# Proof of Vogan’s conjecture on Arthur packets: irreducible parameters of
$p$-adic general linear groups

Clifton Cunningham and Mishty Ray

###### Abstract.

In this paper we prove Vogan’s conjecture on local Arthur packets, as recalled
in [CFMMX]*Section 8.3, Conjecture 1(a), for irreducible Arthur parameters of
$p$-adic general linear groups. This result shows that these Arthur packets
may be characterized by properties of simple perverse sheaves on a moduli
space of Langlands parameters.

##  1\. Introduction

The local Langlands correspondence for a connected reductive algebraic group
$G$ over a $p$-adic field $F$ partitions the set of equivalence classes of
smooth irreducible representations of $G(F)$ into $L$-packets using
equivalence classes of Langlands parameters. Following [Vogan:Langlands, ABV],
the correspondence may be viewed as a bijection between smooth irreducible
representations that share a common infinitesimal parameter of a $p$-adic
group $G(F)$, along with its pure inner forms, and simple objects in the
category of equivariant perverse sheaves on a moduli space of Langlands
parameters. This perspective leads to the notion of an ABV-packet, as
articulated in [CFMMX]*Section 8, which conjecturally generalizes the local
Arthur packet - we call this as Vogan’s conjecture on local Arthur packets. In
this paper, we prove this conjecture for irreducible Arthur parameters of the
$p$-adic group $GL_{n}(F)$.

###  1.1. Simple and irreducible parameters

Let $F$ be a non-archimedean local field with residue characteristic $p$ and
residue field $\mathbb{F}_{q}$, so $q=p^{f}$. Let $W_{F}$ be the Weil group of
$F$ and $W_{F}^{\prime}$ denote the group
$W_{F}\times\operatorname{SL}_{2}(\mathbb{C})$. Recall that a Langlands
parameter is a homomorphism

$\phi:W^{\prime}_{F}\to\prescript{L}{}{G}=\widehat{G}\rtimes W_{F}$

commuting with the natural projections to $W_{F}$, satisfying conditions
adapted from [Borel:Corvallis]. In particular, this means that all the
information of a Langlands parameter $\phi$ is captured by a $1$-cocycle

$\phi^{0}:W^{\prime}_{F}\to\widehat{G}$

such that $\phi(w,g)=\phi^{0}(g,w)\rtimes w$. In the case
$G=\operatorname{GL}_{n}$, the cocycle $\phi^{0}$ is a representation of the
group $W^{\prime}_{F}$.

Now set
$W_{F}^{\prime\prime}:=W_{F}\times\operatorname{SL}_{2}(\mathbb{C})\times\operatorname{SL}_{2}(\mathbb{C})$
and recall that an Arthur parameter [Arthur:book]*Chapter 1 is a homomorphism

$\psi:W^{\prime\prime}_{F}\to\prescript{L}{}{G}=\widehat{G}\rtimes W_{F},$

likewise defined by a $1$-cocycle

$\psi^{0}:W^{\prime}_{F}\to\widehat{G}$

satisfying certain conditions, notably that the restriction of $\psi^{0}$ to
$W_{F}$ is bounded. Again, when $G=\operatorname{GL}_{n}$, $\psi^{0}$ is a
representation. We say that $\psi$ is a simple Arthur parameter of
$\operatorname{GL}_{n}(F)$ if $\psi^{0}$ is an irreducible representation and
the restriction of $\psi^{0}$ to $W_{F}$ is trivial. We say that $\psi$ is
irreducible if $\psi^{0}$ is an irreducible representation of
$W_{F}^{\prime\prime}$.

Every Arthur parameter determines a Langlands parameter by the rule

$\phi_{\psi}(w,x)=\psi\left(w,x,\begin{pmatrix}|w|^{\frac{1}{2}}&0\\\
0&|w|^{-\frac{1}{2}}\end{pmatrix}\right).$

The infinitesimal parameter $\lambda_{\phi}$ of a Langlands parameter $\phi$
is a homomorphism

$\lambda_{\phi}:W_{F}\xrightarrow{}\prescript{L}{}{G}$

defined by the rule

$\lambda_{\phi}(w)=\phi\left(w,\begin{pmatrix}|w|^{\frac{1}{2}}&0\\\
0&|w|^{-\frac{1}{2}}\end{pmatrix}\right).$

See [CFMMX]*Section 4.1 for definition. As before, it is determined by

$\lambda_{\phi}^{0}:W_{F}\to\widehat{G}$

which is a representation when $G=\operatorname{GL}_{n}$. The function
$\phi\mapsto\lambda_{\phi}$ defined on $\widehat{G}$-equivalence classes of
these parameters is not injective in general. The infinitesimal parameters
$\lambda_{\phi_{\psi}}$ for a simple Arthur parameter $\psi$ is unramified as
a representation of $W_{F}$, i.e., it are trivial on the inertia group
$I_{F}$.

###  1.2. Local Langlands correspondence

The local Langlands conjecture for a connected reductive algebraic group $G$
over $F$ gives a map between the set $\Pi(G(F))$ of equivalence classes of
smooth irreducible representations of $G(F)$ and the set
$\Phi(\prescript{L}{}{G})$ of equivalence classes of Langlands parameters.
This map can be refined [Vogan:Langlands, ABV] as a bijection between
$\Pi(G(F))$ and the set $\Xi(\prescript{L}{}{G})$ of equivalence classes of
pairs $\xi=(\phi,\epsilon)$, called enhanced Langlands parameters, where
$\epsilon$ is an irreducible representation of the finite group

$S_{\phi}:=Z_{\widehat{G}}(\phi)/Z_{\widehat{G}}(\phi)^{0}Z(\widehat{G})^{\Gamma_{F}},$

where $\Gamma_{F}$ is the absolute Galois group of $F$. We emphasize the
distinction between Langlands parameters and $L$-parameters - Langlands
parameters are maps and $L$-parameters are $\widehat{G}$-equivalence classes
of Langlands parameters. These $L$-parameters partition $\Pi(G(F))$ into
disjoint finite sets $\Pi_{\phi}(G(F))$ called $L$-packets. This
correspondence must satisfy a list of functoriality properties and should be
compatible with $L$-functions and $\epsilon$-factors on both sides (See
[Borel:Corvallis] or [Kal], for example). When $G=\operatorname{GL}_{n}$, the
local Langlands correspondence is a theorem [Hen, Harris-Taylor]. In this
case, $S_{\phi}=1$ (see [Arthur:book], for example) and the correspondence can
be viewed as a bijection

$\Pi(G(F))\xrightarrow{}\Phi(G),$

where $\Phi(G)$ is the set of $L$-parameters for $G$. Thus, $L$-packets are
singletons for $\operatorname{GL}_{n}$.

###  1.3. A-packets

In 1989, Arthur introduced [arthur1989unipotent] what are now known as Arthur
packets, or $A$-packets, which enjoy some nice properties that non-tempered
$L$-packets fail to satisfy. This was a global theory and the local analogue
was clarified by Arthur’s subsequent work [Arthur:book] in 2013. Local
$A$-packets $\Pi_{\psi}(G(F))$ are finite sets of smooth irreducible
representations of $G(F)$ which are not necessarily disjoint and consist of
unitary admissible representations. The representations that appear in
$A$-packets are said to be of Arthur type. Every A-packet $\Pi_{\psi}(G(F))$
contains a distinguished $L$-packet $\Pi_{\phi_{\psi}}(G(F))$ through the
injection injection $\psi\mapsto\phi_{\psi}$. In the 1990s, Adams, Barbasch,
and Vogan suggested [ABV] a geometric characterization of $A$-packets by using
microlocal analysis on certain stratified complex varieties built out of
Langlands parameters. When $G=\operatorname{GL}_{n}$, $A$-packets are
singletons [Arthur:book]; in particular, the $A$-packet for $\psi$ coincides
with the $L$-packet for $\phi_{\psi}$.

###  1.4. Vogan-Langlands correspondence

Vogan’s perspective [Vogan:Langlands] on the local Langlands correspondence
suggests a beautiful geometric approach to $A$-packets. This perspective
concerns a moduli space $X_{\lambda}$ of Langlands parameters that have the
same infinitesimal parameter $\lambda$. Vogan observed that simple objects in
the category $\operatorname{Per}_{\widehat{G}}(X_{\lambda})$ of equivariant
perverse sheaves on $X_{\lambda}$ are naturally identified with the pairs
$(\phi,\epsilon)$, when we properly interpret the finite group attached to
$\phi$ and consider representations of $G(F)$ together with its pure inner
forms. Viewed in this way, the local Langlands correspondence takes the form
of a bijection

$\Pi_{\lambda}(G/F)\to\operatorname{Per}_{H_{\lambda}}(V_{\lambda})_{\operatorname{/iso}}^{\operatorname{simple}}$

between the set $\Pi_{\lambda}(G/F)$ of equivalence classes of irreducible
objects in the category $\operatorname{Rep}_{\lambda}(G/F)$ of smooth
representations of $G(F)$ and its pure rational forms which have a matching
infinitesimal parameter $\lambda$ and the set
$\operatorname{Per}_{H_{\lambda}}(V_{\lambda})_{\operatorname{/iso}}^{\operatorname{simple}}$
of isomorphism classes of simple objects in the category
$\operatorname{Per}_{\widehat{G}}(X_{\lambda})$.

Instead of working with $\operatorname{Per}_{\widehat{G}}(X_{\lambda})$, we
work with an equivalent category
$\operatorname{Per}_{H_{\lambda}}(V_{\lambda})$ (see [CFMMX]*Section 4.5). The
variety $V_{\lambda}$ of Langlands parameters and the group $H_{\lambda}$ are
described in 2. In the case $G=\operatorname{GL}_{n}$, the correspondence can
be viewed as a bijection

$\Pi_{\lambda}(G(F))\to\operatorname{Per}_{H_{\lambda}}(V_{\lambda})_{\operatorname{/iso}}^{\operatorname{simple}},$

where $\Pi_{\lambda}(G(F))$ is the set of equivalence classes of smooth
irreducible representations of $G(F)$ with a matching infinitesimal parameter
$\lambda$ and

$\operatorname{Per}_{H_{\lambda}}(V_{\lambda})_{\operatorname{/iso}}^{\operatorname{simple}}=\\{\mathcal{IC}({\mathbbm{1}}_{C}):C\subseteq
V_{\lambda}\text{ is an }H_{\lambda}\text{-orbit}\\},$

where ${\mathbbm{1}}_{C}$ denotes the constant local system on $C$. In this
way, every $\pi\in\Pi_{\lambda}(G(F))$ corresponds to
$\mathcal{P}(\pi)=\mathcal{IC}({\mathbbm{1}}_{C})\in\operatorname{Per}_{H_{\lambda}}(V_{\lambda})_{\operatorname{/iso}}^{\operatorname{simple}}$,
for a unique $H_{\lambda}$-orbit $C\subseteq V_{\lambda}$. In other words, in
this case there is a bijection between elements of $\Pi_{\lambda}(G(F))$ and
$H_{\lambda}$-orbits $C$ in $V_{\lambda}$.

