myquote ($\dagger$) __ myquote2  ($*$) part

# On $p$-refined Friedberg–Jacquet integrals and the classical symplectic
locus in the $\GL_{2n}$ eigenvariety

Daniel Barrera Salazar, Andrew Graham and Chris Williams

###### Abstract

Friedberg–Jacquet proved that if $\pi$ is a cuspidal automorphic
representation of $\mathrm{GL}_{2n}(\mathbf{A})$, $\pi$ is a functorial
transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a global zeta integral
$Z_{H}$ over $H=\mathrm{GL}_{n}\times\mathrm{GL}_{n}$ is non-vanishing on
$\pi$. We conjecture a $p$-refined analogue: that any $P$-parahoric
$p$-refinement $\tilde{\pi}^{P}$ is a functorial transfer from
$\mathrm{GSpin}_{2n+1}$ if and only if a $P$-twisted version of $Z_{H}$ is
non-vanishing on the $\tilde{\pi}^{P}$-eigenspace in $\pi$. This twisted
$Z_{H}$ appears in all constructions of $p$-adic $L$-functions via Shalika
models.

We prove various results towards the conjecture by connecting it to the study
of classical symplectic families in the $\mathrm{GL}_{2n}$ eigenvariety. If
$\pi$ is spherical at $p$, there are $(2n)!$ attached $p$-refinements to
Iwahori level; we conjecture the dimensions of such families through each of
these, prove the upper bound unconditionally, and (by constructing the
families) prove the lower bound under a non-critical slope assumption. For
example, for $\operatorname{GL}_{4}$, we conjecture that (modulo 1-dimensional
trivial variation) 8 refinements vary in 2-dimensional symplectic families, 8
in 1-dimensional symplectic families, and prove 8 do not vary in any
symplectic family.

††footnotetext: . _2020 MSC:_ Primary 11F33, 11F67; Secondary 11R23, 11G22

###### Contents

  1. 1 Introduction
  2. I $P$-spin refinements
    1. 2 Structure theory and parahoric $p$-refinements
    2. 3 $P$-spin refinements
  3. II Dimensions of Symplectic Components
    1. 4 The symplectic locus in the eigenvariety
    2. 5 Weight obstructions to symplectic families
    3. 6 Existence of $P$-spin families
    4. 7 Explicit examples for $\operatorname{GL}_{4}$
  4. III $p$-refined Friedberg–Jacquet Integrals
    1. 8 $p$-refined Friedberg–Jacquet integrals: Statements
    2. 9 Proof of Theorem 8.5
    3. 10 Proof of Theorem 8.6

### 1\. Introduction

Let $G$ be a reductive group. In this paper, we prove new connections between
two areas in the study of automorphic forms for $G$:

  * (1)

Non-vanishing of global period integrals for $G$ over a subgroup $H$ such that
$G/H$ is a spherical variety, described in terms of Langlands functoriality
and $L$-values; and

  * (2)

Congruences between systems of Hecke eigenvalues appearing in automorphic
representations for $G$, captured through classical families in the
eigenvariety for $G$.

The period integrals in (1) appear in the Gan–Gross–Prasad conjectures and are
closely connected to the relative Langlands program. The families in (2) have
been centrally important in number theory and arithmetic geometry for decades,
essential to breakthroughs in the Langlands program (through modularity
theorems, constructions of Galois representations, recent instances of
Langlands functoriality, and proofs of local-global compatibility) and Iwasawa
theory (in work on the Birch–Swinnerton-Dyer, Bloch–Kato and Iwasawa main
conjectures).

We consider these questions when $G=\operatorname{GL}_{2n}$ and
$H=\operatorname{GL}_{n}\times\operatorname{GL}_{n}$. In particular:

  * •

We prove upper bounds on the dimensions of symplectic classical families for
$G$. This shows that any systematic congruences between Hecke eigensystems in
symplectic representations are induced (by Langlands functoriality) from
congruences for $\mathrm{GSpin}_{2n+1}$, providing evidence for a folklore
expectation on classical families in eigenvarieties.

  * •

We show that non-vanishing of twisted period integrals for an eigenform for
$G$ implies existence of a symplectic family of ‘large’ dimension through the
corresponding eigensystem in the eigenvariety. We use our results on
symplectic families to prove this forces the eigensystem to arise from
Langlands functoriality from $\mathrm{GSpin}_{2n+1}$.

#### 1.1. Classical families

A system $\alpha$ of Hecke eigenvalues for $G$ is _classical (cuspidal)_ if it
appears in a (cuspidal) automorphic representation $\pi$ of $G(\mathbf{A})$. A
_classical (cuspidal) family_ is any subspace of the eigenvariety in which the
classical (cuspidal) points are Zariski-dense. A fundamental question is:

######  Question 1.

In how many dimensions does $\alpha$ vary in a classical cuspidal family?

In other words: let $\lambda$ be the weight of $\alpha$. In which weight
directions can one deform $\lambda$ and always find classical cuspidal
eigensystems $\alpha_{m}\equiv\alpha\hskip 2.0pt(\mathrm{mod}\hskip
2.0ptp^{m})$ for arbitrarily large $m$?

A folklore expectation, described below, says every non-trivial classical
family for $\operatorname{GL}_{N}$ arises from some form of self-duality. A
family that is essentially self-dual on $\operatorname{GL}_{N}$ is either
_orthogonal_ or _symplectic_ (containing a Zariski-dense set of orthogonal or
symplectic points).

In this paper, we consider Question 1 for symplectic families of
$\operatorname{GL}_{N}(\mathbf{A})$. This forces $N$ to be even (see [AS14]),
so let $G=\operatorname{GL}_{2n}$, and let $\alpha$ be attached to a regular
algebraic cuspidal automorphic representation (RACAR) $\pi$ of
$\operatorname{GL}_{2n}(\mathbf{A})$ that admits a Shalika model (equivalent
to $\pi$ symplectic). We assume that $\pi_{p}$ is unramified, and the Satake
parameter of $\pi_{p}$ is regular, in which case there are $(2n)!$ possible
$p$-refinements $\tilde{\pi}=(\pi,\alpha)$ of $\pi$. Here a _$p$ -refinement_
is a Hecke eigensystem $\alpha$ appearing in the Iwahori-invariants of
$\pi_{p}$.

In this paper, we define a stratification on the $(2n)!$ $p$-refinements
$\alpha$ in terms of parabolic subgroups in $\operatorname{GL}_{2n}$, and we
predict (in Conjecture 4.12) the dimension of any symplectic family through a
given $\alpha$ depends on its position in the stratification. We prove:

  * •

the upper bound on the dimension unconditionally;

  * •

and the lower bound when $\alpha$ has non-critical slope.

We also give theoretical justification for the lower bound in general.

We find that (modulo trivial variation, coming from twists by the norm) there
can exist such symplectic families of exact dimension $d$ for any
$d=0,1,...,n$. This seems striking given that every component of the
eigenvariety through any such $\alpha$ conjecturally has dimension $n$; so
there should be classical families sitting inside ‘generically non-classical’
components of the eigenvariety.

######  Example.

For $\operatorname{GL}_{4}$, there are 24 $p$-refinements $\tilde{\pi}$. By
[Han17, Thm. 1.1.5], every irreducible cuspidal component of the
$\operatorname{GL}_{4}$-eigenvariety is 2-dimensional (modulo trivial
variation). Then:

  * •

8 of the $\tilde{\pi}$ are essentially self-dual, and should vary in
2-dimensional symplectic families, each of which is then an irreducible
component of the eigenvariety.

  * •

8 are ‘symplectic rigid’ – we prove they do not vary in _any_ symplectic
family. In any component through these points in the eigenvariety, the
classical points should be discrete.

  * •

8 of them should vary in a 1-dimensional symplectic family, sitting in a
2-dimensional component of the eigenvariety, which should be generically non-
classical.

In §7 we give explicit examples of $(\pi,\alpha)$ in each of these cases,
showing that ‘generically non-symplectic but with a positive-dimensional
symplectic locus’ cases do indeed occur.

#### 1.2. Previous work on classical families

To put our results into context, we summarise the previous work on Question 1,
which broadly falls into two cases:

  * (I)

$G(\mathbf{R})$ admits discrete series (including e.g. if $G$ forms part of a
Shimura datum),

  * (II)

$G(\mathbf{R})$ does _not_ admit discrete series.

In case (I), Question 1 is fairly well-understood: Urban [Urb11] has shown
that a (non-critical) cohomological cuspidal $\alpha$ _always_ varies
‘maximally’, in all possible weight directions. This generalises the theory of
Hida/Coleman families for modular forms ($G=\operatorname{GL}_{2}$).

However, many fundamental cases – e.g. $\operatorname{GL}_{n}$ for $n\geqslant
3$, and $\operatorname{GL}_{2}$ over non-totally-real fields – are case (II),
where our understanding of Question 1 is extremely poor. Ash–Pollack–Stevens
[APS08] and Calegari–Mazur [CM09] considered the cases of
$\operatorname{GL}_{3}$ and $\mathrm{Res}_{F/\mathbf{Q}}\operatorname{GL}_{2}$
respectively, for $F$ an imaginary quadratic field, and conjectured that: For
$G=\operatorname{GL}_{3}$ or
$\mathrm{Res}_{F/\mathbf{Q}}\operatorname{GL}_{2}$, $\alpha$ varies in a
positive-dimensional classical family if and only if $\alpha$ is essentially
self-dual.

In [Xia18], Xiang has studied one direction of $(\dagger)$ more generally,
proving that if $\alpha$ is essentially self-dual on $\operatorname{GL}_{n}$
(that is, both $\pi$ _and_ $\alpha$ are essentially self-dual) then $\alpha$
varies in a classical family in all ‘self-dual/pure’ directions in weight
space. Since every RACAR, hence every $\alpha$, has pure weight, this
variation is ‘maximal’ in the strongest possible sense.

One goal of this paper is to find analogues of $(\dagger)$ in higher-
dimensional settings, where the picture is more subtle. Even when $\pi$ itself
is essentially self-dual, it admits non-essentially-self-dual refinements
$\alpha$, and we show that some of these can be varied in positive-dimensional
classical families of smaller dimension.

#### 1.3. Philosophy on classical families

Case (I) groups $G$ yield many classical families. A folklore expectation
predicts this accounts for _all_ classical families, in the sense that every
classical family is a $p$-adic Langlands transfer of a case (I) family. For
example, conjecturally:

  * –

For $\operatorname{GL}_{3}$, all classical families are twists of symmetric
square families for $\operatorname{GL}_{2}$;

  * –

For $\mathrm{Res}_{F/\mathbf{Q}}\operatorname{GL}_{2}$, all classical families
are twists of base-change families for $\operatorname{GL}_{2}$, or CM
transfers of families for $\mathrm{Res}_{F^{\prime}/F}\operatorname{GL}_{1}$,
for $F^{\prime}/F$ quadratic.

Before we describe our results precisely, let us explain why they fit strongly
into this philosophy. We hesitantly suggest they provide further evidence
towards it.

Any RACAR $\pi$ of $\operatorname{GL}_{2n}(\mathbf{A})$ that admits a Shalika
model is essentially self-dual, and a Langlands transfer of some RACAR $\Pi$
for $\mathrm{GSpin}_{2n+1}(\mathbf{A})$. Note that
$\mathcal{G}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{GSpin}_{2n+1}$
is a case (I) group. There are $2^{n}n!$ Iwahori $p$-refinements of $\Pi$. By
Urban’s case (I) theorem, each of these varies in a maximal family over weight
space (of dimension $n$, modulo trivial variation). In the style of Chenevier,
each of these families should admit a transfer to $\operatorname{GL}_{2n}$
interpolating Langlands functoriality on classical points. These
$n$-dimensional classical $\operatorname{GL}_{2n}$-families were constructed
and studied in [BDG+], and fall in the case studied by Xiang, corresponding
exactly to the essentially self-dual eigensystems in $\pi$.

