# Quotient branching law for $p$-adic $(\mathrm{GL}_{n+1},\mathrm{GL}_{n})$ I:
generalized Gan-Gross-Prasad relevant pairs

Kei Yuen Chan  Department of Mathematics, The University of Hong Kong
Shanghai Center for Mathematical Sciences, Fudan University
<EMAIL_ADDRESS>
###### Abstract.

Let $G_{n}=\mathrm{GL}_{n}(F)$ be the general linear group over a non-
Archimedean local field $F$. We formulate and prove a necessary and sufficient
condition on determining when

$\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})\neq 0$

for irreducible smooth representations $\pi$ and $\pi^{\prime}$ of $G_{n+1}$
and $G_{n}$ respectively. This resolves the problem of the quotient branching
law.

We also prove that any simple quotient of a Bernstein-Zelevinsky derivative of
an irreducible representation can be constructed by a sequence of derivatives
of essentially square-integrable representations. This result transferred to
affine Hecke algebras of type A gives a generalization of the classical
Pieri’s rule of symmetric groups.

One key new ingredient is a characterization of the layer in the Bernstein-
Zelevinsky filtration that contributes to the branching law, obtained by the
multiplicity one theorem for standard representations, which also gives a
refinement of the branching law. Another key new ingredient is constructions
of some branching laws and simple quotients of Bernstein-Zelevinsky
derivatives by taking certain highest derivatives.

###### Contents

  1. I Introduction
    1. 1 Background
    2. 2 Generalized GGP relevance
    3. 3 The Bernstein-Zelevinsky filtration
    4. 4 Main results and techniques
    5. 5 Outline of proofs
    6. 6 Summary of some key results and their relations
    7. 7 Acknowledgments
  2. II Equivalent definitions of strongly commutative triples
    1. 8 Notations on $\eta$-invariants
    2. 9 Strong commutativity
    3. 10 Combinatorial commutativity
  3. III Generalized GGP relevance from sequences of strongly commutative triples
    1. 11 Unlinked segments for strongly commutative triples
    2. 12 Intersection-union operations for integrals in commutative triples
    3. 13 Intersection-union operations for derivatives in commutative triples
    4. 14 Strong commutation for multisegments
    5. 15 Minimal strongly commutative triples
    6. 16 Generalized GGP relevant pairs and their uniqueness
    7. 17 Double derivatives and double integrals
    8. 18 Symmetry property of relevant pairs
  4. IV Proof of sufficiency of generalized relevance
    1. 19  $\Delta$-reduced representations
    2. 20 Completing to $\Delta$-reduced triples
    3. 21 Smallest derivatives for a strongly commutative triple
    4. 22 Characterizing the supporting layer in BZ filtration
    5. 23 Some results on computing the layer supporting a branching law
    6. 24 Constructing branching laws via Ranking-Selberg integrals
    7. 25 The smallest derivative controlling some BZ-layers supporting branching laws
    8. 26 Proof of sufficiency of the generalized relevant pairs
  5. V Proof of necessity of generalized relevance
    1. 27 Taking highest derivatives in branching laws
    2. 28 Branching law from BZ derivatives
    3. 29 Deforming and thickening simple quotients of BZ derivatives
    4. 30 Exhaustion and relevance conditions
    5. 31 Necessity of generalized relevance and exhaustion theorem
  6. VI Appendices
    1. 32 Appendix A: Specialization map under Bernstein-Zelevinsky filtrations
    2. 33 Appendix B: A short proof for the asymmetry property of left-right BZ derivatives

##  Part I Introduction

###  1\. Background

Let $G_{n}=\mathrm{GL}_{n}(F)$, the general linear group over a local non-
Archimedean field $F$. All representations of $G_{n}$ are smooth and over
$\mathbb{C}$. We regard $G_{n}$ as a subgroup of $G_{n+1}$ via the embedding
$g\mapsto\begin{pmatrix}g&\\\ &1\end{pmatrix}$. The quotient branching law
asks: for an irreducible representation $\pi$ of $G_{n+1}$ and an irreducible
representation $\pi^{\prime}$ of $G_{n}$, determine the space
$\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})$, where $\pi$ is viewed as
$G_{n}$-representation via restriction to the subgroup. The multiplicity-at-
most-one theorem, first fully established by Aizenbud-Gourevitch-Rallis-
Schiffmann [AGRS10] using distributions, asserts that:

$\mathrm{dim}~{}\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})\leq 1.$

An alternate proof heavily using Bernstein-Zelevinsky (BZ) theory and the
multiplicity one theorem [GK71, Sh74] of Whittaker models is given in [Ch23].
As mentioned in [AGRS10, Page 1411], a more difficult question is to find when
the dimension of the Hom space is one. We give a practical solution to this
question in terms of the Langlands correspondence and in full generality (for
$p$-adic general linear groups).

We review some previous known cases of the quotient branching law in the
literature:

  1. (1)

When $\pi$ is cuspidal and $\pi^{\prime}$ is arbitrary, it is known by the
work of Bernstein-Zelevinsky [BZ77] by restricting to a mirabolic subgroup.

  2. (2)

When both $\pi$ and $\pi^{\prime}$ are generic, the Hom is always non-zero by
the work Jacquet–Piateski-Shapiro–Shalika [JPSS83] using Rankin-Selberg
integrals. We also point out a related study in constructing Shintani
functions for spherical representations by Murase-Sugano [MS96].

  3. (3)

When $\pi^{\prime}$ is the trivial representation and $\pi$ is arbitrary, it
is established for $n=2$ by D. Prasad [Pr93] and completed for general $n$ by
his student Venketasubramanian [Ve13].

  4. (4)

When $\pi$ is a generalized Steinberg representation and $\pi^{\prime}$ is
arbitrary, it is known in the work of Chan-Savin [CS21], in which we use both
left and right BZ theory.

  5. (5)

When both $\pi$ and $\pi^{\prime}$ are Arthur-type representations, a
conjecture determining their quotient branching law is formulated by Gan-
Gross-Prasad [GGP20]. Earlier evidence dealing with one direction is
established by M. Gurevich [Gu22] using results in quantum groups. The
conjecture is now settled by Chan [Ch22], in which we use some functorial
properties of parabolic inductions [Ch22+b].

We shall give a condition unifying all above cases. The condition for Arthur
type representations in [GGP20] is called relevant. Following their
terminology, we shall call our condition (Definition 2.4) to be generalized
GGP relevant (or simply relevant).

In a more general framework of the relative Langlands program developed by
Sakellaridis-Venkatesh [SV18], our main result also solves the smooth
distinction problem for the homogeneous spherical variety $\Delta
G_{n}\setminus(G_{n+1}\times G_{n})$.

We remark that the unitary branching law from $G_{n+1}$ to $G_{n}$ could
follow from the Kirillov conjecture [Ki62] for restricting to the mirabolic
subgroup, proved by Bernstein [Be84]. On the other hand, the constituents in
the restriction are also described as a special case of Clozel’s conjecture
[Cl04], proved by Venkatesh [Ve05], which is also a motivation for the non-
tempered conjecture in [GGP20].

We now turn to discussions on BZ derivatives. For more discussions, see the
introduction of [Ch22+]. Here we only mention some classical important
results: the highest BZ derivatives of irreducible representations by
Zelevinsky [Ze80], partial BZ derivatives of ladder representations by Tadić
[Ta87] and Lapid-Mínguez [LM14], and an Hecke algebra approach of BZ
derivatives for Speh representations in [CS19]. As seen in [Ch21] and [Ch22+]
as well as in this article, there is a large amount of interplay between BZ
derivatives and branching laws, as suggested by their theory [BZ77].

Our another main result asserts that any simple quotient of a BZ derivative of
an irreducible representation can be obtained by a sequence of derivatives of
essentially square-integrable representations. Translating this to the setup
of affine Hecke algebras, this result essentially generalizes Pieri’s rule
[Pi93] from symmetric groups to affine Hecke algebras of type A with generic
parameters. We refer the reader to [CS19, Ch22+] for the details of
transferring those results, including the multiplicity-freeness for socles and
cosocles in [Ch22+]. We plan to discuss the combinatorics aspect of such
generalized rule in the sequel.

A special case in this theme is obtained by Grojnowski-Vazirani [GV01] and
Vazirani [Va02]. Indeed, our terminologies in defining the generalized
relevance combinatorially (more precisely, Definition 8.1) and the use of
derivatives also reflect our original viewpoint from using Hecke algebras,
started from the work [CS19, CS21, Ch21] (c.f. symplectic root number and
$L$-function perspective in [GGP12, GGP22+]). It seems that the proof of
[GV01] cannot be directly adapted to proving our generalized Pieri’s rule (but
see some related ideas for $\eta$-invariants), and our proof essentially uses
some nature of branching laws specific to $p$-adic groups.

As we are working representations over $\mathbb{C}$, the modular quotient
branching laws for symmetric groups and finite Hecke algebras of type A by
Kleshchev [Kl95] and Brundan [Br98] respectively are not in our scope at this
point while one may hope that developments from Vigneras’ school on mod $\ell$
representations [Vi96] shed light on this aspect, see e.g. the work of
Sécherre-Venketasubramanian [SV17] on the distinguished case.

Indeed, it is known in [GGP20] that the original GGP relevance is not a
necessary condition for the quotient branching law for other classical groups,
and so we hope this work also sheds light on the quotient branching law
outside GL. The work on generic case by Mœglin-Waldspurger [MW12] for
orthogonal groups, adapted to unitary groups by Beuzart-Plessis [BP16], has
similar spirit to the use of BZ filtrations. For more background and results
on the GGP problems, see a survey [Gr21+]. On the other hand, the works of
Chan-Savin [CS20] and Bakić-Savin [BS22] have started the line of research for
orthogonal groups along the GL ones from Hecke algebra viewpoint [CS19, Ch21].

An irreducible representation of $G_{n+1}$ restricting to a
$G_{n}$-representation is far from being semisimple in general. In [Pr18], D.
Prasad initiated to study some higher structure in the branching law. In
particular, he conjectured the higher Ext groups for generic representations,
which is now proved in [CS21] and extended to standard representations in
[Ch23]. In [Ch21], we determine all the indecomposable components of an
irreducible representation restricted from $G_{n+1}$ to $G_{n}$, classify
irreducible representations which are projective under restriction and
determine the Ext-group at the cohomological degree in quotient branching law,
marking some substantial progress in homological branching laws. However, a
more difficult question, in determining when
$\mathrm{Ext}^{i}_{G_{n}}(\pi,\pi^{\prime})$ is non-zero or even determining
the precise dimensions, remains largely open.

###  2\. Generalized GGP relevance

####  2.1. Basic notations

All representations are smooth and over $\mathbb{C}$ and we shall usually omit
those descriptions. We sometimes do not distinguish representations in the
same isomorphism class. Let $\mathrm{Alg}(G)$ be the category of
representation of a reductive group $G$ over $\mathbb{C}$. For
$\pi\in\mathrm{Alg}(G)$, let $\pi^{\vee}$ be its smooth dual. Let
$\mathrm{Irr}(G)$ be the set of irreducible representations of $G$.

Let $\mathrm{Irr}=\sqcup_{n}\mathrm{Irr}(G_{n})$. Let
$\mathrm{Irr}^{c}(G_{n})$ be the set of irreducible cuspidal representations
of $G_{n}$ and let $\mathrm{Irr}^{c}=\sqcup_{n}\mathrm{Irr}^{c}(G_{n})$. Let
$S_{n}$ be the permutation group on $n$ elements.

For $n_{1}+\ldots+n_{r}=n$, let $P_{n_{1},\ldots,n_{r}}$ be the associated
standard parabolic subgroup in $G_{n}$ i.e. the parabolic subgroup containing
upper triangular matrices and matrices $\mathrm{diag}(g_{1},\ldots,g_{r})$ for
$g_{i}\in G_{n_{i}}$. Denote by $\mathrm{Ind}_{P}^{G_{n}}$ the normalized
parabolic induction from a parabolic subgroup $P$ of $G_{n}$. As usual, for
$\pi_{1}\in\mathrm{Alg}(G_{n_{1}})$ and $\pi_{2}\in\mathrm{Alg}(G_{n_{2}})$,
let
$\pi_{1}\times\pi_{2}=\mathrm{Ind}_{P_{n_{1},n_{2}}}^{G_{n}}\pi_{1}\boxtimes\pi_{2}$.

For $n_{1}+\ldots+n_{r}=n$, let $N_{n_{1},\ldots,n_{r}}$ be the unipotent
radical in $P_{n_{1},\ldots,n_{r}}$. For $\pi\in\mathrm{Alg}(G_{n})$, denote
by $\pi_{N_{n_{1},\ldots,n_{r}}}$ its associated Jacquet module, viewed as a
$G_{n_{1}}\times\ldots\times G_{n_{r}}$-representation. We shall sometimes
abbreviate $N_{n_{1},n_{2}}$ by $N_{n_{2}}$ to lighten notations. (The choice
for $n_{2}$ is compatible with our convention to usually consider derivatives
taken on the ’right’, see Section 8.1 below.) Let $U_{n}=N_{1,\ldots,1}$, the
subgroup of all upper triangular unipotent matrices in $G_{n}$.

We now introduce some combinatorial objects following [Ze80]. A segment
$\Delta$ is a data of the form $[a,b]_{\rho}$ for $a,b\in\mathbb{Z}$ and
$\rho\in\mathrm{Irr}^{c}$. Let $\nu(g)=|\mathrm{det}(g)|_{F}$ be the non-
Archimedean absolute value of the determinant. We also write
$a(\Delta)=\nu^{a}\rho$ and $b(\Delta)=\nu^{b}\rho$. We sometimes write
$[a]_{\rho}$ for $[a,a]_{\rho}$, and write $[a,b]$ for $[a,b]_{1}$ (here $1$
is the trivial representation of $G_{1}$). As standard, we also consider
$[a,a-1]_{\rho}$ as the empty set. We shall also regard a segment
$[a,b]_{\rho}$ as a set $\left\\{\nu^{a}\rho,\ldots,\nu^{b}\rho\right\\}$ so
that one can consider the intersection and union of two segments. A
multisegment is a multiset of non-empty segments. (For our convention and
convenience, we also consider an empty set to be a segment and a
multisegment.) For a segment, let $l_{a}(\Delta)=(b-a+1)n(\rho)$, called the
absolute length of $\Delta$; and let $l_{r}(\Delta)=b-a+1$, called the
relative length of $\Delta$. For a multisegment
$\mathfrak{m}=\left\\{\Delta_{1},\ldots,\Delta_{r}\right\\}$, let
$l_{a}(\mathfrak{m})=\sum_{i}l_{a}(\Delta_{i})$, and let
$\mathrm{csupp}(\mathfrak{m})=\cup_{\Delta}\Delta$. Let $\mathrm{Mult}$ be the
set of multisegments.

Two segments $\Delta$ and $\Delta^{\prime}$ are linked if
$\Delta\cup\Delta^{\prime}$ is still a segment. For two cuspidal
representations $\rho_{1}$ and $\rho_{2}$, we write $\rho_{1}<\rho_{2}$ if
there exists a positive integer $c$ such that $\nu^{c}\rho_{1}\cong\rho_{2}$,
and write $\rho_{1}\leq\rho_{2}$ if $\rho_{1}<\rho_{2}$ or
$\rho_{1}\cong\rho_{2}$. We write $\Delta_{1}<\Delta_{2}$ if $\Delta_{1}$ and
$\Delta_{2}$ are linked and $b(\Delta_{1})<b(\Delta_{2})$, and write
$\Delta_{1}\leq\Delta_{2}$ if $\Delta_{1}<\Delta_{2}$ or
$\Delta_{1}=\Delta_{2}$.

For $\pi\in\mathrm{Irr}$, there exists a unique collection of cuspidal
representations $\rho_{1},\ldots,\rho_{k}$ such that $\pi$ is a simple
composition factor $\rho_{1}\times\ldots\times\rho_{k}$. Let
$\mathrm{csupp}(\pi)=\left\\{\rho_{1},\ldots,\rho_{k}\right\\}$ (as a
multiset), called the cuspidal support of $\pi$.

For each segment $\Delta$, we denote by $\langle\Delta\rangle$ the Zelevinsky
segment representation [Ze80, Section 3.1]. Denote by $\mathrm{St}(\Delta)$
the generalized Steinberg representation [Ze80, Section 9.1]. In particular,
$\mathrm{csupp}(\langle\Delta\rangle)=\mathrm{csupp}(\mathrm{St}(\Delta))=\Delta$.

For a multisegment
$\mathfrak{m}=\left\\{\Delta_{1},\ldots,\Delta_{r}\right\\}$, we label the
segments such that $\Delta_{1}\not\leq\ldots\not\leq\Delta_{r}$. Let
$\zeta(\mathfrak{m})=\langle\Delta_{1}\rangle\times\ldots\times\langle\Delta_{r}\rangle$.
Let
$\lambda(\mathfrak{m})=\mathrm{St}(\Delta_{1})\times\ldots\times\mathrm{St}(\Delta_{r})$,
which we shall refer to a standard module (in the sense of Langlands). Let
$\langle\mathfrak{m}\rangle$ be the unique submodule of $\zeta(\mathfrak{m})$
[Ze80, Section 6.5] and let $\mathrm{St}(\mathfrak{m})$ be the unique simple
quotient of $\lambda(\mathfrak{m})$.

####  2.2. Generalized GGP relevance

We first define a notion of derivatives and integrals:

######  Definition 2.1.

Let $\pi\in\mathrm{Irr}$.

  * (1)

Denote by $I_{\Delta}^{R}(\pi)$ (resp. $I_{\Delta}^{L}(\pi)$) to be the unique
submodule of $\pi\times\mathrm{St}(\Delta)$ (resp.
$\mathrm{St}(\Delta)\times\pi$). We shall call $I_{\Delta}^{L}$ and
$I^{R}_{\Delta}$ to be integrals.

  * (2)

Suppose $l_{a}(\Delta)\leq n$. Let $N$ (resp. $N^{\prime}$) be the unipotent
subgroup of the standard parabolic subgroup corresponding to the partition
$(n-l_{a}(\Delta),l_{a}(\Delta))$ (resp. ($l_{a}(\Delta),n-l_{a}(\Delta))$).
Let $D^{R}_{\Delta}(\pi)$ (resp. $D^{L}_{\Delta}(\pi)$) be the unique
irreducible representation (if exists) such that

$D^{R}_{\Delta}(\pi)\boxtimes\mathrm{St}(\Delta)\hookrightarrow\pi_{N},\quad\mathrm{St}(\Delta)\boxtimes
D^{L}_{\Delta}(\pi)\hookrightarrow\pi_{N^{\prime}}.$

If such module does not exist, set $D^{R}_{\Delta}(\pi)=0$ (resp.
$D^{L}_{\Delta}(\pi)=0$). We shall call those $D^{L}_{\Delta}$ and
$D^{R}_{\Delta}$ to be derivatives.