###  1.5. Vogan’s conjecture on A-packets

Following an approach similar to [ABV], Vogan attached to any Langlands
parameter $\phi$ the set of irreducible representations of $G(F)$ and its pure
inner forms for which the characteristic cycles of the corresponding simple
perverse sheaf contains the conormal bundle of the $\widehat{G}$-orbit of
$\phi$ in the moduli space mentioned above. This notion is revised in [CFMMX]
in terms of vanishing cycles. This set is referred to as the ABV-packet
$\Pi_{\phi}^{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}(G/F)$.

Vogan’s perspective on the local Langlands correspondence introduces pure
A-packets $\Pi_{\psi}(G/F)$ which are unions of $A$-packets for $G$ and its
pure inner forms. This perspective also identifies $\Pi_{\psi}(G/F)$ with a
finite set of simple perverse sheaves on $V_{\lambda_{\psi}}$, where
$\lambda_{\psi}$ is the infinitesimal parameter of $\phi_{\psi}$. What
property do these perverse sheaves share, and how are they determined by the
Arthur parameter $\psi$? This question is answered by the functor

$\operatorname{Evs}_{\psi}:\operatorname{Per}_{H_{\lambda_{\psi}}}(V_{\lambda_{\psi}})\xrightarrow{}\operatorname{Rep}(A_{\psi})$

where $A_{\psi}=Z_{\widehat{G}}(\psi)/Z_{\widehat{G}}(\psi)^{0}$, constructed
in [CFMMX]*Section 7.10. In fact, this functor is defined in terms of a more
general functor
$\operatorname{Ev}_{C_{\psi}}:\operatorname{D}_{H_{\lambda_{\psi}}}(V_{\lambda_{\psi}})\to\operatorname{D}_{H_{\lambda_{\psi}}}(\Lambda^{\text{reg}}_{C_{\psi}})$
where $\Lambda_{C_{\psi}}\subset\Lambda_{\lambda_{\psi}}$ is the conormal
bundle above $C_{\psi}$ in the conormal variety for $V_{\lambda}$
[CFMMX]*Section 7.3. This $\operatorname{Ev}_{C_{\psi}}$ functor is defined in
such a way that

$\left(\operatorname{Ev}_{C_{\psi}}\mathcal{P}\right)_{(x,\xi)}=\left(\operatorname{R\hskip-1.0pt\Phi}_{\xi}\mathcal{P}\right)_{x},$

for all $(x,\xi)\in\Lambda^{\text{reg}}_{C_{\psi}}$, where
$\operatorname{R\hskip-1.0pt\Phi}_{\xi}$ is Deligne’s vanishing cycles functor
for $\xi:V_{\lambda_{\psi}}\to\mathbb{C}$ [CFMMX]*Equation 7.6. When
restricted from $\operatorname{D}_{H_{\lambda_{\psi}}}(V_{\lambda_{\psi}})$ to
$\operatorname{Per}_{H_{\lambda_{\psi}}}(V_{\lambda_{\psi}})$, the functor
$\operatorname{Ev}_{C_{\psi}}$ lands in
$\operatorname{Per}_{H_{\lambda_{\psi}}}(\Lambda^{\text{reg}}_{C_{\psi}})$,
which is equivalent to the semisimple category
$\operatorname{Loc}_{H_{\lambda_{\psi}}}(\Lambda^{\text{reg}}_{C_{\psi}})$ of
equivariant local systems on $\Lambda^{\text{reg}}_{C_{\psi}}$, by
[CFMMX]*Proposition 6.9. This proposition also gives an equivalence

$\operatorname{Loc}_{H_{\lambda_{\psi}}}(\Lambda^{\text{reg}}_{C_{\psi}})\cong\operatorname{Rep}(A_{\psi});$

see [CFMMX]*Equation 7.30. As explained in [CFMMX]*Equations 7.22, 7.24, the
precise relation between $\operatorname{Evs}_{\psi}$ and
$\operatorname{Ev}_{C_{\psi}}$ is given by

(1)
$\operatorname{Evs}_{\psi}\mathcal{P}=\mathcal{H}^{1+\operatorname{codim}C_{\psi}}_{(x_{\psi},\xi_{\psi})}\left(\operatorname{Ev}_{C_{\psi}}\mathcal{P}\right),$

as a vector space with $A_{\psi}$-action, where
$(x_{\psi},\xi_{\psi})\in\Lambda_{\lambda_{\psi}}$ is defined in
[CFMMX]*Equation 6.12, 6.13. For $G=\operatorname{GL}_{n}$, $A_{\psi}=1$ by
[Arthur:book]*Chapter 1, for example; so in this case,
$\operatorname{Rep}(A_{\psi})$ is simply the category of vector spaces.
Following [CFMMX], the ABV-packet for $\psi$ is defined by

(2)
$\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F)):=\\{\pi\in\Pi_{\lambda_{\psi}}(G(F)):\operatorname{Evs}_{\psi}(\mathcal{P}(\pi))\neq
0\\},$

where $\mathcal{P}(\pi)$ is the simple perverse sheaf on $V_{\lambda_{\psi}}$
corresponding to $\pi\in\Pi_{\lambda_{\psi}}(G(F))$.

The main result of this paper, below, gives a proof of [CFMMX]*Section 8.3,
Conjecture 1(a) adapted to general linear groups, for irreducible Arthur
parameters:

######  Theorem 1 (Theorem 6.4).

Let $\psi$ be an irreducible Arthur parameter of $G=\operatorname{GL}_{n}(F)$,
let $\phi_{\psi}$ be its corresponding Langlands parameter, let
$\Pi_{\psi}(G(F))$ be the Arthur packet attached to $\psi$. Then

$\Pi_{\psi}(G(F))=\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F)).$

We briefly overview the structure of the paper. Henceforth,
$G=\operatorname{GL}_{n}$. In Section 2, we write down the Vogan variety
$V_{\lambda}$, the group $H_{\lambda}$, and the explicit action of
$H_{\lambda}$ on $V_{\lambda}$. Our perspective on the local Langlands
correspondence is that of representations corresponding to orbits in the Vogan
variety. Both sides of this correspondence are encoded in combinatorial
objects - multisegments for representations and rank triangles for orbits. We
reiterate these notions and write down an algorithm to go from multisegments
to rank triangles and vice-verse in Section 3. In Section 4, we discuss the
Zelevinsky involution on multisegments and explain how it corresponds to an
involution on orbits. We write down the Moeglin-Waldspurger algorithm to
compute the involution on a multisegment, and use this notion to prove Lemma
4.8. In section 5, we adapt Lemma 4.8 to a statement about orbits in Lemma
5.1. Using this lemma and the properties of the functor
$\operatorname{Evs_{\psi}}$, we prove the conjecture for simple $A$-parameters
in Theorem 5.3. In section 6, we use the process of hyper-unramification to
generalize the result to an irreducible $A$-parameter.

In upcoming work, we expect to generalize Theorem 1 to arbitrary
$A$-parameters for $\operatorname{GL}_{n}(F)$ for a non-archimedean field $F$.

Since ABV-packets are expected to generalize Arthur packets, it is natural to
ask - are all ABV-packets are singletons for $\operatorname{GL}_{n}$? The
answer is no; one can see an ABV-packet of size $2$ for a non Arthur-type
Langlands parameter of $\operatorname{GL}_{16}$ in [KS_sing].

### Acknowledgements

The authors would like to acknowledge Andrew Fiori for sharing his insights on
the combinatorics of multisegments, which was crucial for Lemma 4.8. The
second named author would like to extend her gratitude to Kristaps Balodis and
James Steele for helpful discussions.

##  2\. A moduli space of Langlands parameters

Recall that $F$ is a non-archimedean local field of residue characteristic
$q=p^{f}$ with $p$-adic absolute value denoted by $|\cdot|$ and the
uniformizer denoted by $\varpi$ . The Weil group is
$W_{F}=I_{F}\rtimes\mathbb{Z}$, where $I_{F}$ is the inertia subgroup and
$\mathbb{Z}$ is identified with the cyclic subgroup generated by the
frobenius. From local class field theory, we have the local Artin map
$\alpha:W_{F}\xrightarrow{}F^{\times}$ that maps $I_{F}$ to the group of units
$\mathfrak{o}^{\times}$ in $F^{\times}$ and the geometric frobenius
$\mathfrak{f}$ to $1/\varpi$. We abuse notation and use $|\cdot|$ to denote
the absolute value on $W_{F}$, where $|w|:=|\alpha(w)|$. Thus,
$|\mathfrak{f}|=|1/\varpi|=q$. We have that $G=\operatorname{GL}_{n}(F)$, and
we can work with $\widehat{G}=\operatorname{GL}_{n}(\mathbb{C})$ instead of
$\prescript{L}{}{G}$.

We consider simple Arthur parameters, i.e., of the form

$\displaystyle\psi:W_{F}^{\prime\prime}$
$\displaystyle\longrightarrow\operatorname{GL}_{n}(\mathbb{C})$ (3)
$\displaystyle(w,x,y)$
$\displaystyle\mapsto\operatorname{Sym}^{d}(x)\otimes\operatorname{Sym}^{a}(y),$

where $a$ and $d$ are positive integers 111Here we use $d$ and $a$ in our
notation to refer to the fact that the first $\operatorname{SL}_{2}$ in
$W_{F}^{\prime\prime}=W_{F}\times\operatorname{SL}_{2}(\mathbb{C})\times\operatorname{SL}_{2}(\mathbb{C})$
is the “Deligne-$\operatorname{SL}_{2}$”, i.e., it comes from Weil-Deligne
representations, and the second $\operatorname{SL}_{2}$ is the
“Arthur-$\operatorname{SL}_{2}$”, i.e., it comes from the definition of Arthur
parameters.. For $w\in W_{F}$, let $d_{w}$ denote the matrix
$\begin{pmatrix}|w|^{\frac{1}{2}}&0\\\ 0&|w|^{-\frac{1}{2}}\end{pmatrix}$. The
corresponding Langlands parameter is given by

$\displaystyle\phi_{\psi}:W_{F}^{\prime}$
$\displaystyle\longrightarrow\operatorname{GL}_{n}(\mathbb{C})$ (4)
$\displaystyle(w,x)$ $\displaystyle\mapsto\psi(w,x,d_{w}).$

One can simplify $\psi(w,x,d_{w})$ to get

$\phi_{\psi}(w,x)=|w|^{\frac{a}{2}}\operatorname{Sym}^{d}(x)\oplus|w|^{\frac{a-2}{2}}\operatorname{Sym}^{d}(x)\cdots\oplus|w|^{\frac{-a}{2}}\operatorname{Sym}^{d}(x).$

The infinitesimal parameter associated to this Langlands parameter is given by

$\displaystyle\lambda_{\phi_{\psi}}:W_{F}$
$\displaystyle\longrightarrow\operatorname{GL}_{n}(\mathbb{C})$ (5)
$\displaystyle(w,x)$ $\displaystyle\mapsto\phi_{\psi}(w,d_{w}).$

One can simplify $\phi_{\psi}(w,d_{w})$ to get

(6)
$\lambda_{\phi_{\psi}}(w)=\bigoplus_{i=0}^{a}\bigoplus_{j=0}^{d}|w|^{\frac{a+d}{2}-(i+j)}$

We fix this infinitesimal parameter for the rest of this work and call it
$\lambda$ for simplicity. The element
$\lambda(\mathfrak{f})\in\operatorname{GL}_{n}(\mathbb{C})$ is semisimple with
eigenvalues $\lambda_{i}:=q^{e_{i}}$ occurring with multiplicities $m_{i}$,
where $0\leq i\leq k$, with $e_{i}=e_{i-1}-1$. Let us rewrite the matrix
$\lambda(\mathfrak{f})$ in a more instructive way:

(7) $\lambda(\mathfrak{f})=\begin{bmatrix}q^{e_{0}}I_{m_{0}}&0&\dots\\\
0&q^{e_{1}}I_{m_{1}}&\dots\\\ \vdots&\ddots&\\\
0&&q^{e_{k}}I_{m_{k}}\end{bmatrix}.$

We denote the corresponding eigenspaces by $E_{i}$. If we consider the
$n$-dimensional vector space which is the standard representation of
$\operatorname{Lie}(\widehat{G})$ spanned by the usual basis
$\\{f_{1},...,f_{n}\\}$, then $E_{0}$ is spanned by
$\\{f_{1},...,f_{m_{0}}\\}$, $E_{1}$ by $\\{f_{m_{0}+1},...,f_{m_{1}}\\}$, and
in general $E_{i}$ by $\\{f_{m_{i-1}+1},...,f_{m_{i}}\\}$.