This only accounts, however, for $2^{n}n!$ of the $(2n)!$ possible
$p$-refinements of $\pi$; even for $\operatorname{GL}_{6}$ this is only 48 out
of 720. To look for classical families through the other refinements, we
consider _parabolic_ families for $\mathcal{G}$, as constructed and studied,
for example, in [HL11, BW21]. For any standard parabolic
$\mathcal{P}\subset\mathcal{G}$, one can study $\mathcal{P}$-parahoric
refinements of $\Pi$. We show that for every refinement $\alpha$ of $\pi$,
there exists a unique smallest parabolic $\mathcal{P}\subset\mathcal{G}$ such
that $\alpha$ ‘is a functorial transfer of a $\mathcal{P}$-refinement
$\alpha^{\mathcal{G},\mathcal{P}}$ of $\Pi$’. Under a natural correspondence,
$\mathcal{P}$ corresponds to a unique ‘spin’ parabolic $P\subset G$, and we
call $\alpha$ an _optimally $P$-spin refinement_. If $B$ is the corresponding
Borel, the optimally $B$-spin refinements are exactly the $2^{n}n!$
essentially self-dual ones studied in [BDG+, Xia18]. All of this is defined in
§3, where we give Weyl group, Hecke algebra, and combinatorial definitions of
being $P$-spin, proving they are all equivalent.

Let $\alpha$ be an optimally $P$-spin refinement with associated spin
eigensystem $\alpha^{\mathcal{G},\mathcal{P}}$. Under a non-criticality
assumption, [BW21] shows $\alpha^{\mathcal{G},\mathcal{P}}$ varies in a family
in the $\mathcal{P}$-parabolic $\mathcal{G}$-eigenvariety over a smaller-
dimensional weight space. Again, conceptually, this family should admit a
transfer to the (Iwahoric) $G$-eigenvariety interpolating Langlands
functoriality on classical points. This would produce a classical symplectic
family in the $G$-eigenvariety through $\alpha$, of some smaller dimension
depending on $\mathcal{P}$ (hence $P$).

It is not clear how one should construct these transfer maps in general. There
is a natural map of (abstract) Hecke algebras
$\jmath^{\vee}:\mathcal{H}^{G}\to\mathcal{H}^{\mathcal{G}}$ at Iwahoric level
(see (3.1)), which should induce a map

$\text{[Iwahoric-$\mathcal{G}$-eigenvariety]}\longrightarrow\text{[Iwahoric-$G$-eigenvariety]}.$

However, one needs detailed automorphic information about classical points in
the $\mathcal{G}$-eigenvariety to control this, and in any case this recovers
families already known to exist by [BDG+, Xia18]. At parahoric level the
situation is worse: a transfer map

$\text{[$\mathcal{P}$-parahoric-$\mathcal{G}$-eigenvariety]}\longrightarrow\text{[Iwahoric-$G$-eigenvariety]}$

should be induced from a map
$\jmath^{\vee}_{\mathcal{P}}:\mathcal{H}^{G}\to\mathcal{H}^{\mathcal{G},\mathcal{P}}$
on abstract Hecke algebras, but now there is no natural map, as
$\mathcal{H}^{\mathcal{G},\mathcal{P}}\subsetneq\mathcal{H}^{G}$, but
$\jmath^{\vee}$ is surjective111On the Galois representation side, this should
correspond to the problem of consistently choosing triangulations of the
$(\varphi,\Gamma)$-modules attached to every classical point in a paraboline
family.. To construct even a candidate $\jmath^{\vee}_{\mathcal{P}}$, it seems
necessary to _presuppose_ the existence of the family for $G$ one wants to
construct. As such, we do not pursue this approach to families in this paper.

#### 1.4. Our results on symplectic families

To a spin parabolic $P$, in Definition 3.11 we associate a subset
$X_{P}\subset\\{1,...,n\\}$. Here $X_{B}=\\{1,...,n\\}$ and
$X_{G}=\varnothing$. Let $\pi$ be a symplectic RACAR, and let $\tilde{\pi}$ be
an optimally $P$-spin refinement. We prove:

######  Theorem A.

  1. (i)

Any symplectic family $\mathscr{C}$ through $\tilde{\pi}$ has dimension at
most $\\#X_{P}+1$.

  2. (ii)

When $\alpha$ has non-critical slope and regular weight, there exists a unique
symplectic family through $\tilde{\pi}$, of dimension exactly $\\#X_{P}+1$.

(Here we include, as in the main text, the 1-dimensional trivial variation).

In particular, if $\tilde{\pi}$ is optimally $G$-spin, then $\tilde{\pi}$ is
‘symplectic-rigid’, varying in _no_ non-trivial symplectic family. There are,
for example, 8 such refinements in the $\operatorname{GL}_{4}$ case.

Part (i) is Theorem 4.9, which actually says more: that the weight support of
such a family must lie in a $P$-parahoric weight space, which has dimension
$\\#X_{P}+1$. To prove this, we show first that every classical point in
$\mathscr{C}$ is also optimally $P$-spin, and then obtain obstructions to the
existence of optimally $P$-spin families varying outside the $P$-parabolic
weight space.

Part (ii) is Theorem 4.10. We show further that this unique component is étale
over its image in weight space. To construct these families, we use a
‘refinement-switching’ argument to move between points on the
$\operatorname{GL}(2n)$-eigenvariety attached to a single $\pi$. The proof
highlights interdependencies between the symplectic families through the
different $p$-refinements, with implications for a hypothetical ‘infinite
fern’ construction for $\operatorname{GL}_{2n}$ (see Remark 6.13).

We remark how Theorem A fits into the philosophy above. Writing $\mathscr{E}$
for the $\operatorname{GL}_{2n}$ eigenvariety of some fixed level, we expect
there are an infinite number of closed embeddings
$\\{\iota_{i}:\mathscr{C}_{i}\hookrightarrow\mathscr{E}:i\in I\\}$, where the
$\mathscr{C}_{i}$ are classical families in parabolic $\mathrm{GSpin}_{2n+1}$
eigenvarieties. Each $\mathscr{C}_{i}$ is flat over the relevant parabolic
weight space, and cannot be varied in higher dimension at the level of
$\mathrm{GSpin}_{2n+1}$ eigensystems. However, $\mathscr{E}$ varies over a
higher-dimensional weight space, and in general $\iota_{i}(\mathscr{C}_{i})$
will sit properly inside some larger irreducible component of $\mathscr{E}$.
Theorem A says that this irreducible component cannot have any further
symplectic variation; that is, the subspaces $\iota_{i}(\mathscr{C}_{i})$ of
$\mathscr{E}$ cannot be assembled together into any classical family of higher
dimension. In other words, all classical symplectic variation, and systematic
congruences, should be accounted for by families in (parabolic)
$\mathrm{GSpin}_{2n+1}$ eigenvarieties. This is predicted by our guiding
philosophy on classical families in the eigenvariety, suggesting our results
provide some further evidence for it. Indeed, motivated by the above theorem
and the guiding philosophy, we conjecture:

######  Conjecture B.

Every symplectic family through $\tilde{\pi}$ is the transfer of a classical
parabolic family for $\mathrm{GSpin}_{2n+1}$ and has dimension $\\#X_{P}+1$.

In §7, we give explicit examples for $\operatorname{GL}_{4}$ illustrating
Theorem A and Conjecture B.

#### 1.5. Non-vanishing of twisted period integrals

We give an application to the study of non-vanishing of period integrals. Let
$\pi$ be a RACAR of $G(\mathbf{A})$, and let
$H=\operatorname{GL}_{n}\times\operatorname{GL}_{n}\subset G$. If $\chi$ is an
algebraic Hecke character and $\varphi\in\pi$, then in (8.1) we define an
attached global period integral for $H\subset G$, denoted
$Z_{H}(\varphi,\chi,s)$. The same kind of period integral appears in the GGP
conjectures, and is related to the relative Langlands program.

A result of Friedberg–Jacquet [FJ93] says that for any $s\in\mathbf{C}$, the
following are equivalent:

  1. (1)

There exists $\varphi\in\pi$ such that $Z_{H}(\varphi,\chi,s+1/2)\neq 0$;

  2. (2)

$\pi$ is a functorial transfer of some $\Pi$ on
$\mathrm{GSpin}_{2n+1}(\mathbf{A})$, and $L(\pi\times\chi,s+1/2)\neq 0$.

This is related to the relative Langlands program [SV17]; $G/H$ is a spherical
variety, and $Z_{H}$ is an $H$-period integral (that appears, for example, in
the GGP conjectures in related settings). This phenomenon is also explained in
great generality in [JLR99, p.174].

We propose a $p$-refined analogue of this. Let $P\subsetneq G$ be a proper
spin parabolic, let $\beta\geqslant 1$, and let $ut_{P}^{\beta}\in
G(\mathbf{Q}_{p})$ be the element defined in Notation 8.2. Here $u$ is a
representative for the open orbit of the action of $B$ on $G/H$ and $t_{P}$
defines the Hecke operator at $P$. Let $\tilde{\pi}$ be a $P$-parahoric
$p$-refinement of $\pi$.

######  Conjecture C.

Suppose $\chi$ is finite order and has conductor $p^{\beta}>1$. For any
$s\in\mathbf{C}$, the following are equivalent:

  1. (1)

There exists an eigenvector $\varphi\in\tilde{\pi}^{P}$ such that
$Z_{H}(ut_{P}^{\beta}\cdot\varphi,\chi,s+1/2)\neq 0$.

  2. (2)

All of the following hold:

     * –

$\tilde{\pi}^{P}$ is a functorial transfer of some $\mathcal{P}$-refined
$\tilde{\Pi}^{\mathcal{P}}$ on $\mathrm{GSpin}_{2n+1}(\mathbf{A})$,

     * –

$L(\pi\times\chi,s+1/2)\neq 0$,

     * –

$P$ is contained in the $(n,n)$-parabolic (in the sense of Notation 2.6).

The quantity $Z_{H}(ut_{P}^{\beta}\cdot\varphi,\chi,s+1/2)$ appears in
constructions of $p$-adic $L$-functions via Shalika models [DJR20, BDW, BDG+,
Wil]. Conjecture C highlights a close relationship between the $P$-spin
conditions defined in this paper, and settings where we can expect to
construct non-zero $p$-adic $L$-functions via Shalika models. In this light,
the requirement in (2) that $P$ is contained in the $(n,n)$-parabolic $Q$ is
natural; the Panchishkin condition [Pan94] predicts that to be able to attach
a $p$-adic $L$-function to $\tilde{\pi}^{P}$, one requires $P\subset Q$.

As evidence towards this conjecture, we use Theorem A to prove:

######  Theorem D.

  * (i)

(2) $\Rightarrow$ (1) holds in Conjecture C.

  * (ii)

Suppose $\pi$ has regular weight and there is a non-critical slope further
refinement $\tilde{\pi}$ of $\tilde{\pi}^{P}$ to Iwahori level. Then (1)
$\Rightarrow$ (2) holds in Conjecture C.

In particular, the conjecture holds in full for a large class of
$\tilde{\pi}^{P}$. We actually show (ii) (and deduce the full conjecture)
under weaker assumptions on $\tilde{\pi}^{P}$, which we cautiously imagine
could hold for _all_ $\tilde{\pi}^{P}$; see Theorem 8.6 and Remarks 8.7.

