We are therefore lead to the following simple definition of Breuil-Schraen
$\mathscr{L}$-invariants for general $\mathrm{GL}_{n}$, based on Theorem 1.1

######  Definition 1.3.

A _Breuil-Schraen $\mathscr{L}$-invariant_ for $\mathrm{GL}_{n}$ is a
hyperplane $W\subseteq\mathbf{E}_{\Delta,\emptyset}$ such that

  * •

$W\cap\widehat{\mathbf{E}}_{I,I^{\prime}}\subsetneq W$ for each
$I^{\prime}\subseteq I\subseteq\Delta$ (and we write
$W_{I,I^{\prime}}\subseteq\mathbf{E}_{I,I^{\prime}}$ for the unique hyperplane
such that $\mathbf{E}_{\Delta,I}^{\infty}\otimes
W_{I,I^{\prime}}\otimes\mathbf{E}_{I^{\prime},\emptyset}^{\infty}$ is the
preimage of $W$ under (1.13));

  * •

(1.3) induces an isomorphism

$\mathbf{E}_{I,I^{\prime}}/W_{I,I^{\prime}}\otimes\mathbf{E}_{I^{\prime},I^{\prime\prime}}/W_{I^{\prime},I^{\prime\prime}}\xrightarrow{\sim}\mathbf{E}_{I,I^{\prime\prime}}/W_{I,I^{\prime\prime}}$

for each $I^{\prime\prime}\subseteq I^{\prime}\subseteq I\subseteq\Delta$.

As the first (resp. the second) condition is an open condition (resp. a closed
condition), the moduli space of Breuil-Schraen $\mathscr{L}$-invariants is
given by an explicit locally closed subscheme of the projective space
$\mathbf{P}(\mathbf{E}_{\Delta,\emptyset})$. We prove in Theorem 5.9 that this
moduli space is non-canonically isomorphic to $U^{+}$, the unipotent radical
of $B^{+}$. Note that $U^{+}$ is naturally isomorphic (via Fontaine’s
$\mathbf{D}_{\rm{st}}$) to the moduli space of $\rho$ of the form (1.1) that
satisfies $N^{n-1}\neq 0$. Compared to the approach of [Schr11], our
definition of $\mathscr{L}$-invariants has the advantage of being simple and
independent of choices of bases of various $\mathbf{E}_{I,I^{\prime}}$. When
$n=3$, the first condition reduces to $W_{\alpha_{1}}\neq E\mathrm{val}\neq
W_{\alpha_{2}}$ and (1.9), and the second condition reduces to (1.12).

We conjecture that, for each Breuil-Schraen $\mathscr{L}$-invariant
$W\subseteq\mathbf{E}_{\Delta,\emptyset}$, one should be able to define

  * •

a locally analytic representation $\mathcal{W}$ of
$\mathrm{PGL}_{n}(\mathbb{Q}_{p})$ with
$\operatorname{dim}_{E}\mathrm{Hom}_{\mathrm{PGL}_{n}(\mathbb{Q}_{p})}(\mathrm{St}_{n}^{\rm{an}},\mathcal{W})=1$
and any embedding $\mathrm{St}_{n}^{\rm{an}}\hookrightarrow\mathcal{W}$
induces a surjection

$\mathbf{E}_{\Delta,\emptyset}=\mathrm{Ext}_{\mathrm{PGL}_{n}(\mathbb{Q}_{p})}^{n-1}(1_{n},\mathrm{St}_{n}^{\rm{an}})\twoheadrightarrow\mathrm{Ext}_{\mathrm{PGL}_{n}(\mathbb{Q}_{p})}^{n-1}(1_{n},\mathcal{W})$

with kernel $W$;

  * •

a unique semi-stable
$\rho_{W}:\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})\rightarrow\mathrm{GL}_{n}(E)$
such that $\mathcal{W}$ embeds into the conjectural representation
$\Pi(\rho_{W})$ given by $p$-adic Langlands correspondence.

######  Remark 1.4.

The existence of such $\mathcal{W}$ and $\rho_{W}$ is known for $n\leq 3$
thanks to the following work.

  * •

When $n=2$, we can simply take $\mathcal{W}$ to be $V_{W}$ as in (1.5), which
was firstly introduced by Breuil in [Bre04] (see the paragraph in _loc.it._
before Conjecture 1.1.1) as $\Sigma(2,\mathcal{L})$ (with $\mathcal{L}\in E$
explicitly determined by $W$). Let $f$ be a cusp new form of weight $2$ whose
attached Galois representation is $\rho_{f}$. Assume that
$\rho_{f,p}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\rho_{f}|_{\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})}$
is semi-stable non-crystalline. In Corollaire 1.1.6 of [Bre10], Breuil proves
that (up to an unramified twist) $\Sigma(2,\mathcal{L})$ (or rather its
completion $B(2,\mathcal{L})$) embeds into the $f$-isotypic Hecke eigenspace
inside the completed cohomology of a tower of modular curves, if and only if
the equality $\mathscr{L}(f)=\mathcal{L}$ holds with $\mathscr{L}(f)$ being
the Fontaine-Mazur $\mathscr{L}$-invariant attached to $\rho_{f,p}$. In other
words, $\mathscr{L}(f)$ is determined by $W$ and we can take
$\rho_{W}=\rho_{f,p}$. Note that we can also take $\mathcal{W}$ to be another
uniserial locally analytic representation $V_{W}^{+}$ of length four that fits
into the short exact seqnece

$V_{W}\hookrightarrow V_{W}^{+}\twoheadrightarrow\widetilde{I}(s\cdot 0)$

where $\widetilde{I}(s\cdot 0)$ is another irreducible locally analytic
principal series (cf. the paragraph before Lemma 3.13 of [BD20]). In terms of
terminology of [BD20], our $V_{W}$ (resp. $V_{W}^{+}$) is isomorphic to
$\pi(0,\psi)^{-}$ (resp. $\pi(0,\psi)$) with
$\psi\in\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)$ satisfying
$W=E\psi$ (see (3.23) and (3.27) of _loc.it._). According to [Col10] and
Theorem 1.2.1 of [Eme11], the locally analytic representation $\Pi(\rho_{W})$
attached to $\rho_{W}$ via $p$-adic local Langlands correspondence is actually
isomorphic to $V_{W}^{+}$. Therefore $V_{W}$ and $V_{W}^{+}$ are the only two
possible choices of $\mathcal{W}$ here.

  * •

When $n=3$, following his $\mathrm{Ext}^{1}$-conjecture in [Bre19], Breuil
constructs an explicit family of locally analytic representations (Théorème
1.2 of _loc.it._) and proves a weaker version of local-global compatibility
(Théorème 1.3 of _loc.it._) for them, compared with the strong version in
Conjecture 1.4 of _loc.it._. We write $D$ for the rank three
$(\varphi,\Gamma)$-module (over Robba ring) associated with an ordinary Galois
representation $\rho$ of the form (1.1) with $n=3$. In [BD20], Breuil-Ding
attach to $D$ two locally analytic representations $\Pi(D)^{-}\subseteq\Pi(D)$
of $\mathrm{GL}_{3}(\mathbb{Q}_{p})$ (see (3.111) and (3.112) of _loc.it._).
Then they prove Conjecture 1.4 of [Bre19] (see Theorem 1.1 of [BD20])
crucially using the $p$-adic local Langlands correspondence of
$\mathrm{GL}_{2}(\mathbb{Q}_{p})$. We write $\widetilde{\Pi}(D)^{-}$ (resp.
$\widetilde{\Pi}(D)$) for the amalgamate sum of $\Pi(D)^{-}$ (resp. $\Pi(D)$)
and $\mathrm{St}_{3}^{\rm{an}}$ over the length five subrepresentation with
Jordan–Hölder factors $\mathrm{St}_{3}^{\infty}$, $C_{1,1}$, $C_{2,1}$,
$\widetilde{C}_{1,2}$, $\widetilde{C}_{2,2}$ (see (3.111) of _loc.it._ for the
illustration of this subrepresentation with each factor defined in paragraphs
before it). Then we can combine Proposition 6.8, Theorem 7.1 of [Qian21] with
Theorem 1.1 of [BD20] and obtain

(1.14)
$\mathbf{E}_{\Delta,\emptyset}=\mathrm{Ext}_{\mathrm{PGL}_{3}(\mathbb{Q}_{p})}^{2}(1_{3},\mathrm{St}_{3}^{\rm{an}})\twoheadrightarrow\mathrm{Ext}_{\mathrm{PGL}_{3}(\mathbb{Q}_{p})}^{2}(1_{3},\widetilde{\Pi}^{-}(D))\cong\mathrm{Ext}_{\mathrm{PGL}_{3}(\mathbb{Q}_{p})}^{2}(1_{3},\widetilde{\Pi}(D))$

with the last two terms being one dimensional. Consequently, if we write $W$
for the kernel of (1.14), then we can let $\mathcal{W}$ be either
$\widetilde{\Pi}(D)^{-}$ or $\widetilde{\Pi}(D)$, and $\rho_{W}$ be the $\rho$
that defines $D$. The minimal possible choice of $\mathcal{W}$ (under
inclusion between representations) is actually

$\mathrm{ker}(\widetilde{\Pi}(D)^{-}\twoheadrightarrow
v_{\\{\alpha\\}}^{\infty})$

for any $\alpha\in\Delta$, and in particular is not unique. On the other hand,
under further restriction on $\mathcal{W}$ specified in Conjecture 5.11, it is
possible that there exists a unique maximal choice of $\mathcal{W}$.

We refer the conjectural realization of $\rho_{W}$ inside Drinfeld upper half
space to Conjecture 5.18.

######  Remark 1.5.

It is natural to ask if Theorem 1.1 admits natural generalization to other
reductive groups (with $v_{I}^{\rm{an}}$ defined exactly the same way for each
$I\subseteq\Delta$). Indeed, most combinatorial construction in this paper
works for more general reductive groups and many $\mathrm{Ext}$-groups between
$v_{I}^{\rm{an}}$ can be computed. However, for general group, it seems
unclear how to extend the definition of Breuil-Schraen
$\mathscr{L}$-invariants using cup products involving only
$\mathrm{Ext}$-groups between $v_{I}^{\rm{an}}$.

######  Remark 1.6.

The space $\mathrm{Hom}_{\rm{cont}}(\mathbb{Q}_{p}^{\times},E)$ contains a
unique canonical $E$-line which is $E\mathrm{val}$, so
$\mathbf{E}_{I,I^{\prime}}$ contains a unique canonical $E$-line
$E\mathrm{val}_{\alpha}$ whenever $I\setminus I^{\prime}=\\{\alpha\\}$. For
general $I^{\prime}\subseteq I\subseteq\Delta$, there exists a minimal lattice
of _canonical subspaces_ of $\mathbf{E}_{I,I^{\prime}}$ generated by the image
of the cup product of various
$\mathbf{E}_{I^{\prime\prime},I^{\prime\prime\prime}}$ (for some choices of
$I^{\prime}\subseteq I^{\prime\prime\prime}\subseteq I^{\prime\prime}\subseteq
I$) with various $E\mathrm{val}_{\alpha}$ satisfying $\alpha\in I\setminus
I^{\prime}$. There exists an even more interesting filtration on
$\mathbf{E}_{I,I^{\prime}}$ induced from the layer structure of
$v_{I^{\prime}}^{\rm{an}}$ as a finite length admissible locally analytic
representation. We will study this second filtration as well as its relative
position with respect to the aforementioned lattice of canonical subspaces in
a forthcoming work, which should lead to a proof of Conjecture 5.18 which
generalizes Théorème 6.23 and Remarque 6.24 of [Schr11]. These computations
also shed light on the construction (in the $\mathrm{GL}_{n}$ case) of locally
analytic representations in Breuil’s $\mathrm{Ext}^{1}$ conjecture as well as
those in Theorem 1.1 of [Qian21].

###  1.2. Organization of the paper

In Section 2, we systematically compute various locally analytic cohomology
groups of locally analytic generalized Steinberg representations (LAGS for
short). As each LAGS is derived equivalent to a complex of (direct sum of)
parabolically induced principal series, we reduce our problem to the
computation of a spectral sequence called
$E_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}$ (see Section 2.2). The first page
$E_{1,I_{0},I_{1}}^{\bullet,\bullet}$ can be easily written down with an
explicit basis using results from Section 2.1, so our main task is to
explicitly compute the second page $E_{2,I_{0},I_{1}}^{\bullet,\bullet}$ and
show that the spectral sequence actually degenerates at its second page. To
achieve this, we construct an explicit (and much smaller) combinatorial
subcomplex $E_{1,I_{0},I_{1},\diamond}^{\bullet,k}$ of
$E_{1,I_{0},I_{1}}^{\bullet,k}$ for each $k\in\mathbb{Z}$, based on the notion
of _$(I_{0},I_{1})$ -atomic tuples_ (see Definition 2.13). The key technical
ingredient is to show that the
$E_{1,I_{0},I_{1},\diamond}^{\bullet,k}\rightarrow
E_{1,I_{0},I_{1}}^{\bullet,k}$ is actually a quasi-isomorphism (see
Proposition 2.23), which reduces our problem to the computation of cohomology
of the complex $E_{1,I_{0},I_{1},\diamond}^{\bullet,k}$ for each
$k\in\mathbb{Z}$. In Section 2.5, we complete the proof of Theorem 2.28 which
is the main result of Section 2. In Section 2.6, we consider a variant of
$(I_{0},I_{1})$-atomic tuples which will be crucially used in Section 4.3.

In Section 3, we prove a decomposition theorem for certain space of locally
analytic distributions in Proposition 3.9, which is technically fundamental
for computation of $N$-homology (with $N$ being unipotent radical of some
parabolic subgroup) of locally analytic representations using Bruhat
decomposition. This generalizes results from Section 4.5 of [Schr11].

In Section 4, we compute various (higher) $\mathrm{Ext}$-groups between LAGS.
Using results from Section 3, we prove a generalization of
Bernstein–Zelevinsky geometric lemma (see Section 2.12 of [BZ77]) to locally
analytic setting, and in particular prove some vanishing results on
$N$-(co)homology in Lemma 4.14 and Lemma 4.15. The main result of Section 4.1
is Proposition 4.16, which computes all (higher) $\mathrm{Ext}$-groups between
all $i_{I}^{\rm{an}}$. Proposition 4.16 allows us to compute
$\mathrm{Ext}$-groups between LAGS by a spectral sequence, which turns out to
be among the ones already treated in Section 2, so Theorem 4.22 easily
follows. Section 4.3 is devoted to explicit computation of the cup product map
using the commutative diagram (4.20) and (4.23), which ends up with Theorem
4.31.

In Section 5, we define and study what we call _Breuil-Schraen
$\mathscr{L}$-invariants_. In particular, we propose Conjecture 5.11 that
relate them to Breuil-Ding’s approach of higher $\mathscr{L}$-invariants, and
propose Conjecture 5.18 which generalizes [Schr11] on potential realization of
$\mathscr{L}$-invariants inside the de Rham complex of Drinfeld upper half
spaces. Some expectation on this realization as well as its relation to other
objects are sketched in Remark 5.19.

###  1.3. Acknowledgements

The author is very grateful to Christophe Breuil for his guidance into this
area and his constant encouragement. We apparently benefit a lot from Benjamin
Schraen’s PhD thesis [Schr11] on $\mathrm{GL}_{3}(\mathbb{Q}_{p})$ case, and
this paper is the first step towards full generalization of [Schr11] to
$\mathrm{GL}_{n}$. We would like to thank Yiwen Ding for his papers with
Breuil and extensive discussions around higher $\mathscr{L}$-invariants and to
thank Florian Herzig for his constant interest and regular communication. We
thank Weibo Fu, Yongquan Hu, Ma Luo, Liang Xiao, Daxin Xu and Lizao Ye for
many useful discussions. Finally, we would like to thank Lennart Gehrmann for
discussion around [Geh21], which together with [Schr11] makes it clear that
$\mathrm{Ext}$-groups such as $\mathbf{E}_{\Delta,\emptyset}$ are crucial to
the study of (higher) $\mathscr{L}$-invariants. This work is mainly done when
the author was a Posdoctoral Fellow at University of Toronto, and the author
would like to thank them for their hospitality.

###  1.4. Notation and preliminary

Throughout this paper, $p$ is prime number and $K$ is a finite extension of
$\mathbb{Q}_{p}$. We fix another $p$-adic field $E$ as well as an embedding
$E\hookrightarrow\overline{\mathbb{Q}_{p}}$ such that each embedding
$\iota:K\hookrightarrow\overline{\mathbb{Q}_{p}}$ factors through it. Note
that we write $S$ for the set of all embeddings $\iota:K\hookrightarrow E$ and
identify it with the set of all embeddings
$K\hookrightarrow\overline{\mathbb{Q}_{p}}$ via the fixed embedding
$E\hookrightarrow\overline{\mathbb{Q}_{p}}$.

We recall some standard facts on (admissible) locally analytic representations
of $p$-adic analytic groups. Let $G$ be a $p$-adic analytic group and
$D(G)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}D(G,E)$ be the space of
locally analytic distributions on $G$ (see [ST03] for its definition). We
write $\mathrm{Mod}_{D(G)}$ for the abelian category of (abstract)
$D(G)$-modules, and
$\mathcal{M}(G)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}D(\mathrm{Mod}_{D(G)})$
for its derived category. Given two bounded strict chain complexes of
admissible ($\mathbb{Q}_{p}$-)locally analytic representations
$\mathbf{C}_{1}$, $\mathbf{C}_{2}$ of $G$, we consider their strong dual
$\mathbf{C}_{1}^{\prime}$, $\mathbf{C}_{2}^{\prime}$, and then define the
locally analytic $\mathrm{Ext}$-groups

$\mathrm{Ext}^{\bullet}_{G}(\mathbf{C}_{1},\mathbf{C}_{2})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ext}^{\bullet}_{\mathcal{M}(G)}(\mathbf{C}_{2}^{\prime},\mathbf{C}_{1}^{\prime})$

by $\mathrm{Ext}$ in the derived category $\mathcal{M}(G)$. In particular, we
use the notation

$H^{\bullet}(G,\Pi)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ext}^{\bullet}_{G}(1,\Pi)$

for each admissible locally analytic representation $\Pi$ of $G$.