For the above uniqueness, see [KKKO15, LM16] and also see [Ch22+, Ch22+b].

We define a notion of strongly commutative triples in terms of how a certain
structure of Jacquet module is embedded in the layers of the geometric lemma
of the induced module.

######  Definition 2.2.

(c.f. [Ch22+d]) Let $\Delta,\Delta^{\prime}$ be two segments. Let
$\pi\in\mathrm{Irr}$ and let $\tau=I^{L}_{\Delta^{\prime}}(\pi)$. We say that
$(\Delta,\Delta^{\prime},\pi)$ is a pre-RdLi-commutative triple if
$D^{R}_{\Delta}(\pi)\neq 0$ and the following composition

(2.1) $\displaystyle
D^{R}_{\Delta}(\tau)\boxtimes\mathrm{St}(\Delta)\hookrightarrow\tau_{N}\hookrightarrow(\mathrm{St}(\Delta^{\prime})\times\pi)_{N}\twoheadrightarrow\mathrm{St}(\Delta^{\prime})\dot{\times}^{1}\pi_{N^{\prime}},$

where $N$ and $N^{\prime}$ are corresponding nilpotent orbits, and the last
term is the top layer in the geometric lemma (see, Section 9.1, for precise
notations). Here the first map comes from the unique map in Definition 2.1(2)
and the second map is induced from the embedding in Definition 2.1(1). The
last map is the natural projection from the geometric lemma.

We say that $(\Delta,\Delta^{\prime},\pi)$ is a strongly RdLi-commutative
triple if the triple is pre-RdLi-commutative and moreover, the map in (2.1)
factors through the embedding:

$(\mathrm{St}(\Delta^{\prime})\times
D^{R}_{\Delta_{1}}(\pi))\boxtimes\mathrm{St}(\Delta)\hookrightarrow\mathrm{St}(\Delta^{\prime})\dot{\times}^{1}\pi_{N^{\prime}},$

which is induced from the unique map
$D^{R}_{\Delta}(\pi)\boxtimes\mathrm{St}(\Delta)\hookrightarrow\pi_{N^{\prime}}$.

In [Ch22+d], we formulate and show other equivalent definitions for the strong
commutation in terms of $\eta$-invariants (See Definitions 10.1, 10.2 and
Theorem 10.3). Such triple can then be checked combinatorially in terms of
Langlands parameters (as well as Zelevinsky parameters).

For a multisegment $\mathfrak{m}$, we label the segments in $\mathfrak{m}$ as:
$\Delta_{i}\not>\Delta_{j}$ (resp. $\Delta_{i}\not<\Delta_{j}$) for any $i<j$.
For such an ordering, we shall call it an ascending order (resp. descending
order) (c.f. [Ze80, Theorem 6.1]). Define

$D^{R}_{\mathfrak{m}}(\pi)=D^{R}_{\Delta_{r}}\circ\ldots\circ
D^{R}_{\Delta_{1}}(\pi),\quad(\mathrm{resp.}\
D^{L}_{\mathfrak{n}}(\pi)=D^{L}_{\Delta_{r}}\circ\ldots\circ
D^{L}_{\Delta_{1}}(\pi)),$
$I^{L}_{\mathfrak{m}}(\pi)=I^{L}_{\Delta_{r}}\circ\ldots\circ
I^{L}_{\Delta_{1}}(\pi),\quad(\mathrm{resp.}\
I^{R}_{\mathfrak{n}}(\pi)=I^{R}_{\Delta_{r}}\circ\ldots\circ
I^{R}_{\Delta_{1}}(\pi)).$

We now define the notion of strongly RdLi-commutative triples for
multisegments:

######  Definition 2.3.

Let $\mathfrak{m},\mathfrak{n}$ be multisegments. We write
$\mathfrak{m}=\left\\{\Delta_{1},\ldots,\Delta_{r}\right\\}$ and
$\mathfrak{n}=\left\\{\Delta_{1}^{\prime},\ldots,\Delta_{s}^{\prime}\right\\}$
in an ascending order. Let
$\mathfrak{m}_{i}=\left\\{\Delta_{1},\ldots,\Delta_{i}\right\\}$ and let
$\mathfrak{n}_{j}=\left\\{\Delta_{1}^{\prime},\ldots,\Delta_{s}^{\prime}\right\\}$.
We say that $(\mathfrak{m},\mathfrak{n},\pi)$ is a strongly RdLi-commutative
triple if for any $r-1\geq i\geq 0$ and $s-1\geq j\geq 0$,

$(\Delta_{i+1},\Delta_{j+1}^{\prime},I_{\mathfrak{n}_{j}}\circ
D_{\mathfrak{m}_{i}}(\pi))$

is a strongly RdLi-commutative triple. (When $\mathfrak{m}$ or $\mathfrak{n}$
is empty, the triple $(\mathfrak{m},\mathfrak{n},\pi)$ is automatically
relevant.)

It follows from Propositions 11.2 and 11.3 that the above definition is
independent of a choice of orderings.

######  Definition 2.4.

(c.f. Definition 16.1) (Generalized GGP relevant pair) Let
$\pi\in\mathrm{Irr}(G_{n})$ and let $\pi^{\prime}\in\mathrm{Irr}(G_{m})$. We
say that $(\pi,\pi^{\prime})$ is (generalized) relevant if there exist
multisegments $\mathfrak{m}$ and $\mathfrak{n}$ such that

  1. (1)

$D^{R}_{\mathfrak{m}}(\nu^{1/2}\cdot\pi)\cong
D^{L}_{\mathfrak{n}}(\pi^{\prime})$;

  2. (2)

$(\mathfrak{m},\mathfrak{n},\nu^{1/2}\cdot\pi)$ is a strongly RdLi-commutative
triple.

The original GGP relevant pair [GGP20] (for Arthur type representations and
all classical groups) is defined in terms of Langlands parameters, and
reformulated by Zhiwei Yun more geometrically in terms of moment maps [GGP12,
Section 4]. Our main result, Theorem 4.1 below, shows in a rather indirect way
that the original GGP relevance coincides with Definition 2.4 for Arthur type
representations.

The first main property is the symmetry of such notion:

######  Theorem 2.5.

(=Theorem 18.1) Let $\pi\in\mathrm{Irr}(G_{n})$ and let
$\pi^{\prime}\in\mathrm{Irr}(G_{m})$. Then $(\pi,\pi^{\prime})$ is relevant if
and only if $(\pi^{\prime},\pi)$ is relevant.

Theorem 2.5 can be viewed as a compatibility with the left-right BZ
filtrations (see discussions in Section 4.3).

To state the second property, we need more notations. For two multisegments
$\mathfrak{m}$ and $\mathfrak{m}^{\prime}$, we write
$\mathfrak{m}^{\prime}\leq_{Z}\mathfrak{m}$ if either $\mathfrak{m}^{\prime}$
is obtained from $\mathfrak{m}$ by a sequence of intersection-union operations
(see Section 12.1) or $\mathfrak{m}^{\prime}=\mathfrak{m}$.

######  Definition 2.6.

Let $\pi\in\mathrm{Irr}$. A multisegment $\mathfrak{m}$ is said to be Rd-
minimal (Li-minimal) to $\pi$ if $D^{R}_{\mathfrak{m}}(\pi)\neq 0$ and
$D^{R}_{\mathfrak{m}}(\pi)\not\cong D^{R}_{\mathfrak{n}}(\pi)$ (resp.
$I^{L}_{\mathfrak{m}}(\pi)\not\cong I^{L}_{\mathfrak{n}}(\pi)$) for any other
multisegment $\mathfrak{n}$ with $\mathfrak{n}\lneq_{Z}\mathfrak{m}$. We have
analogous notions for Ld-minimal and Ri-minimal.

One has uniqueness for such minimal multisegments [Ch22+], see Theorem 15.3.
We have the following uniqueness statement.

######  Theorem 2.7.

(=Theorem 16.2) Let $(\pi,\pi^{\prime})$ be a relevant pair in Definition 2.4.
There exist unique multisegments $\mathfrak{m}$ and $\mathfrak{n}$ such that
$\mathfrak{m}$ is $Rd$-minimal to $\nu^{1/2}\cdot\pi$ and $\mathfrak{n}$ is
$Ld$-minimal to $\pi^{\prime}$, and the conditions in Definition 2.4 are
satisfied.

The notion of a strongly commutative triple has good properties with respect
to the intersection-union process, see Propositions 12.1 and 13.2. This
roughly gives the existence part in Theorem 2.7.

The uniqueness of Theorem 2.7 is related to the layers of right BZ filtration
that governs the branching law. More precisely, the uniqueness allows one to
define a refined terminology: $i^{*}$-relevant, where $i^{*}$ is equal to
$l_{a}(\mathfrak{m})$.

###  3\. The Bernstein-Zelevinsky filtration

The Bernstein-Zelevinsky filtration (a.k.a. Mackey theory approach) is our
primary tool for branching law. Such approach has recently been found in much
success for dealing with branching laws for non-tempered cases [GGP20, CS21,
Ch21, Gu22, Ch22, Ch22+, Ch22+d] (also see [Pr93, MW12, Ve13, BP16, SV17,
Pr18]). In an ongoing work of R. Chen and C. Wang [CW], such approach is also
considered to deal with some non-tempered cases for unitary groups. For other
approaches such as using theta correspondence and relative trace formulae, one
sees [Gr21+].

####  3.1. Bernstein-Zelevinsky functors

Let $V_{n}$ be the unipotent radical of $M_{n}$. Let $\overline{\psi}$ be a
non-degenerate character on $F$. Let $\psi:V_{n}\rightarrow\mathbb{C}$ given
by $\psi(v)=\overline{\psi}(v_{n-1})$, where $v_{n-1}$ is the $(n-1,n)$-entry
in $v$. For $\pi\in\mathrm{Alg}(G_{n})$, let

$\pi_{V_{n},\psi}=\delta^{-1/2}\pi/\langle n.v-\psi(n)v:n\in
V_{n},v\in\pi\rangle$

where $\delta$ is a modular character of $M_{n}$.

Following [BZ76, BZ77],

$\Phi^{+}:\mathrm{Alg}(M_{n-1})\rightarrow\mathrm{Alg}(M_{n}),\Phi^{-}:\mathrm{Alg}(M_{n})\rightarrow\mathrm{Alg}(M_{n-1}),$
$\Psi^{+}:\mathrm{Alg}(G_{n-1})\rightarrow\mathrm{Alg}(M_{n}),\Psi^{-}:\mathrm{Alg}(M_{n})\rightarrow\mathrm{Alg}(G_{n-1})$

given by:

$\Phi^{+}(\pi)=\mathrm{Ind}_{M_{n-1}V_{n}}^{M_{n}}\pi\boxtimes\psi,\quad\Phi^{-}(\pi)=\pi_{V_{n},\psi},$

and let

$\Psi^{+}(\pi)=\mathrm{Ind}_{G_{n-1}V_{n}}^{M_{n}}\pi\boxtimes
1,\quad\Psi^{-}(\pi)=\pi_{V_{n},1}.$

The $i$-th right Bernstein-Zelevinsky derivative of $\pi$ is defined as:

$\pi^{(i)}=\Psi^{-}\circ(\Phi^{-})^{i-1}(\pi).$

Let $\theta=\theta_{n}:G_{n}\rightarrow G_{n}$ given by $\theta(g^{-t})$, the
inverse transpose (a.k.a Gelfand-Kazhdan involution). It induces a categorical
auto-equivalence on $\mathrm{Alg}(G_{n})$, still denoted by $\theta$. The left
$i$-the BZ derivatives is defined as:

${}^{(i)}\pi=\theta(\theta(\pi)^{(i)}).$

Their shifted derivatives are defined as:

$\pi^{[i]}=\nu^{1/2}\cdot\pi^{(i)},\quad{}^{[i]}\pi=\nu^{-1/2}\cdot{}^{(i)}\pi.$

Let $\pi\in\mathrm{Irr}$. The level of $\pi$, denoted $\mathrm{lev}(\pi)$, is
the largest integer $i^{*}$ such that $\pi^{(i^{*})}\neq 0$. Let
$\mathfrak{m}\in\mathrm{Mult}$ such that $\pi\cong\langle\mathfrak{m}\rangle$.
The following statements are equivalent, due to [Ze80]:

  1. (1)

$i^{*}$ is the level of $\pi$;

  2. (2)

$i^{*}$ is the largest integer such that ${}^{(i^{*})}\pi\neq 0$;

  3. (3)

$i^{*}$ is equal to the number of segments in $\mathfrak{m}$.

We shall call $\pi^{(i^{*})}$ to be the highest right derivative, denoted by
$\pi^{-}$, and call ${}^{(i^{*})}\pi$ to be the highest left derivative,
denoted by ${}^{-}\pi$. Similarly, set $\pi^{[-]}=\pi^{[i^{*}]}$ and
${}^{[-]}\pi={}^{[i^{*}]}\pi$.

####  3.2. Bernstein-Zelevinsky filtrations

For $\tau\in\mathrm{Alg}(G_{k})$ and $i\geq 1$, define, as
$M_{k+i}$-representations,

$\Gamma^{i}(\tau)=(\Phi^{+})^{i-1}\circ\Psi^{+}(\tau).$

For $\pi\in\mathrm{Alg}(G_{n+1})$, define

$\Lambda_{i}(\pi)=(\Phi^{+})^{i}\circ(\Phi^{-})^{i}(\pi),\quad\Sigma_{i}(\pi)=\Gamma^{i}(\pi^{(i)}).$

Bernstein-Zelevinsky [BZ76, BZ77] shows that there are natural inclusions in
$\mathrm{Alg}(M_{n+1})$:

$\Lambda_{n+1}(\pi):=0\hookrightarrow\Lambda_{n}(\pi)\hookrightarrow\ldots\Lambda_{1}(\pi)\hookrightarrow\Lambda_{0}(\pi)=\pi.$

Furthermore,

$\Lambda_{i-1}(\pi)/\Lambda_{i}(\pi)\cong\Sigma_{i}(\pi).$

We shall refer it to be the right BZ filtration of $\pi$. We shall frequently
use the following for computations: for $\pi$ and $\pi^{\prime}$ to be
admissible, and for any $j$,

$\mathrm{Ext}^{j}_{G_{n}}(\Sigma_{i}(\pi),\pi^{\prime})\cong\mathrm{Ext}^{j}_{G_{n+i-i}}(\pi^{[i]},{}^{(i-1)}\pi^{\prime}),$

see e.g. [CS21, Lemma 2.4].

Note that $\theta$ also induces a categorical equivalence from
$\mathrm{Alg}(M_{n+1})$ to $\mathrm{Alg}(M_{n+1}^{t})$, again denoted by
$\theta$. For $\tau\in\mathrm{Alg}(G_{k})$,

${}^{i}\Gamma(\tau):=\theta(\Gamma^{i}(\theta(\tau))).$

For $\pi\in\mathrm{Alg}(G_{n+1})$, define

${}_{i}\Lambda(\pi):=\theta(\Lambda_{i}(\theta(\pi))),\quad{}_{i}\Sigma(\pi):=\theta(\Sigma_{i}(\theta(\pi))).$

We shall refer

${}_{n+1}\Lambda(\pi)\hookrightarrow{}_{n}\Lambda(\pi)\hookrightarrow\ldots\hookrightarrow{}_{0}\Lambda(\pi)=\pi$

to be the left BZ filtration. For more discussions on left-right BZ
filtrations, see [CS21].

When restricted to $G_{n}$ (via the embedding $g\mapsto\mathrm{diag}(g,1)$),
the left and right BZ filtrations give two filtrations, as
$G_{n}$-representations, on $\pi$.

####  3.3. Variants of Bernstein-Zelevinsky filtrations

For $\pi\in\mathrm{Alg}(G_{n})$, we denote by
$\pi\otimes\zeta_{F}\in\mathrm{Alg}(G_{n})$ its Fourier-Jacobi model of $\pi$,
and $\mathrm{RS}_{k}(\pi)\in\mathrm{Alg}(G_{n+k+1})$ its Rankin-Selberg model.
We shall not recall (and need) the explicit definition (see [Ch22, Ch23]), and
the main property we need is the following:

######  Lemma 3.1.

Let $\lambda$ be a quotient of a standard module of $G_{n}$. Then,

  1. (1)

for any quotient $\lambda^{\prime}$ of a standard module of $G_{n}$,

$\mathrm{dim}~{}\mathrm{Hom}_{G_{n}}(\lambda\otimes\zeta_{F},\lambda^{\prime}{}^{\vee})\leq
1,$

  2. (2)

for any quotient $\lambda^{\prime}$ of a standard module of $G_{n+k}$,

$\mathrm{dim}~{}\mathrm{Hom}_{G_{n+k}}(\mathrm{RS}_{k-1}(\lambda),\lambda^{\prime}{}^{\vee})\leq
1.$

###### Proof.

This follows from the multiplicity one theorem for standard modules in [Ch21]
and [Ch22, Section 5] (also see [GGP12]). ∎

We recall that we have the following filtration:

######  Proposition 3.2.

[Ch22, Proposition 5.13] Let $\pi_{1}\in\mathrm{Alg}(G_{n_{1}})$ and let
$\pi_{2}\in\mathrm{Alg}(G_{n_{2}})$. Then
$(\pi_{1}\times\pi_{2})|_{G_{n_{1}+n_{2}-1}}$ admits a filtration with
successive subquotients as follows:

$\pi_{1}^{[0]}\times(\pi_{2}|_{G_{n_{2}-1}}),\quad\pi_{1}^{[1]}\times(\pi_{2}\otimes\zeta^{F})$

and, for $k\geq 2$,

$\pi_{1}^{[k]}\times\mathrm{RS}_{k-2}(\pi_{2}).$

We remark that one can also consider the filtration as a
$M_{n_{1}+n_{2}}$-representation as in [Ch22], and the filtration in
Proposition 3.2 is then obtained under restriction to $G_{n_{1}+n_{2}-1}$.
More precisely, the corresponding layers, as a $M_{n_{1}+n_{2}}$-filtration,
takes the form:

$\pi_{1}\times(\pi_{2}|_{M_{n_{2}}}),\quad\pi_{1}^{(1)}\times(\pi_{2}\otimes\zeta^{F}),\quad\pi_{1}^{(k)}\times((\Phi^{+})^{k-1}(\nu^{-1/2}\cdot\mathrm{ind}_{G_{n_{2}}}^{M_{n_{2}+1}}\pi_{2})).$

Here the product is the one for mirabolic subgroups, see [BZ77, Section 4.12]
and [Ch22, Section 3.1]. Viewing as $M_{n_{1}+n_{2}}$-representations, one can
then talk about the layer supporting properties in Definition 22.3. One can
further take a $M_{n_{2}+1}$-filtration on
$\nu^{-1/2}\cdot\mathrm{ind}_{G_{n_{2}}}^{M_{n_{2}+1}}\pi_{2}$ to give layers
of the form $(\Phi^{+})^{r-1}\circ\Psi^{+}(\pi_{2}^{(r)})$.