We remark here that whilst Conjecture C is stated globally, when combined with
Friedberg–Jacquet’s original result it can be reduced to a local statement at
$p$ (given in Conjecture 8.10). Our proof of Theorem D(i) is purely local:
given (2), we directly exhibit an eigenvector satisfying (1) using methods
developed in [BDG+]. To prove (ii), we also deploy global methods, using ideas
from [BDW, BDG+] to show that if (1) holds, then we can construct a symplectic
family through $\tilde{\pi}$ over the $P$-parahoric weight space. By (the
stronger form of) Theorem A(i), this forces $\tilde{\pi}^{P}$ to be $P$-spin,
hence $\tilde{\pi}^{P}$ is a functorial transfer.

We expect that this relationship between non-vanishing of twisted period
integrals attached to a $p$-refinement, and the refinement being a functorial
transfer, should be true much more generally. In future work with Lee, we hope
to treat the case of twisted Flicker–Rallis integrals for
$\operatorname{GL}_{n}$ over a CM field, showing non-vanishing implies
transfer from a unitary group.

#### Acknowledgements

We thank our co-authors on [BDG+], Mladen Dimitrov and Andrei Jorza, for many
stimulating discussions whilst preparing our earlier joint paper. We also
thank Valentin Hernandez for interesting discussions on Zariski-density of
crystalline points and the infinite fern. AG was supported by
ERC-2018-COG-818856-HiCoShiVa. CW was supported by EPSRC Postdoctoral
Fellowship EP/T001615/1. DBS was supported by FONDECYT 11201025.

### Set-up and notation

Let $n\geqslant 1$ and let
$G\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{GL}_{2n}$.
We write $B=B_{2n}$ for the Borel subgroup of upper triangular matrices,
$\overline{B}=\overline{B}_{2n}$ for the opposite Borel of lower triangular
matrices and $T=T_{2n}$ for the maximal split torus of diagonal matrices. Let
$\delta_{B}$ be the standard modulus character.

If $\pi$ is a regular algebraic symplectic cuspidal automorphic representation
(RASCAR), then $\pi$ is essentially self-dual (i.e. there exists a Hecke
character $\eta$ such that $\pi^{\vee}\cong\pi\otimes\eta^{-1}$). Moreover by
[AS06, FJ93] $\pi$ is a functorial transfer of a RACAR $\Pi$ on
$\mathrm{GSpin}_{2n+1}(\mathbf{A})$ and admits an $(\eta,\psi)$-Shalika model,
where $\psi$ is the standard additive character of
$\mathbf{Q}\backslash\mathbf{A}$.

## Part I. $P$-spin refinements

### 2\. Structure theory and parahoric $p$-refinements

#### 2.1. Root systems and spin parabolics

Our study of ‘spin’ refinements is rooted in the structure theory of
$\operatorname{GL}_{2n}$ and $\mathrm{GSpin}_{2n+1}$. We recall the following
from [BDG+, §6].

The spaces of algebraic characters/cocharacters of the torus $T\subset
G=\operatorname{GL}_{2n}$ are

$X=\mathbf{Z}e_{1}\oplus\mathbf{Z}e_{2}\oplus\cdots\mathbf{Z}e_{2n},\hskip
12.0ptX^{\vee}=\mathbf{Z}e_{1}^{*}\oplus\mathbf{Z}e_{2}^{*}\oplus\cdots\mathbf{Z}e_{2n}^{*}.$

The root system for $G$ is $A_{2n-1}$, with roots
$R=\\{\pm(e_{i}-e_{j}):1\leqslant i<j\leqslant 2n\\}$, positive roots
$\\{e_{i}-e_{j}:i<j\\}$, and simple roots
$\Delta_{G}=\\{a_{i}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e_{i}-e_{i+1}:i=1,...,2n-1\\}$.
The Weyl group $\mathcal{W}_{G}=\mathrm{S}_{2n}$ acts by permuting the
$e_{i}$. We set this up so that $\sigma\in\mathcal{W}_{G}$ sends $e_{i}$ to
$e_{\sigma^{-1}(i)}$, hence $\sigma$ acts on a character
$\mu=(\mu_{1},...,\mu_{2n})\in X$ as
$\mu^{\sigma}=(\mu_{\sigma(1)},...,\mu_{\sigma(2n)})$.

Let $X_{0}\subset X$ be the space of _pure characters_ $X_{0}=\\{\lambda\in
X:\exists{\sf w}(\lambda)\in\mathbf{Z}\text{ such that
}\lambda_{i}+\lambda_{2n-i+1}={\sf w}(\lambda)\ \forall 1\leqslant i\leqslant
n\\}$, and let

$\mathcal{W}_{G}^{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{\sigma\in\mathcal{W}_{G}:\sigma(X_{0})\subset
X_{0}\\}\subset\mathcal{W}_{G}.$ (2.1)

There is a splitting $\mathcal{W}_{G}^{0}=\\{\pm
1\\}^{n}\rtimes\mathrm{S}_{n},$ where:

  * •

for $1\leqslant i\leqslant n$, $\sigma\in\mathrm{S}_{n}$ sends $e_{i}$ to
$e_{\sigma^{-1}(i)}$, and $e_{2n+1-i}$ to $e_{2n+1-\sigma^{-1}(i)}$;

  * •

and the $i$th copy of $\\{\pm 1\\}$ acts by swapping $e_{i}\leftrightarrow
e_{2n+1-i}$.

Identifying $i\leftrightarrow e_{i}$, we view $\mathcal{W}_{G}^{0}$ as a
subgroup of $\mathrm{S}_{2n}$, and have the following easy fact:

######  Lemma 2.1.

If $\sigma\in\mathcal{W}_{G}^{0}$, then $\sigma(i)+\sigma(2n+1-i)=2n+1$ for
all $1\leqslant i\leqslant n$.

Now fix a standard upper Borel subgroup $\mathcal{B}$ and maximal split torus
$\mathcal{T}$ in $\mathcal{G}=\mathrm{GSpin}_{2n+1}$. This has rank $n+1$
[Asg02, Thm. 2.7]. We use calligraphic letters to denote objects for GSpin,
whilst keeping other notational conventions as before.

######  Proposition 2.2.

The root system for $\mathcal{G}$ is
$(\mathcal{X},\mathcal{R},\mathcal{X}^{\vee},\mathcal{R}^{\vee})$, where

$\mathcal{X}=\mathbf{Z}f_{0}\oplus\mathbf{Z}f_{1}\oplus\cdots\oplus\mathbf{Z}f_{n},\hskip
12.0pt\mathcal{X}^{\vee}=\mathbf{Z}f_{0}^{*}\oplus\mathbf{Z}f_{1}^{*}\oplus\cdots\oplus\mathbf{Z}f_{n}^{*},$

with roots $\mathcal{R}=\\{\pm f_{i}\pm f_{j}:1\leqslant i<j\leqslant
n\\}\cup\\{f_{i}:1\leqslant i\leqslant n\\}$, simple roots

$\Delta_{\mathcal{G}}=\\{b_{i}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f_{i}-f_{i+1}:i=1,...,n-1\\}\cup\\{b_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f_{n}\\},$

and positive roots $\\{f_{i}:1\leqslant i\leqslant n\\}\cup\\{f_{i}\pm
f_{j}:1\leqslant i<j\leqslant n\\}$. The Weyl group
$\mathcal{W}_{\mathcal{G}}$ is isomorphic to $\\{\pm
1\\}^{n}\rtimes\mathrm{S}_{n}$, generated by permutations
$\sigma\in\mathrm{S}_{n}$ and sign changes $\mathrm{sgn}_{i}$, which act on
roots and coroots respectively as (for $j\neq i$)

$\sigma f_{0}=f_{0},\ \ \sigma f_{i}=f_{\sigma^{-1}(i)},\ \ \
\mathrm{sgn}_{i}f_{0}=f_{0}+f_{i},\ \ \mathrm{sgn}_{i}(f_{i})=-f_{i},\ \
\mathrm{sgn}_{j}(f_{i})=f_{i},$ (2.2) $\sigma f_{0}^{*}=f_{0}^{*},\ \ \sigma
f_{i}^{*}=f_{\sigma^{-1}(i)}^{*},\ \ \ \mathrm{sgn}_{i}f_{0}^{*}=f_{0}^{*},\ \
\mathrm{sgn}_{i}(f_{i}^{*})=f_{0}^{*}-f_{i}^{*},\ \
\mathrm{sgn}_{j}(f_{i}^{*})=f_{i}^{*}.$

###### Proof.

The first part is [Asg02, Prop. 2.4], and the second [HS16, Lem. 13.2.2]. ∎

Write $\langle-,-\rangle_{G}$ (resp. $\langle-,-\rangle_{\mathcal{G}}$) for
the natural pairing on $X\times X^{\vee}$ (resp.
$\mathcal{X}\times\mathcal{X}^{\vee}$).

There is a natural injective map $\jmath:\mathcal{X}\hookrightarrow X$ given
by

$f_{i}\longmapsto e_{i}-e_{2n-i+1}\text{ for }1\leqslant i\leqslant n,\qquad
f_{0}\longmapsto e_{n+1}+\cdots+e_{2n},$

with $X_{0}=\jmath(\mathcal{X})$ by [BDG+, Prop. 6.5]. If $\rho_{G}$ and
$\rho_{\mathcal{G}}$ are half the sum of the positive roots for $G$ and
$\mathcal{G}$ respectively, a simple check shows
$\jmath(\rho_{\mathcal{G}})=\rho_{G}$. We also have:

######  Proposition 2.3 ([BDG+], Proposition 6.6).

There is a map $\mathcal{W}_{\mathcal{G}}\to\mathcal{W}_{G}$ of Weyl groups,
also denoted $\jmath$, such that:

  * (i)

$\jmath$ induces an isomorphism
$\mathcal{W}_{\mathcal{G}}\cong\mathcal{W}_{G}^{0}\subset\mathcal{W}_{G}$;

  * (ii)

for all $\sigma\in\mathcal{W}_{\mathcal{G}}$ and $\mu\in\mathcal{X}$, we have
$\jmath(\mu^{\sigma})=\jmath(\mu)^{\jmath(\sigma)}$.

Dually, define also a map $\jmath^{\vee}:X^{\vee}\to\mathcal{X}^{\vee}$ by
sending $\nu\in X^{\vee}$ to

$\jmath^{\vee}(\nu)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{i=0}^{n}\big{\langle}\jmath(f_{i}),\nu\big{\rangle}_{G}\cdot
f_{i}^{*}.$

Then for all $\mu\in\mathcal{X}$ and $\nu\in X^{\vee}$, we have

$\langle\mu,\jmath^{\vee}(\nu)\rangle_{\mathcal{G}}=\langle\jmath(\mu),\nu\rangle_{G}$
(2.3)

by construction. Also let
$\jmath^{\vee}:\mathcal{W}_{G}^{0}\to\mathcal{W}_{\mathcal{G}}$ denote the
inverse to $\jmath:\mathcal{W}_{\mathcal{G}}\cong\mathcal{W}_{G}^{0}$.

######  Proposition 2.4 ([BDG+], Proposition 6.7).

For all $\nu\in X^{\vee}$ and $\sigma\in\mathcal{W}_{G}^{0}$, we have

$\jmath^{\vee}(\nu^{\sigma})=\jmath^{\vee}(\nu)^{\jmath^{\vee}(\sigma)}.$

We take a brief general intermission. For any quasi-split reductive group
$\mathrm{G}$ with a fixed choice of Borel pair $(\mathrm{B},\mathrm{T})$,
there is a well-known inclusion-preserving correspondence between standard
parabolic subgroups $\mathrm{P}$ of $\mathrm{G}$ and subsets
$\Delta_{\mathrm{P}}$ of the set $\Delta$ of simple roots (see e.g. [BW21,
§2.3]). Here $\mathrm{B}$ corresponds to the empty set, and any proper maximal
standard parabolic corresponds to $\Delta\backslash\\{a\\}$ for some simple
root $a\in\Delta$. Further, for any such $\mathrm{P}$ we have a Levi subgroup
$L_{\mathrm{P}}$, with Weyl group $\mathcal{W}_{L_{\mathrm{P}}}$, which is
naturally a subgroup of $\mathcal{W}_{\mathrm{G}}$ (namely, the subgroup that
preserves the $\mathbf{Z}$-span of $\Delta_{\mathrm{P}}$).