Following [Vogan:Langlands] and [CFMMX], we describe the variety $V_{\lambda}$
and group $H_{\lambda}$ below:

(8)
$V_{\lambda}:=\\{X\in\mathfrak{gl}_{n}(\mathbb{C}):\operatorname{Ad}(\lambda(w))X=|w|X,\text{
}\forall w\in W_{F}\\}.$

As the absolute value on $I_{F}$ is trivial, any $X\in V_{\lambda}$ is
determined by the equation

(9) $\operatorname{Ad}(\lambda(\mathfrak{f}))X=qX.$

$V_{\lambda}$ is referred to as the Vogan variety. In this case, it is easy to
see the structure of this variety. Any matrix in $V_{\lambda}$ is strictly
upper triangular or trace zero so it can be identified as a vector subspace of
the $\mathfrak{sl}_{n}$. $\lambda(\mathfrak{f})$ is an element inside the
maximal torus in $\operatorname{SL}_{n}$. Using the Cartan decomposition of
$\mathfrak{sl_{n}}$, $V_{\lambda}$ can be identified as a sum of root spaces
of $\mathfrak{sl_{n}}$ such that the roots evaluate to $q$ at
$\lambda(\mathfrak{f})$. This shows that $V_{\lambda}$ is simply an affine
space. This fact is true more generally, as seen in [CFMMX]*Lemma 5.5.

######  Lemma 2.1.

The Vogan variety $V_{\lambda}$ can be decomposed as

(10)
$V_{\lambda}=\operatorname{Hom}(E_{k},E_{k-1})\times\operatorname{Hom}(E_{k-1},E_{k-2})\times\cdots\times\operatorname{Hom}(E_{1},E_{0}).$

###### Proof.

Let $X\in V_{\lambda}$ and $v\in E_{i}$. By definition,
$\lambda(\mathfrak{f})v=q^{e_{i}}v$ and
$\lambda(\mathfrak{f})X\lambda(\mathfrak{f})^{-1}=qX$. We show that $Xv\in
E_{i-1}$. Indeed

$\displaystyle\lambda(\mathfrak{f})Xv$
$\displaystyle=\lambda(\mathfrak{f})X\lambda(\mathfrak{f})^{-1}\cdot\lambda(\mathfrak{f})v,$
$\displaystyle=qX\cdot q^{e_{i}}v,$ $\displaystyle=q^{e_{i}+1}Xv.$

As $E_{i-1}$ is the eigenspace for $q^{e_{i}+1}$,

$V_{\lambda}\subseteq\operatorname{Hom}(E_{k},E_{k-1})\times\operatorname{Hom}(E_{k-1},E_{k-2})\times\cdots\times\operatorname{Hom}(E_{1},E_{0}).$

Now we show the reverse containment. Let
$T\in\operatorname{Hom}(E_{i},E_{i-1})$. If we view this as an endomorphism of
$\mathbb{C}^{n}$, then the $m_{i-1}\times m_{i}$ matrix associated with $T$
can be thought of as an $n\times n$ matrix $(a_{ij})$ such that $a_{ij}\neq 0$
if $m_{i-2}+1\leq i\leq m_{i-1}\text{ and }m_{i-1}+1\leq j\leq m_{i}$, and
$a_{ij}=0$ otherwise. However, the adjoint (conjugation) action of
$\lambda(\mathfrak{f})$ simply multiplies all the nonzero entries $a_{ij}$ by
$q$. Thus,
$\operatorname{Ad}(\lambda(\mathfrak{f}))\cdot(a_{ij})=q\cdot(a_{ij})$ and the
matrix $(a_{ij})$ can be identified with an element of $V_{\lambda}$. This
shows the reverse containment. ∎

The group $H_{\lambda}$ is $Z_{\widehat{G}}(\lambda)$, the centralizer of the
image of $\lambda$ in $\widehat{G}$. Again, any $g\in H_{\lambda}$ is
determined by the equation

(11) $\operatorname{Ad}(\lambda(\mathfrak{f}))\cdot g=g.$

Using the description of $\lambda(\mathfrak{f})$ in $\eqref{eq:inf}$, we can
directly see that

$Z_{\widehat{G}}(\lambda(\mathfrak{f}))=\begin{bmatrix}\operatorname{GL}_{m_{0}}(\mathbb{C})&0&\dots\\\
0&\operatorname{GL}_{m_{1}}(\mathbb{C})&\dots\\\ \vdots&\ddots&\\\
0&&\operatorname{GL}_{m_{k}}(\mathbb{C})\end{bmatrix}$

We may identify $H_{\lambda}$ as follows.

(12)
$H_{\lambda}=\operatorname{Aut}(E_{k})\times\operatorname{Aut}(E_{k-1})\times\cdots\times\operatorname{Aut}(E_{0})$

$H_{\lambda}$ acts on $V_{\lambda}$ via conjugation. Using the interpretation
from $\eqref{vlambda}$ and $\eqref{hlambda}$, we describe the action as
follows. For elements in $V_{\lambda}$ let us use the notation

$x=(x_{kk-1},\cdots,x_{21},x_{10}),\text{ where
}x_{ii-1}\in\operatorname{Mat}_{m_{i-1},m_{i}}(\mathbb{C}).$

For elements of $H_{\lambda}$, we write

$h=(h_{k},\cdots h_{1},h_{0}),\text{ where
}h_{i}\in\operatorname{GL}_{m_{i}}(\mathbb{C}).$

In this notation, the action $H_{\lambda}$ on $V_{\lambda}$ is given by

(13)
$(h_{k},\ldots,h_{1},h_{0})\cdot(x_{kk-1},\ldots,x_{21},x_{10})=(h_{k-1}x_{kk-1}h_{k}^{-1},\ldots,h_{1}x_{21}h_{2}^{-1},h_{0}x_{10}h_{1}^{-1}).$

Observe that $\text{rank}(h_{i-1}x_{ii-1}h_{i}^{-1})=\text{rank}(x_{ii-1}),$
thus any orbit preserves the ranks of $x_{ii-1}$s. More generally every orbit
is determined by the ranks of $x_{ii-1}$s and all their permitted products.
Realizing elements of $V_{\lambda}$ as a matrix as in (10), we have

$x=\begin{bmatrix}0&x_{10}&0&\dots\\\ 0&0&x_{21}&\dots\\\ \vdots&&\ddots&\\\
&&&x_{kk-1}\\\ 0&0&\dots&0\end{bmatrix}.$

The orbit of any $x$ is determined by its Jordan canonical form. Notice that
$0$ is the only eigenvalue, so calculating dimensions of the generalized
eigenspaces will involve computing ranks of powers of this matrix $x$, which
in turn involves computing ranks of all permitted products

(14) $x_{ij}:=x_{ii-1}x_{i-1i-2}\cdots x_{j+1j}$

where where $0\leq j<i-1\leq k$. Writing down these rank equations exactly
realizes an orbit as an algebraic variety.

With emphasis on the fact that $\lambda$ determines $H_{\lambda}$ and
$V_{\lambda}$, we drop the subscript and use $H$ and $V$ instead.

######  Example 2.2.

For $\operatorname{GL}_{4}$, set
$\psi(w,x,y)=\operatorname{Sym}^{1}(x)\otimes\operatorname{Sym}^{1}(y)$. Then
$\phi_{\psi}=\psi(w,x,d_{w})=|w|^{1/2}x\oplus|w|^{-1/2}x$ and
$\lambda(w)=\psi(w,d_{w},d_{w})$, so

$\lambda(\mathfrak{f})=\begin{bmatrix}q&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&q^{-1}\\\ \end{bmatrix}.$

By direct calculation using (9) and (11)

$\displaystyle V$ $\displaystyle=\left\\{\begin{bmatrix}0&u_{1}&u_{2}&0\\\
0&0&0&v_{1}\\\ 0&0&0&v_{2}\\\ 0&0&0&0\\\
\end{bmatrix}:u_{1},u_{2},v_{1},v_{2}\in\mathbb{C}\right\\}\simeq\mathbb{A}^{4}_{\mathbb{C}},$
$\displaystyle H$ $\displaystyle=\left\\{\begin{bmatrix}t_{1}&0&0&0\\\
0&t_{2}&r&0\\\ 0&s&t_{3}&0\\\ 0&0&0&t_{4}\\\
\end{bmatrix}:t_{1},t_{4}\in\mathbb{C}^{\times},\begin{bmatrix}t_{2}&r\\\
s&t_{3}\end{bmatrix}\in\operatorname{GL}_{2}(\mathbb{C})\right\\}\simeq\operatorname{GL}_{1}(\mathbb{C})\times\operatorname{GL}_{2}(\mathbb{C})\times\operatorname{GL}_{1}(\mathbb{C}).$

The eigenvalues of $\lambda(\mathfrak{f})$ are $q,1,q^{-1}$ with
multiplicities $1,2,$ and $1$ and eigenspaces
$E_{0}=\operatorname{span}\\{(1,0,0,0)\\}$,
$E_{1}=\operatorname{span}\\{(0,1,0,0),(0,0,1,0)\\}$ and
$E_{2}=\operatorname{span}\\{(0,0,0,1)\\}$, respectively. In the formulation
of (10) and (12), we may write

$\displaystyle V$
$\displaystyle=\operatorname{Hom}(E_{2},E_{1})\times\operatorname{Hom}(E_{1},E_{0}),$
$\displaystyle H$
$\displaystyle=\operatorname{Aut}(E_{2})\times\operatorname{Aut}(E_{1})\times\operatorname{Aut}(E_{0}).$

Following (13), the action of $H$ on $V$ is given by

$\left(t_{4},\begin{bmatrix}t_{2}&r\\\
s&t_{3}\end{bmatrix},t_{1}\right)\cdot\left(\begin{bmatrix}v_{1}\\\
v_{2}\end{bmatrix},[u_{1},u_{2}]\right)=\left(\begin{bmatrix}t_{2}&r\\\
s&t_{3}\end{bmatrix}\begin{bmatrix}v_{1}\\\
v_{2}\end{bmatrix}t^{-1}_{4},t_{1}[u_{1},u_{2}]\begin{bmatrix}t_{2}&r\\\
s&t_{3}\end{bmatrix}^{-1}\right).$

##  3\. From multisegements to rank triangles and vice-versa

As outlined in Section 1.4, there is a bijection between equivalence classes
of smooth irreducible representations that share the infinitesimal parameter
$\lambda$ and $H$-orbits in $V$. We know that smooth irreducible
representations of $G$ can be parameterized using multisegments [Z2]. The
relationship between multisegments and orbits appears in [zelevinskii1981p].
In this section, we introduce the combinatorial gadget of rank triangles to
parametrize the orbits and write down an algorithm compute the multisegment
corresponding to a rank triangle and vice-versa. This has been worked out for
some specific cases in $\operatorname{GL}_{16}$ in [KS_sing]; here we present
it with updated notation for $\operatorname{GL}_{n}$.