###  4\. Main results and techniques

####  4.1. Main results

We now state two main results. The first one shows that the generalized GGP
relevant pair governs the branching law.

######  Theorem 4.1.

(= Part of Theorem 26.4+Theorem 31.2) Let $\pi\in\mathrm{Irr}(G_{n+1})$ and
let $\pi^{\prime}\in\mathrm{Irr}(G_{n})$. Then $(\pi,\pi^{\prime})$ is
relevant if and only if $\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})\neq 0$.

######  Theorem 4.2.

(=Theorem 31.3) Let $\pi\in\mathrm{Irr}(G_{n})$. Let $\tau$ be a simple
quotient of $\pi^{(i)}$ for some $i$. Then there exists a multisegment
$\mathfrak{m}$ such that

$D^{R}_{\mathfrak{m}}(\pi)\cong\tau.$

We give some remarks.

  * •

Indeed, we prove a refinement for Theorem 4.1, which also determines the BZ
layer contributing the branching law from $i$-relevance.

  * •

We shall refer Theorem 4.2 to the exhaustion theorem. The construction part is
largely studied in [Ch22+].

  * •

The $n$ and $m$ are arbitrary in Definition 2.4. We shall explain briefly the
corresponding problems without introducing new terminologies. When $n-m$ is
positive and odd (resp. non-negative and even), the related restriction
problem arises from so-called Bessel models (resp. Fourier-Jacobi models)
[GGP12]. When $m-n$ is negative, the related problem can be phrased as an
induction (by taking the smooth duals and Frobenius reciprocity in the
restriction problem of the corresponding positive case).

In [GGP12], it is shown that the quotient branching law for Bessel models and
Fourier-Jacobi models, follows from the corank one case. Some variants of
those models, see Chen-Sun [ChSu15] and Y. Liu [Li14, Section 1.4], are also
now established in [Ch22, Section 5] by relating various models via BZ theory.

  * •

The Langlands correspondence for GL has now been established by Laumon-
Rapoport-Stuhler [LRS93], Harris-Taylor [HT01], Henniart [He00] and Scholze
[Sc13] (also a recent geometric construction of Fargues-Scholze [FS21+]). The
quotient branching law can be checked using the combinatorial formulation for
strongly RdLi-commutative triple (Theorem 10.3).

More precisely, the cuspidal support between $\pi\in\mathrm{Irr}(G_{n+1})$ and
$\pi^{\prime}\in\mathrm{Irr}(G_{n})$ reduces to check finitely many
$\mathfrak{m}$ and $\mathfrak{n}$ for the relevance. Thus, our Theorem 4.1
provides a finite deterministic algorithm to the quotient branching law. The
proofs give hints on how to find a more effective algorithm for checking the
relevance, and we plan to deal with that in the sequel.

####  4.2. Examples

We compare some known examples with Theorem 4.1.

  1. (1)

(Example from distinguished representations [Pr93, Ve13]) Let $\pi^{\prime}$
be the trivial representation of $G_{n}$. Let
$\pi=\langle[-(n-2)/2,n/2],[(n+2)/2]\rangle$. Let $\Delta=[(n+1)/2,(n+3)/2]$
and let $\Delta^{\prime}=[-(n-1)/2]$. Then, it follows from a simple cuspidal
condition that $(\Delta_{1},\Delta_{2},\pi)$ is a strongly RdLi-commutative
triple. This gives the relevance for $(\pi,\pi^{\prime})$. On the other hand,
it is shown in [Pr93, THEOREM 2] and [Ve13, Corollary 6.15] that $\pi$ is
$G_{n}$-distinguished.

  2. (2)

(Example from relatively projective representations [BZ77, CS21, Ch21]) Let
$\pi$ be the Steinberg representation of $G_{n+1}$. Assume $n\geq 2$. Let
$\pi^{\prime}=\mathrm{St}([-(n-5)/2,(n+1)/2]+[-(n-3)/2])$. Let

$\Delta_{1}=[-(n-1)/2,-(n-3)/2],\quad\Delta_{2}=[-(n-3)/2].$

We shall use the notations in Definition 8.1. In such case, we have:

$\eta_{\Delta_{1}}(\pi)=(1,0),\quad\eta_{\Delta_{1}}(I_{\Delta_{2}}(\pi))=(1,1).$

Hence, $\eta_{\Delta_{1}}(\pi)\neq\eta_{\Delta_{1}}(I_{\Delta_{2}}(\pi))$.
Using the equivalent condition in Definition 10.1 and checking other possible
strong commutations, one has that $(\pi,\pi^{\prime})$ is not relevant. On the
other hand, it is shown in [CS21, Theorem 3.3] that
$\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})=0$.

  3. (3)

(Example from non-tempered GGP [GGP20, Gu22, Ch22]) Let
$\pi=\langle[0]\rangle\times\langle[0]\rangle\times\langle[-1,1]\rangle$ and
let $\pi^{\prime}=\langle[-1/2,1/2]\rangle\times\mathrm{St}([-1/2,1/2])$. Let
$\mathfrak{m}=\left\\{[1/2],[1/2],[3/2]\right\\}$ and let
$\mathfrak{n}=\left\\{[-1/2,1/2]\right\\}$. One checks that $\mathfrak{m}$ and
$\mathfrak{n}$ define the relevance for $\pi$ and
$D^{R}_{\mathfrak{m}}(\nu^{1/2}\pi)\cong D^{L}_{\mathfrak{n}}(\pi^{\prime})$.
On the other hand, from [GGP20] and [Ch22], we have
$\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})\neq 0$.

####  4.3. Techniques in branching laws and derivatives

We highlight that the fifth one for standard modules below is recently
established in [Ch23], and the seventh and last ones below are mainly
developed in [Ch22+], and the sixth, eighth and ninth ones below seem to be
new in studying branching law. We consider the sixth one provides more
fundamental structure while in view of Appendix B, the eighth one provides
structural information refining the asymmetry property in [Ch21].

  1. (1)

Bernstein-Zelevinsky theory. The use of BZ filtration in branching laws goes
back to the earlier work of D. Prasad [Pr93] and Flicker [Fl93] for studying
distinguished representations.

  2. (2)

Left-Right BZ filtrations. It turns out that the left and right BZ filtrations
in Section 3.2 give complementary information in most of cases. Substantial
uses include branching law for the Steinberg representation [CS21] and for
indecomposability [Ch21].

  3. (3)

Rankin-Selberg integrals for constructing branching laws. One can realize
irreducible generic representations $\pi$ and $\pi^{\prime}$ of $G_{n+1}$ and
$G_{n}$ as Whittaker models. Then one can roughly define $G_{n}$-invariant
linear functionals on $\pi\otimes\pi^{\prime}{}^{\vee}$ via the Rankin-Selberg
integrals on their Whittaker functions. A detailed argument appears in [Pr93].
We shall use some variations from Cogdell–Piatetski-Shapiro [CPS17] to
construct branching laws in Section 24.

  4. (4)

Gan-Gross-Prasad type reduction and duality. The GGP type reduction relates
restriction problems of different coranks. With similar sort of ideas, one
has: for any representation $\pi$ of $G_{n+1}$ and $\pi^{\prime}$ of $G_{n}$

(4.2)
$\displaystyle\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime}{}^{\vee})\cong\mathrm{Hom}_{G_{n+1}}(\pi^{\prime}\times\sigma,\pi^{\vee})$

for some cuspidal representation $\sigma$ of $G_{2}$. Switching the dual can
sometimes combine with left-right filtration above, see for example the proof
of Theorem 26.4 and Proposition 31.1.

  5. (5)

Multiplicity one theorems for irreducible and standard representations. The
multiplicity one theorem for irreducible and standard representations [AGRS10,
Ch23] provides an uniqueness and thus one can combine with constructing
branching laws (e.g. Rankin-Selberg integral method above) to identify the
required maps. For example, this is used in Lemma 24.1.

We also remark that the multiplicity one for irreducible ones for fields of
positive characteristics is also established in the work of Aizenbud-Avni-
Gourevitch [AAG12] and Mezer [Me21].

  6. (6)

Characterizing the layer supporting a branching law from BZ derivatives.
Another application of the multiplicity one theorem for standard
representations (combined with some homological properties of standard
representations, see Lemma 22.5) is to give a characterization in Corollary
22.7. Such characterization combined with the computation for smallest
derivatives in the relevance pair (Section 21) automatically gives the refined
branching law, once we know the original branching law from relevance. In
sufficiency part, the refined branching law is crucial in singling out a
quotient that contributes the branching law in Lemma 25.2, and such quotient
satisfies certain multiplicity one property allowing to use the strategy in
Lemma 26.1. In necessity part, the refined branching law is crucial in showing
certain admissibility for some multisegments to get a relevance. Another use
is to give a refinement on the standard trick in Lemma 24.1 while it could
possibly be avoided.

  7. (7)

Minimal sequences in constructing simple quotients of BZ derivatives. As
suggested from the definition of generalized relevant pairs and Theorem 2.7,
minimal sequences of derivatives are crucial in the study. Remarkable
properties of the minimal sequences include [Ch22+]: Let $\mathfrak{m}$ be Rd-
minimal to $\pi\in\mathrm{Irr}$. Then

     * •

(Subsequent property) for any submultisegment $\mathfrak{n}$ of
$\mathfrak{m}$, $\mathfrak{n}$ is also Rd-minimal to $\pi$;

     * •

(Commutativity) for any submultisegment $\mathfrak{n}$ of $\mathfrak{m}$,
$\mathfrak{m}-\mathfrak{n}$ is also Rd-minimal to $D^{R}_{\mathfrak{n}}(\pi)$.

It turns out that both properties behave well in the content of strong
commutation, see Corollaries 15.9 and 15.10.

  8. (8)

Deforming branching laws. Let $\pi\in\mathrm{Irr}(G_{n+1})$ and let $\tau$ be
a simple quotient of $\pi^{(i)}$ for some $i$. Then $\tau\times\sigma$ appears
in the quotient of $\pi$ for some good choice of a cuspidal representation
$\sigma\in\mathrm{Irr}(G_{i-1})$. When $\pi$ is thickened (Definition 27.3),
one can carry out certain highest derivatives to construct branching laws, see
more discussions in the beginning of Section 27.

  9. (9)

Deforming simple quotients of BZ derivatives. For an irreducible
representation $\pi$, the Zelevinsky theory [Ze80] implies that there exists
an irreducible thickened representation such that ${}^{-}\pi^{\prime}\cong\pi$
and $\mathrm{lev}(\pi^{\prime})=\mathrm{lev}(\pi)$. It is shown in Theorem
29.3 that there is a natural one-to-one correspondence between the set of
simple quotients of $\pi^{\prime}{}^{(i)}$ and the set of simple quotients of
$\pi^{(i)}$ via taking the highest left derivatives.

The two ideas (8) and (9) of deformations are useful since thickened cases can
be achieved by simple combinatorics. They are also closely related.

  10. (10)

Double derivatives and integrals. Let $\pi\in\mathrm{Irr}$ and let
$\mathfrak{m}\in\mathrm{Mult}$ with $D^{R}_{\mathfrak{m}}(\pi)\neq 0$. Then
there exists a multisegment $\mathfrak{m}^{\prime}$ such that
$D^{R}_{\mathfrak{m}^{\prime}}\circ D^{R}_{\mathfrak{m}}(\pi)\cong\pi^{-}$
[Ch22+]. The integral analogue is established in Theorem 17.4. It is an
important step in Proposition 30.3.

####  4.4. Standard trick

We are going to outline the proofs which are inductive in nature. We first
explain some more notations and a standard trick below in doing induction. We
shall call it a standard trick since idea of this sort has been used in e.g.
[GGP12, Ch22].

Let $\Delta=[a,b]_{\rho}$ be a segment. A segment $\Delta^{\prime}$ is said to
be R-$\Delta$-saturated if $\Delta^{\prime}$ takes the form
$[a^{\prime},b]_{\rho}$ for some $a\leq a^{\prime}\leq b$ and a multisegment
$\mathfrak{p}$ is said to be R-$\Delta$-saturated if any segment in
$\mathfrak{p}$ is $R$-$\Delta$-saturated. A representation $\pi$ is said to be
R-$\Delta$-reduced if $D^{R}_{\Delta^{\prime}}(\pi)=0$ (i.e.
$\varepsilon_{\Delta^{\prime}}(\pi)=0$) for any R-$\Delta$-saturated segment
$\Delta^{\prime}$. These notions indicate a reduction technique. One also has
analogous left version of those notions, and we only remark that we use the
segments of the form $[a,b^{\prime}]_{\rho}$ (for $a\leq b^{\prime}\leq b$) in
the left version.

For a cuspidal representation $\rho$, we say that $\pi$ is strongly
R-$\rho$-reduced (resp. strongly L-$\rho$-reduced) if for any $\Delta$ with
$b(\Delta)\cong\rho$ (resp. $a(\Delta)\cong\rho$), $D^{R}_{\Delta}(\pi)=0$
(resp. $D^{L}_{\Delta}(\pi)=0$). A multisegment $\mathfrak{m}$ is said to be
strongly R-$\rho$-saturated (resp. strongly L-$\rho$-saturated) if any segment
$\Delta$ in $\mathfrak{m}$ satisfies $b(\Delta)\cong\rho$ (resp.
$a(\Delta)\cong\rho$).

For $\pi\in\mathrm{Irr}$, define

$\mathrm{csupp}_{\mathbb{Z}}(\pi)=\left\\{\nu^{c}\cdot\rho:\rho\in\mathrm{csupp}(\pi),\quad
c\in\mathbb{Z}\right\\}.$

An irreducible cuspidal representation $\sigma$ is said to be good to an
irreducible representation $\pi$ if $\sigma$ is not in
$\mathrm{csupp}_{\mathbb{Z}}(\pi)$. In particular, we have $\sigma\times\pi$
is irreducible for such $\sigma$ [Ze80].

######  Lemma 4.3.

Let $\Delta=[a,b]_{\rho}$ be a segment. Let $\pi\in\mathrm{Irr}(G_{n+1-k})$
and let $\pi^{\prime}\in\mathrm{Irr}(G_{n})$. Let $\mathfrak{p}$ be a
R-$\Delta$-saturated multisegment. Let $k=l_{a}(\mathfrak{p})$. Suppose
$\nu^{b}\rho$ is good to $\nu^{-1/2}\pi^{\prime}$. Let
$\sigma\in\mathrm{Irr}^{c}(G_{k})$ good to $\pi$ and $\nu^{-1/2}\pi^{\prime}$.
We also assume that $\nu^{b}\rho$ is $\leq$-maximal in
$\mathrm{csupp}(\mathfrak{p})+\mathrm{csupp}(\pi)$. Then

$\mathrm{Hom}_{G_{n}}(\mathrm{St}(\mathfrak{p})\times\pi,\pi^{\prime})\cong\mathbb{C}$

if and only if

$\mathrm{Hom}_{G_{n}}(\sigma\times\pi,\pi^{\prime})\cong\mathbb{C}.$

We shall prove an alternate and stronger form in Lemma 23.3. We remark that
the assumption $\nu^{b}\rho$ is $\leq$-maximal in
$\mathrm{csupp}(\mathfrak{p})+\mathrm{csupp}(\pi)$ is not essential in the
above lemma, but it is more convenient to consider the stronger form since we
then have a standard module projecting to
$\mathrm{St}(\mathfrak{p})\times\pi$.

###  5\. Outline of proofs

####  5.1. Outline of the proof of sufficiency

We start with $\pi\in\mathrm{Irr}(G_{n+1})$ and
$\pi^{\prime}\in\mathrm{Irr}(G_{n})$ such that $(\pi,\pi^{\prime})$ is
relevant. We mainly explain steps on showing the (non-refined) branching law,
and as mentioned before, the refined one follows from Proposition 21.2 and
Corollary 22.7.

Let $\rho^{\prime}$ be a $\leq$-minimal element in
$\mathrm{csupp}(\pi^{\prime})$ (see Section 2.1 for the ordering $\leq$). We
shall illustrate the idea by only considering the case that
$\rho^{\prime}\notin\mathrm{cuspp}(\nu^{1/2}\cdot\pi)$. Then we find a
multisegment $\mathfrak{p}^{\prime}$ such that
$D^{L}_{\mathfrak{p}^{\prime}}(\pi^{\prime})$ is strongly
L-$\rho^{\prime}$-reduced.

Now the compatibility of commutativity of minimal sequences and strongly
commutative sequences gives that
$(\pi,D^{L}_{\mathfrak{p}^{\prime}}(\pi^{\prime}))$ is still relevant (see
Corollary 15.10 and Lemma 19.3), and so for a suitable choice of cuspidal
representation $\sigma$ of $G_{l_{a}(\mathfrak{p}^{\prime})}$, one can use
induction to obtain

$\mathrm{Hom}_{G_{n}}(\pi,\sigma\times
D^{L}_{\mathfrak{p}^{\prime}}(\pi^{\prime}))\cong\mathbb{C}$

Now, let $\tau^{\prime}=\mathrm{St}(\mathfrak{p}^{\prime})\times
D^{L}_{\mathfrak{p}^{\prime}}(\pi^{\prime})$ (so that $\pi^{\prime}$ is a
submodule of $\tau^{\prime}$) and the standard trick gives that:

(5.3) $\displaystyle\mathrm{Hom}_{G_{n}}(\pi,\tau^{\prime})\cong\mathbb{C}.$

(Note that the strong form of the standard trick in Lemma 23.3 also determines
the layer in the BZ filtration contributing the branching law, say the one
from $i^{*}$-th one. Thus, this gives a non-zero map in

$\mathrm{Hom}_{G_{n}}(\pi^{[i^{*}]},{}^{(i^{*}-1)}\tau^{\prime}).$

Then uniqueness of the Hom (see Corollary 22.6) forces such map factors
through the submodule map from ${}^{(i^{*}-1)}\pi^{\prime}$ to
${}^{(i^{*}-1)}\tau^{\prime}$. This gives a strong evidence for
$\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})\neq 0$. One may also compare with some
arguments of this type in the necessity part below.)