Returning to our specific set-up, note that $\jmath$ acts on simple roots by
sending

$b_{1}\mapsto a_{1}+a_{2n-1},\ \ b_{2}\mapsto a_{2}+a_{2n-2},\ \ \dots,\ \
b_{n-1}\mapsto a_{n-1}+a_{n+1},\ \ b_{n}\mapsto a_{n}.$

######  Definition 2.5.

Let $P\subset G=\operatorname{GL}_{2n}$ be a standard parabolic, corresponding
to a subset $\Delta_{P}\subset\Delta_{G}$. We say $P$ is a _spin parabolic_
if, for any $i$, $a_{i}\in\Delta_{P}$ implies $a_{2n-i}\in\Delta_{P}$; that
is, $\Delta_{P}$ is a union of some of the sets

$A_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{a_{1},a_{2n-1}\\},\
\
A_{2}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{a_{2},a_{2n-2}\\},\dots,\
\
A_{n-1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{a_{n-1},a_{n+1}\\},\
\ A_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{a_{n}\\}.$

If $P$ is a spin parabolic, then there is a corresponding parabolic
$\mathcal{P}\subset\mathcal{G}$, defined by

$b_{i}\in\Delta_{\mathcal{P}}\iff A_{i}\subset\Delta_{P}.$

Under this correspondence the Borel subgroups $B\subset G$ and
$\mathcal{B}\subset\mathcal{G}$ are identified.

######  Notation 2.6.

We call the parabolic $P$ with Levi
$\operatorname{GL}_{n_{1}}\times\cdots\times\operatorname{GL}_{n_{r}}$ the
$(n_{1},...,n_{r})$-parabolic. Note that $P$ is a spin parabolic if and only
if $(n_{1},...,n_{r})$ is symmetric around the middle (so the
(1,4,1)-parabolic is spin, but the (1,3,2)-parabolic is not).

#### 2.2. Parahoric $p$-refinements for $G$

Let $\pi$ be a $p$-spherical RASCAR of $\operatorname{GL}_{2n}(\mathbf{A})$.
We can write $\pi_{p}=\operatorname{Ind}_{B}^{G}\theta$ as an unramified
principal series representation (via normalised induction, i.e. the induction
of $\delta_{B}^{1/2}\theta$). The choice of
$\theta=(\theta_{1},...,\theta_{2n})$ is not unique: we may replace $\theta$
with $\theta^{\sigma}$ for any $\sigma\in\mathcal{W}_{G}=\mathrm{S}_{2n}$ in
the Weyl group of $G$. Here $\theta_{i}^{\sigma}=\theta_{\sigma(i)}$.

######  Definition 2.7.

We say $\theta$ is _spin_ if

$\theta_{1}\theta_{2n}=\theta_{2}\theta_{2n-1}=\cdots=\theta_{n}\theta_{n+1}=\eta_{p}.$
(2.4)

Since $\pi_{p}$ admits an $(\eta_{p},\psi_{p})$-Shalika model, using [AG94]
and [BDG+, §6.1] we may (and will) choose $\theta$ to be spin. This is the
‘Asgari–Shahidi’ convention on $\theta$ described in [BDG+, §6.1]. This is
still not unique: we could replace $\theta$ with $\theta^{\sigma}$, for any
$\sigma\in\mathcal{W}_{G}^{0}\subset\mathcal{W}_{G}$.

Note this is _different_ from how we chose $\theta$ in [BDG+], where we
assumed $\theta_{i}\theta_{n+i}=\eta_{p}$. The two choices are exchanged by
$\tau=\left(\begin{smallmatrix}1&\\\
&w_{n}\end{smallmatrix}\right)\in\mathcal{W}_{G}$ (see §6.1 and Remark 6.12
_op. cit_.).

Now let $B\subset P\subset\operatorname{GL}_{2n}$ be a standard parabolic,
with associated parahoric subgroup
$J_{P}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{g\in\operatorname{GL}_{2n}(\mathbf{Z}_{p}):g\hskip
2.0pt(\mathrm{mod}\hskip 2.0ptp)\in P(\mathbf{F}_{p})\\}$. Note
$J_{B}=\mathrm{Iw}_{G}$. Fix an isomorphism
$i_{p}:\mathbf{C}\to\overline{\mathbf{Q}}_{p}$.

######  Definition 2.8.

  * –

For $1\leqslant r\leqslant 2n$, let
$t_{p,r}=\left(\begin{smallmatrix}pI_{r}&\\\
&I_{n-r}\end{smallmatrix}\right)=(e_{1}^{*}+\cdots+e_{r}^{*})(p)\in
T(\mathbf{Q}_{p})$. Let $U_{p,r}=[J_{P}t_{p,r}J_{P}]$ be the associated double
coset operator.

  * –

Let
$\mathcal{H}_{p}^{P}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{Q}_{p}[U_{p,r},U_{p,2n}:1\leqslant
r\leqslant 2n-1,a_{r}\not\in\Delta_{P}]$ be the Hecke algebra at $p$.

  * –

A _$P$ -parahoric $p$-refinement_ of $\pi$, or _$P$ -refinement_ for short, is
a system $\alpha^{P}:\mathcal{H}_{p}^{P}\to\overline{\mathbf{Q}}_{p}$ of Hecke
eigenvalues such that $i_{p}^{-1}\circ\alpha_{P}$ appears in
$\pi_{p}^{J_{P}}$. As the $U_{p,r}$-eigenvalues on $\pi^{J_{P}}$ are
algebraic, this does not depend on $i_{p}$. We denote this as
$\tilde{\pi}^{P}=(\pi,\alpha^{P})$.

  * –

If $P=B$, then we write ‘$p$-refinement’ or ‘Iwahori $p$-refinement’ instead
of ‘$B$-parahoric $p$-refinement’. We drop the superscript $B$, writing
$\mathcal{H}_{p}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{H}_{p}^{B}$,
$\alpha\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\alpha^{B}$,
$\tilde{\pi}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\tilde{\pi}^{B}$,
etc.

######  Remarks 2.9.

  * (i)

Note $\mathcal{H}_{p}=\mathbf{Q}_{p}[U_{p,1},...,U_{p,2n}]$ is everything. If
$Q$ is the $(n,n)$-parabolic, then
$\mathcal{H}_{p}^{Q}=\mathbf{Q}_{p}[U_{p,n},U_{p,2n}]$ is the Hecke algebra
from [BDW].

  * (ii)

If $P^{\prime}\subset P$ are two parabolics, then we have a natural inclusion
$\mathcal{H}_{p}^{P}\subset\mathcal{H}_{p}^{P^{\prime}}$, so
$\mathcal{H}_{p}^{P}\subset\mathcal{H}_{p}$ for all $P$. Via [OST, Cor. 3.16]
(see also Proposition 2.10 and (3.2) below), any (Iwahori) $p$-refinement
$\tilde{\pi}=(\pi,\alpha)$ restricts to a unique $P$-parahoric $p$-refinement
$\tilde{\pi}^{P}=(\pi,\alpha^{P})$, with
$\alpha^{P}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\alpha|_{\mathcal{H}_{p}^{P}}$.
Here $\tilde{\pi}$ is a further refinement of/extends $\tilde{\pi}^{P}$.

  * (iii)

We wrote $U_{p,r}$ independent of the parabolic $P$; this abuse of notation is
justified by (ii).

The following describes the possible $p$-refinements in terms of the Weyl
group $\mathcal{W}_{G}=\mathrm{S}_{2n}$.

######  Proposition 2.10.

Suppose the Satake parameter of $\pi_{p}=\operatorname{Ind}_{B}^{G}\theta$ is
regular semisimple.

  * (i)

There is a bijection (that depends on $\theta$)

$\Psi_{\theta}:\\{\text{Iwahori }p\text{-refinements of
}\pi\\}\longrightarrow\mathcal{W}_{G},$ (2.5)

such that if $\tilde{\pi}=(\pi,\alpha)$ is a $p$-refinement with
$\Psi_{\theta}(\tilde{\pi})=\sigma$, then for each $r$ we have

$\alpha(U_{p,r})=\delta_{B}^{-1/2}\theta^{\sigma}(t_{p,r})=\prod_{j=1}^{r}p^{-\tfrac{2n-2j+1}{2}}\theta_{\sigma(j)}(p)\neq
0.$ (2.6)

  * (ii)

If $P$ is a standard parabolic with Levi subgroup $L_{P}$, there is a
bijection

$\Psi_{\theta}^{P}:\\{P\text{-refinements of
}\pi\\}\longrightarrow\mathcal{W}_{G}/\mathcal{W}_{L_{P}},$

such that if $\tilde{\pi}^{P}=(\pi,\alpha^{P})$ is a $P$-refinement with
$\Psi_{\theta}^{P}(\tilde{\pi}^{P})=[\sigma]$ for $\sigma\in\mathcal{W}_{G}$,
then $\alpha^{P}(U_{p,r})$ is given by (2.6) whenever
$U_{p,r}\in\mathcal{H}_{p}^{P}$.

  * (iii)

If $\tilde{\pi}^{P}$ is a $P$-refinement, then the possible extensions to
Iwahori level are exactly the $p$-refinements $\tilde{\pi}$ with
$\Psi_{\theta}(\tilde{\pi})=\Psi_{\theta}^{P}(\tilde{\pi}^{P})\hskip
2.0pt(\mathrm{mod}\hskip 2.0pt\mathcal{W}_{L_{P}})$.

###### Proof.

(i) is [Che04, Lem. 4.8.4]. (ii) is [OST, Cor. 3.16]. (iii) is immediate. ∎

######  Remark 2.11.

For any $\nu\in\mathcal{W}_{G}$ and any $p$-refinement $\tilde{\pi}$, we have
$\Psi_{\theta^{\nu}}(\tilde{\pi})=\nu\Psi_{\theta}(\tilde{\pi})$. In [BDG+] we
denoted $\theta$ for what would be $\theta^{\tau}$ here, where
$\tau=\mathrm{diag}(1,w_{n})$, where $w_{n}$ is the longest Weyl element for
$\operatorname{GL}_{n}$. Thus our bijection $\Psi_{\theta}$ is denoted
$\Delta_{\theta^{\tau}}$ there.

The $U_{p,r}$-eigenvalues will not, in general, vary $p$-adic analytically.
For $p$-adic interplation, we must instead use normalised analogues
$U_{p,r}^{\circ}$ of $U_{p,r}$.

######  Definition 2.12.

If $\lambda$ is the weight of $\pi$, we define

$U_{p,r}^{\circ}=\lambda(t_{p,r})U_{p,r}=p^{\lambda_{1}+\cdots+\lambda_{r}}U_{p,r}\in\mathcal{H}_{p}.$

Let $\pi$ be a RASCAR of weight $\lambda$, and $P$ a spin parabolic.