Let
$\mathbf{C}_{1}=[\mathbf{C}_{1}^{\ell}]_{\ell\in\mathbb{Z}},\mathbf{C}_{2}=[\mathbf{C}_{2}^{\ell}]_{\ell\in\mathbb{Z}}$
be two bounded strict chain complexes of admissible locally analytic
representations, and we abuse $\mathbf{C}_{1},\mathbf{C}_{2}$ for the
corresponding objects in $\mathcal{M}(G)$. For each $k,\ell\in\mathbb{Z}$, we
can consider $\mathrm{Ext}^{k}_{G}(\mathbf{C}_{1},\mathbf{C}_{2}^{\ell})$ as
well as $\mathrm{Ext}^{k}_{G}(\mathbf{C}_{1}^{\ell},\mathbf{C}_{2})$.

######  Lemma 1.7.

  1. (i)

There exists a spectral sequence converge to
$\mathrm{Ext}^{\bullet}_{G}(\mathbf{C}_{1},\mathbf{C}_{2})$ whose first page
has $k$-th row given by

$\cdots\rightarrow\mathrm{Ext}^{k}_{G}(\mathbf{C}_{1},\mathbf{C}_{2}^{\ell})\rightarrow\mathrm{Ext}^{k}_{G}(\mathbf{C}_{1},\mathbf{C}_{2}^{\ell+1})\rightarrow\cdots.$

  2. (ii)

There exists a spectral sequence converge to
$\mathrm{Ext}^{\bullet}_{G}(\mathbf{C}_{1},\mathbf{C}_{2})$ whose first page
has $k$-th row given by

$\cdots\rightarrow\mathrm{Ext}^{k}_{G}(\mathbf{C}_{1}^{\ell},\mathbf{C}_{2})\rightarrow\mathrm{Ext}^{k}_{G}(\mathbf{C}_{1}^{\ell-1},\mathbf{C}_{2})\rightarrow\cdots.$

###### Proof.

This is standard inside the derived category $\mathcal{M}(G)$. ∎

Now assume that $G$ is the set of $\mathbb{Q}_{p}$-points of a $p$-adic
reductive group, and $P\subseteq G$ is a parabolic subgroup with unipotent
radical $N_{P}$ and Levi quotient $L$.

######  Lemma 1.8.

Let $M\in\mathrm{Mod}_{D(G)}$ and $M_{L}\in\mathrm{Mod}_{D(L)}$ with $M_{L}$
satisfying the (FIN) condition in Section 6 of [ST05]. Then we have the
following spectral sequence

$\mathrm{Ext}_{D(L)}^{k_{1}}(M_{L},H^{k_{2}}(N_{P},M))\Rightarrow\mathrm{Ext}_{D(G)}^{k_{1}+k_{2}}(D(G)\otimes_{D(P)}M_{L},M).$

###### Proof.

This is equation (43) of [Bre19]. ∎

Let $n\geq 1$ be an integer. We set
$G_{n}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{GL}_{n/K}$ and
let $B_{n}$ (resp. $B_{n}^{+}$, resp. $T_{n}$) be its lower triangular Borel
subgroup (resp. its upper triangular Borel subgroup, resp. its diagonal
maximal torus). We consider the set $\Delta_{n}$ of positive simple roots and
fix a bijection $\Delta_{n}\cong\\{1,2,\dots,n-1\\}$ which sends $(i,i+1)$ to
$i$ for each $1\leq i\leq n-1$. There is a natural bijection between the set
of subsets $I\subseteq\Delta_{n}$ and parabolic subgroups $P_{n,I}\subseteq
G_{n}$ containing $B_{n}$ (with $P_{n,\Delta_{n}}=G_{n}$ and
$P_{n,\emptyset}=B_{n}$). We write $L_{n,I}\subseteq P_{n,I}$ for the standard
Levi subgroup. Let $Z_{n}$ be the center of $G_{n}$ and $Z_{n,I}$ be the
center of $L_{n,I}$. We set
$\overline{G}_{n}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}G_{n}/Z_{n}$,
$\overline{L}_{n,I}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}L_{n,I}/Z_{n}$
and
$\overline{Z}_{n,I}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}Z_{n,I}/Z_{n}$
for each $I\subseteq\Delta_{n}$. We use the notation $\mathfrak{g}_{n}$ (resp.
$\overline{\mathfrak{g}}_{n}$, resp. $\mathfrak{l}_{n,I}$, resp.
$\overline{\mathfrak{l}}_{n,I}$) for the Lie algebra associated with $G_{n}$
(resp. $\overline{G}_{n}$, resp. $L_{n,I}$, resp. $\overline{L}_{n,I}$).

Throughout this paper, we abuse the notation for group schemes (resp. Lie
algebras) for their set of $K$-points, and therefore view them as $p$-adic Lie
groups (resp. $p$-adic Lie algebras). For a Lie algebra $\mathfrak{g}$ over
$K$, we set
$\mathfrak{g}_{E,\iota}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathfrak{g}\otimes_{K,\iota}E$
for each embedding $\iota:K\hookrightarrow E$, and note that
$\mathfrak{g}_{E}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathfrak{g}\otimes_{\mathbb{Q}_{p}}E\cong\prod_{\iota:K\hookrightarrow
E}\mathfrak{g}_{E,\iota}$. We write $G_{n,E}$ for the base change of
$\mathrm{Res}_{K/\mathbb{Q}_{p}}G_{n}$ to $E$, and similarly for other groups
above.

For each $p$-adic reductive group $G$ containing a parabolic subgroup $P$, we
write $(\mathrm{Ind}_{P}^{G}\cdot)^{\rm{an}}$ (resp.
$(\mathrm{Ind}_{P}^{G}\cdot)^{\infty}$) for the locally analytic (resp.
smooth) parabolic induction functor. For each pair of subsets $I\subseteq
I^{\prime}\subseteq\Delta_{n}$, we set

$i_{n,I,I^{\prime}}^{\rm{an}}(\pi_{I})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\left(\mathrm{Ind}_{P_{n,I}\cap
L_{n,I^{\prime}}}^{L_{n,I^{\prime}}}\pi_{I}\right)^{\rm{an}}$

for each locally analytic representation $\pi_{I}$ of $L_{n,I}$ and note that
$i_{n,I,I^{\prime}}^{\rm{an}}(\pi_{I})$ is admissible if $\pi_{I}$ is. If
$\pi_{I}$ is furthermore smooth, we define
$i_{n,I,I^{\prime}}^{\rm{\infty}}(\pi_{I})$ similarly.

Let $\lambda\in X(T_{n,E})$ be a weight which is dominant with respect to
$B_{n,E}^{+}$. For each $I\subseteq\Delta_{n}$, we write $F_{n,I}(\lambda)$
for the algebraic representation of $L_{n,I,E}$ with highest weight $\lambda$
(with respect to $B_{n,E}^{+}$), which induces a finite dimensional
$E$-representation of the $p$-adic Lie group $L_{n,I}$. We define
$i_{n,I}^{\infty}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}i_{n,I,\Delta_{n}}^{\infty}(1_{\overline{L}_{n,I}})$
and a _smooth generalized Steinberg representation_
$V_{n,I}^{\infty}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}i_{n,I}^{\infty}/\sum_{I\subsetneq
I^{\prime}\subseteq\Delta_{n}}i_{n,I^{\prime}}^{\infty}$ for each
$I\subseteq\Delta_{n}$. We define
$i_{n,I}^{\rm{alg}}(\lambda)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}F_{n,\Delta_{n}}(\lambda)\otimes_{E}i_{n,I}^{\infty}$
and the _locally algebraic generalized Steinberg representation_
$V_{n,I}^{\rm{alg}}(\lambda)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}F_{n,\Delta_{n}}(\lambda)\otimes_{E}V_{n,I}^{\infty}$.
For each $I\subseteq\Delta_{n}$, we know that $V_{n,I}^{\infty}$ and thus
$V_{n,I}^{\rm{alg}}(\lambda)$ is irreducible. We write
$\mathrm{Ext}_{G_{n},\lambda}^{\bullet}$ for the $\mathrm{Ext}$-groups fixing
the central character which equals that of $F_{n,\Delta_{n}}(\lambda)$ (see
Remarque 5.1.3 of [Bre19] for similar notation).

Similarly, we define

(1.15)
$V_{n,I,I^{\prime}}^{\rm{an}}(\lambda)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}i_{n,I,I^{\prime}}^{\rm{an}}(F_{n,I}(\lambda))/\sum_{I\subsetneq
I^{\prime\prime}\subseteq
I^{\prime}}i_{n,I,I^{\prime\prime}}^{\rm{an}}(F_{n,I^{\prime\prime}}(\lambda))$

for each $I\subseteq I^{\prime}\subseteq\Delta_{n}$. In particular, we have
$i_{n,I}^{\rm{an}}(\lambda)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}i_{n,I,\Delta_{n}}^{\rm{an}}(F_{n,I}(\lambda))$
and a _locally analytic generalized Steinberg representation_
$V_{n,I}^{\rm{an}}(\lambda)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}V_{n,I,\Delta_{n}}^{\rm{an}}(\lambda)$
for each $I\subseteq\Delta_{n}$ with
$\mathrm{St}_{n}^{\rm{an}}(\lambda)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}V_{n,I}^{\rm{an}}(\lambda)$.
It is clear that $V_{n,\Delta_{n}}^{\rm{an}}(\lambda)\cong
F_{n,\Delta_{n}}(\lambda)$.

We have a natural embedding $i_{n,I}^{\rm{alg}}(\lambda)\hookrightarrow
i_{n,I}^{\rm{an}}(\lambda)$, which induces an embedding
$V_{n,I}^{\rm{alg}}(\lambda)\hookrightarrow V_{n,I}^{\rm{an}}(\lambda)$ that
identifies $V_{n,I}^{\rm{alg}}(\lambda)$ with the locally algebraic vectors of
$V_{n,I}^{\rm{an}}(\lambda)$ (as we know the set of Jordan–Hölder factors
(with multiplicity) of $V_{n,I}^{\rm{an}}(\lambda)$ thanks to [OS13] and
[OS15]). We emphasize that, for each $I^{\prime}\subseteq
I\subseteq\Delta_{n}$, we have a distinguished embedding
$\kappa_{I,I^{\prime}}^{\infty}:i_{n,I}^{\infty}\hookrightarrow
i_{n,I^{\prime}}^{\infty}$, (resp.
$\kappa_{I,I^{\prime}}^{\rm{alg}}(\lambda):i_{n,I}^{\rm{alg}}(\lambda)\hookrightarrow
i_{n,I^{\prime}}^{\rm{alg}}(\lambda)$, resp.
$\kappa_{I,I^{\prime}}^{\rm{an}}(\lambda):i_{n,I}^{\rm{an}}(\lambda)\hookrightarrow
i_{n,I^{\prime}}^{\rm{an}}(\lambda)$) that sends various locally constant
(resp. locally algebraic, resp. locally analytic) functions on
$\overline{G}_{n}$ to themselves (concerning the definition of parabolic
induction).

##  2\. Cohomology of generalized Steinberg

In this section, we compute cohomologies of locally analytic generalized
Steinberg representations via a careful combinatorial study of the first and
second pages of a spectral sequence
$E_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}$ (for some $I_{0}\subseteq
I_{1}\subseteq\Delta_{n}$) induced from the locally analytic Tits complex
$\mathbf{C}_{I_{0},I_{1}}(\lambda)$ (see (2.10) for its definition and [OS13]
for this terminology). The key combinatorial notion is that of
$(I_{0},I_{1})$-atomic tuples (see Definition 2.13), and the main results of
this section are Proposition 2.23 and Theorem 2.28.

###  2.1. Standard bases of group cohomology

Let $\mathfrak{g}$ be a (split) reductive lie algebra. It follows from
Théorème 9.3 of [Kos50] that we have an isomorphism of graded algebra

$H_{\ast}(\mathfrak{g},1)\cong(\wedge^{\ast}\mathfrak{g})^{\mathfrak{g}}.$

Let $D(\mathfrak{g})$ be the subspace of
$(\wedge^{\ast}\mathfrak{g})^{\mathfrak{g}}$ spanned by elements of the form
$u\wedge v$ (namely decomposable). Note that we have a pairing
$H_{\ast}(\mathfrak{g},1)\times H^{\ast}(\mathfrak{g},1)$ and let
$P(\mathfrak{g})^{\prime}$ be the subspace of $H^{\ast}(\mathfrak{g},1)$ given
by the orthogonal complement of $D(\mathfrak{g})$. We set
$P^{k}(\mathfrak{g})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}P(\mathfrak{g})^{\prime}\cap
H^{k}(\mathfrak{g},1)$ for each $k\geq 0$.

######  Theorem 2.1 (Koszul, [Kos50]).

We have an isomorphism of graded algebra

(2.1) $\wedge^{\ast}(P(\mathfrak{g})^{\prime})\cong
H^{\ast}(\mathfrak{g},1_{\mathfrak{g}}).$

Moreover, (2.1) is functorial in the following sense: for each morphism of
(split) reductive lie algebra
$\varphi:~{}\mathfrak{h}\rightarrow\mathfrak{g}$, we have an induced morphism
$\varphi^{\ast}:~{}P(\mathfrak{g})^{\prime}\rightarrow
P(\mathfrak{h})^{\prime}$ which determines the morphism
$\varphi^{\ast}:~{}H^{\ast}(\mathfrak{g},1_{\mathfrak{g}})\rightarrow
H^{\ast}(\mathfrak{h},1_{\mathfrak{h}})$ completely, via (2.1).

The following classical theorem reduces various group cohomologies to Lie
algebra cohomologies.

######  Theorem 2.2 (Casselman–Wigner, [CW74]).

The canonical morphism

$H^{\ast}(G,1_{G})\rightarrow H^{\ast}(\mathfrak{g},1_{\mathfrak{g}})$

is an isomorphism of graded algebra if $G$ is the set of
$\mathbb{Q}_{p}$-points of a semisimple group over $\mathbb{Q}_{p}$.

Note that we do not require the semisimple group over $\mathbb{Q}_{p}$ in
Theorem 2.2 to be split. Also note that similar results hold for compact
$p$-adic analytic groups due to [La65].

We recall $K$, $E$, $S=\\{\iota:K\hookrightarrow E\\}$ and other notation from
1.4. For each $I\subseteq\Delta_{n}$, we set
$r_{I}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}n-\\#I$ and there
exists a partition $n_{I}=(n_{d})_{1\leq d\leq r_{I}}$ of $n$ such that we
have a decomposition into Levi blocks

$L_{n,I}\cong G_{n_{1}}\times G_{n_{2}}\times\cdots\times G_{n_{r_{I}}}$

which induces

(2.2)
$\overline{L}_{n,I}\cong\overline{G}_{n_{1}}\times\overline{G}_{n_{2}}\times\cdots\times\overline{G}_{n_{r_{I}}}\times\overline{Z}_{n,I}.$

with $\overline{Z}_{n,I}\cong(K^{\times})^{r_{I}-1}$.

For each $I\subseteq\Delta_{n}$ and each $1\leq d\leq r_{I}$, we have an
embedding

(2.3)
$\overline{\mathfrak{g}}_{n_{d}}\hookrightarrow\overline{\mathfrak{l}}_{n,I}\hookrightarrow\overline{\mathfrak{g}}_{n}$

which induces an embedding
$\overline{\mathfrak{g}}_{n_{d},E}\hookrightarrow\overline{\mathfrak{g}}_{n,E}$
and therefore a morphism

(2.4)
$\mathrm{Res}_{n,I}^{k,d}:~{}P^{k}(\overline{\mathfrak{g}}_{n,E})\rightarrow
P^{k}(\overline{\mathfrak{g}}_{n_{d},E}).$

The embedding
$\overline{\mathfrak{g}}_{n_{d},E}\hookrightarrow\overline{\mathfrak{g}}_{n,E}$
is the direct sum of
$\overline{\mathfrak{g}}_{n_{d},E,\iota}\hookrightarrow\overline{\mathfrak{g}}_{n,E,\iota}$
for each $\iota\in S$, and thus (2.4) decomposes into direct sum of

$\mathrm{Res}_{n,I,\iota}^{k,d}:~{}P^{k}(\overline{\mathfrak{g}}_{n,E,\iota})\rightarrow
P^{k}(\overline{\mathfrak{g}}_{n_{d},E,\iota})$

for all $\iota\in S$.

The following result on primitive classes for Lie algebra cohomology is well-
known.

######  Theorem 2.3.

For each $n\geq 2$ and $\iota\in S$, we have
$\dim_{E}P^{k}(\overline{\mathfrak{g}}_{n,E,\iota})=1$ if $k=2m-1$ for some
$2\leq m\leq n$ and $P^{k}(\overline{\mathfrak{g}}_{n,E,\iota})=0$ otherwise.
Moreover, the morphism $\mathrm{Res}_{n,I,\iota}^{k,d}$ is an isomorphism if
$k=2m-1$ for some $2\leq m\leq n_{d}$ and is zero otherwise.

######  Lemma 2.4.

The map $\mathrm{Res}_{n,I,\iota}^{k,d}$ depends only on $n$, $k$, $n_{d}$ and
$\iota$.