To illustrate how to conclude the proof, we first consider a simpler
situation. Let $\rho$ be a $\leq$-maximal element in $\mathrm{csupp}(\pi)$.
Suppose $\rho$ is good to $\nu^{-1/2}\cdot\pi^{\prime}$. (This situation may
not always happen and we shall explain how to resolve this next.) Then we can
find a multisegment $\mathfrak{p}$ such that $D^{R}_{\mathfrak{p}}(\pi)$ is
strongly R-$\rho$-reduced (for the existence of such multisegment, see [LM22,
Section 7] and [Ch22+b, Section 2]).

Thus, let $\tau=\mathrm{St}(\mathfrak{p})\times D^{R}_{\mathfrak{p}}(\pi)$ (so
that $\pi$ is a quotient of $\tau$) and similar reasoning as before with the
standard trick gives that:

$\mathrm{Hom}_{G_{n}}(\tau,\pi^{\prime})\cong\mathbb{C}$

A variation of the standard trick also gives that

$\mathrm{Hom}_{G_{n}}(\tau,\tau^{\prime})\cong\mathbb{C}.$

Now $\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})\cong\mathbb{C}$ follows from some
constraints in module theory, see Lemma 26.1.

Now the technical issue is to deal with when such $\rho$ does not exist. To do
so, we introduce more notations. Let
$\mathfrak{m},\mathfrak{n}\in\mathrm{Mult}$ such that:

  * •

$D^{R}_{\mathfrak{m}}(\nu^{1/2}\cdot\pi)\cong
D^{L}_{\mathfrak{n}}(\pi^{\prime})$;

  * •

$(\mathfrak{m},\mathfrak{n},\nu^{1/2}\cdot\pi)$ is a minimal strongly RdLi-
commutative triple (in the sense of Theorem 2.7).

Let $\rho$ be a $\leq$-maximal element in $\mathrm{csupp}(\mathfrak{m})$. Then
find a longest segment $\Delta$ in $\mathfrak{m}_{b=\rho}$ (see Section 8.4
for unexplained notation). Let $\mathfrak{p}\in\mathrm{Mult}$ such that
$D_{\mathfrak{p}}^{R}(\pi)$ is R-$\Delta$-reduced.

A main problem is that it is unclear that

$\mathrm{Hom}_{G_{n}}(\mathrm{St}(\mathfrak{p})\times
D_{\mathfrak{p}}^{R}(\pi),\pi^{\prime})$

has dimension $1$. We take a roundabout by taking some quotients of
$\mathrm{St}(\mathfrak{p})\times D_{\mathfrak{p}}^{R}(\pi)$ and $\pi$.

In order to do so, let $\omega=D_{\mathfrak{p}}^{R}(\pi)$ and we consider a
filtration on $\mathrm{St}(\mathfrak{p})\times\omega$ via a variant of
Bernstein-Zelevinsky filtration in Proposition 3.2 of the form:

(5.4)
$\displaystyle\mathrm{St}(\mathfrak{p})^{[0]}\times(\omega|_{G_{n-l_{a}(\mathfrak{p})}}),\quad\mathrm{St}(\mathfrak{p})^{[1]}\times(\omega\otimes\zeta_{F}),$
(5.5)
$\displaystyle\mathrm{St}(\mathfrak{p})^{[k]}\times\mathrm{RS}_{k-2}(\omega),\quad\mbox{
for $k\geq 2$ }.$

Let
$k^{*}=l_{a}(\mathfrak{p})-l_{a}(\mathfrak{mx}(\nu^{1/2}\Delta,\pi^{\prime}))$
(see Section 8.3 for unexplained notions). We let $\tau$ be the quotient of
$(\mathrm{St}(\mathfrak{p})\times\xi)|_{G_{n}}$ module out the submodule
containing those factors

$\kappa:=\mathrm{St}(\mathfrak{p}^{\prime})^{[k]}\times\mathrm{RS}_{k-2}(\xi),\quad
k>k^{*}\geq 1.$

and let

$\widetilde{\tau}=\tau/\kappa.$

Let

$\widetilde{\pi}=\mathrm{pr}(\tau)/\mathrm{pr}(\kappa),$

where $\mathrm{pr}$ is the quotient map from $\tau$ to $\pi$. Let
$\tau^{\prime}=\mathrm{St}(\mathfrak{p})\times\omega$. It turns out that

(5.6)
$\displaystyle\mathrm{Hom}_{G_{n}}(\widetilde{\tau},\pi^{\prime})\cong\mathbb{C},\quad\mathrm{Hom}_{G_{n}}(\widetilde{\pi},\tau^{\prime})\cong\mathbb{C},\quad\mathrm{Hom}_{G_{n}}(\widetilde{\tau},\tau^{\prime})\cong\mathbb{C}.$

Then one applies Lemma 26.1 to obtain that
$\mathrm{Hom}_{G_{n}}(\widetilde{\pi},\pi^{\prime})\cong\mathbb{C}$, and so
$\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})\cong\mathbb{C}$.

We finally give comments on (5.6). The uniqueness part for
$\mathrm{Hom}_{G_{n}}(\widetilde{\tau},\pi^{\prime})$ and
$\mathrm{Hom}_{G_{n}}(\widetilde{\tau},\tau^{\prime})$ follow from analysing
the layers in (5.4) and (5.5) and seeing that the only possible layer
contributing the non-zero Hom is

$\mathrm{St}(\mathfrak{p}^{\prime})^{[k^{*}]}\times\mathrm{RS}_{k^{*}-2}(\omega)$

Then one applies Frobenius reciprocity and induction. (The uniqueness part for
$\mathrm{Hom}_{G_{n}}(\widetilde{\pi},\tau^{\prime})$ follows from (5.3).)

For the existence part of
$\mathrm{Hom}_{G_{n}}(\widetilde{\pi},\tau^{\prime})$, one first considers the
non-zero map $f$ (by (5.3)) in

$\quad\mathrm{Hom}_{G_{n}}(\pi,\tau^{\prime})$

and then shows that $f|_{\mathrm{pr}(\kappa)}=0$, which one analyses the
possible layer contributing the branching law in $\mathrm{pr}(\kappa)$ and
$\kappa$ (using the technique of smallest derivatives). The details of
computation is done in Section 21 (particularly Corollary 21.3). For the
existence part of $\mathrm{Hom}_{G_{n}}(\widetilde{\tau},\pi^{\prime})$, the
reasoning is similar, but one uses a Rankin-Selberg construction (with control
on the layer supporting the refined branching law, see Lemma 24.1) to replace
(5.3). (The existence for
$\mathrm{Hom}_{G_{n}}(\widetilde{\tau},\tau^{\prime})$ follows from any of the
above two cases.)

####  5.2. Outline of the proof of necessity

We start with $\pi\in\mathrm{Irr}(G_{n+1})$ and
$\pi^{\prime}\in\mathrm{Irr}(G_{n})$ with
$\mathrm{Hom}_{G_{n}}(\pi,\pi^{\prime})\neq 0$. Again, we mainly outline how
to show the relevance and the refined relevance will follow from Proposition
21.2 and Corollary 22.7.

Let $\rho$ be a $\leq$-maximal element in $\mathrm{csupp}(\pi)$. We shall only
deal with the case that $\rho\notin\mathrm{csupp}(\nu^{-1/2}\pi^{\prime})$.
The remaining case can be dealt with using the technique of left-right BZ
filtrations and the symmetry of relevant pairs in Theorem 2.5. This technique
has been used in [CS21, Ch21, Ch22, Ch23] before and so we also refer the
reader to those articles.

One can again find a strongly R-$\rho$-reduced multisegment $\mathfrak{p}$
such that

$\mathrm{St}(\mathfrak{p})\times
D^{R}_{\mathfrak{p}}(\pi)\twoheadrightarrow\pi.$

This gives

$\mathrm{Hom}_{G_{n}}(\mathrm{St}(\mathfrak{p})\times
D^{R}_{\mathfrak{p}}(\pi),\pi^{\prime})\neq 0.$

Thus, the standard trick in Lemma 4.3 gives that

$\mathrm{Hom}_{G_{n}}(\sigma\times D^{R}_{\mathfrak{p}}(\pi),\pi^{\prime})\neq
0$

for some suitable choice of a cuspidal representation $\sigma$. This
inductively gives the relevance for
$(D_{\mathfrak{p}}^{R}(\pi),\pi^{\prime})$. In other words, there exist
multisegments $\mathfrak{m}$ and $\mathfrak{n}$ such that
$(\mathfrak{m},\mathfrak{n},\nu^{1/2}\cdot D_{\mathfrak{p}}^{R}(\pi))$ is a
minimal strongly RdLi-commutative triple and

$D_{\mathfrak{m}}^{R}(\nu^{1/2}\cdot D_{\mathfrak{p}}^{R}(\pi))\cong
D_{\mathfrak{n}}^{L}(\pi^{\prime}).$

Let $\mathfrak{p}^{\prime}=\nu^{1/2}\mathfrak{p}$. Now showing the relevance
of $(\pi,\pi^{\prime})$ can be accomplished by the following steps:

  1. (1)

First, show that $\mathfrak{m}+\mathfrak{p}^{\prime}$ is admissible to
$\nu^{1/2}\cdot\pi$ and

(5.7) $\displaystyle
D^{R}_{\mathfrak{m}+\mathfrak{p}^{\prime}}(\nu^{1/2}\cdot\pi)\cong
D^{L}_{\mathfrak{n}}(\pi^{\prime}).$

The main idea is that the multiplicity one theorem for standard modules (see
Theorem 22.2 for a precise form) forces that

$\mathrm{dim}~{}\mathrm{Hom}_{G_{n}}((\mathrm{St}(\mathfrak{p})\times
D^{R}_{\mathfrak{p}}(\pi))^{[i]},{}^{(i-1)}\pi^{\prime})\leq 1$

and

$\mathrm{dim}~{}\mathrm{Hom}_{G_{n}}(\pi^{[i]},{}^{(i-1)}\pi^{\prime})\leq 1,$

where $i=l_{a}(\mathfrak{m}+\mathfrak{p}^{\prime})$. Indeed, both inequalities
are equation. The first one follows from the induction and a stronger form of
the standard trick (see Lemma 23.3). The second one follows from some control
in multiplicity one theorems (see Corollary 22.6).

The multiplicity-ones above then give the simple quotient
$D^{R}_{\mathfrak{m}}(\nu^{1/2}\cdot D^{R}_{\mathfrak{p}}(\pi))$ in

$(\mathrm{St}(\mathfrak{p})\times D^{R}_{\mathfrak{p}}(\pi))^{[i]}$

must come from $\pi^{[i]}$. (We remark that one has to use $i$-relevance to
show the appearance of the simple quotient.) The exhaustion theorem then
forces the admissibility (see Lemma 30.2). It turns out that
$\mathfrak{m}+\mathfrak{p}^{\prime}$ is also Rd-minimal to $\nu^{1/2}\cdot\pi$
from the theory of minimal sequences and hence (5.7) follows from the
commutativity of a minimal sequence.

  2. (2)

It remains to show that
$(\mathfrak{m}+\mathfrak{p}^{\prime},\mathfrak{n},\nu^{1/2}\cdot\pi)$ is a
strongly RdLi-commutative triple. This part again relies on the commutativity
of minimal sequences of derivatives and the compatibility with strong
commutation in Corollary 15.8.

####  5.3. Outline of the exhaustion part

We now discuss the exhaustion theorem of simple quotients of BZ derivatives
(Theorem 4.2). Note that our proof of Theorem 4.1 also relies on Theorem 4.2.
We shall also use Theorem 4.1 in our proof of Theorem 4.2, but only in a
smaller rank case. We shall refer the reader to Section 31.2 for a more
precise argument for the interactions of these two theorems.

Let $\tau$ be a simple quotient of $\pi^{[i]}$ for some $i$. Let
$\sigma^{\prime}\in\mathrm{Irr}^{c}(G_{n-i-1})$ be good to
$\nu^{1/2}\cdot\pi$. Then, a standard computation using right BZ filtration
gives that

$\mathrm{Hom}_{G_{n}}(\pi,\sigma^{\prime}\times\tau)\neq 0.$

One can now switch to the left BZ filtration, and an intriguing point here is
that the layer in the BZ filtration contributing to the branching law comes
from the bottom layer one, that is Theorem 28.1. (We remark that in order to
prove Theorem 28.1, one first has to consider a thickened case in Definition
27.3, which can be done with some combinatorial arguments. Then one deduces
the non-thickened case by the technique of deformation above.)

Let $\widetilde{\pi}=\nu^{-1/2}\cdot\pi$. The upshot of using the left BZ
filtration is that one is then in the position for applying the standard
trick. Now let $\mathfrak{h}$ be the highest left derivative multisegment of
$\pi$, which we mean
$D^{L}_{\mathfrak{h}}(\widetilde{\pi})={}^{-}\widetilde{\pi}$ (also see
Definition 8.3). Then Theorem 4.1 (which becomes available in the induction
process after the standard trick) with Theorem 2.5 and some commutativity for
minimal sequences (with using a duality of Proposition 14.2 switching to a
LdRi-version) gives that

$D_{\mathfrak{h}}^{L}\circ I^{R}_{\mathfrak{n}}(\widetilde{\pi})\cong
I^{R}_{\mathfrak{n}}\circ D_{\mathfrak{h}}^{L}(\widetilde{\pi})\cong\tau.$

The remaining step is to apply double integrals (Theorem 17.4), but for that
we first have to use deformation of derivatives to show that
$\mathrm{lev}(I^{R}_{\mathfrak{n}}(\pi))=\mathrm{lev}(\pi)$, that is Corollary
29.6.

Then, using double integrals, one finds a multisegment $\mathfrak{n}^{\prime}$
such that

$D_{\mathfrak{h}}^{L}\circ I^{R}_{\mathfrak{n}^{\prime}}\circ
I^{R}_{\mathfrak{n}}(\widetilde{\pi})\cong{}^{-}(I^{R}_{\mathfrak{n}^{\prime}}\circ
I^{R}_{\mathfrak{n}}(\widetilde{\pi}))\cong\nu\cdot\widetilde{\pi}$

(the first isomorphism uses the
$\mathrm{lev}(I^{R}_{\mathfrak{n}^{\prime}}\circ
I^{R}_{\mathfrak{n}}(\widetilde{\pi}))=\mathrm{lev}(I^{R}_{\mathfrak{n}}(\widetilde{\pi}))=\mathrm{lev}(\widetilde{\pi})$).
With the commutativity from level-preserving integrals (Proposition 17.2), we
have

$I^{R}_{\mathfrak{n}^{\prime}}\circ D^{L}_{\mathfrak{h}}\circ
I^{R}_{\mathfrak{n}}(\widetilde{\pi})\cong D_{\mathfrak{h}}^{L}\circ
I^{R}_{\mathfrak{n}^{\prime}}\circ I^{R}_{\mathfrak{n}}(\widetilde{\pi}).$

The RHS is isomorphic to $\nu\cdot\widetilde{\pi}$ as discussed above.
Applying $D_{\mathfrak{n}^{\prime}}^{R}$ to give

$D_{\mathfrak{n}^{\prime}}^{R}(\nu^{1/2}\cdot\pi)\cong\tau.$

###  6\. Summary of some key results and their relations

We shall abbreviate BL for branching law and also abbreviate the following
results:

  * •

(StCo): Properties for (sequences of) strongly commutative triples (Sections
11-14)

  * •

(BZd): Characterize BZ layers determining branching law in terms of
derivatives (Corollary 22.7)

  * •

(UqR): Uniqueness of relevant pairs (Theorem 16.2)

  * •

(SmD): The smallest derivative for relevant pairs (Proposition 21.2)

  * •

(RS): Rankin-Selberg construction (Lemma 24.1)

  * •

(DdDi): Double derivatives and double integrals (Theorems 17.3 and 17.4)

  * •

(SyR): Symmetry of relevant pairs (Theorem 18.1) + duality (Proposition 14.2)

  * •

(Su): Sufficiency of relevant pairs (Theorem 26.4)

$\textstyle{\mathrm{(BZd)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$relation
to relevanceBL-relevance
interpretation$\textstyle{\mathrm{(UqR)}}$$\textstyle{\mathrm{(SmD)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$characterize
BLcharacterize
BL$\textstyle{\mathrm{(StCo)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$level-
preservingtool$\textstyle{\mathrm{(Su)}}$$\textstyle{\mathrm{(RS)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$construct
BL$\textstyle{\mathrm{(SyR)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$duality$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{(DdDi)}}$derivative
interpretation

  * •

(BDth): Left-right branching law from BZ derivatives in thickened cases
(Section 28.3)

  * •

(BD): Left-right branching law from some BZ derivatives (Theorem 28.1)

  * •

(DDer): Deform simple quotients of BZ derivatives by taking highest
derivatives (Theorem 29.3)

  * •

(ExT): Exhaustion theorem for BZ derivatives (Theorem 31.3)

  * •

(Ne): Necessity of relevant pairs (Theorem 31.2)

$\textstyle{\mathrm{(BDth)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$deform
(Section
27)$\textstyle{\mathrm{(BD)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$BL
interpretationconstruct derivatives from
BL$\textstyle{\mathrm{(SyR)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$duality$\textstyle{\mathrm{(Ne)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{(ExT)}}$$\textstyle{\mathrm{(DdDi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$construct
derivatives$\textstyle{(\mathrm{StCo})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$tooltool$\textstyle{(\mathrm{DDer})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$construct/characterize
derivatives

In Appendix B, we also have $(\mathrm{BD})+(\mathrm{BZd})\Rightarrow$ the
asymmetry property of left-right BZ-derivatives in [Ch21].

We also summarize key techniques in constructing (sequences of) strongly
commutative triples:

  * •

Unlinked pairs (Section 11)

  * •

Intersection-union process (Sections 12 and 13)

  * •

Commutativity from minimality (Sections 12 and 13)

  * •

Level-preserving integrals (Proposition 17.2)

  * •

Completing to $\Delta$-reduced triples (Corollary 20.4)

For example, level-preserving integrals play an important role in proving the
symmetry property while the commutativity from minimality is usually combined
with induction from the standard trick.

We finally remark that the above relations mainly follow from our logical
development. One can also reverse some arrows once the main results are
established.

###  7\. Acknowledgments

This article is benefited from communications with Wee Teck Gan, Erez Lapid,
Dipendra Prasad and Gordan Savin. The author would like to thank them all.

##  Part II Equivalent definitions of strongly commutative triples

We review some key results in [Ch22+] and [Ch22+d] in this part.

###  8\. Notations on $\eta$-invariants

####  8.1. Notations

Most of time, we shall usually give/prove statements involving right
derivatives and/or left integrals. The analogous version switching between
left and right can be formulated/proved similarly and will be left to the
reader.

For lightening notation, set $D_{\Delta}=D^{R}_{\Delta}$ and
$I_{\Delta}=I^{L}_{\Delta}$ for a segment $\Delta$.