######  Definition 2.13.

Let $\tilde{\pi}^{P}=(\pi,\alpha^{P})$ be a $P$-refinement of $\pi$. We say
$\tilde{\pi}^{P}$ has _non- $P$-critical slope_ if

$v_{p}(\alpha^{P}(U_{p,r}^{\circ}))<\lambda_{r}-\lambda_{r+1}+1\qquad\text{for
all }1\leqslant r\leqslant 2n-1\text{ with }a_{r}\not\in\Delta_{P}.$

(Note that $\lambda_{r}-\lambda_{r+1}=\lambda_{2n-r}-\lambda_{2n-r+1}$ by
purity, so the bounds for $U_{p,r}^{\circ}$ and $U_{p,2n-r}^{\circ}$ agree).

We say a $p$-refinement $\tilde{\pi}$ has non-$P$-critical slope if its
associated $P$-refinement $\tilde{\pi}^{P}$ does. We say $\tilde{\pi}$ has
non-critical slope if it has non-$B$-critical slope.

### 3\. $P$-spin refinements

Let $P\subset G=\operatorname{GL}_{2n}$ be a spin parabolic. We now generalise
[BDG+, §6] to an arbitrary such $P$. Let
$\pi=\operatorname{Ind}_{B}^{G}\theta$ be a RASCAR of
$\operatorname{GL}_{2n}(\mathbf{A})$ that is spherical and regular at $p$,
recalling we have fixed a spin $\theta$ satisfying
$\theta_{1}\theta_{2n}=\cdots=\theta_{n}\theta_{n+1}=\eta_{p}$ (2.4). Recall
$\Psi_{\theta}$ from (2.5).

######  Definition 3.1.

  * •

We say an Iwahori $p$-refinement $\tilde{\pi}=(\pi,\alpha)$ is a _$P$ -spin
refinement_ if

$\Psi_{\theta}(\tilde{\pi})\in\mathcal{W}_{G}^{0}\cdot\mathcal{W}_{L_{P}}\subset\mathcal{W}_{G}.$

  * •

We say a $P$-refinement $\tilde{\pi}^{P}$ is $P$-spin if

$\Psi_{\theta}^{P}(\tilde{\pi}^{P})\in\mathrm{Im}\Big{(}\mathcal{W}_{G}^{0}\to\mathcal{W}_{G}\to\mathcal{W}_{G}/\mathcal{W}_{L_{P}}\Big{)}\subset\mathcal{W}_{G}/\mathcal{W}_{L_{P}}.$

######  Lemma 3.2.

A $P$-refinement $\tilde{\pi}^{P}$ is $P$-spin if and only if all of its
extensions to Iwahori $p$-refinements are $P$-spin.

###### Proof.

Immediate from the definitions and Proposition 2.10(iii). ∎

######  Remarks 3.3.

  * (i)

The cases of $B$-spin and $Q$-spin refinements, for $Q$ the $(n,n)$-parabolic,
were defined in [BDG+, Lem. 6.12, Rem. 6.14].

  * (ii)

Since any two choices of spin $\theta$ differ by an element of
$\mathcal{W}_{G}^{0}$, this definition is independent of such a choice of
$\theta$ by Remark 2.11.

#### 3.1. $P$-spin refinements via Hecke algebras

Recall objects for $\mathcal{G}=\mathrm{GSpin}_{2n+1}$ (e.g. Borel
$\mathcal{B}$, parabolics $\mathcal{P}$) are written as calligraphic versions
of objects for $G=\mathrm{GL}_{2n}$ (e.g. $B,P$).

As $\pi$ is symplectic, it is the functorial transfer of a RACAR $\Pi$ of
$\mathcal{G}(\mathbf{A})$. Moreover
$\Pi_{p}=\operatorname{Ind}_{\mathcal{B}}^{\mathcal{G}}\theta_{\mathcal{G}}$
is an unramified principal series for $\mathcal{G}(\mathbf{Q}_{p})$, for
$\theta_{\mathcal{G}}$ an unramified character of $\mathcal{T}$ satisfying
$\jmath(\theta_{\mathcal{G}})=\theta$ (by [AS06, p.177(i)] and [AS14, Prop.
5.1]).

Our primary motivation for $P$-spin refinements is that they interact well
with this functoriality, as we will show in Proposition 3.7.

##### 3.1.1. Parahoric refinements for $\mathrm{GSpin}_{2n+1}$

######  Definition 3.4.

Let $\mathcal{B}\subset\mathcal{P}\subset\mathcal{G}$ be a parabolic, with
parahoric subgroup
$\mathcal{J}_{\mathcal{P}}\subset\mathcal{G}(\mathbf{Z}_{p})$.

  * •

For $1\leqslant r\leqslant n$, let
$\mathcal{U}_{p,r}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[\mathcal{J}_{p}\cdot\jmath^{\vee}(t_{p,r})\cdot\mathcal{J}_{p}]$,
where $\jmath^{\vee}(t_{p,r})=(f_{1}^{*}+\cdots+f_{r}^{*})(p)$. Let
$\mathcal{V}_{p}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[\mathcal{J}_{p}\cdot
f_{0}^{*}(p)\cdot\mathcal{J}_{p}]$, which acts on $\Pi^{\mathcal{J}_{p}}_{p}$
via the central action of $p\in\mathbf{Q}_{p}$.

  * •

Define a Hecke algebra
$\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{Q}_{p}[\mathcal{U}_{p,r},\mathcal{V}_{p}:b_{r}\notin\Delta_{\mathcal{P}}]$.

  * •

A _$\mathcal{P}$ -parahoric $p$-refinement
$\tilde{\Pi}^{\mathcal{P}}=(\Pi,\alpha^{\mathcal{G},\mathcal{P}})$ of $\Pi$_
is an eigensystem
$\alpha^{\mathcal{G},\mathcal{P}}:\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}}\to\overline{\mathbf{Q}}_{p}$
appearing in $\Pi_{p}^{\mathcal{J}_{\mathcal{P}}}$. We sometimes write
$\mathcal{P}$-refinement for short.

##### 3.1.2. Functoriality for parahoric refinements

Let $P\subset G$ be a spin parabolic, with associated
$\mathcal{P}\subset\mathcal{G}$. Note that $a_{r}\not\in\Delta_{P}\iff
b_{r}\not\in\Delta_{\mathcal{P}}$, so

$\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}}=\mathbf{Q}_{p}[\mathcal{U}_{p,r},\mathcal{V}_{p}:a_{r}\not\in\Delta_{P}].$

We now relate $P$\- and $\mathcal{P}$-refinements. The map
$\jmath^{\vee}:X^{\vee}\to\mathcal{X}^{\vee}$ induces a map

$\jmath^{\vee}:\mathcal{H}_{p}^{P}\longrightarrow\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}}$
(3.1)

(cf. [BDG+, §6.4]). If $1\leqslant r\leqslant n$ with
$a_{r}\not\in\Delta_{P}$, then $\jmath^{\vee}$ sends

$U_{p,r}\longmapsto\mathcal{U}_{p,r},\qquad
U_{p,2n-r}\longmapsto\mathcal{U}_{p,r}\mathcal{V}_{p}^{n-r},\qquad
U_{p,2n}\longmapsto\mathcal{V}_{p}^{n}.$

For $1\leqslant r\leqslant 2n$, consider the characteristic polynomials

$\mathcal{F}_{\mathcal{G},r}(T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\det\big{(}T-\jmath^{\vee}(U_{p,r})|\Pi_{p}^{\mathcal{J}_{\mathcal{P}}}\big{)},\qquad
F_{G,r}(T)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\det\big{(}T-U_{p,r}|\pi_{p}^{J_{P}}\big{)}.$

######  Lemma 3.5.

Let $1\leqslant r\leqslant 2n$. If $U_{p,r}\in\mathcal{H}_{p}^{P}$, then
$\mathcal{F}_{\mathcal{G},r}(T)$ divides $F_{G,r}(T)$.

###### Proof.

Let
$\nu_{p,r}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e_{1}^{*}+\cdots+e_{r}^{*}\in
X^{\vee}$. By [OST, Cor. 3.16], we may write

$F_{G,r}(T)=\prod_{[\sigma]\in\mathcal{W}_{G}/\mathcal{W}_{L_{P}}}\Big{(}T-p^{\langle\rho_{G},\nu_{p,r}\rangle_{G}}p^{\langle\theta^{\sigma},\nu_{p,r}\rangle_{G}}\Big{)}$
(3.2)

where we identify
$\theta^{\sigma}(t_{p,r})=\theta^{\sigma}(\nu_{p,r}(p))=p^{\langle\theta^{\sigma},\nu_{p,r}\rangle_{G}}$
under the natural extension of $\langle-,-\rangle_{G}$.

For $\mathcal{G}$, [OST, Cor. 3.16] again gives

$\displaystyle\mathcal{F}_{\mathcal{G},r}(T)$
$\displaystyle=\prod_{\omega\in\mathcal{W}_{\mathcal{G}}/\mathcal{W}_{\mathcal{L}_{\mathcal{P}}}}\Big{(}T-p^{\langle\rho_{\mathcal{G}},\jmath^{\vee}(\nu_{p,r})\rangle_{\mathcal{G}}}p^{\langle\theta^{\omega}_{\mathcal{G}},\jmath^{\vee}(\nu_{p,r})\rangle_{\mathcal{G}}}\Big{)}$
$\displaystyle=\prod_{[\sigma]\in\mathcal{W}_{G}^{0}/\mathcal{W}_{L_{P}}^{0}}\Big{(}T-p^{\langle\rho_{G},\nu_{p,r}\rangle_{G}}p^{\langle\theta^{\sigma},\nu_{p,r}\rangle_{G}}\Big{)},$

where we identify $\sigma=\jmath(\omega)$, we write
$\mathcal{W}_{L_{P}}^{0}=\jmath(\mathcal{W}_{\mathcal{L}_{\mathcal{P}}})$, and
we have used $\jmath(\rho_{\mathcal{G}})=\rho_{G}$, Proposition 2.4, and
(2.3).

Now note that
$\mathcal{W}_{L_{P}}^{0}=\mathcal{W}_{L_{P}}\cap\mathcal{W}_{G}^{0}$, so that
$\mathcal{W}_{G}^{0}/\mathcal{W}_{L_{P}}^{0}$ is naturally a subset of
$\mathcal{W}_{G}/\mathcal{W}_{L_{P}}$. It follows immediately that
$\mathcal{F}_{\mathcal{G},r}$ divides $F_{G,r}$. ∎

######  Definition 3.6.

Let $P$ be a spin parabolic and $\tilde{\pi}^{P}=(\pi,\alpha^{P})$ a
$P$-refinement. We say $\tilde{\pi}^{P}$ is _the functorial transfer of a
$\mathcal{P}$-refinement
$\tilde{\Pi}^{\mathcal{P}}=(\Pi,\alpha^{\mathcal{G},\mathcal{P}})$ of $\Pi$_
if $\alpha^{P}$ factors as

$\mathcal{H}_{p}^{P}\xrightarrow{\ \ \jmath^{\vee}\ \
}\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}}\xrightarrow{\ \
\alpha^{\mathcal{G},\mathcal{P}}\ \ }\overline{\mathbf{Q}}_{p}.$

######  Proposition 3.7.

Let $\tilde{\pi}^{P}$ be a $P$-refinement. Then

$\tilde{\pi}^{P}\text{ is $P$-spin}\iff\tilde{\pi}^{P}\text{ is the functorial
transfer of some $\tilde{\Pi}^{\mathcal{P}}$.}$

###### Proof.

Let $\tilde{\pi}^{P}=(\pi,\alpha^{P})$ with
$\Psi_{\theta}^{P}(\tilde{\pi})=[\sigma]\in\mathcal{W}_{G}/\mathcal{W}_{L_{P}}$.
By the proof of the above lemma, and the fact that

$\alpha^{P}(U_{p,r})=\delta_{B}^{-1/2}\theta^{\sigma}(t_{p,r})=p^{\langle\rho_{G},\nu_{p,r}\rangle_{G}}p^{\langle\theta^{\sigma},\nu_{p,r}\rangle_{G}},$

we see that $\alpha^{P}$ factors through $\jmath^{\vee}$ if and only if
$[\sigma]$ is in
$\mathcal{W}_{G}^{0}/\mathcal{W}_{L_{P}}^{0}\subset\mathcal{W}_{G}/\mathcal{W}_{L_{P}}$;
that is, if and only if $\tilde{\pi}^{P}$ is a $P$-spin refinement. ∎

#### 3.2. Optimally $P$-spin refinements

Above, we studied when a $P$-refinement was $P$-spin (for the same $P$). An
Iwahori $p$-refinement $\tilde{\pi}$, however, can be $P$-spin for many
different $P$’s.