###### Proof.

We fix $n$, $n^{\prime}\leq n$, $3\leq k\leq 2n^{\prime}-1$, and $\iota\in S$
throughout the proof. Let $f_{1}$ be the embedding

$\overline{\mathfrak{g}}_{n^{\prime},E}\hookrightarrow\overline{\mathfrak{l}}_{n,I,E}\hookrightarrow\overline{\mathfrak{g}}_{n,E}$

when $I=\\{1,\dots,n^{\prime}-1\\}$ and $d=1$. We consider the adjoint action
of $\overline{G}_{n}$ on $\overline{\mathfrak{g}}_{n,E}$ and set
$f_{g}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ad}(g)(f_{1}):\overline{\mathfrak{g}}_{n^{\prime},E}\hookrightarrow\overline{\mathfrak{g}}_{n,E}$.
For each $g\in\overline{G}_{n}$, $f_{g}$ induces a map
$f_{g}^{\ast}:P^{k}(\overline{\mathfrak{g}}_{n,E,\iota})\rightarrow
P^{k}(\overline{\mathfrak{g}}_{n^{\prime},E,\iota})$ which is an isomorphism
between one dimensional spaces thanks to Theorem 2.3. Consequently, there
exists a unique morphism $\chi:\overline{G}_{n}\rightarrow\mathbb{G}_{m}$ such
that $f_{g}^{\ast}=\chi(g)f_{1}^{\ast}$ for each $g\in\overline{G}_{n}$. As
$\overline{G}_{n}$ is adjoint, $\chi$ must be constant and thus
$f_{g}^{\ast}=f_{1}^{\ast}$ for each $g\in\overline{G}_{n}$. We finally notice
that the map $\mathrm{Res}_{n,I,\iota}^{k,d}$ constructed from different
choices of $I$ (with the same $n$, $k$, $n_{d}=n^{\prime}$ and $\iota$) are
just $f_{g}^{\ast}$ for different choices of $g$, and therefore are all equal.
∎

For each $n\geq 2$, $k\geq 3$ and $\iota\in S$ satisfying
$P^{k}(\overline{\mathfrak{g}}_{n,E,\iota})\neq 0$, we choose a basis
$\\{v_{n,\iota}^{k}\\}$ of $P^{k}(\overline{\mathfrak{g}}_{n,E,\iota})$ such
that they are compatible under all morphisms of the form
$\mathrm{Res}_{n,I}^{k,d}$. For technical convenience, we set
$v_{n}^{0}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}1\in
H^{0}(\mathfrak{g}_{n},1_{\mathfrak{g}_{n}})$ for each $n\geq 1$. For each
$n\geq 2$ and $k\geq 3$, we define $\Sigma_{n,k}$ as the set of subsets

$\Lambda=\\{(m_{1},\iota_{1}),\dots,(m_{r},\iota_{r})\\}\subseteq\\{3,5,\dots,2n-3,2n-1\\}\times
S$

for some $r\geq 1$ such that $\sum_{s=1}^{r}m_{s}=k$. We define
$P^{\Lambda}_{n}$ as the one dimensional subspace of
$\wedge^{\ast}\left(P(\overline{\mathfrak{g}}_{n,E})^{\prime}\right)$ given by

$P^{m_{1}}(\overline{\mathfrak{g}}_{n,E,\iota_{1}})\wedge\cdots\wedge
P^{m_{r}}(\overline{\mathfrak{g}}_{n,E,\iota_{r}})$

for each $\Lambda\in\Sigma_{n,k}$, which clearly does not depend on the choice
of order on $\Lambda$. For each $\Lambda\in\Sigma_{n,k}$, we write
$\widetilde{\Lambda}$ for an enhancement of $\Lambda$ to an ordered set
$\\{(m_{1},\iota_{1}),\cdots,(m_{r},\iota_{r})\\}$, and then set
$v_{n}^{\widetilde{\Lambda}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}v_{n,\iota_{1}}^{m_{1}}\wedge\cdots\wedge
v_{n,\iota_{r}}^{m_{r}}$. So different $v_{n}^{\widetilde{\Lambda}}$ for the
same $\Lambda$ differ by an explicit sign. For each $n\geq 1$, we also set
$\Sigma_{n,0}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{\emptyset\\}$
and
$v_{n}^{\emptyset}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}v_{n}^{0}=1\in
H^{0}(\overline{\mathfrak{g}}_{n,E},1_{\overline{\mathfrak{g}}_{n,E}})$ for
later convenience.

######  Corollary 2.5.

We have canonical isomorphisms

(2.5) $H^{k}(\overline{G}_{n},1_{\overline{G}_{n}})\cong
H^{k}(\overline{\mathfrak{g}}_{n},1_{\overline{\mathfrak{g}}_{n}})\cong
H^{k}(\overline{\mathfrak{g}}_{n,E},1_{\overline{\mathfrak{g}}_{n,E}})\cong\bigoplus_{\Lambda\in\Sigma_{n,k}}P^{\Lambda}_{n}$

for each $k\geq 0$ (with $1_{\overline{G}_{n}}$ and
$1_{\overline{\mathfrak{g}}_{n}}$ understood to be $E$-representation of
$\overline{G}_{n}$ and $\overline{\mathfrak{g}}_{n}$ respectively). In
particular,
$H^{k}(\overline{G}_{n},1_{\overline{G}_{n}})=H^{k}(\overline{\mathfrak{g}}_{n},1_{\overline{\mathfrak{g}}_{n}})=0$
if $k>(n^{2}-1)[K:\mathbb{Q}_{p}]$.

###### Proof.

The first isomorphism follows from Theorem 2.2. The second isomorphism follows
from Theorem 2.1 and Theorem 2.3. The third isomorphism follows from the
Chevalley-Eilenberg complexes that compute the source and the target. The
vanishing is clear from
$\dim\overline{G}_{n}=\dim\overline{\mathfrak{g}}_{n}=(n^{2}-1)[K:\mathbb{Q}_{p}]$,
and can also be seen from the fact that

$n^{2}-1=\sum_{s=2}^{n}2s-1$

which implies that $\Sigma_{n,k}=\emptyset$ if
$k>(n^{2}-1)[K:\mathbb{Q}_{p}]$. ∎

We will use without explanation the isomorphisms in Theorem 2.1 and Theorem
2.2 and view $P^{\Lambda}_{n}$ as a subspace of
$H^{k}(\overline{G}_{n},1_{\overline{G}_{n}})$ from now on.

######  Lemma 2.6.

We have a canonical isomorphism

$H^{k}(\overline{Z}_{n,I},1_{\overline{Z}_{n,I}})\cong\wedge^{k}(\mathrm{Hom}(\overline{Z}_{n,I},E))$

for each $n\geq 1$, $I\subseteq\Delta_{n}$ and $k\geq 0$.

###### Proof.

This is standard as $\overline{Z}_{n,I}$ is abelian (cf. Corollaire 3.11 of
[Schr11]). ∎

Note that we have an embedding
$\overline{Z}_{n,\Delta_{n}\setminus\\{i\\}}\hookrightarrow\overline{Z}_{n,I}$
for each $i\in\Delta_{n}\setminus I$, which actually induces an isomorphism

(2.6) $\overline{Z}_{n,I}\cong\prod_{i\in\Delta_{n}\setminus
I}\overline{Z}_{n,\Delta_{n}\setminus\\{i\\}}.$

Let $\mathrm{val}:K^{\times}\rightarrow\mathbb{Z}\hookrightarrow E$ be the
$p$-adic valuation function with $\mathrm{val}(p)=1$. We fix a choice of
$p$-adic logarithm $\mathrm{log}:\overline{\mathbb{Q}_{p}}^{\times}\rightarrow
E$ satisfying $\mathrm{log}(p)=0$ and write
$\mathrm{log}_{\iota}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{log}\circ\iota$
for each $\iota\in S$, so that
$\\{\mathrm{val}\\}\sqcup\\{\mathrm{log}_{\iota}\mid\iota\in S\\}$ forms a
basis of $\mathrm{Hom}(K^{\times},E)$. Using the standard isomorphism
$\overline{Z}_{n,\Delta_{n}\setminus\\{i\\}}\cong K^{\times}$, we obtain a
basis
$\mathcal{B}_{n,\Delta_{n}\setminus\\{i\\}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{\mathrm{val}_{i}\\}\sqcup\\{\mathrm{log}_{i,\iota}\mid\iota\in
S\\}$ of $\mathrm{Hom}(\overline{Z}_{n,\Delta_{n}\setminus\\{i\\}},E)$. Hence,
we deduce from (2.6) that
$\mathcal{B}_{n,I}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\bigsqcup_{i\in\Delta_{n}\setminus
I}\mathcal{B}_{n,\Delta_{n}\setminus\\{i\\}}$ forms a basis of
$\mathrm{Hom}(\overline{Z}_{n,I},E)$. We fix a total order on
$\mathcal{B}_{n,\emptyset}$ which induces a total order on
$\mathcal{B}_{n,I}\subseteq\mathcal{B}_{n,\emptyset}$ for each
$I\subseteq\Delta_{n}$. We write $v\subseteq\mathcal{B}_{n,I}$ for a subset,
which together with the fixed total order determines a unique element of
$\wedge^{k}\mathrm{Hom}(\overline{Z}_{n,I},E)$. Through this way, the set of
subsets of $\mathcal{B}_{n,I}$ with cardinality $k$ naturally corresponds to a
basis of $\wedge^{k}\mathrm{Hom}(\overline{Z}_{n,I},E)$. Note that each
$v\subseteq\mathcal{B}_{n,I}$ determines a unique maximal
$I_{v}\subseteq\Delta_{n}$ containing $I$ such that
$v\subseteq\mathcal{B}_{n,I_{v}}$. From now on, we will abuse the notation $v$
for both a subset of $\mathcal{B}_{n,I}$ and the corresponding element of
$\wedge^{\\#v}\mathrm{Hom}(\overline{Z}_{n,I},E)$ (using the fixed total order
on $\mathcal{B}_{n,I}$).

######  Lemma 2.7.

The isomorphism (2.2) induces an isomorphism

$H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})\cong\bigoplus_{k_{0}+k_{1}+\cdots+k_{r_{I}}=k}\wedge^{k_{0}}(\mathrm{Hom}(\overline{Z}_{n,I},E))\otimes_{E}\bigotimes_{d=1}^{r_{I}}H^{k_{d}}(\overline{G}_{n_{d}},1_{\overline{G}_{n_{d}}}).$

###### Proof.

This is simply Kunneth formula (cf. Théorème 3.10 of [Schr11]) combined with
Lemma 2.6. ∎

For each $i\in\Delta_{n}$, the embedding
$L_{n,\Delta\setminus\\{i\\}}\subseteq G_{n}$ induces a canonical morphism

$\mathrm{Res}_{n,i}^{k}:~{}H^{k}(\overline{G}_{n},1_{\overline{G}_{n}})\rightarrow
H^{k}(\overline{L}_{n,\Delta_{n}\setminus\\{i\\}},1_{\overline{L}_{n,\Delta_{n}\setminus\\{i\\}}}).$

For each $\Lambda\in\Sigma_{n,k}$ and $i\in\Delta_{n}$, we write
$\mathbf{D}_{n,\Lambda,i}$ for those subsets
$\Lambda^{\prime}\subseteq\Lambda$ satisfying
$\max\\{m\mid(m,\iota)\in\Lambda^{\prime}\\}\leq 2i-1$ and
$\max\\{m\mid(m,\iota)\in\Lambda\setminus\Lambda^{\prime}\\}\leq 2(n-i)-1$.
Note that both $\Lambda^{\prime}$ and $\Lambda\setminus\Lambda^{\prime}$ are
allowed to be empty. In the following, we fix a total order on $S$, and then
fix the choice of enhancement $\widetilde{\Lambda}$ to be the one with
decreasing order on integers and satisfying
$(m,\iota)<(m^{\prime},\iota^{\prime})$ whenever $\iota<\iota^{\prime}$, and
write $v_{n}^{\Lambda}$ instead of $v_{n}^{\widetilde{\Lambda}}$.

######  Lemma 2.8.

Let $\Lambda\in\Sigma_{n,k}$ be a partition and $\widetilde{\Lambda}$ be an
enhancement of $\Lambda$. Then $\mathrm{Res}_{n,i}^{k}(P^{\Lambda}_{n})\neq 0$
if and only if $\mathbf{D}_{n,\Lambda,i}\neq\emptyset$. Moreover, we have

$\mathrm{Res}_{n,i}^{k}(v_{n}^{\Lambda})=\sum_{\Lambda^{\prime}\in\mathbf{D}_{n,\Lambda,i}}\varepsilon(\Lambda^{\prime})v_{i}^{\Lambda^{\prime}}\otimes_{E}v_{n-i}^{\Lambda\setminus\Lambda^{\prime}}.$

###### Proof.

This follows from Theorem 2.1 and Theorem 2.3. ∎

More generally, for each $I^{\prime}\subseteq I\subseteq\Delta_{n}$, the
embedding $L_{n,I^{\prime}}\subseteq L_{n,I}$ induces a canonical morphism

(2.7)
$\mathrm{Res}_{n,I,I^{\prime}}^{k}:~{}H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})\rightarrow
H^{k}(\overline{L}_{n,I^{\prime}},1_{\overline{L}_{n,I^{\prime}}}).$

Let $i\in I$ and $1\leq d\leq r_{I}$ such that
$1+\sum_{d^{\prime}=1}^{d-1}n_{d^{\prime}}\leq
i\leq\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$, then
$\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}$ can be clearly recovered from
$\mathrm{Res}_{n_{d},\overline{i}}^{k_{d}}$ (with
$\overline{i}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}i-\sum_{d^{\prime}=1}^{d-1}n_{d^{\prime}}$)
by tensoring with identity morphisms on cohomologies of other Levi blocks of
$L_{n,I}$ and that of $\overline{Z}_{n,I}$, and then sum over all
decompositions $k=\sum_{d=0}^{r_{I}}k_{d}$ as in Lemma 2.7.

Let $I\subseteq\Delta_{n}$ be a subset and $\underline{k}=\\{k_{d}\\}_{0\leq
d\leq r_{I}}$ be a tuple of non-negative integers satisfying
$|\underline{k}|\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d=0}^{r_{I}}k_{d}=k$.
We choose an element $\Lambda_{d}\in\Sigma_{n_{d},k_{d}}$ for each $1\leq
d\leq r_{I}$ and then write $\underline{\Lambda}=\\{\Lambda_{d}\\}_{1\leq
d\leq r_{I}}$ for the tuple. We set

(2.8)
$\Sigma_{n,I,\underline{k}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\prod_{d=1}^{r_{I}}\Sigma_{n_{d},k_{d}}$

and

$v_{n,I,\underline{k}}^{\underline{\Lambda}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\bigotimes_{d=1}^{r_{I}}v_{n_{d}}^{\Lambda_{d}}\in\bigotimes_{d=1}^{r_{I}}H^{k_{d}}(\overline{G}_{n_{d}},1_{\overline{G}_{n_{d}}})$

for each $\underline{\Lambda}\in\Sigma_{n,I,\underline{k}}$. It is clear that
$\\{v_{n,I,\underline{k}}^{\underline{\Lambda}}\\}_{\underline{\Lambda}\in\Sigma_{n,I,\underline{k}}}$
is a basis of
$\bigotimes_{d=1}^{r_{I}}H^{k_{d}}(\overline{G}_{n_{d}},1_{\overline{G}_{n_{d}}})$.
Moreover, Lemma 2.7 implies that

(2.9) $\\{v\otimes_{E}v_{n,I,\underline{k}}^{\underline{\Lambda}}\mid
v\subseteq\mathcal{B}_{n,I},~{}\\#v=k_{0},~{}\underline{\Lambda}\in\Sigma_{n,I,\underline{k}},~{}|\underline{k}|=k\\}$

forms a basis of $H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$ for each
$I\subseteq\Delta_{n}$ and $k\geq 0$.

###  2.2. Standard spectral sequences

For each pair of subsets $I_{0}\subseteq I_{1}\subseteq\Delta_{n}$, we
consider the following complex

(2.10)
$\mathbf{C}_{I_{0},I_{1}}(\lambda):~{}i_{n,I_{1}}^{\rm{an}}(\lambda)\rightarrow\cdots\rightarrow\bigoplus_{I_{0}\subseteq
I^{\prime}\subseteq
I_{1},\\#I=\ell}i_{n,I}^{\rm{an}}(\lambda)\rightarrow\cdots\rightarrow
i_{n,I_{0}}^{\rm{an}}(\lambda)$

with $i_{n,I_{0}}^{\rm{an}}(\lambda)$ placed at degree $-\\#I_{0}$. Now we
specify each map of the complex (2.10). Recall that we have a distinguished
embedding
$\kappa_{I,I^{\prime}}^{\rm{an}}(\lambda):i_{n,I}^{\rm{an}}(\lambda)\hookrightarrow
i_{n,I^{\prime}}^{\rm{an}}(\lambda)$ for each $I^{\prime}\subseteq
I\subseteq\Delta_{n}$. For each $i\in I\subseteq\Delta_{n}$, we define
$m(I,i)$ to be the number of $i^{\prime}\in I$ satisfying $i^{\prime}<i$. Then
the map $i_{n,I}^{\rm{an}}(\lambda)\hookrightarrow
i_{n,I\setminus\\{i\\}}^{\rm{an}}(\lambda)$ in (2.10) is given by
$(-1)^{m(I,i)}\kappa_{I,I^{\prime}}^{\rm{an}}(\lambda)$, for each
$I_{0}\subseteq I\subseteq I_{1}$ and $i\in I\setminus I_{0}$. One can easily
check that such maps actually define a complex as illustrated by (2.10).

######  Remark 2.9.

There is an easy way to understand the choice of the sign $(-1)^{m(I,i)}$ that
appears in the definition of the complex. We assume $I_{0}=\emptyset$,
$I_{1}=\Delta_{n}$ and $\lambda=0$ for simplicity, then each
$\kappa_{I,I^{\prime}}^{\rm{an}}(0)$ restrict to the identity map on the space
of constant functions on $\overline{G}_{n}$ (as a natural subspace of both
$i_{n,I}^{\rm{an}}(0)$ and $i_{n,I^{\prime}}^{\rm{an}}(0)$ by definition of
parabolic induction). We consider the subcomplex $\mathbf{C}$ of
$\mathbf{C}_{\emptyset,\Delta_{n}}(0)$ consisting of constant functions on
$\overline{G}_{n}$, then our choice of sign $(-1)^{m(I,i)}$ above guarantees
that $\mathbf{C}$ is exact. In fact, the exact sequence $\mathbf{C}$ is simply
the $\\#\Delta_{n}=n-1$-th tensor power of the exact sequence $0\rightarrow
E\xrightarrow{\mathrm{Id}}E\rightarrow 0$ that is supported in degree
$[-1,0]$.

For later convenience, we also set
$\mathbf{C}_{I_{0},I_{1}}(\lambda)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}0$
if $I_{0}\not\subseteq I_{1}$. If we consider two extra sets
$I_{0}^{\prime},I_{1}^{\prime}$ satisfying $I_{0}\subseteq
I_{0}^{\prime}\subseteq I_{1}^{\prime}\subseteq I_{1}$, then we have a natural
commutative diagram by considering truncation of complex

$\displaystyle{\mathbf{C}_{I_{0},I_{1}}(\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\mathbf{C}_{I_{0}^{\prime},I_{1}}(\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle{\mathbf{C}_{I_{0},I_{1}^{\prime}}(\lambda)}$$\displaystyle{\mathbf{C}_{I_{0}^{\prime},I_{1}^{\prime}}(\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Note that $\mathbf{C}_{I_{0},I_{1}}(\lambda)\cong
i_{n,I_{1},\Delta_{n}}^{\rm{an}}(V_{n,I_{0},I_{1}}^{\rm{an}}(\lambda))[-\\#I_{0}]$
(and in particular $\mathbf{C}_{I_{0},\Delta_{n}}(\lambda)\cong
V_{n,I_{0}}^{\rm{an}}(\lambda)[-\\#I_{0}]$) in the derived sense.