###  3.1. Multisegments

The Langlands classification using multisegments is summarized beautifully in
[kudla1994local]*Section 1. We recall this classification in more detail
following the notation in [kudla1994local]. For any representation $\pi$ of
$\operatorname{GL}_{m}(F)$, let $\pi(i):=|\text{det}(\cdot)|^{i}\pi$. For a
partition $n=\underbrace{m+m+\ldots+m}_{r\text{-times}}$ and a supercuspidal
representation $\sigma$ of $\operatorname{GL}_{m}(F)$, we call

(15) $(\sigma,\sigma(1),\ldots,\sigma(r-1))=[\sigma,\sigma(r-1)]=\Delta$

a segment. This segment determines a representation on a parabolic subgroup of
$G$ whose levi is
$\underbrace{\operatorname{GL}_{m}(F)\times\operatorname{GL}_{m}(F)\times\cdots\times\operatorname{GL}_{m}(F)}_{r\text{-times}}$.
More precisely, we have the representation
$\sigma\otimes\sigma(1)\otimes\cdots\otimes\sigma(r-1)$ on the levi, inflated
to a representation of the parabolic, which is trivial on the unipotent. We
can then carry out parabolic induction to obtain the induced representation
$I_{P}^{G}(\Delta)$ of $G$ which has a unique irreducible quotient denoted by
$Q(\Delta)$.

A multisegment is a collection of segments with repetitions allowed. The
Langlands classification theorem tells us that any smooth irreducible
representation of $G$ occurs as a unique irreducible quotient inside a
parabolically induced representation determined by a multisegment
$\\{\Delta_{1},\Delta_{2},\ldots,\Delta_{r}\\}$ subject to certain conditions
- we denote that quotient by $Q(\Delta_{1},\Delta_{2},\ldots,\Delta_{r})$.
Next, for integers $i<j$ we introduce the notation

(16) $[i,j]:=(|\cdot|^{i},|\cdot|^{i+1},\ldots,|\cdot|^{j})$

for a segment which is the special case of (15) when we consider the partition
$1+1+\cdots+1$ and $\sigma$ to be the character $|\cdot|^{i}$ of $F^{\times}$.
This notation may be extended to half integers $i<j$ as long as $j-i+1$ is a
positive integer (this is the length of the segment). A segment of length 1 of
the form $\\{|\cdot|^{i}\\}$ is just denoted $[i]$. There is a partial order
on multisegments which we recall below.

######  Definition 3.1.

Let $\alpha$ and $\beta$ be any two multisegments. We say that
$\alpha\leq\beta$ if we can form $\beta$ by performing elementary operations
on segments in $\alpha$. More precisely, $\alpha\leq\beta$ if we can form
$\beta$ by replacing any two segments $\Delta_{1}$ and $\Delta_{2}$ in
$\alpha$ with

$\begin{cases}\Delta_{1}\cup\Delta_{2}\text{ and
}\Delta_{1}\cap\Delta_{2}&\text{ if }\Delta_{1}\cap\Delta_{2}\neq\emptyset,\\\
\Delta_{1}\cup\Delta_{2}&\text{ if }\Delta_{1}\cup\Delta_{2}=\emptyset\text{
and }\Delta_{1}\cup\Delta_{2}\text{ is a segment},\\\ \Delta_{1}\text{ and
}\Delta_{2}&\text{ otherwise. }\end{cases}$

######  Example 3.2.

In the case of $\operatorname{GL}_{4}$, the Steinberg representation can be
written as $Q([-\frac{3}{2},\frac{3}{2}])$ and the trivial representation is
given by $Q([\frac{3}{2}],[\frac{1}{2}],[-\frac{1}{2}],[-\frac{3}{2}])$. The
multisegment for $\phi_{\psi}$ from Example 2.2 is {[0,1],[-1,0]}.

###  3.2. Rank triangles

We know from Section 2 that any orbit of
$(x_{kk-1},\ldots,x_{21},x_{10})\in V$ is determined by the ranks of $x_{ij}$s
as defined in (14). We arrange these ranks in a rank triangle. This
arrangement helps us write down a visual algorithm to compute the
correspondence between multisegments and rank triangles in Section 3.3. To
that end we set

$r_{ij}:=\operatorname{rank}(x_{ij}),$

where $0\leq j<i\leq k$. Recall that $E_{i}$ is an eigenspace for
$\lambda(\mathfrak{f})$ with eigenvalue $q^{e_{i}}$ with multiplicity $m_{i}$.
We arrange these exponents, multiplicities, and ranks into a triangle to
reflect the corresponding combinations of the $x_{ij}$ and refer to this as a
rank triangle:

$e_{k}$$e_{k-1}$$\cdots$$e_{1}$$e_{0}$$m_{k}$$m_{k-1}$$\cdots$$m_{1}$$m_{0}$$r_{kk-1}$$\hskip
8.5359ptr_{(k-1)(k-2)}\cdots$$r_{21}$$r_{10}$$r_{k(k-2)}$$\cdots$$r_{20}$$\cdots$$\cdots$$r_{k0}$

From left to right, the values in the top row of the rank triangle correspond,
respectively, to the exponents $e_{i}$ of the eigenvalues $q^{e_{i}}$. The
values in the second row correspond respectively to the multiplicities $m_{i}$
of the eigenvalues $q^{e_{i}}$. The row below these multiplicities shows ranks
$r_{ij}$ of $x_{ij}$, which are subject to the condition that every rank is
less than or equal to the two eigenvalue multiplicities above it. For the
other rows, the ranks are subject to exactly two conditions:

  * •

Every rank is less than or equal to the two ranks above it. This is a
consequence of a basic fact from linear algebra: if $A$ and $B$ are matrices
such that $AB$ and $BC$ are defined then
$\operatorname{rank}(AB)\leq\operatorname{rank}(A)$ and
$\operatorname{rank}(AB)\leq\operatorname{rank}(B)$.

  * •

The ranks satisfy

$r_{is}+r_{lj}\leq r_{ij}+r_{ls},\qquad j\leq s<l\leq i.$

This condition is a consequence of the Frobenius inequality from linear
algebra: if $A,B$ and $C$ are matrices such that $AB$, $BC$ and $ABC$ are
defined then
$\operatorname{rank}(AB)+\operatorname{rank}(BC)\leq\operatorname{rank}(ABC)+\operatorname{rank}(B)$.

The set of $H$-orbits in $V$ is naturally in bijection with rank triangles
subject to these conditions.

The set of $H$-orbits in $V$ carries a partial order defined by the Zariski
topology:

######  Definition 3.3.

There is a partial order on the set of orbits in $V_{\lambda}$: For any two
orbits $C$ and $C^{\prime}$, $C\leq C^{\prime}$ if and only if
$C\subseteq\overline{C^{\prime}}$.

This partial order can be read easily from the corresponding rank triangles as
follows: If the ranks for $C$ are $r_{ij}$ and $C^{\prime}$ are
$r^{\prime}_{ij}$ then $C\leq C^{\prime}$ if and only if $r_{ij}\leq
r^{\prime}_{ij}$ for all $i,j$.

######  Example 3.4.

For $\lambda$ in Example 2.2, the template for the rank triangles will look as
follows.

$e_{2}$$e_{1}$$e_{0}$$m_{2}$$m_{1}$$m_{0}$$r_{21}$$r_{10}$$r_{20}$

The orbit corresponding to the case $r_{21}=r_{10}=1$ and $r_{20}=0$ will have
the following rank triangle.

$-1$$0$$1$$1$$2$$1$$1$$1$$0$

###  3.3. Correspondence between rank triangles and multisegments

We write down an algorithm to compute a rank triangle from a multisegment and
vice-versa.

To each segment $\Delta=[e_{j},e_{i}]$ we associate the rank triangle
$T_{\Delta}$ with

$r_{ls}=\begin{cases}1,&j\leq s<l\leq i;\\\
0,&\text{otherwise.}\end{cases}\qquad m_{s}=\begin{cases}1,&j\leq s\leq i;\\\
0,&\text{otherwise}.\end{cases}$

A multisegment $\underline{m}$ then determines the triangle

$T_{\underline{m}}=\sum_{\Delta\in\underline{m}}T_{\Delta}.$

We now explain how to pass from a rank triangle with eigenvalues $q^{e_{i}}$
and multiplicities (top row) $m_{i}$ to a multisegment with support
$\underline{m}_{\lambda}$ by the following inductive rule:.

  1. (1)

Set $\underline{m}:=\emptyset$.

  2. (2)

Let $r_{ij}$ be the lowest and most to the left non-zero entry in the rank
triangle. Note that $j<i$.

  3. (3)

Add the segment $\Delta=[e_{i},e_{j}]$ to $\underline{m}$.

  4. (4)

Replace $r_{kl}$ by $r_{kl}-1$ for each value of $k,l$ with $j\leq l<k\leq i$.
Replace $m_{k}$ by $m_{k}-1$ for each $j\leq k\leq i$. That is, subtract
$T_{\Delta}$ from the rank triangle.

  5. (5)

Repeat steps (ii) through (iv) until all $r_{ij}$ are zero.

  6. (6)

If any $m_{k}\neq 0$, add the singleton $\\{e_{k}\\}$ with multiplicity
$m_{k}$ to $\underline{m}$.

Now $\underline{m}$ is the multisegment determined by the rank triangle. These
procedures, $\underline{m}\leftrightarrow T_{\underline{m}}$ establish a
bijection between rank triangles and multisegments.

In Definition 3.1 we define a parital order on multisegments and in Definition
3.3 we have a partial order on orbits (and therefore rank triangles). From
[zelevinskii1981p] (Theorem 2.2, Contiguity theorem) these partial orders are
compatible under the correspondence. More precisely, let
$\underline{m},T_{\underline{m}}$, and $C_{\underline{m}}$ be a multisegment,
its corresponding rank triangle and orbit respectively. Then the above
statement can be rephrased as: If $\alpha$ and $\beta$ are multisegments, then
$\alpha\leq\beta$ if and only if $C_{\alpha}\leq C_{\beta}$ if and only if
$T_{\alpha}\leq T_{\beta}$.