####  8.2. $\eta$-invariants

######  Definition 8.1.

Let $\Delta=[a,b]_{\rho}$ be a segment. Let $\pi\in\mathrm{Irr}$.

  * •

Define $\varepsilon_{\Delta}(\pi)=\varepsilon_{\Delta}^{R}(\pi)$ (resp.
$\varepsilon_{\Delta}^{L}(\pi)$) to be the largest non-negative integer $k$
such that $D^{k}_{\Delta}(\pi)\neq 0$ (resp. $(D^{L}_{\Delta})^{k}(\pi)\neq
0$). (The power $k$ means doing compositions $k$-times.)

  * •

Define

$\eta_{\Delta}(\pi)=\eta^{R}_{\Delta}(\pi)=(\varepsilon_{[a,b]_{\rho}}(\pi),\varepsilon_{[a+1,b]_{\rho}}(\pi),\ldots,\varepsilon_{[b,b]_{\rho}}(\pi));$
$(resp.\quad\eta_{\Delta}^{L}(\pi)=(\varepsilon^{L}_{[a,b]_{\rho}}(\pi),\varepsilon^{L}_{[a,b-1]_{\rho}}(\pi),\ldots,\varepsilon^{L}_{[a,a]_{\rho}}(\pi))).$

  * •

We may simply write $\eta_{\Delta}(\pi)=0$ if
$\varepsilon_{[c,b]_{\rho}}(\pi)=0$ for all $c=a,\ldots,b$. Similarly, we may
write $\eta_{\Delta}(\pi)\neq 0$ if $\varepsilon_{[c,b]_{\rho}}(\pi)\neq 0$
for some $c=a,\ldots,b$.

  * •

We write $\eta_{\Delta}(\pi)?\ \eta_{\Delta}(\pi^{\prime})$ for
$?\in\left\\{=,\leq,<,\geq,>\right\\}$ if
$\varepsilon_{[c,b]_{\rho}}(\pi)?\varepsilon_{[c,b]_{\rho}}(\pi^{\prime})$ for
all $c$ satisfying $a\leq c\leq b$.

  * •

We similarly define the left version terminologies for
$\varepsilon^{L}_{\Delta}$ and $\eta^{L}_{\Delta}$ if one uses the left
derivatives $D^{L}_{\Delta}$.

####  8.3. Multisegment counterpart of the $\eta$-invariant

######  Definition 8.2.

For $\pi\in\mathrm{Irr}$ and a segment $\Delta=[a,b]_{\rho}$, define

$\mathfrak{mx}(\pi,\Delta):=\mathfrak{mx}^{R}(\pi,\Delta):=\sum_{k=0}^{b-a}\varepsilon_{[a+k,b]_{\rho}}(\pi)\cdot[a+k,b]_{\rho},$
$\mathfrak{mx}^{L}(\pi,\Delta):=\sum_{k=0}^{b-a}\varepsilon^{L}_{[a,b-k]_{\rho}}(\pi)\cdot[a,b-k]_{\rho}.$

Here the numbers $\varepsilon_{[a+k,b]_{\rho}}(\pi)$ and
$\varepsilon^{L}_{[a,b-k]_{\rho}}(\pi)$ are the multiplicities of those
segments.

For $\pi\in\mathrm{Irr}$ and $\rho\in\mathrm{Irr}^{c}$, similarly define:

$\mathfrak{mxpt}(\pi,\rho):=\mathfrak{mxpt}^{R}(\pi,\rho):=\sum_{\Delta:b(\Delta)\cong\rho}\varepsilon_{\Delta}(\pi)\cdot\Delta,$
$\mathfrak{mxpt}^{L}(\pi,\rho)=\sum_{\Delta:a(\Delta)\cong\rho}\varepsilon^{L}_{\Delta}(\pi)\cdot\Delta$

We have that $D_{\mathfrak{mx}(\pi,\Delta)}(\pi)$ is R-$\Delta$-reduced (see
Section 4.4), and similarly $D_{\mathfrak{mx}^{L}(\pi,\Delta)}^{L}(\pi)$ is
L-$\Delta$-reduced (i.e. $\eta_{\Delta}^{L}(\pi)=0$). For the details, one
sees [Ch22+, Corollary 7.23] (also see [Ch22+b, LM22]). Moreover, if
$\mathfrak{m}$ is a $\Delta$-saturated multisegment such that
$D_{\mathfrak{m}}(\pi)\neq 0$, then $\mathfrak{m}$ is a submultisegment of
$\mathfrak{mx}(\pi,\Delta)$.

####  8.4. More notations on multisegments

For $\mathfrak{m}\in\mathrm{Mult}$ and $\rho\in\mathrm{Irr}^{c}$, let

$\mathfrak{m}_{a=\rho}=\left\\{\Delta\in\mathfrak{m}:a(\Delta)\cong\rho\right\\},\quad\mathfrak{m}_{b=\rho}=\left\\{\Delta\in\mathfrak{m}:b(\Delta)\cong\rho\right\\}.$

We remark that $\varepsilon_{\Delta}(\pi)$ is equal to the cardinality of:

$\left\\{\widetilde{\Delta}\in\mathfrak{hd}(\pi)_{a=a(\Delta)}:\Delta\subset\widetilde{\Delta}\right\\}.$

####  8.5. Highest derivative multisegments $\mathfrak{hd}$

Following results in [Ch22+], we shall use the following quick definition of
the highest derivative multisegment.

######  Definition 8.3.

[Ch22+] Let $\pi\in\mathrm{Irr}$. Define the highest right (resp. left)
derivative multisegment, denoted by
$\mathfrak{hd}(\pi):=\mathfrak{hd}^{R}(\pi)$ (resp. $\mathfrak{hd}^{L}(\pi)$),
of $\pi$ to be the unique minimal multisegment such that

$D_{\mathfrak{hd}(\pi)}(\pi)\cong\pi^{-},\quad(\mbox{resp.
$D_{\mathfrak{hd}^{L}(\pi)}^{L}(\pi)\cong{}^{-}\pi$}).$

######  Definition 8.4.

[Ch22+, Section 7] Let $\mathfrak{h}\in\mathrm{Mult}$. Let
$\Delta=[a,b]_{\rho}$. The removal process for $(\Delta,\mathfrak{h})$ is an
algorithm to carry out the following steps:

  1. (1)

Find the shortest segment $[a,b^{\prime}]_{\rho}$ in $\mathfrak{h}$ with
$b^{\prime}\geq b$. Name the segment to be $\Delta_{1}$.

  2. (2)

We inductively define the shortest segment $\Delta_{i}=[a_{i},b_{i}]_{\rho}$
(for $i\geq 2$) in $\mathfrak{h}$ satisfying $a_{i-1}<a_{i}$ and
$b_{i}<b_{i-1}$. The process terminates when no more such segment can be found
and we denote $\Delta_{2},\ldots,\Delta_{r}$ to be all those segments.

  3. (3)

Define new segments as follows:

     * •

$\Delta^{tr}_{i}=[a_{i+1},b_{i}]_{\rho}$ for $i<r$

     * •

$\Delta^{tr}_{r}=[b_{r-1}+1,b_{r}]_{\rho}$ (possibly empty).

  4. (4)

Define

$\mathfrak{r}(\Delta,\mathfrak{h})=\mathfrak{h}-\sum_{i=1}^{r}\Delta_{i}+\sum_{i=1}^{r}\Delta_{i}^{tr}.$

We shall call $\Delta_{1},\ldots,\Delta_{r}$ to be the removal sequence for
$(\Delta,\mathfrak{h})$.

For a multisegment
$\mathfrak{m}=\left\\{\Delta_{1},\ldots,\Delta_{r}\right\\}$ written in an
ascending order, define

$\mathfrak{r}(\mathfrak{m},\mathfrak{h})=\mathfrak{r}(\Delta_{r},\ldots,\mathfrak{r}(\Delta_{1},\mathfrak{h})\ldots).$

It is shown in [Ch22+, Section 7] that
$\mathfrak{r}(\mathfrak{m},\mathfrak{h})$ is independent of a choice of such
ordering.

For a segment $\Delta$ and a multisegment $\mathfrak{h}$, define
$\varepsilon_{\Delta}(\mathfrak{h})$ to be the number of segments
$\widetilde{\Delta}$ in $\mathfrak{h}_{a=a(\Delta)}$ with
$\widetilde{\Delta}\subset\Delta$.

As shown in [Ch22+], the removal process is a simpler process in keep-tracking
the effect of derivatives. For a multisegment $\mathfrak{h}$, define

$\varepsilon_{[a,b]_{\rho}}(\mathfrak{h})=|\left\\{[a,c]_{\rho}\in\mathfrak{h}:b\leq
c\right\\}|.$

In particular, we have:

######  Theorem 8.5.

[Ch22+, Theorem 7.20] Let $\pi\in\mathrm{Irr}$ and let
$\mathfrak{h}=\mathfrak{hd}(\pi)$. Let $\Delta$ be a segment such that
$D_{\Delta}(\pi)\neq 0$. Then for any segment $\Delta^{\prime}\not<\Delta$,
$\varepsilon_{\Delta^{\prime}}(D_{\Delta}(\pi))$ is equal to
$\varepsilon_{\Delta^{\prime}}(\mathfrak{r}(\Delta,\mathfrak{h}))$.

####  8.6. Change of $\eta_{\Delta}$ by a left integral

The following lemma will be useful later.

######  Lemma 8.6.

Let $\pi\in\mathrm{Irr}$. For any segments $\Delta$ and $\Delta^{\prime}$,
$\mathfrak{mx}(\pi,\Delta)$ is a submultisegment of
$\mathfrak{mx}(I_{\Delta^{\prime}}(\pi),\Delta)$, equivalently

$\eta_{\Delta}(\pi)\leq\eta_{\Delta}(I_{\Delta^{\prime}}(\pi)).$

###### Proof.

Let $\mathfrak{m}=\mathfrak{mx}(\pi,\Delta)$ and let
$\sigma=\mathrm{St}(\Delta^{\prime})$. We have the embedding:

$\pi\hookrightarrow D_{\mathfrak{m}}(\pi)\times\mathrm{St}(\mathfrak{m}).$

Then

$I_{\sigma}(\pi)\hookrightarrow\sigma\times\pi\hookrightarrow\sigma\times
D_{\mathfrak{m}}(\pi)\times\mathrm{St}(\mathfrak{m}).$

Thus $D_{\mathfrak{m}}(I_{\sigma}(\pi))\neq 0$. Then, for instance, one
applies Theorem 8.5 and [Ch22+, Corollary 7.23] to see that
$\mathfrak{m}\subset\mathfrak{mx}(I_{\sigma}(\pi),\Delta)$.

∎

####  8.7. Some properties on $|\eta|_{\Delta}$

######  Definition 8.7.

(c.f. [LM22, Section 7]) For $\pi\in\mathrm{Irr}$ and a segment
$\Delta=[a,b]_{\rho}$, define

$|\eta|_{\Delta}(\pi)=\varepsilon_{[a,b]_{\rho}}(\pi)+\varepsilon_{[a+1,b]_{\rho}}(\pi)+\ldots+\varepsilon_{[b,b]_{\rho}}(\pi).$

We shall use the following properties in Section 16.2.

######  Proposition 8.8.

Let $\pi\in\mathrm{Irr}$ and $\Delta=[a,b]_{\rho}\in\mathrm{Seg}$. Let
$\Delta^{\prime}=[c,d]_{\rho}$ be a segment with $D_{\Delta^{\prime}}(\pi)\neq
0$. Then

  1. (1)

For $c<a$ and $d=b$,
$\eta_{\Delta}(\pi)=\eta_{\Delta}(D_{\Delta^{\prime}}(\pi))$.

  2. (2)

For $b\geq c\geq a$ and $d=b$,
$|\eta|_{\Delta}(\pi)=|\eta|_{\Delta}(D_{\Delta^{\prime}}(\pi))-1$.

  3. (3)

For $b>d\geq c\geq a$,
$|\eta|_{\Delta}(\pi)=|\eta|_{\Delta}(D_{\Delta^{\prime}}(\pi))$.

  4. (4)

For $b>d$ and $a>c$,
$\eta_{\Delta}(D_{\Delta^{\prime}}(\pi))\geq\eta_{\Delta}(\pi)$.

###### Proof.

We shall give a proof using removal sequences. A possible alternative is to do
analysis in the geometric lemma.

For (2), let $\Delta_{1},\ldots,\Delta_{r}$ be the removal sequence for
$(\Delta^{\prime},\pi)$, and let $\Delta_{1}^{tr},\ldots,\Delta_{r}^{tr}$ be
the truncated segments. Note that, by Definition 8.4, those
$\Delta_{1},\ldots,\Delta_{r}$ contribute to $|\eta|_{\Delta}(\pi)$. Only
$\Delta_{1}^{tr},\ldots,\Delta_{r-1}^{tr}$, but not $\Delta_{r}^{tr}$
contribute to $|\eta|_{\Delta}(\pi)$. Now Theorem 8.5 concludes that
$|\eta|_{\Delta}(\pi)$ and $|\eta|_{\Delta}(D_{\Delta^{\prime}}(\pi))$ differs
by $1$.

For (1), it is similar to the one for (2). Use the terminologies above.
Suppose $i$ is the smallest integer such that $\Delta_{i}$ contributes
$\eta_{\Delta}(\pi)$. We have that $i>1$ from the condition of (1). We only
have $\Delta_{i},\ldots,\Delta_{r}$ (among those $\Delta_{k}$’s) contributing
$\eta_{\Delta}(\pi)$ and only have
$\Delta_{i-1}^{tr},\ldots,\Delta_{r-1}^{tr}$ (among those $\Delta_{k}^{tr}$’s)
contributing $\eta_{\Delta}(D_{\Delta^{\prime}}(\pi))$. From the removal
process in Definition 8.4(3), one also has that $\Delta_{k-1}^{tr}$ and
$\Delta_{k}$ contribute to the same $\varepsilon_{\widetilde{\Delta}}$ for
some $\Delta$-saturated segment $\widetilde{\Delta}$. Now Theorem 8.5
concludes (1).

(3) is similar to (2) and (1). Using notations in (2), suppose
$\Delta_{1},\ldots,\Delta_{j}$ (possibly $i=0$) are those all, among those
$\Delta_{k}$’s, contributing to $|\eta|_{\Delta}(\pi)$. Then,
$\Delta_{1}^{tr},\ldots,\Delta_{j}^{tr}$ are all, among those
$\Delta_{k}^{tr}$’s, contributing to
$|\eta|_{\Delta}(D_{\Delta^{\prime}}(\pi))$. Now (3) again follows from
Theorem 8.5.

(4) is also similar. Use the notations in (2). By (3), we may assume that
$a>c$. Suppose $\Delta_{i},\ldots,\Delta_{j}$ are those all contributing to
$\eta_{\Delta}(\pi)$. (If those segments $\Delta_{i},\ldots,\Delta_{j}$ do not
exist, then the case is slightly simpler as the number may increase by the
contribution from a truncated segment.) Then
$\Delta_{i-1}^{tr},\ldots,\Delta_{j}^{tr}$ are those all contributing to
$\eta_{\Delta}(\pi)$. Again, we have that among those, $\Delta_{k-1}^{tr}$ and
$\Delta_{k}$ give contribution to the same $\varepsilon_{\widetilde{\Delta}}$
for some $\Delta$-saturated segment $\widetilde{\Delta}$; for
$\Delta_{j}^{tr}$, it increases $\varepsilon_{\Delta_{j}^{tr}}$ in
$\eta_{\Delta}$ by $1$. One then uses Theorem 8.5 again. ∎

We give a refinement of Proposition 8.8(3).

######  Proposition 8.9.

We use the notations in the previous lemma. Suppose $b>d\geq c\geq a$. Let
$\widetilde{\Delta}=[\widetilde{a},\widetilde{b}]_{\rho}$ be the first segment
in the removal sequence for $(\Delta^{\prime},\mathfrak{hd}(\pi))$.

  1. (1)

If $\widetilde{b}<b$, then
$\eta_{\Delta^{\prime}}(D_{\Delta}(\pi))=\eta_{\Delta^{\prime}}(\pi)$.

  2. (2)

If $\widetilde{b}\geq b$, then there exists a $\Delta$-saturated segment
$\overline{\Delta}=[\overline{a},b]_{\rho}$ with $\overline{a}>c$ such that
$\eta_{\overline{\Delta}}(D_{\Delta}(\pi))\neq\eta_{\overline{\Delta}}(\pi)$
and $\varepsilon_{\overline{\Delta}}(D_{\Delta}(\pi))>0$.

###### Proof.

Let $\Delta_{1},\ldots,\Delta_{r}$ be the removal sequence for
$(\Delta^{\prime},\mathfrak{hd}(\pi))$. Let
$\Delta_{1}^{tr},\ldots,\Delta_{r}^{tr}$ be the truncated ones. For (1), by
Definition 8.4 (2) and (3), none of those segments contributes to
$\eta_{\Delta}(D_{\Delta^{\prime}}(\pi))$ and $\eta_{\Delta}(\pi)$. Thus, we
have the equality.

We now consider (2). Let $k$ be the largest integer such that
$a(\Delta_{k})\geq a(\Delta)$. Such integer exists by the assumption. If
$k\neq r$, we pick $\overline{\Delta}=[a(\Delta_{k+1}),b(\Delta)]$; and if
$k=r$, we pick $\overline{\Delta}=[\nu
b(\Delta^{\prime}),b(\Delta)](\neq\emptyset)$. Then $\Delta_{k}^{tr}$ will add
extra one to the coordinates $\varepsilon_{\overline{\Delta}}(\pi)$ to get
$\varepsilon_{\overline{\Delta}}(D_{\Delta^{\prime}}(\pi))$ while other
coordinates are unchanged. This gives the desired statement. ∎

####  8.8. A criteria on minimality

We first have the following minimality. The following is shown in [Ch22+,
Proposition 17.2], while we demonstrate a different argument (which however
depends on results [Ch22+] as well).

######  Lemma 8.10.

Let $\mathfrak{m}\in\mathrm{Mult}$ be minimal to $\pi$. Let $\Delta$ be a
segment such that

$a(\widetilde{\Delta})<a(\Delta),\quad b(\widetilde{\Delta})<b(\Delta)$

for any segment $\widetilde{\Delta}\in\mathfrak{m}$. We also assume
$D_{\Delta}\circ D_{\mathfrak{m}}(\pi)\neq 0$. Then $\mathfrak{m}+\Delta$ is
minimal to $\pi$ if and only if

$\eta_{\Delta}(D_{\mathfrak{m}}(\pi))=\eta_{\Delta}(\pi).$

###### Proof.

Note that when $\mathfrak{m}$ has only one segment. This is proved in [Ch22+,
Lemma 14.3 and Proposition 14.4].