It follows from Lemma 1.8 and Theorem 7.1 of [Koh11] that we have canonical
isomorphisms

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

for each $I\subseteq\Delta_{n}$. Moreover, for each $I^{\prime}\subseteq
I\subseteq\Delta_{n}$, the map

$\mathrm{Ext}_{G_{n},\lambda}^{\bullet}(F_{n,\Delta_{n}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\rightarrow\mathrm{Ext}_{G_{n},\lambda}^{\bullet}(F_{n,\Delta_{n}}(\lambda),i_{n,I^{\prime}}^{\rm{an}}(\lambda))$

induced from
$\kappa_{I,I^{\prime}}^{\rm{an}}(\lambda):i_{n,I}^{\rm{an}}(\lambda)\hookrightarrow
i_{n,I^{\prime}}^{\rm{an}}(\lambda)$ is simply the restriction map (2.7). It
follows from item (i) of Lemma 1.7 that there exists a standard spectral
sequence $\\{E_{r,I_{0},I_{1}}^{-\ell,k}\\}_{r\geq
0,\\#I_{0}\leq\ell\leq\\#I_{1},k\geq 0}$ which converges to

$\mathrm{Ext}_{G_{n},\lambda}^{\bullet}(F_{n,\Delta_{n}}(\lambda),\mathbf{C}_{I_{0},I_{1}}(\lambda))\cong\mathrm{Ext}_{G_{n},\lambda}^{\bullet+\\#I_{0}}(F_{n,\Delta_{n}}(\lambda),i_{n,I_{1},\Delta_{n}}^{\rm{an}}(V_{n,I_{0},I_{1}}^{\rm{an}}(\lambda))).$

For each $k\geq 0$ and $\\#I_{0}\leq\ell\leq n-1$, we have

$E_{1,I_{0},I_{1}}^{-\ell,k}\cong\bigoplus_{I_{0}\subseteq I\subseteq
I_{1},\\#I=\ell}\mathrm{Ext}_{G_{n},\lambda}^{k}(F_{n,\Delta_{n}}(\lambda),i_{n,I}^{\rm{an}}(\lambda))\cong\bigoplus_{I_{0}\subseteq
I\subseteq I_{1},\\#I=\ell}H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}}),$

and the differential
$d_{1,I_{0},I_{1}}^{-\ell,k}:~{}E_{1,I_{0},I_{1}}^{-\ell,k}\rightarrow
E_{1,I_{0},I_{1}}^{-\ell+1,k}$ has the form

$\bigoplus_{i\in I\setminus
I_{0}}(-1)^{m(I,i)}\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}$

after restricting to $H^{k}(\overline{L}_{n,I},1_{\overline{L}_{n,I}})$, for
each $I_{0}\subseteq I\subseteq I_{1}$ satisfying $\\#I=\ell$.

Given two pair of subsets $I_{0}\subseteq I_{1}\subseteq\Delta_{n}$ and
$I_{0}^{\prime}\subseteq I_{1}^{\prime}\subseteq\Delta_{n}$, if
$I_{0}\subseteq I_{0}^{\prime}$ and $I_{1}\subseteq I_{1}^{\prime}$, then the
natural map
$\mathbf{C}_{I_{0},I_{1}}(\lambda)\rightarrow\mathbf{C}_{I_{0}^{\prime},I_{1}^{\prime}}(\lambda)$
induces a map between spectral sequences
$E_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}\rightarrow
E_{\bullet,I_{0}^{\prime},I_{1}^{\prime}}^{\bullet,\bullet}$.

###  2.3. $(I_{0},I_{1})$-standard and $(I_{0},I_{1})$-atomic tuples

In this section, we introduce $(I_{0},I_{1})$-standard and
$(I_{0},I_{1})$-atomic elements (see Definition 2.13) of the bases introduced
in Section 2.1. These combinatorial notion will be crucial later for the proof
of Theorem 2.28.

######  Definition 2.10.

We say that a subset $I\subseteq\Delta_{n}$ is an _interval_ if there exists
$1\leq i\leq j\leq n-1$ such that $I=\\{i,i+1,\dots,j\\}$. For each
$I\subseteq\Delta_{n}$, there exists a unique decomposition
$I=\bigsqcup_{d=1}^{r_{I}}I^{d}$ such that $I^{d}$ corresponds to the set of
positive simple roots in the $d$-th Levi block of $L_{n,I}$. Hence, $I^{d}$ is
either empty or a maximal subinterval of $I$ with $n_{d}=\\#I^{d}+1$, and
moreover $i<i^{\prime}$ for each $i\in I^{d}$ and $i^{\prime}\in
I^{d^{\prime}}$ satisfying $1\leq d<d^{\prime}\leq r_{I}$.

Let $I_{0}\subseteq I\subseteq I_{1}$ be a subset satisfying $\\#I=\ell$ and
$\underline{k}=\\{k_{d}\\}_{1\leq d\leq r_{I}}$ be a tuple satisfying
$|\underline{k}|=k$. Let $v\subseteq\mathcal{B}_{n,I}$ with $\\#v=k_{0}$ and
$\underline{\Lambda}\in\Sigma_{n,I,\underline{k}}$ (see (2.8)), we use in the
following the shortened notation
$\Theta=(v,I,\underline{k},\underline{\Lambda})$ and write
$x_{\Theta}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}v\otimes_{E}v_{n,I,\underline{k}}^{\underline{\Lambda}}$.
So we always have $I_{0}\subseteq I\subseteq I_{v}\cap I_{1}$.

The space $E_{1,I_{0},I_{1}}^{-\ell,k}$ admits a basis of the form

$\\{x_{\Theta}\mid\Theta=(v,I,\underline{k},\underline{\Lambda}),~{}v\subseteq\mathcal{B}_{n,\emptyset},~{}\underline{\Lambda}\in\Sigma_{n,I,\underline{k}},~{}I_{0}\subseteq
I\subseteq I_{v}\cap I_{1},~{}|\underline{k}|=k,~{}\\#I=\ell,~{}|v|=k_{0}\\}.$

Hence, we have a decomposition (with $E_{1,I_{0},I_{1},v}^{-\ell,k}$ the
$v$-isotypic direct summand)

$E_{1,I_{0},I_{1}}^{-\ell,k}=\bigoplus_{v\subseteq\mathcal{B}_{n,\emptyset}}E_{1,I_{0},I_{1},v}^{-\ell,k}$

which is compatible with the differential $d_{1,I_{0},I_{1}}^{-\ell,k}$. For
each $x\in E_{1,I_{0},I_{1},v}^{-\ell,k}$ and each
$\Theta=(v,I,\underline{k},\underline{\Lambda})$, we write $c_{\Theta}(x)$ for
the coefficient of $x$ attached to $x_{\Theta}$.

We introduce two fundamental construction which (if defined) sends a given
tuple $\Theta=(v,I,\underline{k},\underline{\Lambda})$ to a new tuple.

  * •

For each $i\in\Delta_{n}\setminus I$, there exists a unique $1\leq d\leq
r_{I}$ such that $i=\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$, and we define
$p_{i}^{+}(\Theta)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(v,I\sqcup\\{i\\},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
by the condition that $\Lambda^{\prime}_{d^{\prime}}=\Lambda_{d^{\prime}}$ for
each $1\leq d^{\prime}\leq d-1$,
$\Lambda^{\prime}_{d^{\prime}}=\Lambda_{d^{\prime}+1}$ for each $d+1\leq
d^{\prime}\leq r_{I}-1$ and
$\Lambda^{\prime}_{d}=\Lambda_{d}\sqcup\Lambda_{d+1}$. So $p_{i}^{+}(\Theta)$
is well defined if and only if $i\in\Delta_{n}\setminus I$ and
$\Lambda_{d}\cap\Lambda_{d+1}=\emptyset$.

  * •

For each $i\in I$, there exists a unique $1\leq d\leq r_{I}$ such that $i\in
I^{d}$, and we define
$p_{i}^{-}(\Theta)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(v,I\setminus\\{i\\},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
by the condition that $\Lambda^{\prime}_{d^{\prime}}=\Lambda_{d^{\prime}}$ for
each $1\leq d^{\prime}\leq d-1$,
$\Lambda^{\prime}_{d^{\prime}}=\Lambda_{d^{\prime}-1}$ for each $d+2\leq
d^{\prime}\leq r_{I}+1$, $\Lambda^{\prime}_{d}=\Lambda_{d}$ and
$\Lambda^{\prime}_{d+1}=\emptyset$. So $p_{i}^{-}(\Theta)$ is well defined if
and only if $i\in I$ and $\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}\leq
2(i-\sum_{d^{\prime}=1}^{d-1}n_{d^{\prime}})-1$.

######  Remark 2.11.

For each $1\leq d\leq r_{I}$, we call $\Lambda_{d}$ the _$d$ -th block of the
tuple $\Theta$_. Intuitively speaking, $p_{i}^{+}(\Theta)$ (when defined) is
obtained from $\Theta$ by gluing two adjacent blocks separated by $i$, and
$p_{i}^{-}(\Theta)$ (when defined) is obtained from $\Theta$ by splitting a
block into two adjacent blocks separated by $i$, leaving the second block
trivial.

######  Definition 2.12.

Let $\Theta=(v,I,\underline{k},\underline{\Lambda})$,
$\Theta^{\prime}=(v^{\prime},I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
be two tuple. We say that _$\Theta^{\prime}$ is an improvement of $\Theta$
with level $d$_ for some $1\leq d\leq r_{I}$, if there exists $i\in
I^{d}\setminus I_{0}$ and
$i^{\prime}=\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}\in(I_{v}\cap I_{1})\setminus
I$ such that $\Theta^{\prime}=p_{i^{\prime}}^{+}(p_{i}^{-}(\Theta))$ (which
forces $v=v^{\prime}$, $\underline{k}=\underline{k}^{\prime}$ and
$\underline{\Lambda}=\underline{\Lambda}^{\prime}$). We say that _$\Theta$ is
smaller than $\Theta^{\prime}$ with level $\geq d$_ for some $1\leq d\leq
r_{I}$, written $\Theta<_{d}\Theta^{\prime}$, if there exists a sequence of
tuples $\Theta=\Theta^{0},\dots,\Theta^{m}=\Theta^{\prime}$ such that
$\Theta^{m^{\prime}}$ is an improvement of $\Theta^{m^{\prime}-1}$ with level
$\geq d$ for each $1\leq m^{\prime}\leq m$. We say that $\Theta$ and
$\Theta^{\prime}$ are _equivalent_ (with respect to $(I_{0},I_{1})$) if there
exists $\Theta^{\prime\prime}$ such that $\Theta<_{1}\Theta^{\prime\prime}$
and $\Theta^{\prime}<_{1}\Theta^{\prime\prime}$.

For each $1\leq d\leq r_{I}$ satisfying $I^{d}\cap I_{0}\neq\emptyset$, there
exists a unique maximal possible $1\leq d^{\prime}\leq r_{I_{0}}$ such that
$\emptyset\neq I_{0}^{d^{\prime}}\subseteq I^{d}$ and $I^{d}\setminus
I_{0}^{d^{\prime}}=I^{d,-}\sqcup I^{d,+}$ with
$I^{d,-}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{i\in I^{d}\mid
i<i^{\prime}\text{ for each }i^{\prime}\in I_{0}^{d^{\prime}}\\}$ and
$I^{d,+}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{i\in I^{d}\mid
i>i^{\prime}\text{ for each }i^{\prime}\in I_{0}^{d^{\prime}}\\}$. So we
always have $I^{d,+}\cap I_{0}=\emptyset$ but $I^{d,-}\cap I_{0}$ might be
non-empty. If $I^{d,-}\neq\emptyset$, then we have $I^{d,-}\setminus
I_{0}\neq\emptyset$ and set
$n_{d}^{-}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\min\\{i\mid i\in
I^{d,-}\setminus
I_{0}\\}-\sum_{d^{\prime\prime}=1}^{d-1}n_{d^{\prime\prime}}\leq\\#I^{d,-}$.

Note that the sets $I\subseteq I_{v}\cap I_{1}$ determines a sequence

$0=r_{v,I_{1},I}^{0}<r_{v,I_{1},I}^{1}<\cdots<r_{v,I_{1},I}^{r_{I_{v}\cap
I_{1}}}=r_{I}$

characterized by $\Delta_{n}\setminus(I_{v}\cap
I_{1})=\\{\sum_{d=1}^{r_{v,I_{1},I}^{s}}n_{d}\mid 1\leq s\leq r_{I_{v}\cap
I_{1}}-1\\}$. Given $1\leq d\leq r_{I}$, note that
$\sum_{d^{\prime}=1}^{d-1}n_{d^{\prime}}\in\\{0\\}\sqcup\Delta_{n}\setminus(I_{v}\cap
I_{1})$ if and only if there exists $1\leq s\leq r_{I_{v}\cap I_{1}}$ such
that $d=r_{v,I_{1},I}^{s-1}+1$, and similarly
$\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}\in\\{n\\}\sqcup\Delta_{n}\setminus(I_{v}\cap
I_{1})$ if and only if there exists $1\leq s\leq r_{I_{v}\cap I_{1}}$ such
that $d=r_{v,I_{1},I}^{s}$.

######  Definition 2.13.

For each tuple $\Theta=(v,I,\underline{k},\underline{\Lambda})$, we define its
_sign_ as
$\varepsilon(\Theta)\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(-1)^{\sum_{i\in
I}i}$. We say that $\Theta$ is _$(I_{0},I_{1})$ -standard_ if there exists
$2\leq d_{I_{0},I_{1},\Theta}\leq r_{I}$ such that

  * •

$\sum_{d^{\prime}=1}^{d_{I_{0},I_{1},\Theta}-1}n_{d^{\prime}}\in(I_{v}\cap
I_{1})\setminus I$;

  * •

$\Lambda^{d_{I_{0},I_{1},\Theta}}=\emptyset$ and $\Lambda^{d}\neq\emptyset$
for each $d_{I_{0},I_{1},\Theta}+1\leq d\leq r_{I}$;

  * •

$I^{d_{I_{0},I_{1},\Theta}}\subseteq I_{0}$; and

  * •

there does not exist $\Theta^{\prime}$ such that
$\Theta<_{d_{I_{0},I_{1},\Theta}}\Theta^{\prime}$.

We say that $\Theta$ is _maximally $(I_{0},I_{1})$-atomic_ if

  * •

there does not exist $\Theta^{\prime}$ such that $\Theta<_{1}\Theta^{\prime}$;

  * •

if $k_{d}=0$ (namely $\Lambda_{d}=\emptyset$) for some $1\leq d\leq r_{I}$,
then there exists $1\leq s\leq r_{I_{v}\cap I_{1}}$ such that
$d=r_{v,I_{1},I}^{s-1}+1$;

  * •

for each $1\leq s\leq r_{I_{v}\cap I_{1}}$ satisfying
$I^{r_{v,I_{1},I}^{s}}\cap I_{0}\neq\emptyset$,
$\Lambda_{r_{v,I_{1},I}^{s}}\neq\emptyset$ and
$(\\{2n_{r_{v,I_{1},I}^{s}}-1\\}\times
S)\cap\Lambda_{r_{v,I_{1},I}^{s}}=\emptyset$, we have
$I^{r_{v,I_{1},I}^{s},+}=\emptyset$ and there exists
$n^{\prime}\geq\\#I^{r_{v,I_{1},I}^{s},-}+1$ and $\iota\in S$ such that
$(2n^{\prime}-1,\iota)\in\Lambda_{r_{v,I_{1},I}^{s}}$; and

  * •

for each $1\leq s\leq r_{I_{v}\cap I_{1}}$ satisfying
$I^{r_{v,I_{1},I}^{s}}\cap I_{0}=\emptyset$ and
$\Lambda_{r_{v,I_{1},I}^{s}}\neq\emptyset$, there exists $\iota\in S$ such
that $(2n_{r_{v,I_{1},I}^{s}}-1,\iota)\in\Lambda_{r_{v,I_{1},I}^{s}}$.

We say that $\Theta$ is _$(I_{0},I_{1})$ -atomic_ if there exists a maximally
$(I_{0},I_{1})$-atomic $\Theta^{\prime}$ such that
$\Theta<_{1}\Theta^{\prime}$. For each equivalent class $\Omega$ of
$(I_{0},I_{1})$-atomic tuples, we associate an $(I_{0},I_{1})$-atom

$x_{\Omega}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{\Theta=(v,I,\underline{k},\underline{\Lambda})\in\Omega}\varepsilon(\Theta)x_{\Theta}.$

The _bidegree_ of a $(I_{0},I_{1})$-atom $x_{\Omega}$ is defined to be
$(-\ell_{\Omega},k_{\Omega})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(-\\#I,|\underline{k}|)$
for an arbitrary $\Theta=(v,I,\underline{k},\underline{\Lambda})\in\Omega$.

######  Lemma 2.14.

Let $\Theta=(v,I,\underline{k},\underline{\Lambda})$ be a tuple as before and
$1\leq d\leq r_{I}$ be an integer. If $I^{d}\cap I_{0}=\emptyset$, then the
following statements are equivalent

  1. (i)

there exists $i\in I^{d}$ such that
$\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}(x_{\Theta})\neq 0$;

  2. (ii)

$n_{d}\geq 2$ and $\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}(x_{\Theta})\neq 0$
for
$i\in\\{-1+\sum_{d^{\prime}=1}^{d}n_{d^{\prime}},1+\sum_{d^{\prime}=1}^{d-1}n_{d^{\prime}}\\}$;

  3. (iii)

$n_{d}\geq 2$ and $(\\{2n_{d}-1\\}\times S)\cap\Lambda_{d}=\emptyset$.

If $I^{d}\cap I_{0}\neq\emptyset$, then the following statements are
equivalent

  1. (iv)

there exists $i\in I^{d,+}$ such that
$\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}(x_{\Theta})\neq 0$;

  2. (v)

$I^{d,+}\neq\emptyset$ and
$\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}(x_{\Theta})\neq 0$ for
$i=-1+\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$;

  3. (vi)

$I^{d,+}\neq\emptyset$ and $(\\{2n_{d}-1\\}\times
S)\cap\Lambda_{d}=\emptyset$.

If $I^{d}\cap I_{0}\neq\emptyset$ and $I^{d,+}=\emptyset$, then the following
statements are equivalent

  1. (vii)

there exists $i\in I^{d,-}$ such that
$c_{p_{i}^{-}(\Theta)}(\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}(x_{\Theta}))\neq
0$;

  2. (viii)

$I^{d,-}\neq\emptyset$ and
$c_{p_{i}^{-}(\Theta)}(\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}(x_{\Theta}))\neq
0$ for $i=\\#I^{d,-}+\sum_{d^{\prime}=1}^{d-1}n_{d^{\prime}}$);

  3. (ix)

$I^{d,-}\neq\emptyset$ and
$\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}<2\\#I^{d,-}+1$.