######  Example 3.5.

Let us compute the multisegment for the rank triangle from Example 3.4.

$-1$$0$$1$$1$$2$$1$$1$$1$$0$$[-1,0]$$-1$$0$$1$$0$$1$$1$$0$$1$$0$$[0,1]$$-1$$0$$1$$0$$0$$0$$0$$0$$0$Terminate

The resulting multisegment is $\\{[0,1],[-1,0]\\}$. This orbit corresponds to
the representation $Q([0,1],[-1,0])$. In this way, we can compute the
multisegments for all orbits in the Vogan variety $V$ from in Example 2.2.

$\Pi_{\lambda}(\operatorname{GL}_{4}(F))$ | Rank triangle
---|---
$Q([-1,1],[0])$ | $-1$$0$$1$$1$$2$$1$$1$$1$$1$
$Q([0,1],[-1,0])$ | $-1$$0$$1$$1$$2$$1$$1$$1$$0$
$Q([1],[0],[-1,0])$ | $-1$$0$$1$$1$$2$$1$$1$$0$$0$
$Q([0,1],[0],[-1])$ | $-1$$0$$1$$1$$2$$1$$0$$1$$0$
$Q([1],[0],[0],[-1])$ | $-1$$0$$1$$1$$2$$1$$0$$0$$0$

The open orbit in $V$ consists of elements of full rank. From the table above,
we see that the open orbit corresponds to the representation $Q([-1,1],[0])$.
On the other hand, having directly computed the multisegment {[-1,0],[0,1]}
for $\phi_{\psi}$ from Example 2.2, we now have its rank triangle.

##  4\. Zelevinsky involution on multisegments and orbits

We now work with an involution on orbits, often referred to as the Zelevinsky
involution, which interchanges the Steinberg and trivial representations. This
appears in [Z2], [zelevinskii1981p], and is expanded upon in [MW:involution].
We first write down an algorithm to compute the involution
$\alpha\mapsto\tilde{\alpha}$ following [MW:involution] and then prove an
important combinatorial lemma (Lemma 4.8). Finally, we see the analogous
notions on orbits.

Let $\alpha$ denote a multisegment. This means that $\alpha$ is a collection
of segments of the form $[b,e]$, where $b$ and $e$ are integers or half
integers and are referred to as the base value and end values of the segment
respectively.

###  4.1. Moeglin-Waldspurger algorithm

In this section, write down the Moeglin-Waldpurger algorithm [MW:involution]
to compute the Zelevinsky involution on multisegments for
$\operatorname{GL}_{n}$. For this we recall what it means for one segment to
precede the other. Let $\Delta_{1}=[b_{1},e_{1}]$ and
$\Delta_{2}=[b_{2},e_{2}]$ be two segments.

######  Definition 4.1.

$\Delta_{1}$ is said to precede $\Delta_{2}$ if and only if

$b_{1}<b_{2}\text{, }e_{1}<e_{2},\text{ and }b_{2}\leq e_{1}+1.$

We list the steps to compute $\tilde{\alpha}$ below.

First, we compute the segment $M(\alpha)$ associated to $\alpha$:

  1. (1)

Set $e$ to be the largest end value that appears in any segment. Set $m:=e$.

  2. (2)

Consider all segments in $\alpha$ with end value $m$. Among these, choose a
segment with the largest base value and call it $\Delta_{m}$.

  3. (3)

Consider the set of all segments in $\alpha$ that precede $\Delta_{m}$ with
end value $m-1$. If this is empty, go to step $(5)$. Otherwise, choose a
segment from this set with the largest base value and call it $\Delta_{m-1}$.

  4. (4)

Set $m:=m-1$ and go to step (3).

  5. (5)

Return $M(\alpha):=[m,e]$.

Next, we inherit the following notation from the above procedure: for any
$i\in\\{m,m+1,\cdots,e\\}$, set $\Delta_{i}$ as the segment whose base value
is the highest among segments with end value $i$. Define $\alpha\setminus
M(\alpha)$ to be the multisegment obtained by removing $i$ from $\Delta_{i}$
for all $m\leq i\leq e$. Then, the dual $\tilde{\alpha}$ is recursively
defined via

$\tilde{\alpha}:=(M(\alpha),\widetilde{\alpha\setminus M(\alpha)}).$

The fact that this procedure gives you an involution comes from
[MW:involution]. In particular, we have the useful property that
$\tilde{\tilde{\alpha}}=\alpha$.

######  Example 4.2.

Let us continue working in the setting of Example 2.2 and compute the
involution on multisegments that appear:

$\alpha$ | $\tilde{\alpha}$
---|---
$\\{[-1,1],[0]\\}$ | $\\{[-1],[0],[0],[1]\\}$
$\\{[-1,0],[0,1]\\}$ | $\\{[-1,0],[0,1]\\}$
$\\{[1],[0],[-1,0]\\}$ | $\\{[0,1],[0],[-1]\\}$

###  4.2. Lemma about simple multisegments

The aim of this section is to prove Lemma 4.8. We establish some terminology
and notation associated to a multisegment $\alpha$. The length of a segment
$[b,e]$ is simply the value $e-b+1$.

$L_{\alpha}$:= Length of the longest segment in $\alpha$.
$n_{\alpha}$:= number of segments in $\alpha$
$c_{\alpha}$:= minimum number of segments that
$\bigcup_{\Delta\in\alpha}\Delta$ can be broken into.

######  Lemma 4.3.

Let $\alpha$ and $\beta$ be any multisegments. If $\alpha\leq\beta$, then

  1. (1)

$L_{\alpha}\leq L_{\beta}$.

  2. (2)

$n_{\alpha}\geq n_{\beta}.$

  3. (3)

$n_{\tilde{\alpha}}\geq L_{\alpha}$ and $n_{\alpha}\geq L_{\tilde{\alpha}}$.

###### Proof.

  1. (1)

Since $\beta$ is obtained by performing elementary operations on segments in
$\alpha$, the longest segment in $\beta$ is either a segment from $\alpha$ or
a union of two segments in $\alpha$. Therefore, $L_{\beta}\geq L_{\alpha}.$

  2. (2)

Every pair of segments in $\alpha$ is replaced by at most two segments to form
$\beta$. Therefore $n_{\beta}\leq n_{\alpha}$.

  3. (3)

Fix a segment in $\alpha$. From the Moeglin-Waldspurger algorithm, one can
observe that each element of that segment appears in a distinct segment in
$\tilde{\alpha}$. In particular, this is true for the longest segment of
$\alpha$. Thus, we get $n_{\tilde{\alpha}}\geq L_{\alpha}$. Since
$\tilde{\tilde{\alpha}}=\alpha$, we get the other inequality.

∎

######  Definition 4.4.

$\alpha$ is a simple multisegment if it is of the form

$\\{[b,e],[b+1,e+1],\cdots,[b+n-1,e+n-1]\\}.$

Observe that $n$ in the definition above is equal to $n_{\alpha}$.

######  Lemma 4.5.

If $\alpha$ is a simple multisegment, then

  1. (1)

$\tilde{\alpha}$ is simple.

  2. (2)

$n_{\tilde{\alpha}}=L_{\alpha}$.

  3. (3)

$c_{\alpha}=1$.

###### Proof.

  1. (1)

Applying the Moeglin-Waldspurger algorithm, one can explicitly compute

$\tilde{\alpha}=\\{[e-k,e+n_{\alpha}-k-1]:0\leq k\leq e-b\\}.$

Clearly, it is simple.

  2. (2)

Observe that the length of every segment in $\alpha$ is the same, and
$L_{\alpha}=e-b+1$. Observing how the index $k$ varies in part $(1)$ tells us
that $n_{\alpha}=e-b+1$.

  3. (3)

$c_{\alpha}=1$ because

$\bigcup_{\Delta\in\alpha}\Delta=[b,e+n-1].$

∎

######  Lemma 4.6.

If $L_{\tilde{\alpha}}=n_{\alpha}$, $\alpha\leq\beta$, and
$\tilde{\alpha}\leq\tilde{\beta}$, then

$n_{\alpha}=n_{\beta}=L_{\tilde{\alpha}}=L_{\tilde{\beta}}.$

The above statement also holds true if $\alpha$ is interchanged with
$\tilde{\alpha}$ and $\beta$ with $\tilde{\beta}$.

###### Proof.

By hypothesis, $\tilde{\alpha}\leq\tilde{\beta}$. Therefore,

$\displaystyle L_{\tilde{\alpha}}$ $\displaystyle\leq L_{\tilde{\beta}},\text{
(from Lemma \ref{lemma1})}$ $\displaystyle\leq
n_{\tilde{\tilde{\beta}}},\text{ (from Lemma \ref{lemma1}) }$
$\displaystyle=n_{\beta},\text{ (since $\tilde{\tilde{\beta}}=\beta$)}$
$\displaystyle\leq n_{\alpha},\text{ (hypothesis and Lemma \ref{lemma1})}$
$\displaystyle=L_{\tilde{\alpha}}.\text{ (hypothesis)}$

∎

######  Lemma 4.7.

Let $\alpha$ be a simple multisegment and $\beta$ any multisegment such that
$\alpha\leq\beta$ and $L_{\alpha}\leq L_{\beta}$. Then, $\alpha=\beta$.

###### Proof.

Suppose $\alpha<\beta$. Then, there is a segment in $\beta$ of the form
$\Delta_{1}\cup\Delta_{2}$ strictly containing $\Delta_{1}$ and $\Delta_{2}$,
where $\Delta_{1},\Delta_{2}\in\alpha$. Now $\alpha$ is simple, so every
segment has the same length and that length is equal to $L_{\alpha}$. Thus,
the strict containment above implies

$L_{\beta}\geq\text{ length of }\Delta_{1}\cup\Delta_{2}>\text{ length of
}\Delta_{1}=L_{\alpha},$

which contradicts the hypothesis. ∎

Now we are ready to state the primary result of this section.

######  Lemma 4.8.

Suppose $\alpha$ is a simple multisegment and $\beta$ is any multisegment
satisfying $\alpha\leq\beta$ and $\tilde{\alpha}\leq\tilde{\beta}$, then
$\alpha=\beta$.

###### Proof.

$\alpha$ is simple, so from Lemma 4.5 $n_{\tilde{\alpha}}=L_{\alpha}$. Lemma
4.6 implies $L_{\alpha}=L_{\beta}$. The hypothesis of Lemma 4.7 is satisfied
and thus $\alpha=\beta$. ∎

###  4.3. Involution on orbits

Following [CFMMX]*Section 6, we explain the notion of a dual orbit which is
used to define an analogous involution on orbits $C\mapsto\widehat{C}$ of
$V_{\lambda}$.