Let $\mathfrak{m}=\left\\{\Delta_{1},\ldots,\Delta_{r}\right\\}$ in an
ascending order. Suppose $\mathfrak{m}+\Delta$ is minimal to $\pi$. Then, by
the subsequent property [Ch22+], $\Delta_{1}+\Delta$ is still minimal to
$\pi$. Thus, by the two segment case [Ch22+],
$\eta_{\Delta}(D_{\Delta_{1}}(\pi))=\eta_{\Delta}(\pi)$. We have that
$\left\\{\Delta_{2},\ldots,\Delta_{r}\right\\}+\Delta$ is minimal to
$D_{\Delta_{1}}(\pi)$ by [Ch22+] (also see Lemma 15.6 below) and so by
induction,
$\eta_{\Delta}(D_{\mathfrak{m}}(\pi))=\eta_{\Delta}(D_{\Delta_{1}}(\pi))$ and
so we have the only if direction.

For the if direction, suppose $\mathfrak{m}+\Delta$ is not minimal. Then, by
[Ch22+, Corollary 1.6], we can find a consecutive pair such that the
intersection-union still gives the same derivatives. Then, we must have one
segment to be $\Delta$. Then, by using consecutive pairs, we may relabel the
segments such that another segment is $\Delta_{r}$. Then [Ch22+, Proposition
14.4] and Proposition 8.8(4) imply that

$\eta_{\Delta}(D_{\Delta_{r}}\circ
D_{\mathfrak{m}-\Delta_{r}}(\pi))>\eta_{\Delta}(D_{\mathfrak{m}-\Delta_{r}}(\pi)).$

On the other hand, by Proposition 8.8(4):

$\eta_{\Delta}(D_{\Delta_{r-1}}\circ\ldots\circ
D_{\Delta_{1}}(\pi))\geq\ldots\geq\eta_{\Delta}(D_{\Delta_{1}}(\pi)).$

Thus, combining, we obtain that
$\eta_{\Delta}(D_{\mathfrak{m}}(\pi))>\eta_{\Delta}(\pi)$. ∎

###  9\. Strong commutativity

Since most of results in this section have been shown in [Ch22+d], some parts
will be brief. For generic representations $\sigma,\sigma^{\prime}$, we can
extend the definition of pre-commutativity and strong commutativity for a
triple $(\sigma,\sigma^{\prime},\pi)$ in an obvious manner (see [Ch22+d] for
more discussions). In particular, when $\sigma=\mathrm{St}(\Delta)$ and
$\sigma^{\prime}=\mathrm{St}(\Delta^{\prime})$, their terminologies for
$(\sigma,\sigma^{\prime},\pi)$ coincide with $(\Delta,\Delta^{\prime},\pi)$.

####  9.1. Geometric lemma

For a $G_{k}\times G_{l}$-representation $\omega$ and a $G_{n}$-representation
$\pi$, inflate the $G_{n}\times G_{k}\times G_{l}$-representation
$\pi\boxtimes\omega$ to a $P_{n+k}\times G_{l}$-representation. Denote the
(normalized) parabolically induced module

$\mathrm{Ind}_{P_{n,k}\times G_{l}}^{G_{n+k}\times G_{l}}(\pi\boxtimes\omega)$

by $\pi\dot{\times}^{1}\omega$.

Recall that the geometric lemma is shown in [BZ77], which is important in
formulating the pre-commutativity in Definition 2.2 and [Ch22+d]. For
$\pi\in\mathrm{Alg}(G_{k})$ and $\pi^{\prime}\in\mathrm{Alg}(G_{l})$ and for
$r\leq l$, $P_{k+l}P_{k+l-r,r}$ is the closed set in $G_{k+1}$. This gives
rise the top layer $\pi\dot{\times}^{1}(\pi^{\prime}_{N_{r}})$ in the
geometric lemma on $(\pi\times\pi^{\prime})_{N_{r}}$. Hence, it gives a
natural surjection from $(\pi\times\pi^{\prime})_{N_{r}}$ to
$\pi\dot{\times}^{1}(\pi^{\prime}_{N_{r}})$ .

####  9.2. Strong commutativity $\Rightarrow$ Commutativity

It is not hard to prove the following proposition from definitions:

######  Proposition 9.1.

[Ch22+d, Proposition 5.1] Let $\sigma_{1},\sigma_{2}$ be generic
representations. Let $(\sigma_{1},\sigma_{2},\pi)$ be a strongly RdLi-
commutative triple. Then $I_{\sigma_{2}}\circ D_{\sigma_{1}}(\pi)\cong
D_{\sigma_{1}}\circ I_{\sigma_{2}}(\pi)$.

####  9.3. Examples of pre-commutativity

We give simple examples of pre-commutativity, which can be deduced from a
simple application of the geometric lemma:

######  Example 9.2.

  1. (1)

Suppose $\Delta_{1}\cap\Delta_{2}=\emptyset$. Then
$(\Delta_{1},\Delta_{2},\pi)$ is a pre-RdLi-commutative triple (for any
$\pi\in\mathrm{Irr}$).

  2. (2)

Let $\Delta_{1}=[a_{1},b_{1}]_{\rho}$ and let
$\Delta_{2}=[a_{2},b_{2}]_{\rho}$ be segments. Suppose $a_{2}<a_{1}$ or
$b_{2}<b_{1}$. Then $(\Delta_{1},\Delta_{2},\pi)$ is a pre-RdLi-commutative
triple (for any $\pi\in\mathrm{Irr}$).

####  9.4. Pre-commutativity $\Rightarrow$ Strong commutativity

######  Proposition 9.3.

[Ch22+d, Proposition 6.3] Let
$(\mathrm{St}(\Delta),\mathrm{St}(\Delta^{\prime}),\pi)$ be a pre-RdLi-
commutative triple. Let $\mathfrak{p}=\mathfrak{mx}(\Delta,\pi)$. Then
$(\mathrm{St}(\mathfrak{p}),\mathrm{St}(\Delta^{\prime}),\pi)$ is also a pre-
RdLi-commutative triple.

###### Proof.

We only sketch the proof. Now one labels the segments in $\mathfrak{p}$ such
that

$a(\Delta_{r})\leq\ldots\leq a(\Delta_{1}).$

Let $\mathfrak{p}_{k}=\left\\{\Delta_{1},\ldots,\Delta_{k}\right\\}$. One
proceeds inductively on $k$. When $k=1$, it is automatic from the given
hypothesis. We shall assume that we are not in a case of Example 9.2.

Suppose we have that
$(\mathrm{St}(\mathfrak{p}_{k}),\mathrm{St}(\Delta^{\prime}),\pi)$ is a
strongly RdLi-commutative triple. We now proceed in two cases:

  1. (1)

$a(\Delta_{k+1})>a(\Delta^{\prime})$. This follows from a simple application
of the geometric lemma (or uses Example 9.2(2)) and the inductive hypothesis.

  2. (2)

$a(\Delta_{k+1})\leq a(\Delta^{\prime})$. Suppose
$(\mathrm{St}(\mathfrak{p}),\mathrm{St}(\Delta_{2}),\pi)$ is not strongly
RdLi-commutative triple. Let $l=l_{a}(\mathfrak{p}_{k})$. Write
$\Delta=[a,b]_{\rho}$ and $\Delta^{\prime}=[a^{\prime},b^{\prime}]_{\rho}$.
Let

$\overline{\Delta}^{\prime}=[b+1,b^{\prime}]_{\rho},\quad\underline{\Delta}^{\prime}=[a^{\prime},b]_{\rho}.$

Let $p=l_{a}(\mathfrak{p}_{k})$, $n=l_{a}(\Delta_{2})+n(\pi)$,
$r=l_{a}(\overline{\Delta}^{\prime})$ and
$s=l_{a}(\underline{\Delta}^{\prime})$. Then, there is only one possible
geometric layer can work:

$D_{\mathfrak{p}_{k+1}}(\pi)\boxtimes\mathrm{St}(\mathfrak{p}_{k+1})\hookrightarrow\mathrm{Ind}_{P_{r,n-p-r}\times
P_{s,p-s}}^{G_{n-p}\times
G_{p}}(\mathrm{St}(\overline{\Delta}^{\prime})\boxtimes\mathrm{St}(\underline{\Delta}^{\prime})\boxtimes\pi_{N_{l}})^{\phi},$

where $\phi$ is a twist bringing to a $G_{r}\times G_{n-p-r}\times G_{s}\times
G_{p-s}$-representation. Let $\lambda$ be the rightmost representation. Now
the pre-commutativity for
$(\mathrm{St}(\mathfrak{p}_{k}),\mathrm{St}(\Delta_{2}),\pi)$ forces that the
submodule

$D_{\mathfrak{p}_{k+1}}(\pi)\boxtimes\mathrm{St}(\underline{\Delta}^{\prime})\boxtimes
D^{L}_{\underline{\Delta}^{\prime}}(\mathrm{St}(\mathfrak{p}_{k+1}))\hookrightarrow\lambda_{N_{r,q-r}},$

where $q=l_{a}(\mathfrak{p}_{k+1})$, lies in the top layer from the geometric
lemma on $\lambda_{N_{r,q-r}}$. This contradicts to a structure arising from
the socle-irreducible property of a big derivative in [Ch22+b, Section 9]. See
[Ch22+d] for more details.

∎

######  Theorem 9.4.

[Ch22+d] Let $(\Delta_{1},\Delta_{2},\pi)$ be a pre-RdLi-commutative triple.
Then $(\Delta_{1},\Delta_{2},\pi)$ is a strongly RdLi-commutative triple.

###### Proof.

We sketch main ideas of a proof that relies on more explicit structure. Let
$l=l_{a}(\Delta_{2})$. As in [Ch22+b], we define the big derivative as:

$\mathbb{D}_{\Delta_{2}}(\pi):=\mathrm{Hom}_{G_{l}}(\mathrm{St}(\Delta_{2}),\pi_{N_{l}}),$

where the $G_{l}$-action is defined via embedding to the second factor in
$G_{n(\pi)-l}\times G_{l}$. The big derivative is equipped with
$G_{n(\pi)-l}$-action via embedding to the factor in $G_{n(\pi)-l}\times
G_{l}$.

By the condition of pre-RdLi-commutativity, we have that:

$D_{\Delta_{1}}\circ
I_{\Delta_{2}}(\pi)\hookrightarrow\mathrm{St}(\Delta_{2})\times\mathbb{D}_{\Delta_{1}}(\pi)$

To show strong commutativity, it is equivalent to show that the map factors
through the embedding $\mathrm{St}(\Delta_{2})\times
D_{\Delta_{1}}(\pi)\hookrightarrow\mathrm{St}(\Delta_{2})\times\mathbb{D}_{\Delta_{1}}(\pi)$.

It is shown in [Ch22+b] that $\mathbb{D}_{\Delta_{1}}(\pi)$ satisfies the
socle-irreducible property. Thus, it suffices to show that for any composition
factor $\tau$ in $\mathbb{D}_{\Delta_{1}}(\pi)/D_{\Delta_{1}}(\pi)$, there is
no embedding from $D_{\Delta_{1}}\circ I_{\Delta_{2}}(\pi)$ to
$\mathrm{St}(\Delta_{2})\times\tau$.

We shall not go through all the details. The main idea is to consider the
invariants $\eta_{\Delta_{1}}$ and $\eta_{\Delta_{2}}^{L}$. By the arguments
in the proof of Theorem 10.3 (which we only need pre-commutativity rather than
strong commutativity, and use Proposition 9.3), we have that

$\eta_{\Delta_{1}}(\pi)=\eta_{\Delta_{1}}(I_{\Delta_{2}}(\pi)),\quad\eta_{\Delta_{2}}^{L}(D_{\Delta_{1}}\circ
I_{\Delta_{2}}(\pi))=\eta^{L}_{\Delta_{2}}(I_{\Delta_{2}}(\pi)).$

On the other hand, some computations on Jacquet functors give that

$|\eta|_{\Delta_{1}}(\tau)\leq|\eta|_{\Delta_{1}}(D_{\Delta_{1}}(\pi))=|\eta|_{\Delta_{1}}(\pi)-1,\quad|\eta|_{\Delta_{2}}^{L}(\tau)\leq|\eta|_{\Delta_{2}}^{L}(\pi)$

and $\eta_{\Delta_{1}}(\tau)\neq\eta_{\Delta_{1}}(D_{\Delta_{1}}(\pi))$. Then
one uses those data to analyse and show that the embeddings
$D_{\Delta_{1}}\circ
I_{\Delta_{2}}(\pi)\hookrightarrow\mathrm{St}(\Delta_{2})\times\tau$ cannot
happen. ∎

###  10\. Combinatorial commutativity

######  Definition 10.1.

Let $\Delta_{1},\Delta_{2}$ be segments. Let $\pi\in\mathrm{Irr}$. We say that
$(\Delta_{1},\Delta_{2},\pi)$ is a combinatorially RdLi-commutative triple if
$D_{\Delta_{1}}(\pi)\neq 0$ and

$\eta_{\Delta_{1}}(I_{\Delta_{2}}(\pi))=\eta_{\Delta_{1}}(\pi).$

There is a dual definition:

######  Definition 10.2.

Let $\Delta_{1},\Delta_{2}$ be segments. Let $\pi\in\mathrm{Irr}$. We say that
$(\Delta_{1},\Delta_{2},\pi)$ is a dual combinatorially RdLi-commutative
triple if

$\eta_{\Delta_{2}}^{L}(D_{\Delta_{1}}\circ
I_{\Delta_{2}}(\pi))=\eta_{\Delta_{2}}^{L}(I_{\Delta_{2}}(\pi)).$

######  Theorem 10.3.

[Ch22+d, Theorem 8.4] Let $\Delta_{1},\Delta_{2}$ be segments. Let
$\pi\in\mathrm{Irr}$. Then the followings are equivalent:

  1. (1)

$(\mathrm{St}(\Delta_{1}),\mathrm{St}(\Delta_{2}),\pi)$ is a strongly RdLi-
commutative triple;

  2. (2)

$(\mathrm{St}(\Delta_{2}),\mathrm{St}(\Delta_{1}),D_{\Delta_{1}}\circ
I_{\Delta_{2}}(\pi))$ is a strongly LdRi-commutative triple;

  3. (3)

$(\Delta_{1},\Delta_{2},\pi)$ is a combinatorially RdLi-commutative triple;

  4. (4)

$(\Delta_{1},\Delta_{2},\pi)$ is a dual combinatorially RdLi-commutative
triple.

###### Proof.

We sketch a proof. Let $\mathfrak{p}=\mathfrak{mx}(\Delta_{1},\pi)$. For (1)
$\Rightarrow$ (3), note that Proposition 9.3 holds for any $\Delta$-saturated
$\mathfrak{p}^{\prime}$ such that
$D_{\mathfrak{p}^{\prime}}(I_{\Delta_{2}}(\pi))\neq 0$. One can now deduce (3)
with Lemma 8.6. An alternative proof is given in [Ch22+d, Theorem 8.4].

For (3) $\Rightarrow$ (1). It is simpler to prove that
$(\mathrm{St}(\mathfrak{p}),\mathrm{St}(\Delta_{2}),\pi)$ is a strongly RdLi-
commutative triple [Ch22+d] by using the property that $D_{\mathfrak{p}}\circ
I_{\Delta_{2}}(\pi)\boxtimes\mathrm{St}(\mathfrak{p})$ is a direct summand in
$I_{\Delta_{2}}(\pi)_{N_{l}}$, where $l=l_{a}(\mathfrak{p})$. (For instance,
one may use [Ch22+d, Proposition 4.4] and the decomposition
$\pi\hookrightarrow D_{\mathfrak{p}}(\pi)\times\mathrm{St}(\mathfrak{p})$.)
Then one sees that $(\Delta_{1},\Delta_{2},\pi)$ is pre-RdLi-commutative and
so (1) follows from Proposition 9.3.

We now prove (1) $\Rightarrow$ (2). Let $\sigma_{1}=\mathrm{St}(\mathfrak{p})$
and let $\sigma_{2}=\mathrm{St}(\Delta_{2})$. Let $\tau=I_{\Delta_{2}}\circ
D_{\mathfrak{p}}(\pi)\cong D_{\mathfrak{p}}\circ I_{\Delta_{2}}(\pi)$ (the
commutativity follows from Proposition 9.3 and the strong commutativity of
$(\mathrm{St}(\mathfrak{p}),\mathrm{St}(\Delta_{2}),\pi)$). Let $m=n(\pi)$,
$l_{1}=l_{a}(\sigma_{1})$, $l_{2}=l_{a}(\sigma_{2})$. Let
$N=N_{l_{2},m,l_{1}}$. We consider the following commutative diagram:

$\textstyle{\tau_{N_{l_{2},m}}\boxtimes\sigma_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sigma_{2}\boxtimes
D_{\Delta_{2}}^{L}(\tau)\boxtimes\sigma_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{\Delta_{2}}(\pi)_{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\tau\times\sigma_{1})_{N}}$$\textstyle{\sigma_{2}\boxtimes\pi_{N_{m,l_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

(The commutativity of the diagram follows from some socle irreducible property
of parabolic inductions, see [Ch22+d, Section 2].) Indeed,
$\tau\boxtimes\sigma_{1}$ is a direct summand in
$I_{\Delta_{2}}(\pi)_{N_{l_{1}}}$ (see [Ch22+d, Proposition 7.1]) and so gives
a non-zero composition as follows:

$\tau\boxtimes\sigma_{1}\hookrightarrow
I_{\Delta_{2}}(\pi)_{N_{l_{1}}}\rightarrow(\tau\times\sigma_{1})_{N_{l_{1}}}\twoheadrightarrow\tau\boxtimes\sigma_{1},$

where the last projection comes from the quotient in the geometric lemma. (For
more details, see [Ch22+d].) This gives the layer that the copy
$\sigma_{2}\boxtimes D_{\sigma_{2}}(\tau)\boxtimes\sigma_{1}$ lies in
$(\tau\times\sigma_{1})_{N}$ from geometric lemma. Now from the commutativity
diagram, one must have that $(\sigma_{2},\sigma_{1},\tau)$ is pre-LdRi-
commutative triple. This implies that $(\Delta_{2},\Delta_{1},\tau)$ is also
pre-LdRi-commutative triple (see e.g. [Ch22+d, Proposition 7.2]). Then
$(\Delta_{2},\Delta_{1},\tau)$ is also strongly LdRi-commutative triple by
Theorem 9.4 as desired.