###### Proof.

We only treat the case $I^{d}\cap I_{0}=\emptyset$ and the other two cases are
similar. For each $i\in I^{d}$, it follows from Lemma 2.8 that
$\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}(x_{\Theta})\neq 0$ if and only if
$\mathbf{D}_{n_{d},\Lambda_{d},\overline{i}}\neq\emptyset$ where
$\overline{i}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}i-\sum_{d^{\prime}=1}^{d-1}n_{d^{\prime}}$.
Note that $\mathbf{D}_{n_{d},\Lambda_{d},\overline{i}}\neq\emptyset$ implies
the existence of $\Lambda^{\prime}\subseteq\Lambda$ such that
$\max\\{m\mid(m,\iota)\in\Lambda^{\prime}\\}\leq 2\overline{i}-1$ and
$\max\\{m\mid(m,\iota)\in\Lambda_{d}\setminus\Lambda^{\prime}\\}\leq
2(n_{d}-\overline{i})-1$, which altogether implies that
$\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}\leq 2n_{d}-3$ (which is equivalent to
$(\\{2n_{d}-1\\}\times S)\cap\Lambda_{d}=\emptyset$). In other words, we have
(i) implies (iii). As (iii) clearly implies (ii) (namely
$\mathbf{D}_{n_{d},\Lambda_{d},n_{d}-1}\neq\emptyset\neq\mathbf{D}_{n_{d},\Lambda_{d},1}$)
and (ii) clearly implies (i), we finish the proof. ∎

######  Lemma 2.15.

Let $\Theta=(v,I,\underline{k},\underline{\Lambda})$ be a maximally
$(I_{0},I_{1})$-atomic tuple. Let $1\leq d\leq r_{I}$ be an integer. Then we
have

  * •

If $I^{d}\cap I_{0}=\emptyset$ and $(\\{2n_{d}-1\\}\times
S)\cap\Lambda_{d}=\emptyset$, then $n_{d}=1$, $\Lambda_{d}=\emptyset$ and
there exists $1\leq s\leq r_{I_{v}\cap I_{1}}$ such that
$d=r_{v,I_{1},I}^{s-1}+1$.

  * •

If $I^{d}\cap I_{0}\neq\emptyset$ and $I^{d,+}\neq\emptyset$, then there
exists $\iota\in S$ such that $(2n_{d}-1,\iota)\in\Lambda_{d}$;

  * •

If $I^{d}\cap I_{0}\neq\emptyset$, $I^{d,+}=\emptyset$ and
$\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}<2\\#I^{d,-}+1$, then
$I^{d,-}=\emptyset$, $\Lambda_{d}=\emptyset$ and there exists $1\leq s\leq
r_{I_{v}\cap I_{1}}$ such that $d=r_{v,I_{1},I}^{s-1}+1$.

###### Proof.

We may assume throughout that $d\neq r_{v,I_{1},I}^{s}$ for each $1\leq s\leq
r_{I_{v}\cap I_{1}}$ (see Definition 2.13). We treat the three claims
separately

  * •

Let $d$ satisfy $I^{d}\cap I_{0}=\emptyset$ and $(\\{2n_{d}-1\\}\times
S)\cap\Lambda_{d}=\emptyset$. If $n_{d}\geq 2$, then
$\Theta<p_{i_{2}}^{+}p_{i_{1}}^{-}(\Theta)$ for
$i_{1}=-1+\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$ and
$i_{2}=\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$ (using Lemma 2.14), which
contradicts the maximality of $\Theta$. Hence, we have $n_{d}=1$, $k_{d}=0$
and $\Lambda_{d}=\emptyset$, which gives a $1\leq s\leq r_{I_{v}\cap I_{1}}$
such that $d=r_{v,I_{1},I}^{s-1}+1$ by Definition 2.13.

  * •

Let $d$ satisfy $I^{d}\cap I_{0}\neq\emptyset$ and $I^{d,+}\neq\emptyset$. If
$(\\{2n_{d}-1\\}\times S)\cap\Lambda_{d}=\emptyset$, then Lemma 2.14 implies
that $\Theta<p_{i_{4}}^{+}p_{i_{3}}^{-}(\Theta)$ for
$i_{3}=-1+\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$ and
$i_{4}=\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$, which contradicts the
maximality of $\Theta$.

  * •

Let $d$ satisfy $I^{d}\cap I_{0}\neq\emptyset$, $I^{d,+}=\emptyset$ and
$\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}<2\\#I^{d,-}+1$. If
$I^{d,-}\neq\emptyset$, then Lemma 2.14 implies that
$\Theta<p_{i_{6}}^{+}p_{i_{5}}^{-}(\Theta)$ for

$i_{5}=-1-\\#I^{d}\cap
I_{0}+\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}=\\#I^{d,-}+\sum_{d^{\prime}=1}^{d-1}n_{d^{\prime}}$

and $i_{6}=\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$, which contradicts the
maximality of $\Theta$. Hence, we have $I^{d,-}=\emptyset$, which together
with $\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}<2\\#I^{d,-}+1$ forces $k_{d}=0$
(namely $\Lambda_{d}=\emptyset$), and thus there exists $1\leq s\leq
r_{I_{v}\cap I_{1}}$ such that $d=r_{v,I_{1},I}^{s-1}+1$ by Definition 2.13.

∎

Now we classify $(I_{0},I_{1})$-atomic tuples when
$k\in\\{\ell+\\#I_{1}-2\\#I_{0},\ell+\\#I_{1}-2\\#I_{0}+1\\}$.

######  Lemma 2.16.

We have the following consequences of existence of $(I_{0},I_{1})$-atomic
tuples.

  * •

If $\Theta=(v,I,\underline{k},\underline{\Lambda})$ is $(I_{0},I_{1})$-atomic,
then we have

$\ell-2\\#I_{0}+\\#I_{1}\leq k\leq\ell-n+1+[K:\mathbb{Q}_{p}](n^{2}-1).$

  * •

If $k=\ell+\\#I_{1}-2\\#I_{0}$, then $I_{v}\cup I_{1}=\Delta_{n}$ and a
maximally $(I_{0},I_{1})$-atomic $\Theta$ satisfies

    * –

there does not exist $i\in\Delta_{n}\setminus I_{v}$ such that
$\\#v\cap\mathcal{B}_{n,\Delta_{n}\setminus\\{i\\}}\geq 2$ (which implies that
$k_{0}=\\#\Delta_{n}\setminus I_{v}$);

    * –

if $I^{d}\cap I_{0}=\emptyset$, then $\Lambda_{d}\neq\\{(2n_{d}-1,\iota)\\}$
for any $\iota\in S$ if and only if $n_{d}=1$, $\Lambda_{d}=\emptyset$ and
there exists $1\leq s\leq r_{I_{v}\cap I_{1}}$ such that
$d=r_{v,I_{1},I}^{s-1}+1$;

    * –

if $I^{d}\cap I_{0}\neq\emptyset$, then $I^{d}\cap I_{0}$ is an interval and
$I^{d,+}=\emptyset$, and moreover $\Lambda_{d}\neq\\{(2\\#I^{d,+}+1,\iota)\\}$
for any $\iota\in S$ if and only if $I^{d,-}=\emptyset$,
$\Lambda_{d}=\emptyset$ and there exists $1\leq s\leq r_{I_{v}\cap I_{1}}$
such that $d=r_{v,I_{1},I}^{s-1}+1$.

  * •

If $k=\ell+\\#I_{1}-2\\#I_{0}+1$, then either $\\#I_{v}\cup I_{1}=n-2$ and a
maximally $(I_{0},I_{1})$-atomic $\Theta$ satisfies exactly the same three
conditions as $k=\ell+\\#I_{1}-2\\#I_{0}$ case, or $I_{v}\cup
I_{1}=\Delta_{n}$ and a maximally $(I_{0},I_{1})$-atomic $\Theta$ satisfies
the conditions in $k=\ell+\\#I_{1}-2\\#I_{0}$ case with exactly one of the
following modifications

    * –

there exists exactly one $i\in\Delta_{n}\setminus I_{v}$ such that
$\\#v\cap\mathcal{B}_{n,\Delta_{n}\setminus\\{i\\}}\geq 2$ and the equality
holds for that $i$;

    * –

for exactly one choice of $1\leq s\leq r_{I_{v}\cap I_{1}}$ and
$d=r_{v,I_{1},I}^{s-1}+1$, we have $I^{d,+}=\emptyset\neq I^{d,-}$ and there
exists $\iota\in S$ such that $\Lambda_{d}=\\{(2\\#I^{d,-}+1,\iota)\\}$ if
$I^{d}\cap I_{0}\neq\emptyset$ and $\Lambda_{d}=\\{(2n_{d}-1,\iota)\\}$ if
$I^{d}\cap I_{0}=\emptyset$.

###### Proof.

It is harmless to assume that our $\Theta$ is maximally $(I_{0},I_{1})$-atomic
(see Definition 2.12 and Definition 2.13). We start with the trivial
observation that $r_{I_{v}}-1\leq
k_{0}\leq([K:\mathbb{Q}_{p}]+1)(r_{I_{v}}-1)\leq([K:\mathbb{Q}_{p}]+1)(r_{I}-1)$,
which implies that

$\sum_{d=0}^{r_{I}}k_{d}\leq
k_{0}+\sum_{d=1}^{r_{I}}[K:\mathbb{Q}_{p}](n_{d}^{2}-1)=k_{0}+[K:\mathbb{Q}_{p}](-r_{I}+\sum_{d=1}^{r_{I}}n_{d}^{2})\\\
\leq([K:\mathbb{Q}_{p}]+1)(r_{I}-1)+[K:\mathbb{Q}_{p}](n^{2}-3r_{I}+2)=(1-2[K:\mathbb{Q}_{p}])(r_{I}-1)+[K:\mathbb{Q}_{p}](n^{2}-1)\\\
\leq\ell-n+1+[K:\mathbb{Q}_{p}](n^{2}-1).$

We have the following observations from Lemma 2.15.

  * •

If $I^{d}\cap I_{0}=\emptyset$ and there does not exist $1\leq s\leq
r_{I_{v}\cap I_{1}}$ such that $d=r_{v,I_{1},I}^{s-1}+1$, then $k_{d}\geq
2n_{d}-1$ and the equality holds if and only if
$\Lambda_{d}=\\{(2n_{d}-1,\iota)\\}$ for some $\iota\in S$.

  * •

If $I^{d}\cap I_{0}=\emptyset$ and there exists $1\leq s\leq r_{I_{v}\cap
I_{1}}$ such that $d=r_{v,I_{1},I}^{s-1}+1$, then either $n_{d}=1$ (with
$k_{d}=0$) or $k_{d}\geq 2n_{d}-1$. So we always have $k_{d}\geq 2n_{d}-2$ and
the equality holds if and only if $n_{d}=1$.

  * •

If $I^{d}\cap I_{0}\neq\emptyset$ and there does not exist $1\leq s\leq
r_{I_{v}\cap I_{1}}$ such that $d=r_{v,I_{1},I}^{s-1}+1$, then $k_{d}\geq
2\\#I^{d,-}+1\geq 2n_{d}-2\\#I^{d}\cap I_{0}-1$ and the equality holds if and
only if $I^{d}\cap I_{0}$ is an interval, $I^{d,+}=\emptyset$ and
$\Lambda_{d}=\\{(2n_{d}-2\\#I^{d}\cap I_{0}-1,\iota)\\}$ for some $\iota\in
S$. (Note that $I^{d,+}\neq\emptyset$ would force $k_{d}\geq
2n_{d}-1>2\\#I^{d,-}+1$.)

  * •

If $I^{d}\cap I_{0}\neq\emptyset$ and there exists $1\leq s\leq r_{I_{v}\cap
I_{1}}$ such that $d=r_{v,I_{1},I}^{s-1}+1$, then either
$I^{d,+}=I^{d,-}=\emptyset$ (with $k_{d}\geq 0$) or $I^{d,+}=\emptyset\neq
I^{d,-}$ (with $k_{d}\geq 2n_{d}-2n_{d}^{-}+1$) or $I^{d,+}\neq\emptyset$
(with $k_{d}\geq 2n_{d}-1$). So we always have $k_{d}\geq 2n_{d}-2\\#I^{d}\cap
I_{0}-2$ and the equality holds if and only if $I^{d,+}=I^{d,-}=\emptyset$.

Consequently, for each $1\leq s\leq r_{I_{v}\cap I_{1}}$, we have

$\sum_{d=r_{v,I_{1},I}^{s-1}+1}^{r_{v,I_{1},I}^{s}}k_{d}\geq-1+\sum_{d=r_{v,I_{1},I}^{s-1}+1}^{r_{v,I_{1},I}^{s}}(2n_{d}-2\\#I^{d}\cap
I_{0}-1).$

Summing over $1\leq s\leq r_{I_{v}\cap I_{1}}$, we have

$\sum_{d=1}^{r_{I}}k_{d}\geq-r_{I_{v}\cap I_{1}}+2n-r_{I}-2\\#I_{0},$

which together with $k_{0}\geq r_{I_{v}}-1$ implies that

$k=\sum_{d=0}^{r_{I}}k_{d}\geq r_{I_{v}}-1-r_{I_{v}\cap
I_{1}}+2n-r_{I}-2\\#I_{0}=(n-\\#I_{v})-1-(n-\\#I_{v}\cap
I_{1})+2n-(n-\ell)-2\\#I_{0}\\\ =\ell+n-1+\\#I_{v}\cap
I_{1}-\\#I_{v}-2\\#I_{0}=\ell+n-1+\\#I_{1}-\\#I_{v}\cup
I_{1}-2\\#I_{0}\geq\ell+\\#I_{1}-2\\#I_{0}.$

The precise conditions on $\Theta$ for the equality
$k=\ell+\\#I_{1}-2\\#I_{0}$ to hold are clear from the discussion above. The
case $k=\ell+\\#I_{1}-2\\#I_{0}+1$ is similar and follows from the following
observation: if $\Theta$ is a maximally $(I_{0},I_{1})$-atomic tuple such that
$\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}>2\\#I^{d,-}+1$ for some $1\leq s\leq
r_{I_{v}\cap I_{1}}$ and $r_{v,I_{1},I}^{s-1}+1\leq d\leq r_{v,I_{1},I}^{s}$
satisfying $I^{d}\cap I_{0}\neq\emptyset$, then we must have
$k\geq\ell+\\#I_{1}-2\\#I_{0}+2$. ∎

###  2.4. A combinatorial subcomplex of $E_{1,I_{0},I_{1}}^{\bullet,k}$

In this section, we construct a combinatorial subcomplex of
$E_{1,I_{0},I_{1}}^{\bullet,k}$ for each $k\geq 0$ using
$(I_{0},I_{1})$-atomic tuples. The main result here is Proposition 2.23, where
we prove that this combinatorial subcomplex is quasi-isomorphic to
$E_{1,I_{0},I_{1}}^{\bullet,k}$.

######  Definition 2.17.

We consider all sets of the form
$\mathbf{I}_{>d}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(I^{d^{\prime\prime}})_{d^{\prime\prime}>d}$
for some $I_{0}\subseteq I\subseteq I_{v}\cap I_{1}$ with $\\#I=\ell$ and
$1\leq d\leq r_{I}=n-\ell$. Then the set $\\{\mathbf{I}_{>d}\\}_{I,d}$ admits
a natural partial order described as follows. Given $\mathbf{I}_{>d}$ and
$\mathbf{I}^{\prime}_{>d^{\prime}}$ for some $(I,d)$ and
$(I^{\prime},d^{\prime})$, we say that
$\mathbf{I}^{\prime}_{>d^{\prime}}<\mathbf{I}_{>d}$ if exactly one of the
following holds

  * •

$d^{\prime}<d$ and $(I^{\prime})^{d^{\prime\prime}}=I^{d^{\prime\prime}}$ for
each $d^{\prime\prime}>d$;

  * •

there exists $d_{\flat}\geq\max\\{d,d^{\prime}\\}$ such that
$(I^{\prime})^{d_{\flat}}\subsetneq I^{d_{\flat}}$ and
$(I^{\prime})^{d^{\prime\prime}}=I^{d^{\prime\prime}}$ for each
$d^{\prime\prime}>d_{\flat}$.

It is not difficult to check that this is a well defined partial order on
$\\{\mathbf{I}_{>d}\\}_{I,d}$.

For each $(I_{0},I_{1})$-standard $\Theta$, we set
$i_{I_{0},I_{1},\Theta}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d=1}^{d_{I_{0},I_{1},\Theta}-1}n_{d}\in(I_{v}\cap
I_{1})\setminus I$ and consider
$\Theta^{+}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}p_{i_{I_{0},I_{1},\Theta}}^{+}(\Theta)$
which is clearly well defined. For each $(I_{0},I_{1})$-standard $\Theta$, we
associate the set
$\mathbf{I}_{\Theta}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}(I^{d})_{d>d_{I_{0},I_{1},\Theta}}$.

######  Lemma 2.18.

For each $v\subseteq\mathcal{B}_{n,\emptyset}$ with $\\#v=k_{0}$, the subset

$\\{d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{\Theta^{+}})\mid\Theta=(v,I,\underline{k},\underline{\Lambda})\text{
is }(I_{0},I_{1})\text{-standard },|\underline{k}|=k,\\#I=\ell\\}\subseteq
E_{1,I_{0},I_{1},v}^{-\ell,k}$

is linearly independent.

###### Proof.

Let
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
be a $(I_{0},I_{1})$-standard tuple satisfying $\Theta^{\prime}\neq\Theta$ and

$c_{\Theta^{\prime}}\left(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{\Theta^{+}})\right)\neq
0.$

Then we have the following possibilities

  * •

If $d_{I_{0},I_{1},\Theta^{\prime}}<d_{I_{0},I_{1},\Theta}-1$, then we have
$\Theta^{\prime}<_{d_{I_{0},I_{1},\Theta^{\prime}}}p_{i_{I_{0},I_{1},\Theta}^{\prime}}^{+}(p_{i_{I_{0},I_{1},\Theta}}^{-}(\Theta^{\prime}))$
with
$i_{I_{0},I_{1},\Theta}^{\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d=1}^{d_{I_{0},I_{1},\Theta}}n_{d}\in(I_{v}\cap
I_{1})\setminus I$, which contradicts the fact that $\Theta^{\prime}$ is
$(I_{0},I_{1})$-standard.