From (8), we may write

$V=\\{x\in\mathfrak{gl}_{n}(\mathbb{C}):\operatorname{Ad}(\lambda(\mathfrak{f})x=qx\\}.$

Using the killing form on $\mathfrak{gl}_{n}(\mathbb{C})$, the dual variety
$V^{*}$ may be realized as

$V^{*}:=\\{x\in\mathfrak{gl}_{n}(\mathbb{C}):\operatorname{Ad}(\lambda(\mathfrak{f})x=q^{-1}x\\}.$

We have a presentation of $V^{*}$ analogous to (10):

$V^{*}=\operatorname{Hom}(E_{1},E_{0})\times\operatorname{Hom}(E_{k-1},E_{k-2})\times\cdots\times\operatorname{Hom}(E_{k},E_{k-1}),$

with an $H$-action

$(h_{k},\ldots,h_{1},h_{0})\ldots(y_{10},y_{21},\ldots,y_{kk-1})=(h_{0}y_{10}h_{1}^{-1},h_{1}y_{21}h_{2}^{-1},\ldots,h_{k-1}y_{kk-1}h_{k}^{-1}),$

where $(h_{k},\ldots,h_{1},h_{0})\in H$ and
$(y_{10},y_{21},\ldots,y_{kk-1})\in V^{*}$.

The cotangent variety $T^{*}(V)$ for $V$ is simply $V\times V^{*}$. For $x\in
V$ and $y\in V^{*}$, $[x,y]=xy-yx$ is the usual Lie bracket from
$\mathfrak{gl}_{n}(\mathbb{C})$. The conormal variety, $\Lambda\subseteq
T^{*}(V)$, is defined as the kernel of the map
$[\cdot,\cdot]:T^{*}(V)\xrightarrow{}\operatorname{Lie}(H)$.

By [CFMMX]*Proposition 6.3.1, for each $H$-orbit $C$ in $V$,

(17) $\Lambda_{C}:=\\{(x,y)\in\Lambda:x\in C\\}$

is the conormal bundle to $C$. By identifying $V^{**}\simeq V$, for an
$H$-orbit $C^{*}$ in $V^{*}$ we may write

$\Lambda_{C^{*}}=\\{(x,y)\in\Lambda:x\in C^{*}\\}$

where $\Lambda_{C^{*}}$ is the conormal bundle to $C^{*}$.

For any $C$, there exists a unique orbit $C^{*}$ in $V^{*}$ with
$\bar{\Lambda}_{C}\simeq\bar{\Lambda}_{C^{*}}$; see [Pyasetskii]*Corollary 2.
The map $C\mapsto C^{*}$ defines a bijection the set of $H$-orbits in $V$ and
the set of $H$-orbits in $V^{*}$. Using this, we may define

$\widehat{C}:=\prescript{t}{}{C^{*}},$

where $y\mapsto\prescript{t}{}{y}$ is matrix transposition. Then
$C\mapsto\widehat{C}$ defines an involution on the orbits.

In section 4.1, we wrote down the Zelevinsky involution on multisegments and
in this section we see an involution on orbits. Since multisegments and orbits
correspond to each other via rank triangles as seen in Section 3.3, it is
natural to ask whether this involution is compatible under this
correspondence. Crucially, it is. More precisely, if the multisegment $\alpha$
corresponds to the orbit $C$, then $\tilde{\alpha}$ corresponds to
$\widehat{C}$. See [knight1996representations] for more details.

######  Example 4.9.

From Example 4.2, we see that the involution applied to $\\{[-1,1],[0]\\}$
gives $\\{[1],[0],[0],[-1]\\}$. From Example 3.5, we see that the involution
interchanges the open orbit (with elements of full rank) with the closed point
(origin).

##  5\. Vogan’s conjecture for simple Arthur parameters

Recall the simple Arthur parameter $\psi$ defined in (2), the Langlands
parameter $\phi_{\psi}$ coming from it defined in (2), and its infinitesimal
parameter $\lambda$ defined in (2). We remind the reader that
$G=\operatorname{GL}_{n}$. Now, $\pi\in\Pi(G(F))$ is said to have the
infinitesimal parameter $\lambda$ if its corresponding Langlands parameter
$\phi$ has the infinitesimal parameter $\lambda$. Let $\Pi_{\lambda}(G(F))$
denote all such $\pi$ and $\Phi_{\lambda}(G(F))$ denote equivalence classes of
all such $\phi$. As outlined in sections 1.4 and 2, equivalence classes of
smooth irreducible representations of $G(F)$ with infinitesimal parameter
$\lambda$ correspond to equivalence classes of simple objects in
$\operatorname{Per}_{H}(V)$. This bijection is summarized below:

${{\Pi_{\lambda}(G(F))}}$${{\Phi_{\lambda}(G(F))}}$${{H\operatorname{-orbits}\operatorname{in}V}}$${{\operatorname{Perv}_{H}(V)^{\operatorname{simple}}_{/\operatorname{iso}},}}$${{\pi\hskip
5.69054pt}}$${\phi}$${{\hskip 5.69054ptC_{\phi}\hskip 5.69054pt}}$${{\hskip
5.69054pt\mathcal{P}(\pi)=\mathcal{IC}(\mathbb{1}_{C_{\phi}}).}}$

Thus, there is a unique orbit $C_{\phi_{\psi}}$ in $V$ attached to
$\phi_{\psi}$. We shorten this notation to $C_{\psi}$. This orbit in turn
corresponds to a simple multisegment - one can see this directly by writing
down the multisegment for the Langlands parameter $\phi_{\psi}$ when
$\psi(w,x,y)=\operatorname{Sym}^{d}(x)\otimes\operatorname{Sym}^{a}(y)$:

(18)
$\left\\{\left[\tfrac{-(d+a)}{2},\tfrac{-(d+a)}{2}+d\right],\left[\tfrac{-(d+a)}{2}+1,\tfrac{-(d+a)}{2}+d+1\right],\ldots,\left[\tfrac{-(d+a)}{2}+a,\tfrac{-(d+a)}{2}+d+a\right]\right\\}.$

The dictionary between Langlands parameters and multisegments is written down
in many places, see [kudla1994local]*Section 4.2, for example. In Section 3.3
we saw that rank triangles and multisegments correspond to each other and this
correspondence respects the partial order on both sides. In section 4.3, we
see that this correspondence respects the involution on both sides. Thus, we
may adapt Lemma 4.8 as follows:

######  Lemma 5.1.

Let $C_{\psi}$ be the orbit corresponding to a simple Arthur parameter as
explained above. Let $C$ be any $H$-orbit in $V$. If $C_{\psi}\leq C$ and
$\widehat{C}_{\psi}\leq\widehat{C}$, then $C_{\psi}=C$.

######  Lemma 5.2.

Let $C_{\psi}$ be the orbit corresponding to a simple Arthur parameter. Then

$\widehat{C}_{\psi}=C_{\widehat{\psi}},$

where ${\widehat{\psi}}$ is the the simple Arthur parameter obtained by
interchanging the role of the two $\operatorname{SL}_{2}(\mathbb{C})$’s in
$W_{F}^{\prime\prime}$:

$\widehat{\psi}(w,x,y):=\psi(w,y,x).$

###### Proof.

Since $\psi$ is simple, $C_{\psi}$ corresponds to the multisegment written in
(18). The dual of this multisegment, in the sense of 4.1, is

$\left\\{\left[\tfrac{-(d+a)}{2},\tfrac{-(d+a)}{2}+a\right],\left[\tfrac{-(d+a)}{2}+1,\tfrac{-(d+a)}{2}+a+1\right],\ldots,\left[\tfrac{-(d+a)}{2}+d,\tfrac{-(d+a)}{2}+d+a\right]\right\\}.$

By inspection, this multisegment comes from the simple Arthur parameter
$(w,x,y)\mapsto\operatorname{Sym}^{a}(x)\otimes\operatorname{Sym}^{d}(y)$.
From the discussion in 4.3, this multisegment also corresponds to
$\widehat{C}_{\psi}$, as required. ∎

We recall the functor defined in (1.5):

$\operatorname{Evs}_{\psi}:\operatorname{Per}_{H_{\lambda_{\psi}}}(V_{\lambda_{\psi}})\xrightarrow{}\operatorname{Rep}(A_{\psi}),.$

For the Langlands parameter $\phi_{\psi}$, we recall the ABV-packet from (2):

$\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F)):=\\{\pi\in\Pi_{\lambda}(G(F)):\operatorname{Evs}_{\psi}(\mathcal{P}(\pi))\neq
0\\}.$

We now state and prove the main theorem of this paper.

######  Theorem 5.3.

Let $\psi$ be a simple Arthur parameter of $G=\operatorname{GL}_{n}$,
$\phi_{\psi}$ its corresponding Langlands parameter, $\Pi_{\psi}(G(F))$ the
Arthur packet attached to $\psi$, and
$\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F))$
the ABV-packet attached to $\phi_{\psi}$. Then,

$\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F))=\Pi_{\psi}(G(F)).$

###### Proof.

The local Langlands correspondence tells us that $L$-packets are singletons.
As discussed in 1.3, the Arthur packet $\Pi_{\psi}(G(F))$ is a singleton and
it coincides with the $L$-packet for $\phi_{\psi}$. We may write

$\Pi_{\psi}(G(F))=\Pi_{\phi_{\psi}}(G(F))=\\{\pi_{\psi}\\}.$

Let $C_{\psi}$ be the unique orbit associated to the Langlands parameter
$\phi_{\psi}$. By [CFMMX]*Theorem 7.19,
$\operatorname{Evs}_{\psi}\mathcal{IC}(\mathbb{1}_{C_{\psi}})\neq 0$. By
definition, this implies

$\Pi_{\phi_{\psi}}(G(F))\subseteq\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F)),$

and therefore

$\Pi_{\psi}(G(F))\subseteq\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F)).$

To prove the theorem, it is sufficient to prove that
$\Pi_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F))$
is a singleton, i.e., it contains no representation other that
$\\{\pi_{\psi}\\}$.

Suppose we have a representation
$\pi\in\Pi_{\phi}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F))$.
By definition, this means

(19) $\operatorname{Evs}_{\psi}(\mathcal{P}(\pi))\neq 0.$

From the discussion in sections 1.4 and 2, we know that $\pi$ corresponds to
an $H$-orbit $C$ in $V$ and $\mathcal{P}(\pi)=\mathcal{IC}(\mathbb{1}_{C})$.
We may rephrase (19) as

(20) $\operatorname{Evs}_{\psi}\mathcal{IC}({\mathbbm{1}}_{C})\neq 0.$

Using [KS_sing]*Proposition 3.2.1 and Lemma 5.2, (19) also implies that

(21)
$\operatorname{Evs}_{\widehat{\psi}}\mathcal{IC}({\mathbbm{1}}_{\widehat{C}})\neq
0.$

Again from [KS_sing]*Proposition 3.2.1, (20) implies that $C_{\psi}\leq C$ and
(21) implies $\widehat{C}_{\psi}\leq\widehat{C}$. The hypothesis of Lemma 5.1
is satisfied, so we may conclude that $C_{\psi}=C$. The corresponding
representations are the same, as required.

∎

##  6\. Vogan’s conjecture for irreducible Arthur parameters

We now extend the result in Theorem 5.3 to an irreducible Arthur parameter of
$\operatorname{GL}_{n}$, i.e., of the form

(22)
$\psi(w,x,y)=\rho(w)\otimes\operatorname{Sym}^{d}(x)\operatorname{Sym}^{a}(y),$

where $\rho$ is an irreducible admissible representation of $W_{F}$ of
dimension $m$, and $a$ and $d$ are non-negative integers. First note that
$\rho=\rho_{0}\otimes|\cdot|^{s}$, where $\rho_{0}$ is of Galois type and
$|\cdot|$ is the absolute value on the Weil group as explained in section 2.