Proof for (2)$\Rightarrow$(1) is similar to that of (1)$\Rightarrow$(2). A
proof for (2)$\Leftrightarrow$(4) is similar to that of
(1)$\Leftrightarrow$(3). ∎

##  Part III Generalized GGP relevance from sequences of strongly commutative
triples

In this section, we define and study the strong commutation for multisegments,
extending the segment case in Part II. As shown in [Ch22+] (see Theorem 15.3),
there is a good theory of minimal multisegments for derivatives and integrals.
Sections 11 to 15 show the compatibility of such theory with the strong
commutation for multisegments, which will be useful in the later proofs.
Another two main results in this part are the uniqueness property (Theorem
16.2) and the symmetry property (Theorem 18.1) for relevance.

###  11\. Unlinked segments for strongly commutative triples

####  11.1. Unlinked segments

We first have the following result, see e.g. [Ch22+, Lemma 4.10]:

######  Lemma 11.1.

Let $\Delta_{1},\Delta_{2}$ be unlinked segments. Then $D_{\Delta_{1}}\circ
D_{\Delta_{2}}(\pi)\cong D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)$ and
$I_{\Delta_{1}}\circ I_{\Delta_{2}}(\pi)\cong I_{\Delta_{2}}\circ
I_{\Delta_{1}}(\pi)$.

######  Proposition 11.2.

Let $\Delta_{1}^{\prime},\Delta_{2}^{\prime}$ be unlinked segments. Let
$\sigma=\mathrm{St}(\left\\{\Delta_{1}^{\prime},\Delta_{2}^{\prime}\right\\})$.
Let $\Delta$ be another segment. Let $\pi\in\mathrm{Irr}$. The following
conditions are equivalent:

  1. (1)

$(\Delta,\Delta_{1}^{\prime},\pi)$ and
$(\Delta,\Delta_{2}^{\prime},I_{\Delta_{1}^{\prime}}(\pi))$ are strongly RdLi-
commutative triples;

  2. (2)

$(\Delta,\Delta_{2}^{\prime},\pi)$ and
$(\Delta,\Delta_{1}^{\prime},I_{\Delta_{2}^{\prime}}(\pi))$ are strongly RdLi-
commutative triples.

###### Proof.

Suppose (1). By Theorem 10.3,

$\eta_{\Delta}(\pi)=\eta_{\Delta}(I_{\Delta_{1}^{\prime}}(\pi))=\eta_{\Delta}(I_{\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}}(\pi))$

and, by Lemma 8.6,

$\eta_{\Delta}(\pi)\leq\eta_{\Delta}(I_{\Delta_{2}^{\prime}}(\pi))\leq\eta_{\Delta}(I_{\Delta_{1}^{\prime}}\circ
I_{\Delta_{2}^{\prime}}(\pi))=\eta_{\Delta}(I_{\Delta_{1}^{\prime}}\circ
I_{\Delta_{2}^{\prime}}(\pi)).$

Thus, those inequalities are equalities. This implies (2). The other direction
can be proved similarly. ∎

######  Proposition 11.3.

Let $\Delta_{1},\Delta_{2}$ be unlinked segments. Let
$\sigma=\mathrm{St}(\left\\{\Delta_{1},\Delta_{2}\right\\})$. Let
$\Delta^{\prime}$ be another segment. Let $\pi\in\mathrm{Irr}$. The following
conditions are equivalent:

  1. (1)

$(\Delta_{1},\Delta^{\prime},\pi)$ and
$(\Delta_{2},\Delta^{\prime},D_{\Delta_{1}}(\pi))$ are strongly RdLi-
commutative triples;

  2. (2)

$(\Delta_{2},\Delta^{\prime},\pi)$ and
$(\Delta_{1},\Delta^{\prime},D_{\Delta_{2}}(\pi))$ are strongly RdLi-
commutative triples.

###### Proof.

Let $\tau=I_{\Delta^{\prime}}\circ D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)$.
Note that if $(\Delta^{\prime},\Delta_{2},\tau)$ is strongly LdRi-commutative,
then $D_{\Delta^{\prime}}\circ I_{\Delta_{2}}(\tau)=D_{\Delta_{1}}(\pi)$ by
Proposition 9.1 and so $I_{\Delta_{2}}(\tau)=I_{\Delta^{\prime}}\circ
D_{\Delta_{1}}(\pi)$.

By using the duality in Theorem 10.3, the two conditions can be rephrased as:

  1. (1)

$(\Delta^{\prime},\Delta_{1},I_{\Delta_{2}}(\tau))$ and
$(\Delta^{\prime},\Delta_{2},\tau)$ are strongly LdRi-commutative triples;

  2. (2)

$(\Delta^{\prime},\Delta_{2},I_{\Delta_{1}}(\tau))$ and
$(\Delta^{\prime},\Delta_{1},\tau)$ are strongly LdRi commutative triples.

Now the proposition follows from the LdRi version of the integral one in
Proposition 11.2. ∎

###  12\. Intersection-union operations for integrals in commutative triples

####  12.1. Intersection-union process

Let $\mathfrak{m}_{1},\mathfrak{m}_{2}\in\mathrm{Mult}$. We say that
$\mathfrak{m}_{2}$ is obtained from $\mathfrak{m}_{1}$ by an intersection-
union process if there exists a pair of linked segments
$\Delta_{1},\Delta_{2}$ such that

$\mathfrak{m}_{2}=\mathfrak{m}_{1}-\left\\{\Delta_{1},\Delta_{2}\right\\}+\Delta_{1}\cup\Delta_{2}+\Delta_{1}\cap\Delta_{2}.$

(Drop the last term if $\Delta_{1}\cap\Delta_{2}=\emptyset$.) Recall that
$\leq_{Z}$ is defined in Section 2.2.

####  12.2. Strong commutativity under intersection-union process for
integrals

In this section, we shall prove:

######  Proposition 12.1.

Let $\Delta_{1},\Delta_{2}$ be linked segments. Let $\widetilde{\Delta}$ be
another segment. Let $\Delta_{1}^{\prime}=\Delta_{1}\cup\Delta_{2}$ and
$\Delta_{2}^{\prime}=\Delta_{1}\cap\Delta_{2}$. Let $\pi\in\mathrm{Irr}$.
Suppose $I_{\Delta_{2}}\circ I_{\Delta_{1}}(\pi)\cong
I_{\Delta_{1}^{\prime}}\circ I_{\Delta_{2}^{\prime}}(\pi)$. Then, the
following statements are equivalent:

  1. (1)

$(\widetilde{\Delta},\Delta_{1},\pi)$ and
$(\widetilde{\Delta},\Delta_{2},I_{\Delta_{1}}(\pi))$ are strongly RdLi-
commutative triples;

  2. (2)

$(\widetilde{\Delta},\Delta_{1}^{\prime},\pi)$ and
$(\widetilde{\Delta},\Delta_{2}^{\prime},I_{\Delta_{1}^{\prime}}(\pi))$ are
strongly RdLi-commutative triples;

  3. (3)

$(\widetilde{\Delta},\Delta_{2}^{\prime},\pi)$ and
$(\widetilde{\Delta},\Delta_{1}^{\prime},I_{\Delta_{2}^{\prime}}(\pi))$ are
strongly RdLi-commutative triples.

Moreover, if any of the equivalent conditions holds, then $I_{\Delta_{2}}\circ
I_{\Delta_{1}}\circ D_{\widetilde{\Delta}}(\pi)\cong
I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}\circ
D_{\widetilde{\Delta}}(\pi)$.

###### Proof.

We use the notations in the statement of the proposition. Suppose (1) holds.
By Theorem 10.3,

$(*)\quad\eta_{\widetilde{\Delta}}(\pi)=\eta_{\widetilde{\Delta}}(I_{\Delta_{1}}(\pi))=\eta_{\widetilde{\Delta}}(I_{\Delta_{2}}\circ
I_{\Delta_{1}}(\pi))=\eta_{\widetilde{\Delta}}(I_{\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}}(\pi)).$

On the other hand, by Lemma 8.6,

$\eta_{\widetilde{\Delta}}(\pi)\leq\eta_{\widetilde{\Delta}}(I_{\Delta_{1}^{\prime}}(\pi))\leq\eta_{\widetilde{\Delta}}(I_{\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}}(\pi)).$

Now, (*) forces that the inclusion is actually equations. Hence, we have that
$(\widetilde{\Delta},\Delta_{1}^{\prime},\pi)$ and
$(\widetilde{\Delta},\Delta_{2}^{\prime},I_{\Delta_{2}^{\prime}}(\pi))$ are
combinatorially commutative triples, and so are strongly commutative triples
by Theorem 10.3. This proves (2).

Since $\Delta_{1}^{\prime}$ and $\Delta_{2}^{\prime}$ are unlinked, we have
$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}(\pi)\cong
I_{\Delta_{1}^{\prime}}\circ I_{\Delta_{2}^{\prime}}(\pi)$. Then a similar
argument as above will prove (3).

For (3) $\Rightarrow$ (2), one can argue similarly by using the followings:

$\eta_{\widetilde{\Delta}}(\pi)=\eta_{\widetilde{\Delta}}(I_{\Delta_{1}^{\prime}}(\pi))=\eta_{\widetilde{\Delta}}(I_{\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}}(\pi))=\eta_{\widetilde{\Delta}}(I_{\Delta_{2}}\circ
I_{\Delta_{1}}(\pi)),$
$\eta_{\widetilde{\Delta}}(\pi)\leq\eta_{\widetilde{\Delta}}(I_{\Delta_{1}}(\pi))\leq\eta_{\widetilde{\Delta}}(I_{\Delta_{2}}\circ
I_{\Delta_{1}}(\pi)).$

Proving (3) $\Rightarrow$ (1) is similar. The last assertion follows by
applying $D_{\widetilde{\Delta}}$ on $I_{\Delta_{2}}\circ
I_{\Delta_{1}}(\pi)\cong I_{\Delta_{1}^{\prime}}\circ
I_{\Delta_{2}^{\prime}}(\pi)$ and then using Proposition 9.1. ∎

######  Remark 12.2.

An analogous statement does not hold in general (while it is not so far away,
c.f. Lemma 12.3). For example, consider $\Delta_{1}=[0,1]$ and
$\Delta_{2}=[2,3]$ and $\widetilde{\Delta}=[1,2]$. Let
$\pi=\mathrm{St}([0,3])$. It is clear that $D_{\Delta_{2}}\circ
D_{\Delta_{1}}(\pi)=D_{[0,3]}(\pi)=\mathbb{C}$ as $G_{0}$-representation. We
clearly have that $([0,3],\widetilde{\Delta},\pi)$ is strongly RdLi-
commutative triple, but $([0,1],\widetilde{\Delta},\pi)$ is not even pre-RdLi-
commutative triple.

####  12.3. Commutativity for derivatives and integrals

######  Lemma 12.3.

[Ch22+, Proposition 15.3] Let $\pi\in\mathrm{Irr}$. Let $\Delta_{1}$ and
$\Delta_{2}$ be linked segments with $\Delta_{1}<\Delta_{2}$. If
$D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)\not\cong
D_{\Delta_{1}\cup\Delta_{2}}\circ D_{\Delta_{1}\cap\Delta_{2}}(\pi)$, then

$D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)\cong D_{\Delta_{1}}\circ
D_{\Delta_{2}}(\pi).$

######  Remark 12.4.

It is not true in general if one switches the condition
$\Delta_{1}<\Delta_{2}$ to $\Delta_{2}<\Delta_{1}$. For example, let

$\pi=\langle\left\\{[0,2],[0,1],[1,2]\right\\}\rangle.$

Let $\Delta_{1}=[1]$ and let $\Delta_{2}=[2]$. Note that $D_{[1]}\circ
D_{[2]}(\pi)\cong\langle\left\\{[0,1],[0],[1,2]\right\\}\rangle$, but
$D_{[1,2]}(\pi)\cong D_{[2]}\circ
D_{[1]}(\pi)=\langle\left\\{[0,2],[0],[1]\right\\}\rangle$.

######  Lemma 12.5.

Let $\pi\in\mathrm{Irr}$. Let $\Delta_{1}^{\prime}$ and $\Delta_{2}^{\prime}$
be linked segments with $\Delta_{1}^{\prime}<\Delta_{2}^{\prime}$. If
$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}(\pi)\not\cong
I_{\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}\cap\Delta_{2}^{\prime}}(\pi)$, then

$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}(\pi)\cong
I_{\Delta_{1}^{\prime}}\circ I_{\Delta_{2}^{\prime}}(\pi).$

###### Proof.

Let $\tau=I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}(\pi)$. Then
$D^{L}_{\Delta_{1}^{\prime}}\circ D^{L}_{\Delta_{2}^{\prime}}(\tau)=\pi$ and
$D^{L}_{\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime}}\circ
D^{L}_{\Delta_{1}^{\prime}\cap\Delta_{2}^{\prime}}(\tau)\not\cong
D^{L}_{\Delta_{1}^{\prime}}\circ D^{L}_{\Delta_{2}^{\prime}}(\tau)$. Thus, the
left version of Lemma 12.3 implies that

$D^{L}_{\Delta_{1}^{\prime}}\circ D^{L}_{\Delta_{2}^{\prime}}(\tau)\cong
D^{L}_{\Delta_{2}^{\prime}}\circ D^{L}_{\Delta_{1}^{\prime}}(\tau),$

which implies the lemma. ∎

######  Proposition 12.6.

Let $\pi\in\mathrm{Irr}$. Let $\Delta_{1}^{\prime}$ and $\Delta_{2}^{\prime}$
be linked segments with $\Delta_{1}^{\prime}<\Delta_{2}^{\prime}$. Let
$\Delta$ be another segment. Suppose

$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}(\pi)\not\cong
I_{\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}\cap\Delta_{2}^{\prime}}(\pi).$

Then $(\Delta,\Delta_{1}^{\prime},\pi)$ and
$(\Delta,\Delta_{2}^{\prime},I_{\Delta_{1}^{\prime}}(\pi))$ are strongly RdLi-
commutative triples if and only if $(\Delta,\Delta_{2}^{\prime},\pi)$ and
$(\Delta,\Delta_{1}^{\prime},I_{\Delta_{2}^{\prime}}(\pi))$ are strongly RdLi-
commutative triples. Moreover, if one of the equivalent conditions holds, then

$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}\circ
D_{\Delta}(\pi)\not\cong I_{\Delta_{2}^{\prime}\cup\Delta_{1}^{\prime}}\circ
I_{\Delta_{2}^{\prime}\cap\Delta_{1}^{\prime}}\circ D_{\Delta}(\pi).$

###### Proof.

The strong commutation implies
$\eta_{\Delta}(\pi)=\eta_{\Delta}(I_{\Delta_{1}^{\prime}}(\pi))=\eta_{\Delta}(I_{\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}}(\pi))$. Moreover, Lemma 8.6 implies that
$\eta_{\Delta}(\pi)\leq\eta_{\Delta}(I_{\Delta_{2}^{\prime}}(\pi))\leq\eta_{\Delta}(I_{\Delta_{1}^{\prime}}\circ
I_{\Delta_{2}^{\prime}}(\pi))$. Combining the equation with Lemma 12.5, the
above inequalities have to be equalities. Now the strong commutations of
$(\Delta,\Delta_{2}^{\prime},\pi)$ and
$(\Delta,\Delta_{1}^{\prime},I_{\Delta_{2}^{\prime}}(\pi))$ follow from
Theorem 10.3.

We now prove the last assertion. One approach is to develop those
combinatorial invariants for ladder representations, in which one may use the
Kret-Lapid description for the Jacquet modules of ladder representations
[KL12]. Here we give a proof using module structures directly. Suppose not to
derive a contradiction. Let $\tau=I_{\Delta_{1}^{\prime}}\circ
I_{\Delta_{2}^{\prime}}(\pi)(\cong I_{\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}}(\pi))$. Let $l=l_{a}(\Delta)$. We consider the
following diagram:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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where $s_{1},s_{2},s_{3}$ are the surjections to the top layer in the
geometric lemma; and
$\lambda=(\mathrm{St}(\Delta_{1}^{\prime})\times\mathrm{St}(\Delta_{2}^{\prime})\times
D_{\Delta}(\pi)$;

$\sigma=\mathrm{St}(\Delta_{1}^{\prime}+\Delta_{2}^{\prime}),\quad\sigma^{\prime}=\mathrm{St}(\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime}+\Delta_{1}\cap\Delta_{2}^{\prime}).$

Using the strong commutativity (see [Ch22+d, Proposition 5.5]), we have a map
$p:D_{\Delta}(\tau)\boxtimes\mathrm{St}(\Delta)\rightarrow\lambda\boxtimes\mathrm{St}(\Delta)$
such that $j_{2}\circ p=s_{2}\circ i\circ i^{\prime}$. Using [Ch22+, Appendix
B] and our assumption (and so $D_{\Delta}(\tau)\cong D_{\Delta}\circ
I_{\Delta_{1}^{\prime}}\circ I_{\Delta_{2}^{\prime}}\cong D_{\Delta}\circ
I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}(\pi)\cong
I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}\circ D_{\Delta}(\pi)$),
we have that $t\circ p\neq 0$ and so $j_{3}\circ t\circ p\neq 0$. This implies
that $t^{\prime\prime}\circ i\neq 0$. Hence, this implies that
$\tau\hookrightarrow\sigma^{\prime}\times\pi$. Thus,

$\tau\cong I_{\sigma^{\prime}}(\pi),$

With Lemma 12.5, it gives a contradiction to the given condition. ∎

###  13\. Intersection-union operations for derivatives in commutative triples

####  13.1. Strong commutativity under intersection-union process for
derivatives

######  Lemma 13.1.

Let $\Delta_{1}^{\prime},\Delta_{2}^{\prime}$ be linked segments. Let $\Delta$
be another segment. Let
$\widetilde{\Delta}_{1}^{\prime}=\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime}$
and
$\widetilde{\Delta}_{2}^{\prime}=\Delta_{1}^{\prime}\cap\Delta_{2}^{\prime}$.
Suppose $D_{\Delta}(\pi)\neq 0$ and

$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}\circ
D_{\Delta}(\pi)\cong I_{\widetilde{\Delta}_{2}^{\prime}}\circ
I_{\widetilde{\Delta}_{1}^{\prime}}\circ D_{\Delta}(\pi).$

If $(\Delta,\Delta_{1}^{\prime},\pi)$ and
$(\Delta,\Delta_{2}^{\prime},I_{\Delta_{1}^{\prime}}(\pi))$ are strongly RdLi-
commutative triples, then the following holds:

  1. (1)

$(\Delta,\widetilde{\Delta}_{1}^{\prime},\pi)$ and
$(\Delta,\widetilde{\Delta}_{2}^{\prime},I_{\widetilde{\Delta}_{1}^{\prime}}(\pi))$
are strongly RdLi-commutative triples;

  2. (2)

$(\Delta,\widetilde{\Delta}_{2}^{\prime},\pi)$ and
$(\Delta,\widetilde{\Delta}_{1}^{\prime},I_{\widetilde{\Delta}_{2}^{\prime}}(\pi))$
are strongly RdLi-commutative triples;

  3. (3)

$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}(\pi)\cong
I_{\widetilde{\Delta}_{2}^{\prime}}\circ
I_{\widetilde{\Delta}_{1}^{\prime}}(\pi)$.