  * •

If $d_{I_{0},I_{1},\Theta^{\prime}}=d_{I_{0},I_{1},\Theta}-1$, then we have
$\mathbf{I}_{\Theta^{\prime}}=\mathbf{I}_{\Theta}\sqcup\\{(I^{\prime})^{d_{I_{0},I_{1},\Theta}}\\}$.

  * •

If $d_{I_{0},I_{1},\Theta^{\prime}}\geq d_{I_{0},I_{1},\Theta}$, then
$I^{d_{I_{0},I_{1},\Theta^{\prime}}+1}\cap I_{0}\neq\emptyset$, and
$\Theta^{+}=p_{i}^{+}(\Theta^{\prime})$ with $i\in
I^{d_{I_{0},I_{1},\Theta^{\prime}}+1,-}$. Moreover, the tuple
$\mathbf{I}_{\Theta^{\prime}}=\\{(I^{\prime})^{d}\mid
d>d_{I_{0},I_{1},\Theta^{\prime}}\\}$ satisfies
$(I^{\prime})^{d_{I_{0},I_{1},\Theta^{\prime}}+1}\subsetneq
I^{d_{I_{0},I_{1},\Theta^{\prime}}+1}$ and $(I^{\prime})^{d}=I^{d}$ for each
$d>d_{I_{0},I_{1},\Theta^{\prime}}+1$.

Hence, we always have $\mathbf{I}_{\Theta^{\prime}}<\mathbf{I}_{\Theta}$ for
partial order introduced in Definition 2.17. The proof is thus finished by
induction on this partial order. ∎

######  Lemma 2.19.

For each $v\subseteq\mathcal{B}_{n,\emptyset}$ and $0\neq x\in
E_{1,I_{0},I_{1},v}^{-\ell,k}$, there exists $x^{\prime}\in
E_{1,I_{0},I_{1},v}^{-\ell-1,k}$ such that

$c_{\Theta}(x-d_{1,I_{0},I_{1}}^{-\ell-1,k}(x^{\prime}))=0$

for each $(I_{0},I_{1})$-standard
$\Theta=(v,I,\underline{k},\underline{\Lambda})$.

###### Proof.

This follows immediately from Lemma 2.18. In fact, we can construct
$x^{\prime}$ as a linear combination of various $x_{\Theta^{+}}$ for
$(I_{0},I_{1})$-standard tuples $\Theta$, by an induction on the partial order
introduced in Definition 2.17. ∎

For each tuple $\Theta$, we define an integer

$e_{\Theta}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d=1}^{r_{I}}\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}$

which will be useful in our later induction argument. (Here we use the
convention $\max\\{m\mid(m,\iota)\in\emptyset\\}=0$.) It has the following
simple property.

######  Lemma 2.20.

Let $\Theta$ be a tuple.

  * •

If $i\in(I_{v}\cap I_{1})\setminus I$ and $p_{i}^{+}(\Theta)$ is defined, then
$e_{p_{i}^{+}(\Theta)}\leq e_{\Theta}$. Moreover,
$e_{p_{i}^{+}(\Theta)}=e_{\Theta}$ if and only if either
$\Lambda_{d}=\emptyset$ or $\Lambda_{d+1}=\emptyset$, where $1\leq d\leq
r_{I}$ satisfies $\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}=i$.

  * •

If $i\in I$ and $p_{i}^{-}(\Theta)$ is defined, then
$e_{p_{i}^{-}(\Theta)}=e_{\Theta}$.

###### Proof.

This is immediate from the definition of $p_{i}^{+}(\Theta)$ (resp.
$p_{i}^{-}(\Theta)$). We use the fact that, if
$\Lambda_{d}\cap\Lambda_{d+1}=\emptyset$, then

$\max\\{m\mid(m,\iota)\in\Lambda_{d}\sqcup\Lambda_{d+1}\\}\leq\max\\{m\mid(m,\iota)\in\Lambda_{d}\\}+\max\\{m\mid(m,\iota)\in\Lambda_{d+1}\\},$

and the equality holds if and only if either $\Lambda_{d}=\emptyset$ or
$\Lambda_{d+1}=\emptyset$. ∎

For each bidegree $(-\ell,k)$, we define
$E_{1,I_{0},I_{1},\diamond}^{-\ell,k}\subseteq E_{1,I_{0},I_{1}}^{-\ell,k}$ as
the subspace spanned by $x_{\Omega}$ where $\Omega$ runs through equivalence
classes of $(I_{0},I_{1})$-atomic tuples satisfying
$(-\ell_{\Omega},k_{\Omega})=(-\ell,k)$. As different equivalence classes of
$(I_{0},I_{1})$-atomic tuples do not intersect, it is clear that
$\\{x_{\Omega}\mid(-\ell_{\Omega},k_{\Omega})=(-\ell,k)\\}$ is a basis of
$E_{1,I_{0},I_{1},\diamond}^{-\ell,k}$, which induces a basis of
$E_{1,I_{0},I_{1},\diamond,v}^{-\ell,k}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}E_{1,I_{0},I_{1},\diamond}^{-\ell,k}\cap
E_{1,I_{0},I_{1},v}^{-\ell,k}$ for each $v\subseteq\mathcal{B}_{n,\emptyset}$.

######  Lemma 2.21.

We have
$d_{1,I_{0},I_{1}}^{-\ell,k}(E_{1,I_{0},I_{1},\diamond,v}^{-\ell,k})\subseteq
E_{1,I_{0},I_{1},\diamond,v}^{-\ell+1,k}$ for each $1\leq\ell\leq n-1$, $k\geq
0$ and $v\subseteq\mathcal{B}_{n,\emptyset}$. In particular,
$E_{1,I_{0},I_{1},\diamond,v}^{\bullet,k}$ is a subcomplex of
$E_{1,I_{0},I_{1},v}^{\bullet,k}$ for each $k\geq 0$ and
$v\subseteq\mathcal{B}_{n,\emptyset}$.

###### Proof.

It suffices to prove that $d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Omega})\in
E_{1,I_{0},I_{1},\diamond,v}^{-\ell+1,k}$ for each equivalence class $\Omega$
of $(I_{0},I_{1})$-atomic tuples satisfying
$(-\ell_{\Omega},k_{\Omega})=(-\ell,k)$. Let
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
be a tuple satisfying

$c_{\Theta^{\prime}}(\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}(x_{\Theta}))\neq
0$

for some $\Theta=(v,I,\underline{k},\underline{\Lambda})\in\Omega$ and $i\in
I$. Then we only have the following two possibilities

  * •

The tuple
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
is not $(I_{0},I_{1})$-atomic, and there exists a unique $1\leq s\leq
r_{I_{v}\cap I_{1}}$ and $r_{v,I_{1},I}^{s-1}+2\leq d\leq r_{v,I_{1},I}^{s}-1$
such that $k^{\prime}_{d}=0$ (namely $\Lambda^{\prime}_{d}=\emptyset$).
Moreover, if we write
$i_{1}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d^{\prime}=1}^{d-1}n^{\prime}_{d^{\prime}}$
and
$i_{2}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d^{\prime}=1}^{d}n^{\prime}_{d^{\prime}}$,
then
$\Theta\in\\{p_{i_{1}}^{+}(\Theta^{\prime}),p_{i_{2}}^{+}(\Theta^{\prime})\\}\subseteq\Omega$,
and moreover any $\Theta^{\prime\prime}\in\Omega$ that satisfies
$c_{\Theta^{\prime}}(\mathrm{Res}_{n,I^{\prime\prime},I^{\prime\prime}\setminus\\{i^{\prime\prime}\\}}^{k}(x_{\Theta^{\prime\prime}}))\neq
0$ (with
$\Theta^{\prime\prime}=(v,I^{\prime\prime},\underline{k}^{\prime\prime},\underline{\Lambda}^{\prime\prime})\in\Omega$
and $i^{\prime\prime}\in I^{\prime\prime}$) must be either
$p_{i_{1}}^{+}(\Theta^{\prime})$ or $p_{i_{2}}^{+}(\Theta^{\prime})$. Note
that we have

$\varepsilon(p_{i_{2}}^{+}(\Theta^{\prime}))=(-1)^{i_{2}-i_{1}}\varepsilon(p_{i_{1}}^{+}(\Theta^{\prime}))$

and

$c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{p_{i_{2}}^{+}(\Theta^{\prime})}))=(-1)^{m(I^{\prime}\sqcup\\{i_{2}\\},i_{2})-m(I^{\prime}\sqcup\\{i_{1}\\},i_{1})}c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{p_{i_{1}}^{+}(\Theta^{\prime})}))=(-1)^{i_{2}-i_{1}-1}c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{p_{i_{1}}^{+}(\Theta^{\prime})})).$

Consequently, we have

$c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Omega}))=\varepsilon(p_{i_{1}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{p_{i_{1}}^{+}(\Theta^{\prime})}))+\varepsilon(p_{i_{2}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{p_{i_{2}}^{+}(\Theta^{\prime})}))\\\
=\varepsilon(p_{i_{1}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{p_{i_{1}}^{+}(\Theta^{\prime})}))(1+(-1)^{i_{2}-i_{1}}(-1)^{i_{2}-i_{1}-1})=0.$

  * •

The tuple
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
is $(I_{0},I_{1})$-atomic and we write $\Omega^{\prime}$ for its equivalence
class. For each
$\Theta^{\prime\prime}=(v,I^{\prime\prime},\underline{k}^{\prime\prime},\underline{\Lambda}^{\prime\prime})\in\Omega^{\prime}$,
there exists a unique $i_{\Theta^{\prime\prime}}\in(I_{v}\cap I_{1})\setminus
I^{\prime\prime}$ such that
$\Theta^{\prime\prime\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}p_{i_{\Theta^{\prime\prime}}}^{+}(\Theta^{\prime\prime})=(v,I^{\prime\prime\prime},\underline{k}^{\prime\prime\prime},\underline{\Lambda}^{\prime\prime\prime})\in\Omega$
(by comparing $\underline{\Lambda}^{\prime\prime\prime}=\underline{\Lambda}$
and $\underline{\Lambda}^{\prime\prime}=\underline{\Lambda}^{\prime}$). This
gives a natural map

$\Omega^{\prime}\rightarrow\Omega:~{}\Theta^{\prime\prime}\mapsto
p_{i_{\Theta^{\prime\prime}}}^{+}(\Theta^{\prime\prime}).$

Using the basis of $E_{1,I_{0},I_{1}}^{-\ell+1,k}$ of the form
$\\{x_{\Theta^{\prime\prime}}\\}$ for various tuples $\Theta^{\prime\prime}$,
we define $d_{1,I_{0},I_{1},\Omega^{\prime}}^{-\ell,k}$ as the composition of
$d_{1,I_{0},I_{1}}^{-\ell,k}$ with the projection to the subspace spanned by
$\\{x_{\Theta^{\prime\prime}}\mid\Theta^{\prime\prime}\in\Omega^{\prime}\\}$.
We claim that there exists $\varepsilon(\Omega,\Omega^{\prime})\in\\{1,-1\\}$
such that
$d_{1,I_{0},I_{1},\Omega^{\prime}}^{-\ell,k}(x_{\Omega})=\varepsilon(\Omega,\Omega^{\prime})x_{\Omega^{\prime}}$.
In fact, if we pick up a $\Theta^{\prime\prime}\in\Omega^{\prime}$ (which
determines a
$\Theta^{\prime\prime\prime}=p_{i_{\Theta^{\prime\prime}}}^{+}(\Theta^{\prime\prime})\in\Omega$)
as above, it suffices to show that
$\varepsilon(\Theta^{\prime\prime})\varepsilon(\Theta^{\prime\prime\prime})c_{\Theta^{\prime\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Theta^{\prime\prime\prime}}))$
is independent of the choice of $\Theta^{\prime\prime}\in\Omega^{\prime}$. It
is clear that
$\varepsilon(\Theta^{\prime\prime})\varepsilon(\Theta^{\prime\prime\prime})=(-1)^{i_{\Theta^{\prime\prime}}}$
and

$c_{\Theta^{\prime\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Theta^{\prime\prime\prime}}))=(-1)^{m(I^{\prime\prime\prime},i_{\Theta^{\prime\prime}})}c_{\Theta^{\prime\prime}}(\mathrm{Res}_{n,I^{\prime\prime\prime},I^{\prime\prime}}^{k}(x_{\Theta^{\prime\prime\prime}})).$

Then we observe that both
$i_{\Theta^{\prime\prime}}-m(I^{\prime\prime\prime},i_{\Theta^{\prime\prime}})$
and
$c_{\Theta^{\prime\prime}}(\mathrm{Res}_{n,I^{\prime\prime\prime},I^{\prime\prime}\\}}^{k}(x_{\Theta^{\prime\prime\prime}}))$
depend only on $\underline{\Lambda}^{\prime\prime}$ and
$\underline{\Lambda}^{\prime\prime\prime}$, so does
$\varepsilon(\Theta^{\prime\prime})\varepsilon(\Theta^{\prime\prime\prime})c_{\Theta^{\prime\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Theta^{\prime\prime\prime}}))$.
As all different choices of $\Theta^{\prime\prime}\in\Omega^{\prime}$ share
the same $\underline{\Lambda}^{\prime\prime}$ and
$\underline{\Lambda}^{\prime\prime\prime}$,
$\varepsilon(\Omega,\Omega^{\prime})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\varepsilon(\Theta^{\prime\prime})\varepsilon(\Theta^{\prime\prime\prime})c_{\Theta^{\prime\prime}}(d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Theta^{\prime\prime\prime}}))$
is well defined.

As a summary, we deduce that

(2.11)
$d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Omega})=\sum_{\Omega^{\prime}}\varepsilon(\Omega,\Omega^{\prime})x_{\Omega^{\prime}}\in
E_{1,I_{0},I_{1},\diamond,v}^{-\ell+1,k}$

where $\Omega^{\prime}$ runs through all equivalence classes of
$(I_{0},I_{1})$-atomic tuples that appear in the second possibility above. The
proof is thus finished. ∎

######  Lemma 2.22.

Let $v\subseteq\mathcal{B}_{n,\emptyset}$ and $x\in
E_{1,I_{0},I_{1},v}^{-\ell,k}$ such that $d_{1,I_{0},I_{1}}^{-\ell,k}(x)\in
E_{1,I_{0},I_{1},\diamond,v}^{-\ell+1,k}$ and $c_{\Theta}(x)=0$ for each
$\Theta=(v,I,\underline{k},\underline{\Lambda})$ which is either
$(I_{0},I_{1})$-standard or maximally $(I_{0},I_{1})$-atomic. Then $x=0$.

###### Proof.

For each tuple $\Theta=(v,I,\underline{k},\underline{\Lambda})$, we define
$d_{I_{0},I_{1},\Theta}$ as the maximal integer satisfying
$k_{d_{I_{0},I_{1},\Theta}}=0$ and $r_{v,I_{1},I}^{s-1}+2\leq
d_{I_{0},I_{1},\Theta}\leq r_{v,I_{1},I}^{s}$ for some $1\leq s\leq
r_{I_{v}\cap I_{1}}$ if it exists, and as $1$ otherwise. This clearly extends
the definition of $d_{I_{0},I_{1},\Theta}$ when $\Theta$ is
$(I_{0},I_{1})$-standard. We assume inductively that
$c_{\Theta^{\prime}}(x)=0$ for all $\Theta^{\prime}$ satisfying either
$\Theta<_{d_{I_{0},I_{1},\Theta}}\Theta^{\prime}$ or
$e_{\Theta^{\prime}}<e_{\Theta}.$

It is harmless to assume that $\Theta$ is neither $(I_{0},I_{1})$-standard nor
maximally $(I_{0},I_{1})$-atomic. Hence, there exists $s\leq s_{1}\leq
r_{I_{v}\cap I_{1}}$ and
$\max\\{d_{I_{0},I_{1},\Theta},r_{v,I_{1},I}^{s_{1}-1}+1\\}\leq d_{1}\leq
r_{v,I_{1},I}^{s_{1}}$ and $i_{1}\in I^{d_{1}}$ such that
$p_{i_{1}}^{-}(\Theta)$ is defined. We choose $d_{1}$ and $i_{1}$ to be
maximal possible, and thus $p_{i_{1}}^{-}(\Theta)$ is
$(I_{0},I_{1})$-standard. If $c_{\Theta}(x)\neq 0$, then there exists
$i_{2}\in(I_{v}\cap I_{1})\setminus I$ such that
$c_{p_{i_{2}}^{+}p_{i_{1}}^{-}(\Theta)}(x)\neq 0$. If $i_{2}<i_{1}$, then
$p_{i_{2}}^{+}p_{i_{1}}^{-}(\Theta)$ is $(I_{0},I_{1})$-standard and thus
contradicts our assumption. If
$i_{2}>\sum_{d^{\prime}=1}^{d_{1}}n_{d^{\prime}}$, then as
$\Lambda_{d^{\prime}}\neq\emptyset$ for each
$d^{\prime}>d_{I_{0},I_{1},\Theta}$, we deduce from Lemma 2.20 that
$e_{p_{i_{2}}^{+}p_{i_{1}}^{-}(\Theta)}<e_{\Theta}$, which contradicts our
inductive assumption. Hence, we must have
$i_{2}=\sum_{d^{\prime}=1}^{d_{1}}n_{d^{\prime}}$ and thus
$\Theta<_{d_{I_{0},I_{1},\Theta}}p_{i_{2}}^{+}p_{i_{1}}^{-}(\Theta)$, another
contradiction. The proof is thus finished. ∎

######  Proposition 2.23.

For each $v\subseteq\mathcal{B}_{n,\emptyset}$ and $k\geq 0$, the subcomplex

$E_{1,I_{0},I_{1},\diamond,v}^{\bullet,k}\rightarrow
E_{1,I_{0},I_{1},v}^{\bullet,k}$

induces an quasi-isomorphism.

###### Proof.

We fix an $\\#I_{0}\leq\ell\leq\\#I_{1}$ and $k\geq 0$.