######  Definition 3.8.

We say an Iwahori $p$-refinement $\tilde{\pi}$ is _optimally $P$-spin_ if it
is $P$-spin and there is no spin $P^{\prime}\subsetneq P$ such that it is
$P^{\prime}$-spin.

######  Corollary 3.9.

Let $\tilde{\pi}=(\pi,\alpha)$ be an Iwahori $p$-refinement.

  * (i)

If $P$ and $P^{\prime}$ are spin parabolics and $\tilde{\pi}$ is $P$-spin and
$P^{\prime}$-spin, then $\tilde{\pi}$ is $P\cap P^{\prime}$-spin.

  * (ii)

$\tilde{\pi}$ is optimally $P_{\tilde{\pi}}$-spin for precisely one spin
parabolic $B\subseteq P_{\tilde{\pi}}\subseteq G$.

###### Proof.

(i) By Proposition 3.7, the associated $P$-refinement $\alpha^{P}$ and
$P^{\prime}$-refinement $\alpha^{P^{\prime}}$ both factor through spin Hecke
algebras; that is, there are maps

$\alpha^{\mathcal{G},\mathcal{P}}:\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}}=\mathbf{Q}_{p}[\mathcal{U}_{p,r},\mathcal{V}_{p}:a_{r}\not\in\Delta_{P}]\to\overline{\mathbf{Q}}_{p},\qquad\alpha^{\mathcal{G},\mathcal{P}^{\prime}}:\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}^{\prime}}=\mathbf{Q}_{p}[\mathcal{U}_{p,r},\mathcal{V}_{p}:a_{r}\not\in\Delta_{P^{\prime}}]\to\overline{\mathbf{Q}}_{p}$

such that

$\mathcal{H}_{p}^{P}\xrightarrow{\ \ \jmath^{\vee}\ \
}\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}}\xrightarrow{\ \
\alpha^{\mathcal{G},\mathcal{P}}\ \
}\overline{\mathbf{Q}}_{p},\qquad\mathcal{H}_{p}^{P^{\prime}}\xrightarrow{\ \
\jmath^{\vee}\ \
}\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}^{\prime}}\xrightarrow{\ \
\alpha^{\mathcal{G},\mathcal{P}^{\prime}}\ \ }\overline{\mathbf{Q}}_{p}.$

These extend to a map

$\alpha^{\mathcal{G},\mathcal{P}\cap\mathcal{P}^{\prime}}:\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}\cap\mathcal{P}^{\prime}}=\mathbf{Q}_{p}[\mathcal{U}_{p,r},\mathcal{V}_{p}:a_{r}\not\in\Delta_{P}\cap\Delta_{P^{\prime}}]\to\overline{\mathbf{Q}}_{p}.$

Since $\Delta_{P}\cap\Delta_{P^{\prime}}=\Delta_{P\cap P}$, we find
$\alpha^{P\cap P^{\prime}}$ factors through
$\alpha^{\mathcal{G},\mathcal{P}\cap\mathcal{P}^{\prime}}$, whence $\alpha$ is
a $P\cap P^{\prime}$-spin refinement, as required.

(ii) The unique minimum $P_{\tilde{\pi}}$ is the intersection of all $P$ such
that $\tilde{\pi}$ is $P$-spin. ∎

#### 3.3. $P$-spin refinements combinatorially

We now introduce a convenient combinatorial description of $p$-refinements.
Let $\tilde{\pi}$ be a $p$-refinement, with
$\Psi_{\theta}(\tilde{\pi})=\sigma$. We represent this by the tuple
$\tilde{\pi}\sim\\{\sigma(1)\sigma(2)\cdots\sigma(2n)\\}$ (for example, if
$\sigma$ is the transposition in $\mathrm{S}_{4}$ exchanging 1 and 2, then we
represent $\tilde{\pi}$ as $\\{2134\\}$). From this, we can easily read off
whether $\tilde{\pi}$ is $P$-spin.

######  Definition 3.10.

Let $\sigma\in\mathcal{W}_{G}$. For $1\leqslant r\leqslant n$, we say $\sigma$
is _$r$ -spin_ if

$\forall i\leqslant r,\ \exists j\geqslant 2n+1-r\ \text{s.t.}\
\sigma(i)+\sigma(j)=2n+1.$ (3.3)

We say a $p$-refinement $\tilde{\pi}$ is $r$-spin if
$\sigma=\Psi_{\theta}(\tilde{\pi})$ is $r$-spin.

In particular, to be $r$-spin, in the tuple $\\{\sigma(1)\cdots\sigma(2n)\\}$,
it must be possible to pair off the first $r$ numbers and last $r$ numbers
into pairs that sum to $2n+1$. For example:

  * •

The $p$-refinement $\tilde{\pi}\sim\\{216345\\}$ (for $\operatorname{GL}_{6}$)
is 1-spin (since $\sigma(1)+\sigma(6)=2+5=7$). It is not 2-spin, as
$\\{2,1\\}$ and $\\{4,5\\}$ cannot be paired off into pairs summing to 7.
Similarly it is not 3-spin.

  * •

The $p$-refinement $\tilde{\pi}^{\prime}\sim\\{132456\\}$ is 1-spin and
3-spin, but not 2-spin.

######  Definition 3.11.

For a spin parabolic $P$, define $X_{P}\subset\\{1,...,n\\}$ by

$i\in X_{P}\iff a_{i}\not\in\Delta_{P}\iff A_{i}\not\subset\Delta_{P}.$

This defines an inclusion-reversing bijection between spin parabolics $P$ and
subsets $X_{P}\subset\\{1,...,n\\}$. If $X\subset\\{1,...,n\\}$, we say
$\tilde{\pi}$ is _$X$ -spin_ if it is $r$-spin for all $r\in X$.

######  Proposition 3.12.

Let $P$ be a spin parabolic and $\tilde{\pi}$ a $p$-refinement. Then

$\tilde{\pi}\text{ is }P\text{-spin}\iff\tilde{\pi}\text{ is
}X_{P}\text{-spin}.$ (3.4)

It is optimally $P$-spin if and only if
$X_{P}=\bigcup_{\begin{subarray}{c}X\subset\\{1,...,n\\}\\\ \tilde{\pi}\text{
is $X$-spin}\end{subarray}}X$ is maximal with this property.

######  Example.

Recall $P_{\tilde{\pi}}$ is the unique spin parabolic such that $\tilde{\pi}$
is optimally $P_{\tilde{\pi}}$-spin. The example $\tilde{\pi}\sim\\{216345\\}$
above is 1-spin but not 2- or 3-spin, so $X_{P_{\tilde{\pi}}}=\\{1\\}$, hence
$\Delta_{P_{\tilde{\pi}}}=\\{a_{2},a_{3},a_{4}\\}$, i.e $P_{\tilde{\pi}}$ is
the (1,4,1)-parabolic. Similarly
$\Delta_{P_{\tilde{\pi}^{\prime}}}=\\{a_{2},a_{4}\\}$, so
$P_{\tilde{\pi}^{\prime}}$ is the (1,2,2,1)-parabolic.

###### Proof.

For $1\leqslant r\leqslant n$, let $P_{r}$ be the $(r,2n-2r,r)$-parabolic.
Note that

$\textstyle P=\bigcap_{\begin{subarray}{c}r\in\\{1,...,n\\}\\\
a_{r}\not\in\Delta_{P}\end{subarray}}P_{r},\qquad\text{thus}\qquad
X_{P}=\bigcup_{\begin{subarray}{c}r\in\\{1,...,n\\}\\\
a_{r}\not\in\Delta_{P}\end{subarray}}X_{P_{r}},$

so by Corollary 3.9(i), it suffices to show that

$\tilde{\pi}\text{ is }P_{r}\text{-spin}\iff\tilde{\pi}\text{ is
}r\text{-spin}.$ (3.5)

First suppose $\tilde{\pi}$ is $P_{r}$-spin, so we can write
$\Psi_{\theta}(\tilde{\pi})=\zeta\sigma$, with $\zeta\in\mathcal{W}_{G}^{0}$
and $\sigma\in\mathcal{W}_{L_{P_{r}}}$. Note
$\sigma\in\mathcal{W}_{L_{P_{r}}}=\mathrm{S}_{r}\times\mathrm{S}_{2n-2r}\times\mathrm{S}_{r}$
preserves $\\{1,...,r\\}$ and $\\{2n+1-r,...,2n\\}$, hence $\sigma$ is
$r$-spin.

By Lemma 2.1, as $\zeta\in\mathcal{W}_{G}^{0}$, $\sigma(i)+\sigma(j)=2n+1$ if
and only if $\zeta\sigma(i)+\zeta\sigma(j)=2n+1$, i.e.

$\text{($\sigma$ is $r$-spin) $\iff$ ($\zeta\sigma$ is $r$-spin)}.$ (3.6)

It follows that $\zeta\sigma$, hence $\tilde{\pi}$, is $r$-spin, giving
$\Rightarrow$ in (3.5).

Conversely, suppose $\tilde{\pi}$ is $r$-spin, and let
$\sigma=\Psi_{\theta}(\tilde{\pi})\in\mathcal{W}_{G}$.

######  Claim 3.13.

Without loss of generality we may assume $\sigma$ preserves $\\{1,...,r\\}$.

_Proof of claim:_ We may renormalise $\theta$ by elements of
$\mathcal{W}_{G}^{0}$, as this preserves both being $P_{r}$-spin (Remark 3.3)
and $r$-spin (by Remark 2.11 and (3.6)). We do so repeatedly.

First, without loss of generality we may take

$\\{\sigma(1),...,\sigma(r)\\}\subset\\{1,...,n\\}.$ (3.7)

Indeed, if $\sigma(i)>n$ for $1\leqslant i\leqslant r$, then there exists
$2n+1-r\leqslant j\leqslant 2n$ such that $\sigma(i)+\sigma(j)=2n+1$, so that
$\sigma(j)\leqslant n$; and we may exchange $\sigma(i)$ and $\sigma(j)$ by the
transposition $(\sigma(i),\sigma(j))\in\mathcal{W}_{G}^{0}$.

Given (3.7), after acting by an element of $S_{n}\subset\mathcal{W}_{G}^{0}$,
we may assume $\\{\sigma(1),...,\sigma(r)\\}=\\{1,...,r\\}$, proving the
claim.

As $\tilde{\pi}$ is $r$-spin, if $\sigma$ preserves $\\{1,...,r\\}$, it must
also preserve $\\{2n+1-r,...,2n\\}$. This means
$\sigma\in\mathrm{S}_{r}\times\mathrm{S}_{2n-2r}\times\mathrm{S}_{r}=\mathcal{W}_{L_{P_{r}}}$,
so $\sigma$ (hence $\tilde{\pi}$) is $P_{r}$-spin, giving $\Leftarrow$ in
(3.5), and hence (3.4).

The last statement is immediate as $P\leftrightarrow X_{P}$ is inclusion-
reversing. ∎

#### 3.4. The function $\gamma_{\tilde{\pi}}$

Finally, we introduce one more combinatorial description of being $P$-spin,
which will be useful when we study symplectic families.