We apply the procedure of hyper-unramification as explained in [CFMMX]*Section
5 on the infinitesimal parameter $\lambda$. Note that
$\lambda(w)=\rho_{0}(w)\otimes|w|^{s}\otimes\operatorname{Sym}^{d}(d_{w})\otimes\operatorname{Sym}^{a}(d_{w})$
where $d_{w}$ is defined in Section 2. For any
$X\in\operatorname{SL}_{2}(\mathbb{C})$, set
$\eta(X):=\operatorname{Sym}^{d}(X)\otimes\operatorname{Sym}^{a}(X)$.

######  Lemma 6.1.

The semisimple element $\lambda(\mathfrak{f})$ has the decomposition
$\lambda(\mathfrak{f})=s_{\lambda}t_{\lambda}$ where

$s_{\lambda}=\rho_{0}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\otimes|\mathfrak{f}|^{\operatorname{Re}(s)}\otimes\eta(d_{\mathfrak{f}})$

is the hyperbolic part and

$t_{\lambda}=\rho_{0}(\mathfrak{f})\otimes|\mathfrak{f}|^{i\operatorname{Im}(s)}\otimes\eta\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$

is the elliptic part.

###### Proof.

Clearly,
$s_{\lambda}t_{\lambda}=t_{\lambda}s_{\lambda}=\lambda(\mathfrak{f})$. To show
that $s_{\lambda}$ is hyperbolic, it suffices to show that its eigenvalues are
positive real numbers. Indeed, after $s_{\lambda}$ is identified with a
matrix, its eigenvalues are of the form
$q^{\operatorname{Re}s+\frac{a+d}{2}+j}$ where $j$ is an integer. This can be
seen from direct calculation aided by Equation 6. To show that $t_{\lambda}$
is elliptic, it suffices to show that its eigenvalues have complex norm 1.
Indeed, $t_{\lambda}$ can be identified with the (block) matrix

$\begin{bmatrix}\rho_{0}(\mathfrak{f})q^{i\operatorname{Im}s}&0&\cdots&0\\\
0&\rho_{0}(\mathfrak{f})q^{i\operatorname{Im}s}&\cdots&0\\\ \vdots&&\ddots&\\\
0&\cdots&0&\rho_{0}(\mathfrak{f})q^{i\operatorname{Im}s}\end{bmatrix},$

so complex norm of its eigenvalues are determined by those of
$\rho_{0}(\mathfrak{f})$. However, $\rho_{0}$ is of Galois-type which means it
has finite image. Thus $\rho_{0}(\mathfrak{f})$ is a matrix of finite order,
whose eigenvalues must be roots of unity.

∎

The process of hyper-unramification described in [CFMMX]*Section 5.3 adapted
to our case gives us an unramified parameter
$\lambda_{nr}:W_{F}\xrightarrow[]{}J_{\lambda}$, where $J_{\lambda}$ is
$Z_{\widehat{G}}(\lambda|_{I_{F}},t_{\lambda})$ whose image of frobenius is
the hyperbolic element $s_{\lambda}$. Using [CFMMX]*Lemma 5.3 we have that

$V_{\lambda_{nr}}=V_{\lambda},\text{ and }H_{\lambda_{nr}}=H_{\lambda}^{0}.$

######  Lemma 6.2.

Suppose $\psi$ is an irreducible Arthur parameter as defined in Equation 22.
Then,

  1. (1)

$J_{\lambda}=\operatorname{GL}_{n/m}(\mathbb{C})$.

  2. (2)

$H_{\lambda_{nr}}=H_{\lambda}$

###### Proof.

  1. (1)

By definition, $J_{\lambda}=Z_{\hat{G}}(\lambda|_{I_{F}},t_{\lambda})$. Now
$\lambda(w)|_{I_{F}}=\rho(w)|_{I_{F}}\otimes\eta(d_{w})|_{I_{F}}$. Since
$I_{F}$ consists of elements of absolute value 1,
$\lambda(w)|_{I_{F}}=\rho_{0}(w)|_{I_{F}}\otimes\eta\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$. As $\rho_{0}$ is of Galois-type, it is determined by a
finite list of elements. In particular, there are elements
$\sigma_{1},\sigma_{2},\ldots,\sigma_{k}$ in $I_{F}$ so that their images
(block matrices) of the form

$\begin{bmatrix}\rho_{0}(\sigma_{i})&0&\cdots&0\\\
0&\rho_{0}(\sigma_{i})&\cdots&0\\\ \vdots&&\ddots&\\\
0&0&\cdots&\rho_{0}(\sigma_{i})\end{bmatrix}$

determine $\rho_{0}|_{I_{F}}$.

$\displaystyle J_{\lambda}$
$\displaystyle=Z_{\operatorname{GL}_{n}(\mathbb{C})}(\lambda(w)|_{I_{F}},t_{\lambda})$
$\displaystyle=\\{X\in\operatorname{GL}_{n}(\mathbb{C}):t_{\lambda}Xt_{\lambda}^{-1}=X\text{
and }\lambda(\sigma_{i})X\lambda(\sigma_{i})^{-1}=X\text{ for }1\leq i\leq
k\\}$
$\displaystyle=\\{X=(X_{rs})_{r,s}\in\operatorname{GL}_{n}(\mathbb{C})\text{
where
}X_{rs}\in\operatorname{Mat}_{m}(\mathbb{C}):\rho_{0}(\mathfrak{f})X_{rs}\rho_{0}(\mathfrak{f})^{-1}=X_{rs}\text{
and }\rho_{0}(\sigma_{i})X_{rs}\rho_{0}(\sigma_{i})^{-1}=X_{rs}\\}$
$\displaystyle=\\{X=(X_{rs})_{r,s}\in\operatorname{GL}_{n}(\mathbb{C})\text{
where }X_{rs}\in\operatorname{Mat}_{m}(\mathbb{C}):X_{rs}\in
Z_{\operatorname{Mat}_{n}(\mathbb{C})}(\rho_{0})\\}$
$\displaystyle=\\{X=(c_{rs}\operatorname{Id}_{m})_{r,s}\in\operatorname{GL}_{n}(\mathbb{C})\text{
where }c_{rs}\in\mathbb{C}\\}$
$\displaystyle\simeq\operatorname{GL}_{n/m}(\mathbb{C}).$

The first equality is by definition, the second by the fact that
$\lambda|_{I_{F}}$ is determined by $\sigma_{i}s$ as explained above, the
third follows by writing down $X,t_{\lambda}$, and $\lambda(\sigma_{i})s$ as
block matrices, the fourth follows from the fact that $\rho_{0}$ is completely
determined by $\mathfrak{f}$ and $\sigma_{i}$s, and the fifth follows from
Schur’s lemma applied to the irreducible representation $\rho_{0}$ for every
$r,s$. The isomorphism in the sixth step comes from realizing
$(c_{rs}\operatorname{Id}_{m})_{r,s}$ as
$(c_{rs})_{r,s}\otimes\operatorname{Id}_{m}$ whose determinant is
$\operatorname{det}((c_{rs})_{r,s})^{m}$ using the determinant of the
Kronecker product of two matrices.

  2. (2)

As centralizers in $\operatorname{GL}_{n}(\mathbb{C})$ are connected,
$H_{\lambda}$ is connected so $H_{\lambda_{nr}}=H_{\lambda}^{0}=H_{\lambda}$.

∎

The above lemma shows that if we replace $J_{\lambda}$ with the isomorphic
general linear group, we may interpret the unramified infinitesimal parameter
as a homomorphism $W_{F}\xrightarrow{}\operatorname{GL}_{n/m}(\mathbb{C})$
whose image at the frobenius can now be thought of as
$|\mathfrak{f}|^{\operatorname{Re}s}\otimes\eta(d_{\mathfrak{f}})$, and whose
Vogan variety and group acting on it remain the same as that of $\lambda$.
Also observe that removing the factor $|\mathfrak{f}|^{\operatorname{Re}s}$
from this infinitesimal parameter does not change the variety and the group
acting on it for trivial reasons. We collect these observations on hyper-
unramification of $\psi$ as defined in 22 in the following lemma.

######  Lemma 6.3.

Suppose $\psi$ is an irreducible Arthur parameter of $\operatorname{GL}_{n}$,
i.e.,

$\psi(w,x,y)=\rho(w)\otimes\operatorname{Sym}^{d}(x)\otimes\operatorname{Sym}^{a}(y),$

where $\rho$ is an irreducible representation of $W_{F}$ in
$\operatorname{GL}_{m}(\mathbb{C})$. We obtain the unramified parameter

$\lambda_{nr}:W_{F}\xrightarrow{}\operatorname{GL}_{n/m}(\mathbb{C})$

defined via
$w\mapsto\operatorname{Sym}^{d}(d_{w})\otimes\operatorname{Sym}^{a}(d_{w})$,
with

$V_{\lambda_{nr}}\simeq V_{\lambda},\text{ and }H_{\lambda_{nr}}\simeq
H_{\lambda},$

where these isomorphisms are obtained by an application of Schur’s lemma, as
in the proof of Lemma 6.2 (1).

The upshot is that $\lambda_{nr}$ can be thought of as the infinitesimal
parameter of
$\psi_{nr}(w,x,y):=\operatorname{Sym}^{d}(x)\otimes\operatorname{Sym}^{a}(y)$,
after an application of the isomorphisms $V_{\lambda_{nr}}\simeq V_{\lambda}$
and $H_{\lambda_{nr}}\simeq H_{\lambda}$ that are induced by taking scalar
matrices to their respective scalars in a smaller matrix group. Since the
variety and group action are unchanged, the category
$\operatorname{Per}_{H_{\lambda_{nr}}}(V_{\lambda_{nr}})=\operatorname{Per}_{H_{\lambda}}(V_{\lambda})$.
As a result, we directly see that the isomorphisms give us a bijection
$C\mapsto C_{nr}$ of orbits which preserves the ordering and the Zelevinsky
involution. The geometric side of the Vogan-Langlands correspondence can thus
be reduced to the study of the unramified infinitesimal parameter. This paves
the way for generalizing Theorem 5.3 from simple Arthur parameters to
irreducible Arthur parameters.