We first show that the induced map on cohomology is surjective, namely for
each $x\in E_{1,I_{0},I_{1},v}^{-\ell,k}$ satisfying
$d_{1,I_{0},I_{1},v}^{-\ell,k}(x)=0$, there exists $x^{\prime}\in
E_{1,I_{0},I_{1},v}^{-\ell-1,k}$ and $x^{\prime\prime}\in
E_{1,I_{0},I_{1},\diamond,v}^{-\ell,k}$ such that
$x=x^{\prime\prime}+d_{1,I_{0},I_{1}}^{-\ell-1,k}(x^{\prime})$. In fact, given
such $x$, we can choose $x^{\prime}$ as in Lemma 2.19, and then take

$x^{\prime\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{\Omega}c_{\Theta}(x-d_{1,I_{0},I_{1}}^{-\ell-1,k}(x^{\prime}))\varepsilon(\Theta)x_{\Omega}\in
E_{1,I_{0},I_{1},\diamond,v}^{-\ell,k}$

where $\Omega$ runs through all equivalence classes of $(I_{0},I_{1})$-atomic
tuples with the fixed bidegree $(-\ell,k)$ and $v$ as above, and
$\Theta\in\Omega$ is maximally $(I_{0},I_{1})$-atomic with sign
$\varepsilon(\Theta)\in\\{1,-1\\}$ as in Definition 2.13. The equality
$x=x^{\prime\prime}+d_{1,I_{0},I_{1}}^{-\ell-1,k}(x^{\prime})$ then follows
from Lemma 2.22.

Now we show that the induced map on cohomology is injective, namely for each
$y\in E_{1,I_{0},I_{1},\diamond,v}^{-\ell,k}$, and $y^{\prime}\in
E_{1,I_{0},I_{1},v}^{-\ell-1,k}$ satisfying
$y=d_{1,I_{0},I_{1}}^{-\ell-1,k}(y^{\prime})$, there exists
$y^{\prime\prime}\in E_{1,I_{0},I_{1},\diamond,v}^{-\ell-1,k}$ such that
$y=d_{1,I_{0},I_{1}}^{-\ell-1,k}(y^{\prime\prime})$. By Lemma 2.19 we may
assume that $c_{\Theta}(y^{\prime})=0$ for each $(I_{0},I_{1})$-standard
$\Theta$. Then we take

$y^{\prime\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{\Omega}c_{\Theta}(y^{\prime})\varepsilon(\Theta)x_{\Omega}\in
E_{1,I_{0},I_{1},\diamond,v}^{-\ell-1,k}$

where $\Omega$ runs through all equivalence classes of $(I_{0},I_{1})$-atomic
tuples with the fixed bidegree $(-\ell-1,k)$ and $v$ as above and
$\Theta\in\Omega$ is maximally $(I_{0},I_{1})$-atomic with sign
$\varepsilon(\Theta)\in\\{1,-1\\}$ as in Definition 2.13. Then we have

$d_{1,I_{0},I_{1}}^{-\ell-1,k}(y^{\prime}-y^{\prime\prime})=y-d_{1,I_{0},I_{1}}^{-\ell-1,k}(y^{\prime\prime})\in
E_{1,I_{0},I_{1},\diamond,v}^{-\ell,k},$

which together with Lemma 2.22 (with bidegree $(-\ell,k)$ there replaced with
$(-\ell-1,k)$) implies that $y^{\prime}-y^{\prime\prime}=0$ and thus
$y=d_{1,I_{0},I_{1}}^{-\ell-1,k}(y^{\prime\prime})$. The proof is thus
finished. ∎

######  Remark 2.24.

If $I_{0}=I_{0}^{1}$ (namely $I_{0}=\\{1,\dots,\\#I_{0}\\}$), then each
$(I_{0},I_{1})$-atomic tuple is automatically maximal and thus each
equivalence class of $(I_{0},I_{1})$-atomic tuples contains exactly one
element. Moreover, one can check that
$d_{1,I_{0},I_{1}}^{-\ell,k}(E_{1,I_{0},I_{1},\diamond}^{-\ell,k})=0$ and thus
$E_{1,I_{0},I_{1},\diamond}^{-\ell,k}\cong E_{2,I_{0},I_{1}}^{-\ell,k}$ for
each bidegree $(-\ell,k)$ in this case.

###  2.5. Computations in bottom two degrees

In this section, we finish the computation of
$\mathrm{Ext}_{G_{n},\lambda}^{h}(F_{n,\Delta_{n}}(\lambda),\mathbf{C}_{I_{0},I_{1}}(\lambda))$
when $h\in\\{\\#I_{1}-2\\#I_{0},\\#I_{1}-2\\#I_{0}+1\\}$.

######  Lemma 2.25.

Let $\Omega$ be an equivalence class of $(I_{0},I_{1})$-atomic tuples with
$(-\ell_{\Omega},k_{\Omega})=(-\ell,k)$.

  1. (i)

If $k=\ell+\\#I_{1}-2\\#I_{0}$, then
$d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Omega})=0$.

  2. (ii)

If $k=\ell+\\#I_{1}-2\\#I_{0}+1$, then
$d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Omega})\neq 0$ if and only if $I_{v}\cup
I_{1}=\Delta_{n}$ and there exists a unique $1\leq s\leq r_{I_{v}\cap I_{1}}$
satisfying $\Lambda_{r_{v,I_{1},I}^{s-1}+1}\neq\emptyset$ and $I^{d}\cap
I_{0}\neq\emptyset$ for some $r_{v,I_{1},I}^{s-1}+1\leq d\leq
r_{v,I_{1},I}^{s}$. Moreover, if $d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Omega})\neq
0$, then there exists a unique equivalence class $\Omega^{\prime}$ of
$(I_{0},I_{1})$-atomic tuples such that
$d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Omega})=\pm x_{\Omega^{\prime}}$, and
$\Omega^{\prime}$ also determines $\Omega$ uniquely.

###### Proof.

Let $\Omega^{\prime}$ be an arbitrary equivalence class of
$(I_{0},I_{1})$-atomic tuples that appears in (2.11), then it is clear that
$k_{\Omega^{\prime}}=k_{\Omega}$ and $\ell_{\Omega^{\prime}}=\ell_{\Omega}-1$.
By the choice of $\Omega^{\prime}$ (see the second possibility of Lemma 2.21),
there exists $\Theta=(v,I,\underline{k},\underline{\Lambda})\in\Omega$,
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})\in\Omega^{\prime}$
and $i\in I\setminus I^{\prime}$ such that
$\Theta=p_{i}^{+}(\Theta^{\prime})$. We note from Lemma 2.16 that
$\\#\Lambda_{d}\leq 1$ for each $1\leq d\leq r_{I}$. If
$k=\ell+\\#I_{1}-2\\#I_{0}$, then there must exist $1\leq s\leq r_{I_{v}\cap
I_{1}}$ and $r_{v,I_{1},I^{\prime}}^{s-1}+2\leq d^{\prime}\leq
r_{v,I_{1},I^{\prime}}^{s}$ such that
$\Lambda^{\prime}_{d^{\prime}}=\emptyset$, which contradicts the fact that
$\Theta^{\prime}$ is $(I_{0},I_{1})$-atomic. Item (ii) thus follows. Similar
argument works for the case $k=\ell+\\#I_{1}-2\\#I_{0}+1$ and either
$\\#I_{v}\cup I_{1}=n-2$ or there exists a unique $i\in\Delta_{n}\setminus
I_{v}$ such that $\\#v\cap\mathcal{B}_{n,\Delta_{n}\setminus\\{i\\}}\geq 2$.
It suffices to treat the case $k=\ell+\\#I_{1}-2\\#I_{0}+1$, $I_{v}\cup
I_{1}=\Delta_{n}$ and $k_{0}=\\#\Delta_{n}\setminus I_{v}$, and there exists a
unique $1\leq s\leq r_{I_{v}\cap I_{1}}$ such that
$\Lambda_{r_{v,I_{1},I}^{s-1}+1}\neq\emptyset$. If $I^{d}\cap I_{0}=\emptyset$
for all $r_{v,I_{1},I}^{s-1}+1\leq d\leq r_{v,I_{1},I}^{s}$, the same argument
as in item (ii) proves that $\Omega^{\prime}$ cannot exist. If $I^{d}\cap
I_{0}\neq\emptyset$ for some $r_{v,I_{1},I}^{s-1}+1\leq d\leq
r_{v,I_{1},I}^{s}$, then we have $I^{d,-}\neq\emptyset$ and
$\Lambda_{d}=\\{(2\\#I^{d,-}+1,\iota)\\}$ (for some $\iota\in S$) for each
such $d$, and $\Omega^{\prime}$ is characterized by the following conditions

  * •

$r_{v,I_{1},I^{\prime}}^{s^{\prime}}=r_{v,I_{1},I}^{s^{\prime}}$ for each
$1\leq s^{\prime}\leq s-1$ and
$r_{v,I_{1},I^{\prime}}^{s^{\prime}}=r_{v,I_{1},I}^{s^{\prime}}+1$ for each
$s\leq s^{\prime}\leq r_{I_{v}\cap I_{1}}$;

  * •

$\Lambda^{\prime}_{d^{\prime}}=\Lambda_{d^{\prime}}$ for each $1\leq
d^{\prime}\leq r_{v,I_{1},I^{\prime}}^{s-1}$,
$\Lambda^{\prime}_{r_{v,I_{1},I^{\prime}}^{s-1}+1}=\emptyset$ and
$\Lambda^{\prime}_{d^{\prime}}=\Lambda_{d^{\prime}-1}$ for each
$r_{v,I_{1},I^{\prime}}^{s-1}+2\leq d^{\prime}\leq r_{I^{\prime}}$.

It is then clear that $\Omega$ and $\Omega^{\prime}$ uniquely determine each
other, and thus item (iii) follows. ∎

Let $\\#I_{0}\leq\ell\leq\\#I_{1}$ be an integer. We write
$\Psi_{I_{0},I_{1}}^{-\ell,k}$ for the set of equivalence classes $\Omega$ of
$(I_{0},I_{1})$-atomic tuples satisfying
$(-\ell_{\Omega},k_{\Omega})=(-\ell,\ell+\\#I_{1}-2\\#I_{0})$ if
$k=\ell+\\#I_{1}-2\\#I_{0}$. We write $\Psi_{I_{0},I_{1}}^{-\ell,k}$ for the
set of equivalence classes $\Omega$ of $(I_{0},I_{1})$-atomic tuples
satisfying $(-\ell_{\Omega},k_{\Omega})=(-\ell,\ell+\\#I_{1}-2\\#I_{0}+1)$ and
$d_{1,I_{0},I_{1}}^{-\ell,k}(x_{\Omega})=0$ (see item (iii) of Lemma 2.25) if
$k=\ell+\\#I_{1}-2\\#I_{0}+1$.

Recall that
$E_{2,I_{0},I_{1}}^{-\ell,k}=\mathrm{ker}(d_{1,I_{0},I_{1}}^{-\ell,k})/\mathrm{im}(d_{1,I_{0},I_{1}}^{-\ell-1,k})$
for each $\\#I_{0}\leq\ell\leq\\#I_{1}$ and $k\geq 0$.

######  Proposition 2.26.

For each $\\#I_{0}\leq\ell\leq\\#I_{1}$, the subset

(2.12)
$\\{x_{\Omega}\\}_{\Omega\in\Psi_{I_{0},I_{1}}^{-\ell,k}}\subseteq\mathrm{ker}(d_{1,I_{0},I_{1}}^{-\ell,k})$

induces a basis of $E_{2,I_{0},I_{1}}^{-\ell,k}$ if
$k\in\\{\ell+\\#I_{1}-2\\#I_{0},\ell+\\#I_{1}-2\\#I_{0}+1\\}$.

###### Proof.

This follows clearly from Proposition 2.23 and Lemma 2.25. Note that
$E_{1,I_{0},I_{1},\diamond}^{-\ell,\ell+\\#I_{1}-2\\#I_{0}-1}=0$ by Lemma
2.16. ∎

######  Lemma 2.27.

The spectral sequence $\\{E_{r,I_{0},I_{1}}^{-\ell,k}\\}_{r\geq
1,\\#I_{0}\leq\ell\leq\\#I_{1},k\geq 0}$ degenerates at the second page.

###### Proof.

We set
$\mathcal{B}_{n,\emptyset}^{\infty}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\\{\mathrm{val}_{i}\mid
i\in\Delta_{n}\\}$ and there exists a natural map between power sets
$2^{\mathcal{B}_{n,\emptyset}}\rightarrow
2^{\mathcal{B}_{n,\emptyset}^{\infty}}$ by mapping $v$ to the intersection
$v\cap\mathcal{B}_{n,\emptyset}^{\infty}$. It is clear that we have a
decomposition

$E_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}\cong\bigoplus_{v\subseteq\mathcal{B}_{n,\emptyset}}E_{\bullet,I_{0},I_{1},v}^{\bullet,\bullet}$

into its $v$-isotypic components and we set

${}_{v^{\infty}}E_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}\cong\bigoplus_{v\subseteq\mathcal{B}_{n,\emptyset},v\cap\mathcal{B}_{n,\emptyset}^{\infty}=v^{\infty}}E_{\bullet,I_{0},I_{1},v}^{\bullet,\bullet}$

with differential ${}_{v^{\infty}}d_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}$.
Hence, it suffices to show that
${}_{v^{\infty}}d_{2,I_{0},I_{1}}^{\bullet,\bullet}=0$. The spectral sequence
${}_{v^{\infty}}E_{\bullet,I_{0},I_{1}}^{\bullet,\bullet}$ actually arises
from the double complex ${}_{v^{\infty}}E_{0,I_{0},I_{1}}^{\bullet,\bullet}$
with

${}_{v^{\infty}}E_{0,I_{0},I_{1}}^{-\ell,k}\cong\bigoplus_{I_{0}\subseteq
I\subseteq I_{v^{\infty}}\cap
I_{1},\\#I=\ell}\wedge^{k}\overline{\mathfrak{l}}_{n,I}^{\ast}$

with the column complexes being direct sum of Koszul complexes for various
Levi, and differential of row complexes being direct sum of

$(-1)^{m(I,i)}\mathrm{Res}_{n,I,I\setminus\\{i\\}}^{k}:\wedge^{k}\overline{\mathfrak{l}}_{n,I}^{\ast}\rightarrow\wedge^{k}\overline{\mathfrak{l}}_{n,I\setminus\\{i\\}}^{\ast}$

with $I_{0}\subseteq I\subseteq I_{v^{\infty}}\cap I_{1}$ and $i\in I\setminus
I_{0}$. Following Section 14 of [BT82] we write $\delta$ for the row
differential and $D$ for the column differential of the double complex
${}_{v^{\infty}}E_{0,I_{0},I_{1}}^{\bullet,\bullet}$. Note that
${}_{v^{\infty}}E_{0,I_{0},I_{1}}^{\bullet,\bullet}$ contains a sub double
complex ${}_{v^{\infty},\natural}E_{0,I_{0},I_{1}}^{\bullet,\bullet}$ with

${}_{v^{\infty},\natural}E_{0,I_{0},I_{1}}^{-\ell,k}\cong\bigoplus_{I_{0}\subseteq
I\subseteq I_{v^{\infty}}\cap
I_{1},\\#I=\ell}(\wedge^{k}\overline{\mathfrak{l}}_{n,I}^{\ast})^{\overline{\mathfrak{l}}_{n,I}}.$

Note that $D$ restricts to zero on
${}_{v^{\infty},\natural}E_{0,I_{0},I_{1}}^{\bullet,\bullet}$ and in fact the
embedding
${}_{v^{\infty},\natural}E_{0,I_{0},I_{1}}^{-\ell,\bullet}\hookrightarrow_{v^{\infty}}E_{0,I_{0},I_{1}}^{-\ell,\bullet}$
is a quasi-isomorphism for each $\\#I_{0}\leq\ell\leq\\#I_{1}$. Then we
observe that $x_{\Omega}\in_{v^{\infty}}E_{2,I_{0},I_{1}}^{-\ell,k}$ (for each
equivalence class $\Omega$ of $(I_{0},I_{1})$-atomic tuple that show up) can
be lifted to an element of
$\widetilde{x}_{\Omega}\in~{}_{v^{\infty},\natural}E_{0,I_{0},I_{1}}^{-\ell,k}\subseteq~{}_{v^{\infty}}E_{0,I_{0},I_{1}}^{-\ell,k}$
satisfying $D(\widetilde{x}_{\Omega})=0$ and
$\delta(\widetilde{x}_{\Omega})=0$, which implies that
${}_{v^{\infty}}d_{2,I_{0},I_{1}}^{-\ell,k}(x_{\Omega})=0$ (using the
description of ${}_{v^{\infty}}d_{2,I_{0},I_{1}}^{-\ell,k}$ on page 162 of
[BT82]). The proof is thus finished. ∎

######  Theorem 2.28.

Let $I_{0}\subseteq I_{1}\subseteq\Delta_{n}$ be a pair of subsets.

  * •

If
$\mathrm{Ext}_{G_{n},\lambda}^{h}(F_{n,\Delta_{n}}(\lambda),\mathbf{C}_{I_{0},I_{1}}(\lambda))\neq
0$, then $\\#I_{1}-2\\#I_{0}\leq h\leq n^{2}-n$.

  * •

For each $h\in\\{\\#I_{1}-2\\#I_{0},\\#I_{1}-2\\#I_{0}+1\\}$, the space
$M_{I_{0},I_{1}}^{h}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\mathrm{Ext}_{G_{n},\lambda}^{h}(F_{n,\Delta_{n}}(\lambda),\mathbf{C}_{I_{0},I_{1}}(\lambda))$
admits a canonical decreasing filtration

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

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

###### Proof.

This follows directly from Proposition 2.26 and Lemma 2.27. ∎

###  2.6. Twisted $(I_{0},I_{1})$-atomic tuples

In this section, we consider a variant of $(I_{0},I_{1})$-atomic tuple as in
Definition 2.13. Let $\Theta=(v,I,\underline{k},\underline{\Lambda})$ be a
maximally $(I_{0},I_{1})$-atomic tuple and $\Omega$ be its equivalence class.
We consider an integer $1\leq s_{0}\leq r_{I_{v}\cap I_{1}}$ that satisfies

######  Condition 2.29.

  * •

$I^{r_{v,I_{1},I}^{s_{0}-1}+1}\subseteq I_{0}$ and
$\Lambda_{r_{v,I_{1},I}^{s_{0}-1}+1}=\emptyset$;

  * •

$\Lambda_{d}=\\{(2n_{d}-1,\iota_{d})\\}$ for some $\iota_{d}\in S$, for each
$r_{v,I_{1},I}^{s_{0}-1}+2\leq d\leq r_{v,I_{1},I}^{s_{0}}$ satisfying
$I^{d}\cap I_{0}=\emptyset$;

  * •

$I^{d,+}=\emptyset$, $I^{d,-}\neq\emptyset$ and
$\Lambda_{d}=\\{(2\\#I^{d,-}+1,\iota_{d})\\}$ for some $\iota_{d}\in S$, for
each $r_{v,I_{1},I}^{s_{0}-1}+2\leq d\leq r_{v,I_{1},I}^{s_{0}}$ satisfying
$I^{d}\cap I_{0}\neq\emptyset$.