######  Definition 3.14.

Let $\tilde{\pi}$ be a $p$-refinement and $\sigma=\Psi_{\theta}(\tilde{\pi})$.
Define an injective map

$\gamma_{\tilde{\pi}}:\\{1,...,n\\}\lhook\joinrel\relbar\joinrel\rightarrow\\{1,...,2n\\}$

by setting $\gamma_{\tilde{\pi}}(i)$ to be the unique integer such that

$\sigma(i)+\sigma(2n+1-\gamma_{\tilde{\pi}}(i))=2n+1.$

######  Lemma 3.15.

The map $\gamma_{\tilde{\pi}}$ is independent of the choice of $\theta$
satisfying $\theta_{i}\theta_{2n+1-i}=\eta_{p}$.

###### Proof.

If $\theta^{\prime}$ is another such choice, there exists
$\nu\in\mathcal{W}_{G}^{0}$ such that $\theta^{\prime}=\theta^{\nu}$. Remark
2.11 says
$\Psi_{\theta^{\prime}}(\tilde{\pi})=\nu\Psi_{\theta}(\tilde{\pi})=\nu\sigma$.
By Lemma 2.1, $\gamma_{\tilde{\pi}}$ is unchanged if we replace $\sigma$ with
$\nu\sigma$. ∎

######  Lemma 3.16.

Let $\tilde{\pi}$ be a $p$-refinement. For $1\leqslant r\leqslant n$, we have

$\tilde{\pi}\text{ is $r$-spin }\iff\text{ $\gamma_{\tilde{\pi}}$ sends
$\\{1,...,r\\}$ to itself.}$

###### Proof.

We know $\gamma_{\tilde{\pi}}$ preserves $\\{1,...,r\\}$ if and only if
$2n+1-r\leqslant 2n+1-\gamma_{\tilde{\pi}}(i)\leqslant 2n$ for all $i$. By
definition of $\gamma_{\tilde{\pi}}$, this is if and only if the sets
$\\{\sigma(1),...,\sigma(r)\\}$ and $\\{\sigma(2n+1-r),...,\sigma(2n)\\}$ can
be paired off into pairs summing to $2n+1$. But this is the definition of
$r$-spin. ∎

######  Proposition 3.17.

Let $P$ be a spin parabolic, let $\tilde{\pi}$ be a $p$-refinement, and
$\gamma_{\tilde{\pi}}:\\{1,...,n\\}\hookrightarrow\\{1,...,2n\\}$ the function
from Definition 3.14. Then

$\tilde{\pi}\text{ is $P$-spin }\iff\text{ $\gamma_{\tilde{\pi}}$ preserves
$\\{1,...,r\\}$ whenever $r\in X_{P}$.}$

Additionally, $\tilde{\pi}$ is optimally $P$-spin if $\gamma_{\tilde{\pi}}$
does _not_ preserve $\\{1,...,r\\}$ for all $r\not\in X_{P}$.

###### Proof.

Both statements follow by combining Proposition 3.12 with Lemma 3.16. ∎

## Part II. Dimensions of Symplectic Components

In part II, we focus on full Iwahori refinements $\tilde{\pi}$, and study the
families through such refinements in the Iwahori eigenvariety. In particular,
we conjecture a classification on the dimension of such symplectic families
based on the unique spin parabolic $P_{\tilde{\pi}}$ such that $\tilde{\pi}$
is optimally $P_{\tilde{\pi}}$-spin, prove the upper bound, and prove the
lower bound in special cases.

### 4\. The symplectic locus in the eigenvariety

#### 4.1. The eigenvariety

Recall that $K=K^{p}\mathrm{Iw}_{G}$ is Iwahori at $p$, and let
$\mathscr{W}=\mathscr{W}_{K}$ be the _weight space_ for $G$ of level $K$
(defined e.g. in [BDG+, §10.1]). It is a $2n$-dimensional
$\mathbf{Q}_{p}$-rigid space. Let
$\mathcal{H}=\mathcal{H}^{p}\cdot\mathcal{H}_{p}$, for
$\mathcal{H}^{p}=\otimes_{v\nmid p\infty}\mathcal{H}_{v}$ the tame Hecke
algebra of e.g. [BW21, Def. 2.2].

The central object of study in this paper is the _eigenvariety for $G$_.

######  Theorem 4.1.

([Han17, Thm. 1.1.2]). There exists a canonical separated rigid analytic space
$\mathscr{E}^{G}_{K}$, and a locally finite map
$w:\mathscr{E}_{K}^{G}\to\mathscr{W}$, such that the $L$-points
$x\in\mathscr{E}^{G}$ with $w(x)=\lambda$ biject with finite-slope systems of
$\mathcal{H}$-eigenvalues in the overconvergent cohomology
$\mathrm{H}^{\bullet}_{\mathrm{c}}(S_{K},\mathscr{D}_{\lambda})$.

Here $\mathcal{H}_{p}$ acts on the cohomology via normalised Hecke operators
$U_{p,r}^{\circ}$ [BDW, Rem. 3.13].

A point $x\in\mathscr{E}_{K}^{G}$ is _classical (cuspidal)_ if the
corresponding system of eigenvalues appears in $\pi_{x}^{K}$ for a (cuspidal)
automorphic representation $\pi_{x}$ of $G(\mathbf{A})$ of weight $w(x)$.
Following [Urb11], [Han17, Conj. 1.1.5] predicts:

######  Conjecture 4.2.

Every irreducible component of $\mathscr{E}_{K}^{G}$ containing a non-critical
cuspidal classical point of regular weight has dimension $n+1$.

The notion of non-criticality is [Han17, Def. 3.2.3]. In [Han17, Prop. B.1],
Newton has proved that every such component has dimension at least $n+1$. The
full conjecture – which Urban styles as a ‘non-abelian Leopoldt conjecture’ –
has been proved for $\operatorname{GL}_{N}$ with $N\leqslant 4$, but is wide
open beyond this.

#### 4.2. The classical and symplectic loci

######  Definition 4.3.

The _classical cuspidal locus_ $\mathscr{L}_{K}^{G}\subset\mathscr{E}_{K}^{G}$
is the Zariski closure of the classical cuspidal points in
$\mathscr{E}_{K}^{G}$.

Let $\mathscr{W}_{0}\subset\mathscr{W}$ be the $(n+1)$-dimensional _pure
weight space_ , the Zariski-closure of all pure algebraic weights (that is,
dominant weights $\lambda=(\lambda_{1},...,\lambda_{2n})$ such that
$\lambda_{1}+\lambda_{2n}=\lambda_{2}+\lambda_{2n-1}=\cdots=\lambda_{n}+\lambda_{n+1}={\sf
w}(\lambda)$ for some ${\sf w}(\lambda)\in\mathbf{Z}$). By [Clo90, Lem. 4.9]
any classical cuspidal point $x$ has weight $w(x)\in\mathscr{W}_{0}$, so:

######  Proposition 4.4.

We have $w(\mathscr{L}_{K}^{G})\subset\mathscr{W}_{0}$.

Through any point $x\in\mathscr{L}^{G}_{K}$, there is a ‘trivial’
1-dimensional family, corresponding to twists by the norm. (In the
introduction, for more conceptual statements, we removed this trivial
variation; but here, for cleaner comparisons to other works, we leave it in).

######  Definition 4.5.

Let $x\in\mathscr{L}_{K}^{G}$ be a classical cuspidal point.

  * •

An irreducible neighbourhood of $\mathscr{L}_{K}^{G}$ through $x$ is _trivial_
if it is exactly 1-dimensional, given by twists by the norm and varying over
the weight family $\\{w(x)+(\kappa,....,\kappa)\\}$.

  * •

A _classical family_ through $x$ is a non-trivial irreducible neighbourhood
$\mathscr{C}\subset\mathscr{L}_{K}^{G}$ of $x$.

  * •

We say a point/eigensystem $x\in\mathscr{L}_{K}^{G}$ is _arithmetically rigid_
if it cannot be varied in a classical family (i.e. it varies only in a trivial
family).

Little is known, or even precisely conjectured, about the classical cuspidal
locus. However, there is a folklore expectation that _all_ classical families
should come from discrete series, in the sense described in §1.3. In
particular, all such families should ‘come from self-duality’.

Given the above expectation, it is natural to study RACARs $\pi$ of
$G(\mathbf{A})$ that are essentially self-dual. Such RACARs are either
orthogonal or symplectic. We focus on the latter.

######  Definition 4.6.

Define the _symplectic locus_
$\mathscr{S}_{K}^{G}\subset\mathscr{L}_{K}^{G}\subset\mathscr{E}^{G}_{K}$ to
be the Zariski closure of all classical cuspidal points $x$ such that
$\pi_{x}$ is symplectic. A _symplectic family through $x$_ is a non-trivial
irreducible neighbourhood of $x$ in $\mathscr{S}_{K}^{G}$.

Our main result (Theorem A of the introduction) gives upper/lower bounds for
the dimensions of symplectic families. We state this in the stronger form we
prove in §4.4.

#### 4.3. Parabolic weight spaces

To state the more precise version of Theorem A that we actually prove, we must
introduce parabolic weight spaces.

Recall that if $P\subset G$ is a parabolic, then the _$P$ -parabolic weight
space_ is the subspace $\mathscr{W}^{P}\subset\mathscr{W}$ of characters that
extend to characters of $L_{P}$. If $\lambda_{\pi}\in\mathscr{W}$ is any fixed
weight, we denote its coset

$\mathscr{W}_{\lambda_{\pi}}^{P}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lambda_{\pi}+\mathscr{W}^{P}\subset\mathscr{W},$

and call it the _$P$ -parabolic weight space through $\lambda_{\pi}$_. These
notions are defined in general, and in detail, in [BW21, §3.1]. We also define
the pure subspaces $\mathscr{W}_{0}^{P}$ and
$\mathscr{W}_{0,\lambda_{\pi}}^{P}$ to be the intersections of
$\mathscr{W}^{P}$ and $\mathscr{W}_{\lambda_{\pi}}^{P}$ with
$\mathscr{W}_{0}$. We now compute their dimensions.

######  Lemma 4.7.

If $\lambda_{\pi}=(\lambda_{\pi,1},...,\lambda_{\pi,2n})$ and
$\lambda=(\lambda_{1},...,\lambda_{2n})$ are two weights, then
$\lambda\in\mathcal{W}_{\lambda_{\pi}}^{P}$ if and only if

$\lambda_{i}-\lambda_{i+1}=\lambda_{\pi,i}-\lambda_{\pi,i+1}\qquad\forall
i\text{ such that }a_{i}\in\Delta_{P}.$ (4.1)

###### Proof.

We have $\lambda\in\mathscr{W}_{\lambda_{\pi}}^{G}$ if and only if
$\lambda-\lambda_{\pi}=:\mu=(\mu_{1},...,\mu_{2n})=(\lambda_{1}-\lambda_{\pi,1},...,\lambda_{2n}-\lambda_{\pi,2n})$
factors through $L_{P}$. If
$L_{P}=\operatorname{GL}_{m_{1}}\times\cdots\times\operatorname{GL}_{m_{r}}$,
then this happens if and only if $\mu$ factors through
$\det_{1}\times\cdots\times\det_{r}$. This is equivalent to
$\mu_{1}=\cdots=\mu_{m_{1}}$, …, $\mu_{2n-m_{r}+1}=\cdots=\mu_{2n}$ (i.e. the
$\mu_{i}$’s are constant in each Levi factor); or in other words, that
$\lambda_{i}-\lambda_{\pi,i}=\mu_{i}=\mu_{i+1}=\lambda_{i+1}-\lambda_{\pi,i+1}$
for all $i$ with $a_{i}\in\Delta_{P}$. Rearranging gives (4.1). ∎

In particular, $\lambda_{i}-\lambda_{i+1}$ can vary in a $P$-parabolic weight
family if and only if $a_{i}\not\in\Delta_{P}$. For example, in a
$B$-parabolic weight family weights can vary in all directions (since
$\Delta_{B}=\varnothing$). If $Q$ is the $(n,n)$-parabolic, then
$\Delta_{Q}=\Delta_{B}\backslash\\{a_{n}\\}$, so in a $Q$-parabolic family
$\lambda_{1}-\lambda_{2}$, …, $\lambda_{n-1}-\lambda_{n}$ are fixed,
$\lambda_{n}-\lambda_{n+1}$ can vary, and $\lambda_{n+1}-\lambda_{n+2}$, …,
$\lambda_{2n-1}-\lambda_{2n}$ are fixed, so we get the 2-dimensional variation
of [BDW].