######  Theorem 6.4.

Suppose $\psi$ is an irreducible Arthur parameter of
$G=\operatorname{GL}_{n}$, i.e.,

$\psi(w,x,y)=\rho(w)\otimes\operatorname{Sym}^{d}(x)\otimes\operatorname{Sym}^{a}(y),$

where $\rho$ is an irreducible representation of $W_{F}$ on
$\operatorname{GL}_{m}(\mathbb{C})$. Let $\Pi_{\psi}(G(F))$ denote the local
Arthur packet of $\psi$ and
$\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F))$
denote the ABV-packet attached to $\phi_{\psi}$. Then,

$\Pi_{\psi}(G(F))=\Pi_{\phi_{\psi}}^{{\mbox{\raisebox{1.0pt}{\scalebox{0.5}{$\mathrm{ABV}$}}}}}(G(F)).$

###### Proof.

From Lemma 6.3, $V_{\lambda}\simeq V_{\lambda_{nr}}$ and $H_{\lambda}\simeq
H_{\lambda_{nr}}$. We have a bijection $C\mapsto C_{nr}$ between orbits in
$V_{\lambda}$ and $V_{\lambda_{nr}}$ which preserves partial order and the
Zelevinsky involution on orbits. This gives a bijection $\pi\mapsto\pi_{nr}$
between $\Pi_{\lambda}(G(F))$ and $\Pi_{\lambda_{nr}}(G(F))$ preserving the
partial order and involution on multisegments such that
$(C_{\pi})_{nr}=C_{\pi_{nr}}$. In particular, $(C_{\psi})_{nr}=C_{\psi_{nr}}$.
Since $\operatorname{Evs}_{\psi}$ is purely dependent on the geometry, we have
that $\operatorname{Evs}_{\psi}\mathcal{IC}(\mathbb{1}_{C_{\pi}})\neq 0$ for
$\pi\in\Pi_{\lambda}(G(F))$ if and only if
$\operatorname{Evs}_{\psi_{nr}}\mathcal{IC}(\mathbb{1}_{C_{\pi_{nr}}})\neq 0$
for the corresponding $\pi_{nr}\in\Pi_{\lambda_{nr}}(G(F))$. Since the ABV-
packet for $\psi_{nr}$ is a singleton from Theorem 5.3, same is true for the
ABV packet for $\psi$. ∎

######  Remark 6.5.

The process of hyper-unramification taking $\psi\mapsto\psi_{nr}$ gives us a
bijection between $\Pi_{\lambda}(G(F))$ and $\Pi_{\lambda_{nr}}(G(F))$ which
can be written down explicitly in terms of multisegments. Let $\pi_{\rho}$ be
the supercuspidal representation of $\operatorname{GL}_{m}(F)$ corresponding
to the irreducible representation $\rho$. The correspondence at the level of
multisegments is induced by taking any segment of the form
$[\pi_{\rho}(b),\pi_{\rho}(e)]$ to the segment $[b,e]$.

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Conversational negation of strings of wordssec:cn-sentence In the
distributional approach to natural language processing, a commonly used model
is the bag-of-words that disregards any grammar and treats words as a
structureless bag. Coecke et al. CSC proposed the DisCoCat model, which
combines grammar and distributional word meanings in a categorical framework,
allowing us to compose the meaning of words to get the meaning of a sentence.
In this section, we do the same for conversational negation by introducing the
conversational negation of sentences that is calculated using the
conversational negation of words.

Modelling conversational negation As pointed out by Oaksford and Stenning
oaksford:1992reasoning, the negation of more complex structures consisting of
multiple words may be interpreted as the negation of a subset of the
constituents. For example, a sentence such as “Bob did not drive to Oxford by
car” could be interpreted as:

  1. [label=), topsep=0.1em, itemsep=-0.3em]

  2. 1.

Bob did not drive to Oxford by car - Alice did

  3. 2.

Bob did not drive to Oxford by car - He carpooled

  4. 3.

Bob did not drive to Oxford by car - He drove to London

  5. 4.

Bob did not drive to Oxford by car \- He drove a van

  6. 5.

Bob did not drive to Oxford by car - Alice carpooled to Oxford

where the underlines indicate which words are being negated. The last example
is one of many possible cases that negate multiple constituents. While some of
these alternatives might immediately seem more plausible to the reader, the
correct choice is inherently dependent on the context. Based on this
interpretation, that a negation of multiple words is negating a subset of the
constituents, we extend our conversational negation framework to a string of
words by utilizing conversational negation of individual words (see Section
LABEL:sec:cn-word). As the correct interpretation of which words to negate may
not usually be obvious, we create a mixture of all possible interpretations.
Therefore, the negation of a string of $n$ words $w_{1}\otimes
w_{2}\otimes...\otimes w_{n}$ is a weighted mixture of all the interpretations
where only one word is negated, all the interpretations where two words are
negated, and so on. We have: [height=6.6em, valign=c]figures/CNstring.pdf &
∑_i = 1^n p_{w_i} (w_1 ⊗…⊗(w_i) ⊗…⊗w_n ) +
∑_i = 1^n ∑_j = i + 1^n p_{w_i, w_j} ( w_1 ⊗…⊗(w_i) ⊗…⊗(w_j) ⊗…⊗w_n )
\+ … where in the overall mixture representing the negation, each
interpretation has some weight. Formally, for $S=\\{w_{1},...,w_{n}\\}$ and
non-empty $S^{\prime}\subseteq S$ which we call the negation set, we get:
[height=6.6em, valign=c]figures/CNstring.pdf & ∑_S’ ∈P(S)∖{∅} p_S’ ⨂_i=1^n
{w_i if w_i ／∈S’
(w_i) if w_i ∈S’ To apply this negation to sentences, we adopt the view that
sentences are processes updating wires as presented by Coecke in
[coecke:2020textstructure]. These processes are built from a combination of
meaning states, interacting via updates. We propose the negation of a sentence
to be viewed as the same set of meaning states, first updated by the
conversational negation of the words before updating the wires as if the
negation was absent. This is in line with Kamp and Reyle [kamp:2013discourse,
pg. 104], who also treat the grammatical structure of negated sentences as if
they were not negated. Hence, we represent the conversational negation as a
function called $\cn$ that maps the circuit of a sentence to a circuit where
the sentence is pre-composed with $\cns$. For example, applying $\cn$ to the
sentence “Alice loves Bob”, we get the circuit for the sentence “Alice doesn’t
love Bob”:

$\includegraphics[width=0.9]{figures/AliceDoesntLoveBob.pdf}$

Here, we sum over all possible non-empty subsets of {Alice, love, Bob}, each
of which is weighted by the appropriate scalar.

##  Deriving the weights

The main challenge for the conversational negation of a string is deriving the
weights for the different interpretations of the negation. The choice of
correct interpretation, and therefore the weights, is dependent on context.
Context can be derived from many sources, such as the person who is speaking
and their intentions. In spoken language, intonation could clarify the
speaker’s intent by emphasizing the words meant to be negated. Another source
of context is the grammatical structure of the negated sentence itself. Given
the earlier example “Bob did not drive to Oxford by car,” which mentions the
mode of transport explicitly, intuitively the focus of negation is on this
detail. If the speaker solely wanted to negate Bob’s destination, the sentence
“Bob did not drive to Oxford” would be sufficient, not requiring any
additional detail. The other example, “Alice does not love Bob,” is more
ambiguous as the grammatical structure does not indicate the target of the
negation. Context from surrounding sentences can further determine the
sensibility of interpretations to a negation. Consecutive sentences should
have meanings consistent with each other and the reader’s general
understanding of the world. The statement Bob did not drive to Oxford by car -
He drove to London is reasonable with the knowledge that Oxford and London are
cities an hour’s drive apart. Likewise, Bob did not drive to Oxford by car \-
He drove a van makes sense because car and van are similar vehicles. Overall,
no single source of context is sufficient. A combination of all contextual
information—worldly, textual, physical, grammatical, intonation, etc.—is
required to determine the correct interpretation. While the different weights
for the negation sets are context-dependent, some general observations can be
made. Larger negation sets should tend to have smaller weights. The
psychological reasoning is that humans are less able to focus on a larger
number of details due to “limited information processing capacity”
[evans1989bias, oaksford:1992reasoning]. Considering the previous example, one
would require a lot of context for the interpretation of “Alice doesn’t love
Bob” to sensibly imply “Claire likes Dave”. Secondly, one can observe that the
weight of a negation set should depend on the likelihood of its individual
elements to be the target of the negation. If Alice being the target of the
negation is unlikely, for instance if the entire text is about Alice, then the
negation set of both Alice and love is also unlikely.

###  Determining weights using entailment

As mentioned earlier, one possible source of context can be the surrounding
text. In a text which solely talks about Alice and Bob, the sentence “Alice
doesn’t love Bob” probably intends to negate the word love, therefore
asserting that Alice feels emotions other than love for Bob. Building on this
intuition, we propose to use entailment measures to derive the weights for the
different interpretations of a negation. If the given interpretation of the
negation entails the surrounding sentences to a high degree, then the
interpretation is consistent with the surrounding text, and hence it is more
likely to be the intended meaning of the sentence. We compare each possible
interpretation of the negation with the surrounding sentences, where sentences
closer in the text have more influence towards the final weighting than
sentences that are further away. Let us consider the following, simplified
scenario of a negation, followed immediately by the clarification with both
sentences of the same grammatical structure:

color1 This is not red wine
color2 This is white wine

Here, we colour code the sentences for visual clarity, where the negated
sentence is red. We want to determine the respective weights
$p_{\\{{\color{color1}\texttt{red}}\\}},\
p_{\\{{\color{color1}\texttt{wine}}\\}}$ and
$p_{\\{{\color{color1}\texttt{red},\ \texttt{wine}}\\}}$ of the negation sets
{color1red}, {color1wine} and {color1red, wine}.111For the sake of simplicity,
we ignore the fact that “This” could also be the target of the negation, as in
“This is not red wine. That is!”. As a human reader, the sentence following
the negation clarifies that color1wine is not negated. Thus, the intended
negation set is {color1red}. To mathematically come to the same conclusion, we
calculate the entailment between each interpretation of the negation and the
follow-up sentence, i.e. how much each negation set of color1 not red wine
entails color2 white wine. We compare the two sentences word by word (or more
precisely, treating adjectives and nouns individually) and then take the
product of the results. We consider:

  * •

color1 not red wine $\sqsubseteq$ color2 white wine \- We first calculate the
entailment of $\cnw(\texttt{\color{color1} red})$ with color2 white which is
medium as something that is not red could have many other colors, including
white. The entailment of color1 wine with color2 wine is maximal as a wine is
indeed a wine. Therefore the overall score of this interpretation is high.
_Overall entailment: high_

  * •

color1 not red wine $\sqsubseteq$ color2 white wine \- This interpretation has
medium entailment between
$\cnw(\texttt{\color{color1} wine})$ and color2 wine due to the fact that in
distributional semantics, a word and its conversational negation appear in
similar contexts [saif:2013computingLexicalContrast, oaksford:1992reasoning].
Yet the entailment between color1 red and color2 white is low since something
being red does not entail that it is white.
_Overall entailment: low_

  * •

color1 not red wine $\sqsubseteq$ color2 white wine \- This interpretation has
a medium entailment between $\cnw(\texttt{\color{color1} red})$ and color2
white and a medium entailment between $\cnw(\texttt{\color{color1} wine})$ and
color2 wine.
_Overall entailment: medium_

Comparing the three interpretations, the first option has the highest score,
matching our intuition of being the correct choice. While this entailment
method relies on the sentences having an identical grammatical structure to
compare the sentences word by word, we can also directly compare entailment
between two sentences. This requires composition operations and entailment
measures which interact well to preserve word level entailment on the sentence
level. This is still a field of active research, with some promising results
presented in [kartsaklis-sadrzadeh-2016-distributional, kartsaklis-
etal-2016-distributional, Sadrzadeh_2018, delascuevas:2020catsclimb].