Then we choose a $r_{v,I_{1},I}^{s_{0}-1}+1\leq d_{0}\leq
r_{v,I_{1},I}^{s_{0}}$ and write
$i_{d}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}$
and
$i_{d}^{\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\max\\{i\in
I\setminus I_{0}\mid i<i_{d}\\}$ for each $r_{v,I_{1},I}^{s_{0}-1}+1\leq d\leq
d_{0}$. Then we set

$\Theta^{s_{0},d_{0}}=(v,I^{s_{0},d_{0}},\underline{k}^{s_{0},d_{0}},\underline{\Lambda}^{s_{0},d_{0}})\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}p_{i_{d_{0}}^{\prime}}^{-}p_{i_{d_{0}-1}}^{+}\cdots
p_{i_{2}^{\prime}}^{-}p_{i_{1}}^{+}(\Theta)$

and call it the _$(s_{0},d_{0})$ -twist of $\Theta$_. Note that
$\Theta^{s_{0},d_{0}}$ is well-defined thanks to Condition 2.29 and satisfies
$\Lambda^{s_{0},d_{0}}_{d}=\Lambda_{d+1}$ for each
$r_{v,I_{1},I}^{s_{0}-1}+1\leq d\leq d_{0}-1$,
$\Lambda^{s_{0},d_{0}}_{d_{0}}=\emptyset$ and
$\Lambda^{s_{0},d_{0}}_{d}=\Lambda_{d}$ for each $d_{0}+1\leq d\leq
r_{v,I_{1},I}^{s_{0}}$. In particular, $\Theta^{s_{0},d_{0}}$ satisfies

######  Condition 2.30.

  * •

$(I^{s_{0},d_{0}})^{d_{0}}\subseteq I_{0}$ and $\Lambda_{d_{0}}=\emptyset$;

  * •

$\Lambda_{d}=\\{(2n_{d}-1,\iota_{d})\\}$ for some $\iota_{d}\in S$, for each
$r_{v,I_{1},I}^{s_{0}-1}+1\leq d\leq r_{v,I_{1},I}^{s_{0}}$ satisfying $d\neq
d_{0}$ and $I^{d}\cap I_{0}=\emptyset$;

  * •

$I^{d,+}=\emptyset$, $I^{d,-}\neq\emptyset$ and
$\Lambda_{d}=\\{(2\\#I^{d,-}+1,\iota_{d})\\}$ for some $\iota_{d}\in S$, for
each $d_{0}+1\leq d\leq r_{v,I_{1},I}^{s_{0}}$ satisfying $I^{d}\cap
I_{0}\neq\emptyset$.

  * •

$I^{d,-}=\emptyset$, $I^{d,+}\neq\emptyset$ and
$\Lambda_{d}=\\{(2\\#I^{d,+}+1,\iota_{d})\\}$ for some $\iota_{d}\in S$, for
each $r_{v,I_{1},I}^{s_{0}-1}+1\leq d\leq d_{0}-1$ satisfying $I^{d}\cap
I_{0}\neq\emptyset$.

Given two tuples
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
and
$\Theta^{\prime\prime}=(v,I^{\prime\prime},\underline{k}^{\prime\prime},\underline{\Lambda}^{\prime\prime})$,
we say that $\Theta^{\prime\prime}$ is a _$(s_{0},d_{0})$ -twisted
improvement_ of $\Theta^{\prime}$ of level $d$ if one of the following holds

  * •

$\Theta^{\prime\prime}$ is an improvement of $\Theta^{\prime}$ of level $d$
with $d\leq r_{v,I_{1},I^{\prime}}^{s_{0}-1}$ or $d\geq
r_{v,I_{1},I^{\prime}}^{s_{0}}+1$ (see Definition 2.12);

  * •

$d\geq d_{0}$ and there exists $i\in(I^{\prime})^{d}\setminus I_{0}$ and
$i^{\prime}=\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}\in(I_{v}\cap I_{1})\setminus
I^{\prime}$ such that
$i\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$ and
$\Theta^{\prime\prime}=p_{i^{\prime}}^{+}p_{i}^{-}(\Theta^{\prime})$;

  * •

$d\leq d_{0}-1$ and there exists $i\in(I^{\prime\prime})^{d}\setminus I_{0}$
and $i^{\prime}=\sum_{d^{\prime}=1}^{d}n_{d^{\prime}}\in(I_{v}\cap
I_{1})\setminus I^{\prime\prime}$ such that
$i^{\prime}\leq\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}}_{d^{\prime}}$ and
$\Theta^{\prime}=p_{i^{\prime}}^{+}p_{i}^{-}(\Theta^{\prime\prime})$.

Similar to Definition 2.12, we can use the notion $(s_{0},d_{0})$-twisted
improvement to define _$(s_{0},d_{0})$ -twisted smaller_ and _$(s_{0},d_{0})$
-twisted equivalent_. Then it is easy to see that $\Theta^{s_{0},d_{0}}$ is
$(s_{0},d_{0})$-twisted smaller than no other tuple, and thus is _maximally
$(s_{0},d_{0})$-twisted $(I_{0},I_{1})$-atomic_. Then a tuple is
_$(s_{0},d_{0})$ -twisted $(I_{0},I_{1})$-atomic_ if it is
$(s_{0},d_{0})$-twisted smaller than some tuple of the form
$\Theta^{s_{0},d_{0}}$. Given $\Omega$ above, we write $\Omega^{s_{0},d_{0}}$
for the $(s_{0},d_{0})$-twisted equivalence class of $\Theta^{s_{0},d_{0}}$,
and call it the $(s_{0},d_{0})$-twist of $\Omega$. Finally, we set
$x_{\Omega^{s_{0},d_{0}}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{\Theta\in\Omega^{s_{0},d_{0}}}\varepsilon(\Theta)x_{\Theta}$.

######  Lemma 2.31.

Let
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
which has the same bidegree $(-\ell,k)$ as $\Theta$. Assume that $\Theta$
satisfies Condition 2.29 and $\Theta^{\prime}$ satisfies

  * •

$r_{v,I_{1},I^{\prime}}^{s}=r_{v,I_{1},I}^{s}$ for each $1\leq s\leq
r_{I_{v}\cap I_{1}}$;

  * •

$(I^{\prime})^{d}=I^{d}$ for each $1\leq d\leq r_{I^{\prime}}$ satisfying
either $d\leq r_{v,I_{1},I^{\prime}}^{s_{0}-1}$ or $d\geq
r_{v,I_{1},I^{\prime}}^{s_{0}}+1$.

Then $\Theta^{\prime}\in\Omega^{s_{0},d_{0}}$ if and only if the following
holds

  1. (i)

$\underline{\Lambda}^{\prime}=\underline{\Lambda}^{s_{0},d_{0}}$;

  2. (ii)

$n^{\prime}_{r_{v,I_{1},I^{\prime}}^{s_{0}-1}+1}\leq
n^{s_{0},d_{0}}_{r_{v,I_{1},I^{\prime}}^{s_{0}-1}+1}$ if $d_{0}\geq
r_{v,I_{1},I^{\prime}}^{s_{0}-1}+2$, and
$n^{\prime}_{r_{v,I_{1},I^{\prime}}^{s_{0}}}\leq
n^{s_{0},d_{0}}_{r_{v,I_{1},I^{\prime}}^{s_{0}}}$ if $d_{0}\leq
r_{v,I_{1},I^{\prime}}^{s_{0}}-1$;

  3. (iii)

$\sum_{d^{\prime}=1}^{d_{0}-1}n^{\prime}_{d^{\prime}}\leq\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}}_{d^{\prime}}$
and
$\sum_{d^{\prime}=1}^{d_{0}}n^{\prime}_{d^{\prime}}\geq\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$.

###### Proof.

The ‘only if’ part follows immediately from the definition of a
$(s_{0},d_{0})$-twisted improvement, and it suffices to prove the ‘if’ part.
It is harmless to replace $\Theta^{\prime}$ with a $(s_{0},d_{0})$-twisted
maximal tuple in its $(s_{0},d_{0})$-twisted equivalence class, and we need to
show that $\Theta^{\prime}=\Theta^{s_{0},d_{0}}$. Item (iii) (which implies
$(I^{\prime})^{d_{0}}\supseteq(I^{s_{0},d_{0}})^{d_{0}}$) together with
$\Theta^{\prime}$ being $(s_{0},d_{0})$-twisted maximal forces
$\sum_{d^{\prime}=1}^{d_{0}-1}n^{\prime}_{d^{\prime}}=\sum_{d^{\prime}=1}^{d_{0}-1}n^{s_{0},d_{0}}_{d^{\prime}}$,
$\sum_{d^{\prime}=1}^{d_{0}}n^{\prime}_{d^{\prime}}=\sum_{d^{\prime}=1}^{d_{0}}n^{s_{0},d_{0}}_{d^{\prime}}$
and in particular $(I^{\prime})^{d_{0}}=(I^{s_{0},d_{0}})^{d_{0}}\subseteq
I_{0}$. The fact that $\Theta^{\prime}$ is $(s_{0},d_{0})$-twisted maximal
also implies that $\max\\{m\mid(m,\iota)\in\Lambda^{\prime}_{d}\\}\geq
2n^{\prime}_{d}-2\\#(I^{\prime})^{d}\cap I_{0}-1$ for each
$r_{v,I_{1},I^{\prime}}^{s_{0}-1}+2\leq d\leq
r_{v,I_{1},I^{\prime}}^{s_{0}}-1$ satisfying $d\neq d_{0}$, using arguments in
the proof Lemma 2.16. Using item (i) and Condition 2.30, we have

(2.13) $n^{\prime}_{d}\leq\\#(I^{\prime})^{d}\cap
I_{0}+\frac{1}{2}(1+\max\\{m\mid(m,\iota)\in\Lambda^{\prime}_{d}\\})=n^{s_{0},d_{0}}-\\#(I^{s_{0},d_{0}})^{d}\cap
I_{0}+\\#(I^{\prime})^{d}\cap I_{0}$

for each $r_{v,I_{1},I^{\prime}}^{s_{0}-1}+2\leq d\leq
r_{v,I_{1},I^{\prime}}^{s_{0}}-1$ satisfying $d\neq d_{0}$. Item (ii) implies
that (2.13) holds for
$d\in\\{r_{v,I_{1},I^{\prime}}^{s_{0}-1}+1,r_{v,I_{1},I^{\prime}}^{s_{0}}\\}\setminus\\{d_{0}\\}$
as well. Sum up all the inequalities for
$r_{v,I_{1},I^{\prime}}^{s_{0}-1}+1\leq d\leq r_{v,I_{1},I^{\prime}}^{s_{0}}$
(with $n^{\prime}_{d_{0}}=n^{s_{0},d_{0}}_{d_{0}}$), we obtain
$\sum_{d=r_{v,I_{1},I^{\prime}}^{s_{0}-1}+1}^{r_{v,I_{1},I^{\prime}}^{s_{0}}}n^{\prime}_{d}\leq\sum_{d=r_{v,I_{1},I^{\prime}}^{s_{0}-1}+1}^{r_{v,I_{1},I^{\prime}}^{s_{0}}}n^{s_{0},d_{0}}_{d}$
which has to be an equality. Hence, all inequalities are equalities and we
have $I^{\prime}=I^{s_{0},d_{0}}$ and thus
$\Theta^{\prime}=\Theta^{s_{0},d_{0}}$. ∎

######  Proposition 2.32.

Let $\Theta=(v,I,\underline{k},\underline{\Lambda})$ be a maximally
$(I_{0},I_{1})$-atomic tuple and $\Omega$ its equivalence class. Assume that
Condition 2.29 holds for some $1\leq s_{0}\leq r_{I_{v}\cap I_{1}}$ and thus
$\Theta^{s_{0},d_{0}}$ as well as $\Omega^{s_{0},d_{0}}$ are defined for each
$r_{v,I_{1},I}^{s_{0}-1}+1\leq d_{0}\leq r_{v,I_{1},I}^{s_{0}}$. Then we have

$x_{\Omega^{s_{0},d_{0}}}-x_{\Omega}\in
d_{1,I_{0},I_{1}}^{-\ell-1,k}(E_{1,I_{0},I_{1}}^{-\ell-1,k})$

for each $1\leq s_{0}\leq r_{I_{v}\cap I_{1}}$ and
$r_{v,I_{1},I}^{s_{0}-1}+1\leq d_{0}\leq r_{v,I_{1},I}^{s_{0}}$.

###### Proof.

As $\Theta^{s_{0},r_{v,I_{1},I}^{s_{0}-1}+1}=\Theta$ by definition, it
suffices to assume that $d_{0}\geq r_{v,I_{1},I}^{s_{0}-1}+2$ and prove that

$x_{\Omega^{s_{0},d_{0}}}-x_{\Omega^{s_{0},d_{0}-1}}\in
d_{1,I_{0},I_{1}}^{-\ell-1,k}(E_{1,I_{0},I_{1}}^{-\ell-1,k}).$

Let
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})\in\Omega^{s_{0},d_{0}}$
be a tuple, and then set
$i_{\Theta^{\prime}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d^{\prime}=1}^{d_{0}-1}n^{\prime}_{d^{\prime}}\in(I_{v}\cap
I_{1})\setminus I^{\prime}$. Then we write $\Omega^{s_{0},d_{0},+}$ for the
set of tuples of the form $p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})$ for
some $\Theta^{\prime}\in\Omega^{s_{0},d_{0}}$, and define
$x_{\Omega^{s_{0},d_{0},+}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{\Theta^{\prime\prime}\in\Omega^{s_{0},d_{0},+}}\varepsilon(\Theta^{\prime\prime})x_{\Theta^{\prime\prime}}$.
We claim that

(2.14)
$d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{\Omega^{s_{0},d_{0},+}})=\varepsilon(\Omega,s_{0},d_{0})(x_{\Omega^{s_{0},d_{0}}}-x_{\Omega^{s_{0},d_{0}-1}})$

for some $\varepsilon(\Omega,s_{0},d_{0})\in\\{1,-1\\}$, which is clearly
sufficient for our purpose.

Let $\Theta^{\prime\prime}\in\Omega^{s_{0},d_{0},+}$ and
$\Theta^{\prime}=(v,I^{\prime},\underline{k}^{\prime},\underline{\Lambda}^{\prime})$
be an arbitrary tuple satisfying

(2.15)
$c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{\Theta^{\prime\prime}}))\neq
0.$

There exists a unique $1\leq d\leq r_{I^{\prime\prime}}$ and
$i\in(I^{\prime\prime})^{d}\setminus I^{\prime}$ such that
$\Theta^{\prime\prime}=p_{i}^{+}(\Theta^{\prime})$. Condition 2.29 on $\Omega$
and the definition of $\Omega^{s_{0},d_{0},+}$ imply that
$\\#\Lambda^{\prime\prime}_{d^{\prime}}\leq 1$ for each $1\leq d^{\prime}\leq
r_{I^{\prime\prime}}$ and in fact $\\#\Lambda^{\prime\prime}_{d^{\prime}}=1$
for each $r_{v,I_{1},I^{\prime\prime}}^{s_{0}-1}+1\leq d^{\prime}\leq
r_{v,I_{1},I^{\prime\prime}}^{s_{0}}$. Hence, we have the following two
possibilities

  * •

$\Lambda^{\prime}_{d}=\emptyset$,
$\Lambda^{\prime}_{d+1}=\Lambda^{\prime\prime}_{d}$ and we set
$d_{1}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}d$;

  * •

$\Lambda^{\prime}_{d}=\Lambda^{\prime\prime}_{d}$,
$\Lambda^{\prime}_{d+1}=\emptyset$ and we set
$d_{1}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}d+1$.

In particular, we always have $\Lambda^{\prime}_{d_{1}}=\emptyset$ and thus
$d_{1}$ is uniquely determined by $\Theta^{\prime}$ (as the unique
$r_{v,I_{1},I^{\prime}}^{s_{0}-1}+1\leq d^{\prime}\leq
r_{v,I_{1},I^{\prime}}^{s_{0}}$ satisfying
$\Lambda^{\prime}_{d^{\prime}}=\emptyset$). Now we fix a tuple
$\Theta^{\prime}$ such that there exists
$\Theta^{\prime\prime}\in\Omega^{s_{0},d_{0},+}$ that satisfies (2.15), and
then write
$i_{\Theta^{\prime}}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d^{\prime}=1}^{d_{1}}n^{\prime}_{d^{\prime}}$
and
$i_{\Theta^{\prime}}^{\prime}\stackrel{{\scriptstyle\textrm{\tiny{def}}}}{{=}}\sum_{d^{\prime}=1}^{d_{1}-1}n^{\prime}_{d^{\prime}}$
for short. We note that such $\Theta^{\prime\prime}$ has to be either
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})$ or
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})$. We clearly have
$\varepsilon(p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime}))=(-1)^{n^{\prime}_{d_{1}}}\varepsilon(p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime}))$
and

$c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})}))=(-1)^{n^{\prime}_{d_{1}}-1}c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})})).$

We have the following observations for $\Theta^{\prime}$.

  * •

Any $\Theta^{\prime\prime}\in\Omega^{s_{0},d_{0},+}$ satisfying (2.15) must be
either $p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})$ or
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})$, and at least one of
them is in $\Omega^{s_{0},d_{0},+}$ by our choice of $\Theta^{\prime}$.

  * •

If
$p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime}),p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$,
then we have

(2.16)
$c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{\Omega^{s_{0},d_{0},+}}))=\varepsilon(p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})}))+\varepsilon(p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})}))=0.$

  * •

If $p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$ and
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\notin\Omega^{s_{0},d_{0},+}$,
we have

(2.17)
$c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{\Omega^{s_{0},d_{0},+}}))=\varepsilon(p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})})).$

  * •

If $p_{i_{\Theta^{\prime}}}^{+}(\Theta^{\prime})\notin\Omega^{s_{0},d_{0},+}$
and
$p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})\in\Omega^{s_{0},d_{0},+}$,
we have

(2.18)
$c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{\Omega^{s_{0},d_{0},+}}))=\varepsilon(p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime}))c_{\Theta^{\prime}}(d_{1,I_{0},I_{1}}^{-\ell-1,k}(x_{p_{i_{\Theta^{\prime}}^{\prime}}^{+}(\Theta^{\prime})})).$

  * •

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