###### Proof.

(3) follows from Proposition 12.6. Then (1) and (2) follow from Proposition
12.1. ∎

The following is the derivative analog for Proposition 12.1.

######  Proposition 13.2.

Let $\Delta_{1},\Delta_{2}$ be linked segments. Let $\Delta^{\prime}$ be
another segment. Let $\widetilde{\Delta}_{1}=\Delta_{1}\cup\Delta_{2}$ and let
$\widetilde{\Delta}_{2}=\Delta_{1}\cap\Delta_{2}$. Suppose
$D_{\Delta_{1}}(\pi)\neq 0$ and $D_{\Delta_{2}}(\pi)\neq 0$, and
$D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)\cong D_{\widetilde{\Delta}_{2}}\circ
D_{\widetilde{\Delta}_{1}}(\pi)$. If $(\Delta_{1},\Delta^{\prime},\pi)$ and
$(\Delta_{2},\Delta^{\prime},D_{\Delta_{1}}(\pi))$ are strongly RdLi-
commutative triples, then the following holds:

  1. (1)

$(\widetilde{\Delta}_{1},\Delta^{\prime},\pi)$ and
$(\widetilde{\Delta}_{2},\Delta^{\prime},D_{\widetilde{\Delta}_{1}}(\pi))$ are
strongly RdLi-commutative triple;

  2. (2)

$(\widetilde{\Delta}_{2},\Delta^{\prime},\pi)$ and
$(\widetilde{\Delta}_{1},\Delta^{\prime},D_{\widetilde{\Delta}_{2}}(\pi))$ are
strongly RdLi-commutative triple;

  3. (3)

$D_{\Delta_{2}}\circ D_{\Delta_{1}}\circ I_{\Delta^{\prime}}(\pi)\cong
D_{\widetilde{\Delta}_{2}}\circ D_{\widetilde{\Delta}_{1}}\circ
I_{\Delta^{\prime}}(\pi)$.

###### Proof.

Let

$\tau=D_{\Delta_{2}}\circ D_{\Delta_{1}}\circ I_{\Delta^{\prime}}(\pi)\cong
I_{\Delta^{\prime}}\circ D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)$

Using the given condition and Lemma 11.1, one also has:

$\tau\cong I_{\Delta^{\prime}}\circ D_{\widetilde{\Delta}_{1}}\circ
D_{\widetilde{\Delta}_{2}}(\pi)\cong I_{\Delta^{\prime}}\circ
D_{\widetilde{\Delta}_{2}}\circ D_{\widetilde{\Delta}_{1}}(\pi).$

One first reformulates into equivalent conditions and statements involving
$\tau$ by Theorem 10.3 and Proposition 9.1: e.g. the original two strong
commutations become that $(\Delta^{\prime},\Delta_{1},I_{\Delta_{2}}(\tau))$
and $(\Delta^{\prime},\Delta_{2},\tau)$ are strongly LdRi-commutative triples;
(1) equivalent to
$(\Delta^{\prime},\widetilde{\Delta}_{1},I_{\widetilde{\Delta}_{2}}(\tau))$
and $(\Delta^{\prime},\widetilde{\Delta}_{2},\tau)$ are strongly LdRi-
commutative triples, etc. Then the equivalent statements follow from Lemma
13.1.

∎

####  13.2. Commutativity for derivatives

######  Lemma 13.3.

Let $\pi\in\mathrm{Irr}$. Let $\Delta_{1}^{\prime}$ and $\Delta_{2}^{\prime}$
be linked segments with $\Delta_{1}^{\prime}<\Delta_{2}^{\prime}$. Let
$\Delta$ be another segment. Suppose

$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}\circ
D_{\Delta}(\pi)\not\cong I_{\Delta_{2}^{\prime}\cup\Delta_{1}^{\prime}}\circ
I_{\Delta_{2}^{\prime}\cap\Delta_{1}^{\prime}}\circ D_{\Delta}(\pi).$

Then, $(\Delta,\Delta_{1}^{\prime},\pi)$ and
$(\Delta,\Delta_{2}^{\prime},I_{\Delta_{1}^{\prime}}(\pi))$ are strongly RdLi-
commutative triples if and only if $(\Delta,\Delta_{2}^{\prime},\pi)$ and
$(\Delta,\Delta_{1}^{\prime},I_{\Delta_{2}^{\prime}}(\pi))$ are strongly RdLi-
commutative triples. Moreover, if one of the equivalent conditions holds, then

$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}(\pi)\not\cong
I_{\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}\cap\Delta_{2}^{\prime}}(\pi).$

###### Proof.

The last assertion follows from Proposition 12.1 and then the equivalent
condition follows from Proposition 12.6.

∎

The following is the derivative analog of Proposition 12.6. One may also use
Definition 10.2 to show the last assertion of Proposition 13.4 and then use a
similar proof as in Proposition 12.1. Once we have Proposition 13.4, we can
also deduce back Lemma 13.3 by the duality.

######  Proposition 13.4.

Let $\pi\in\mathrm{Irr}$. Let $\Delta_{1}$ and $\Delta_{2}$ be linked segments
with $\Delta_{1}<\Delta_{2}$. Let $\Delta^{\prime}$ be another segment.
Suppose

$D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)\not\cong
D_{\Delta_{1}\cup\Delta_{2}}\circ D_{\Delta_{1}\cap\Delta_{2}}(\pi).$

Then $(\Delta_{1},\Delta^{\prime},\pi)$ and
$(\Delta_{2},\Delta^{\prime},D_{\Delta_{1}}(\pi))$ are strongly RdLi-
commutative triples if and only if $(\Delta_{2},\Delta^{\prime},\pi)$ and
$(\Delta_{1},\Delta^{\prime},D_{\Delta_{2}}(\pi))$ are strongly RdLi-
commutative triples. Moreover, if one of the equivalent conditions holds, then

$D_{\Delta_{2}}\circ D_{\Delta_{1}}\circ I_{\Delta^{\prime}}(\pi)\not\cong
D_{\Delta_{1}\cap\Delta_{2}}\circ D_{\Delta_{1}\cup\Delta_{2}}\circ
I_{\Delta^{\prime}}(\pi).$

###### Proof.

We only prove the only if direction. Suppose
$(\Delta_{1},\Delta^{\prime},\pi)$ and
$(\Delta_{2},\Delta^{\prime},D_{\Delta_{1}}(\pi))$ are strongly RdLi-
commutative triples.

Let $\omega=I_{\Delta^{\prime}}\circ D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)$.
By Theorem 10.3, $(\Delta^{\prime},\Delta_{1},I_{\Delta_{2}}(\omega))$ and
$(\Delta^{\prime},\Delta_{2},\omega)$ are strongly LdRi-commutative triples.
Note that

$I_{\Delta_{1}}\circ
I_{\Delta_{2}}(D_{\Delta^{\prime}}(\omega))\cong\pi\not\cong
I_{\Delta_{1}\cup\Delta_{2}}\circ
I_{\Delta_{1}\cap\Delta_{2}}(D_{\Delta^{\prime}}(\omega)).$

Hence, by Lemma 13.3, $(\Delta^{\prime},\Delta_{1},\omega)$ and
$(\Delta^{\prime},\Delta_{2},I_{\Delta_{1}}(\omega))$ are strongly LdRi-
commutative triples. Now, by Theorem 10.3,
$(\Delta_{1},\Delta^{\prime},I_{\Delta_{1}}\circ D_{\Delta^{\prime}}(\omega))$
and $(\Delta_{2},\Delta^{\prime},D_{\Delta^{\prime}}\circ I_{\Delta_{2}}\circ
I_{\Delta_{1}}(\omega))$ are strongly RdLi-commutative triples. Now using
$D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)\cong D_{\Delta_{1}}\circ
D_{\Delta_{2}}(\pi)$ in Lemma 12.3, we have that
$(\Delta_{2},\Delta^{\prime},\pi)$ and
$(\Delta_{1},\Delta^{\prime},D_{\Delta_{2}}(\pi))$ are strongly LdRi-
commutative triples.

The last assertion follows by reformulating the last assertion in Lemma 13.3
(with Proposition 9.1 for some commutations). ∎

###  14\. Strong commutation for multisegments

####  14.1. Strongly commutative triples for multisegments

######  Definition 14.1.

Let $\mathfrak{m},\mathfrak{n}\in\mathrm{Mult}$. Let $\pi\in\mathrm{Irr}$.
Suppose $D_{\mathfrak{m}}(\pi)\neq 0$. We write
$\mathfrak{m}=\left\\{\Delta_{1},\ldots,\Delta_{r}\right\\}$ in an ascending
order and
$\mathfrak{n}=\left\\{\Delta_{1}^{\prime},\ldots,\Delta_{s}^{\prime}\right\\}$
in an ascending order. For $1\leq i\leq r$, let
$\mathfrak{m}_{i}=\left\\{\Delta_{1},\ldots,\Delta_{i}\right\\}$. For $1\leq
j\leq s$, let
$\mathfrak{n}_{j}=\left\\{\Delta_{1}^{\prime},\ldots,\Delta_{j}^{\prime}\right\\}$,
and let $\mathfrak{m}_{0}=\emptyset$ and $\mathfrak{n}_{0}=\emptyset$. We say
that $(\mathfrak{m},\mathfrak{n},\pi)$ is a strongly RdLi-commutative triple
if for any $1\leq i\leq r$ and $1\leq j\leq s$,

$(\Delta_{i},\Delta_{j},I_{\mathfrak{n}_{j-1}}\circ
D_{\mathfrak{m}_{i-1}}(\pi))$

is a strongly RdLi-commutative triple in the sense of Definition 2.2.

Note that, by Propositions 11.2 and 11.3, the strong commutation for
multisegments is independent for a choice of an ascending order.

We can similarly define the notion of LdRi-commutative triples. We have the
following duality in view of Theorem 10.3.

######  Proposition 14.2.

Let $(\mathfrak{m},\mathfrak{n},\pi)$ be a strongly RdLi-commutative triple.
Then $(\mathfrak{n},\mathfrak{m},I_{\mathfrak{n}}\circ D_{\mathfrak{m}}(\pi))$
is a strongly LdRi-commutative triple.

###### Proof.

This follows from repeated uses of Theorem 10.3 (1)$\Leftrightarrow$(2). We
omit the details.

∎

In terms of branching law, one may consider this is compatible with the
duality in (4.2).

####  14.2. Combinatorial commutation for a sequence

######  Lemma 14.3.

Let $\mathfrak{m},\mathfrak{n}\in\mathrm{Mult}$. Let $\Delta,\Delta^{\prime}$
be segments. Let $\pi\in\mathrm{Irr}$. The following conditions are
equivalent:

  1. (1)

$(\mathfrak{m},\Delta^{\prime},\pi)$ is strongly RdLi-commutative;

  2. (2)

$\eta^{L}_{\Delta^{\prime}}(D_{\mathfrak{m}}\circ
I_{\Delta^{\prime}}(\pi))=\eta^{L}_{\Delta^{\prime}}(I_{\Delta^{\prime}}(\pi))$.

The following conditions are equivalent:

  1. (1)

$(\Delta,\mathfrak{n},\pi)$ is strongly RdLi-commutative;

  2. (2)

$\eta_{\Delta}(I_{\mathfrak{n}}(\pi))=\eta_{\Delta}(\pi)$.

###### Proof.

Write $\mathfrak{m}=\left\\{\Delta_{1},\ldots,\Delta_{r}\right\\}$ with the
labelling in an ascending order. Let

$\mathfrak{m}_{j}=\left\\{\Delta_{1},\ldots,\Delta_{j}\right\\}.$

The second equivalence is easier by repeatedly using combinatorial
commutativity (and Theorem 10.3) and we only prove the first one. For the
first one, we first have the following inequalities (by the left version of
Lemma 8.6):

$\eta^{L}_{\Delta^{\prime}}(D_{\mathfrak{m}}\circ
I_{\Delta^{\prime}}(\pi))\leq\ldots\leq\eta^{L}_{\Delta^{\prime}}(D_{\mathfrak{m}_{j}}\circ
I_{\Delta^{\prime}}(\pi))\leq\ldots\leq\eta^{L}_{\Delta^{\prime}}(I_{\Delta^{\prime}}(\pi)).$

We now assume (2). Then all the inequalities above are equalities. Thus,
inductively, we have that

$\eta^{L}_{\Delta^{\prime}}(D_{\Delta_{j}}\circ I_{\Delta^{\prime}}\circ
D_{\mathfrak{m}_{j-1}}(\pi))=\eta^{L}_{\Delta^{\prime}}(D_{\Delta_{j}}\circ
D_{\mathfrak{m}_{j-1}}\circ
I_{\Delta^{\prime}}(\pi))=\eta^{L}_{\Delta^{\prime}}(D_{\mathfrak{m}_{j}}\circ
I_{\Delta^{\prime}}(\pi)),$

where the first equality follows from the strong commutativity in previous
$j-1$ cases and Proposition 9.1. Using Theorem 10.3, we now have
$(\Delta_{j},\Delta^{\prime},D_{\mathfrak{m}_{j-1}}(\pi))$ is strongly RdLi-
commutative.

(1) $\Rightarrow$ (2) follows from Theorem 10.3 and Proposition 9.1. ∎

###  15\. Minimal strongly commutative triples

####  15.1. Minimality for pairs

######  Lemma 15.1.

Let $(\Delta,\Delta_{1}^{\prime},\pi)$ and
$(\Delta,\Delta_{2}^{\prime},I_{\Delta_{1}^{\prime}}(\pi))$ be strongly RdLi-
commutative triples. Then $I_{\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}}(\pi)\not\cong
I_{\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}\cap\Delta_{2}^{\prime}}(\pi)$ if and only if
$I_{\Delta_{2}^{\prime}}\circ I_{\Delta_{1}^{\prime}}\circ
D_{\Delta}(\pi)\not\cong I_{\Delta_{1}^{\prime}\cup\Delta_{2}^{\prime}}\circ
I_{\Delta_{1}^{\prime}\cap\Delta_{2}^{\prime}}\circ D_{\Delta}(\pi)$.

###### Proof.

The if direction follows from Proposition 12.1. For the only if direction, it
follows from Lemma 13.1. ∎

######  Lemma 15.2.

Let $(\Delta_{1},\Delta^{\prime},\pi)$ and
$(\Delta_{2},\Delta^{\prime},D_{\Delta_{1}}(\pi))$ be strongly RdLi-
commutative triples. Then $D_{\Delta_{2}}\circ
D_{\Delta_{1}}(I_{\Delta^{\prime}}(\pi))\not\cong
D_{\Delta_{1}\cup\Delta_{2}}\circ
D_{\Delta_{1}\cap\Delta_{2}}(I_{\Delta^{\prime}}(\pi))$ if and only if
$D_{\Delta_{2}}\circ D_{\Delta_{1}}(\pi)\not\cong
D_{\Delta_{1}\cup\Delta_{2}}\circ D_{\Delta_{1}\cap\Delta_{2}}(\pi)$.

###### Proof.

The if part follows from Proposition 13.4. For the only if direction, it
follows from 13.2.

∎

####  15.2. Minimality for a sequence

We recall some theory of minimal sequences established in [Ch22+].

######  Theorem 15.3.

[Ch22+, Theorem 1.7] (Uniqueness of minimality) Let $\pi\in\mathrm{Irr}$.

  1. (1)

For $\mathfrak{m},\mathfrak{n}\in\mathrm{Mult}$ with
$D_{\mathfrak{m}}(\pi)\cong D_{\mathfrak{n}}(\pi)$ (resp.
$D^{L}_{\mathfrak{m}}(\pi)\cong D^{L}_{\mathfrak{n}}(\pi)$), if both
$\mathfrak{m}$ and $\mathfrak{n}$ are Rd-minimal (resp. Ld-minimal) to $\pi$,
then $\mathfrak{m}=\mathfrak{n}$.

  2. (2)

For $\mathfrak{m},\mathfrak{n}\in\mathrm{Mult}$ with
$I_{\mathfrak{m}}(\pi)\cong I_{\mathfrak{n}}(\pi)$ (resp.
$I^{R}_{\mathfrak{m}}(\pi)\cong I^{R}_{\mathfrak{n}}(\pi)$), if both
$\mathfrak{m}$ and $\mathfrak{n}$ are Li-minimal (resp. Ri-minimal) to $\pi$,
then $\mathfrak{m}=\mathfrak{n}$.

###### Proof.

(1) is proved in [Ch22+]. For (2), let $\tau=I_{\mathfrak{m}}(\pi)$. Hence
$D^{R}_{\mathfrak{m}}(\tau)\cong D^{R}_{\mathfrak{n}}(\tau)$. Li-minimality to
$\pi$ implies Ld-minimality to $\tau$. Thus (2) follows from (1). ∎

######  Definition 15.4.

A strongly RdLi-commutative triple $(\mathfrak{m},\mathfrak{n},\pi)$ is said
to be minimal if $\mathfrak{m}$ is Rd-minimal to $\pi$ and $\mathfrak{n}$ is
Li-minimal to $\pi$.

While we only require that the minimality holds for $\pi$ in Definition 15.4,
the following proposition shows that the minimality holds for ’intermediate
terms’.

######  Proposition 15.5.

Let $(\mathfrak{m},\mathfrak{n},\pi)$ be a minimal strongly RdLi-commutative
triple. Write $\mathfrak{m}=\left\\{\Delta_{1},\ldots,\Delta_{r}\right\\}$ in
an ascending order and write
$\mathfrak{n}=\left\\{\Delta_{1}^{\prime},\ldots,\Delta_{s}^{\prime}\right\\}$
in an ascending order. Let
$\mathfrak{m}_{i}=\left\\{\Delta_{1}^{\prime},\ldots,\Delta_{i}^{\prime}\right\\}$
and let $\mathfrak{n}_{i}=\left\\{\Delta_{1},\ldots,\Delta_{i}\right\\}$ (with
$\mathfrak{n}_{0}=\emptyset$ and $\mathfrak{m}_{0}=\emptyset$). Let
$\bar{\mathfrak{m}}_{i}=\mathfrak{m}-\mathfrak{m}_{i}$ and let
$\bar{\mathfrak{n}}_{i}=\mathfrak{n}-\mathfrak{n}_{i}$. Then, for any $i$ and
$j$, $\bar{\mathfrak{m}}_{i}$ (resp. $\bar{\mathfrak{n}}_{j}$) is Rd-minimal