######  Lemma 4.8.

For any spin parabolic $P$ and $\lambda_{\pi}\in X_{0}\subset\mathscr{W}_{0}$,
we have $\mathrm{dim}(\mathscr{W}_{0,\lambda_{\pi}}^{P})=\\#X_{P}+1$.

###### Proof.

By Lemma 4.7, each $\lambda_{i}-\lambda_{i+1}$ is constant in
$\mathscr{W}_{\lambda_{\pi}}^{P}$ if and only if $a_{i}\in\Delta_{P}$, and
each such condition decreases the dimension by 1; so

$\mathrm{dim}(\mathscr{W}_{\lambda_{\pi}}^{P})=2n-\\#\Delta_{P}=\\#\\{1\leqslant
i\leqslant 2n-1:a_{i}\not\in\Delta_{P}\\}+1.$

If $\lambda\in\mathscr{W}_{0,\lambda_{\pi}}^{P}$ and $1\leqslant i\leqslant
n-1$, we must have
$\lambda_{i}+\lambda_{2n+1-i}=\lambda_{i+1}+\lambda_{2n-i}$, whence
$\lambda_{i}-\lambda_{i+1}=\lambda_{2n-i}-\lambda_{2n+1-i}$. (If $i=n$, this
still holds; but then it is vacuous). Thus
$\mathrm{dim}(\mathscr{W}_{0,\lambda_{\pi}}^{P})=\\#\\{1\leqslant i\leqslant
n:a_{i}\not\in\Delta_{P}\\}+1=\\#X_{P}+1$, as required. ∎

#### 4.4. Main results/conjecture: the dimension of symplectic families

We now precisely state the stronger forms of Theorem A that we actually prove.
Let $\pi$ be a RASCAR of weight $\lambda_{\pi}$ that is spherical and regular
at $p$, and let $\tilde{\pi}$ be an optimally $P_{\tilde{\pi}}$-spin
$p$-refinement. In §5, we will show the following ‘upper bound’:

######  Theorem 4.9.

Any symplectic family $\mathscr{C}\subset\mathscr{S}^{G}_{K}$ through
$\tilde{\pi}$ is supported over the $P_{\tilde{\pi}}$-parabolic pure weight
space, i.e.

$w(\mathscr{C})\subset\mathscr{W}_{0,\lambda_{\pi}}^{P_{\tilde{\pi}}}.$

In particular, $\mathrm{dim}(\mathscr{C})\leqslant\\#X_{P_{\tilde{\pi}}}+1$.

Note we make no non-criticality assumption here. The second statement is
Theorem A(i); this follows immediately from the first statement, as $w$ is a
locally finite map and
$\mathrm{dim}(\mathscr{W}_{0,\lambda_{\pi}}^{P_{\tilde{\pi}}})=\\#X_{P_{\tilde{\pi}}}+1$
by Lemma 4.8.

Our second main result, a stronger form of Theorem A(ii), is a ‘lower bound’.
Away from $p$, let $K_{1}(\pi)^{p}\subset G(\mathbf{A}_{f}^{(p)})$ be the
Whittaker new level from [JPSS81] (see e.g. [BDW, (7.2)]). Let
$K_{1}(\tilde{\pi})=K_{1}(\pi)^{p}\mathrm{Iw}_{G}$. In §6, we prove:

######  Theorem 4.10.

Suppose that $\tilde{\pi}$ has non-critical slope and $\lambda_{\pi}$ is
regular. Then there is a unique symplectic family through $\tilde{\pi}$ in
$\mathscr{E}_{K_{1}(\tilde{\pi})}^{G}$. This family has dimension exactly
$\\#X_{P_{\tilde{\pi}}}+1$, and is étale over
$\mathscr{W}_{0,\lambda_{\pi}}^{P_{\tilde{\pi}}}$ at $\tilde{\pi}$.

######  Remark 4.11.

Our guiding expectation is that any classical cuspidal family for $G$ should
be a transfer of a discrete series family. Which discrete series families,
then, should give rise to the families of Theorem 4.10?

Since $\tilde{\pi}$ is an optimally $P_{\tilde{\pi}}$-spin $p$-refinement, by
Proposition 3.7, the associated $P_{\tilde{\pi}}$-refinement
$\tilde{\pi}^{P_{\tilde{\pi}}}$ is a functorial transfer of a
$\mathcal{P}_{\tilde{\pi}}$-refinement
$\tilde{\Pi}^{\mathcal{P}_{\tilde{\pi}}}$ for $\mathrm{GSpin}_{2n+1}$. Then
$\tilde{\Pi}^{\mathcal{P}_{\tilde{\pi}}}$ should vary in a ‘spin family’
$\mathscr{C}^{\mathcal{G}}$ over an $(\\#X_{P_{\tilde{\pi}}}+1)$-dimensional
$\mathcal{P}$-parabolic weight space
$\mathscr{W}^{\mathcal{P}_{\tilde{\pi}}}_{\mathcal{G},\lambda_{\Pi}}$ for
$\mathcal{G}$ (see e.g. [BW21, Cor. 5.16]). The map $\jmath$ from §2.1
isomorphically identifies
$\mathscr{W}_{\mathcal{G},\lambda_{\Pi}}^{\mathcal{P}_{\tilde{\pi}}}$ and
$\mathscr{W}_{0,\lambda_{\pi}}^{P_{\tilde{\pi}}}$, and under Langlands
functoriality, we expect that the family of Theorem 4.10 is exactly a transfer
to $G$ of the expected spin family $\mathscr{C}^{\mathcal{G}}$.

If we _suppose_ the existence of this $p$-adic functoriality map, then Theorem
4.9 implies that the image of $\mathscr{C}^{\mathcal{G}}$ in the Iwahori-level
$\operatorname{GL}_{2n}$-eigenvariety is itself an irreducible component of
the symplectic locus (that is, it is not a proper subspace of some larger
irreducible component).

Remark 4.11, and the philosophy above, suggest the following.

######  Conjecture 4.12.

Let $\tilde{\pi}$ be a $p$-refined RASCAR of $\operatorname{GL}_{2n}$. Every
symplectic family through $\tilde{\pi}$ is the transfer of a classical
parabolic family for $\mathrm{GSpin}_{2n+1}$, varies over
$\mathscr{W}_{0,\lambda_{\pi}}^{P_{\tilde{\pi}}}$, and has dimension
$\\#X_{P_{\tilde{\pi}}}+1$.

#### 4.5. The dimension of classical families

We have predicted the dimension of _symplectic_ families through symplectic
$\tilde{\pi}$. It is desirable to describe more generally the _classical_
families. If the following is true, then these questions are equivalent.

######  Expectation 4.13.

Every classical family through a $p$-refined RASCAR is symplectic. In
particular, Conjecture 4.12 describes all _classical_ families through
RASCARs.

We do not state this as a formal conjecture; without further evidence, we do
not feel confident to rule out ‘strange’ behaviour in higher dimension, where
it is harder to classify all the possible lifts from discrete series. For
example, we do not rule out classical cuspidal families through $\tilde{\pi}$
that are lifts from discrete series but not themselves essentially self-dual.

If we restrict to _essentially self-dual_ families – that is, where the
essentially self-dual points are Zariski-dense – then we are on safer ground.
Any such family should be symplectic or orthogonal. The symplectic/orthogonal
loci should never intersect at classical cohomological points, meaning every
classical essentially self-dual family through a $p$-refined RASCAR should be
symplectic.

In the case of $\operatorname{GL}_{4}$, we expect every classical family to be
essentially self-dual, motivating:

######  Conjecture 4.14.

Let $\tilde{\pi}$ be a $p$-refined RASCAR $\pi$ of $\operatorname{GL}_{4}$.
Every classical family through $\tilde{\pi}$ is the transfer of a classical
family on $\mathrm{GSp}_{4}$, which varies over a $P_{\tilde{\pi}}$-parabolic
weight space and has dimension $\\#X_{P_{\tilde{\pi}}}+1$.

This could be considered a (symplectic) $\operatorname{GL}_{4}$ analogue of
[CM09] (for Bianchi modular forms) and [APS08] (for $\operatorname{GL}_{3})$.
It seems at least as difficult.

### 5\. Weight obstructions to symplectic families

Let $\pi$ be a RASCAR of weight $\lambda_{\pi}$ that is spherical and regular
at $p$, and $\tilde{\pi}$ an optimally $P_{\tilde{\pi}}$-spin $p$-refinement.
In this section, we prove Theorem 4.9. In particular, let $\mathscr{C}$ be any
classical symplectic family through $\tilde{\pi}$. We show that $\mathscr{C}$
varies only over $\mathscr{W}_{0,\lambda_{\pi}}^{P_{\tilde{\pi}}}$, so has
dimension at most $\\#X_{P_{\tilde{\pi}}}+1$.

Recall $\pi_{p}=\operatorname{Ind}_{B}^{G}\theta$, with
$\theta_{i}\theta_{2n+1-i}=\eta_{p}$ (2.4). This fixed
$\Psi_{\theta}:\\{p\text{-refinements}\\}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{W}_{G}$.

#### 5.1. Identities between Hecke eigenvalues

Given a $p$-refinement $\tilde{\pi}=(\pi,\alpha)$, we have so far given
several criteria for it being $P$-spin. The most natural, in terms of transfer
from $\mathrm{GSpin}_{2n+1}$, is conceptually useful but is hard to check. To
study the $P$-spin condition in $p$-adic families, we would prefer a
characterisation purely in terms of eigenvalues that is intrinsic to
$\operatorname{GL}_{2n}$, with no reference to $\mathrm{GSpin}_{2n+1}$. The
following is an easy starting point:

######  Lemma 5.1.

If $\tilde{\pi}=(\pi,\alpha)$ is $r$-spin, then

$\eta_{0}(p)^{n-r}\cdot\alpha(U_{p,r}^{\circ})=\alpha(U_{p,2n-r}^{\circ}).$
(5.1)

###### Proof.

By (3.5), $\tilde{\pi}$ is $P_{r}$-spin for the $(r,2n-2r,r)$-parabolic
$P_{r}$. Applying Proposition 3.7 to $\tilde{\pi}^{P_{r}}$, we see
$\alpha^{P_{r}}$ factors through
$\jmath^{\vee}:\mathcal{H}_{p}^{P_{r}}\to\mathcal{H}_{p}^{\mathcal{G},\mathcal{P}_{r}}$.
Note $\jmath^{\vee}$ sends $U_{p,r}\mapsto\mathcal{U}_{p,r}$ and
$U_{p,2n-r}\mapsto\mathcal{V}_{p}^{n-r}\mathcal{U}_{p,r}$, and that
$\mathcal{V}_{p}$ acts on $\Pi$ via $\eta(p)$; so this factorisation implies
that

$\eta_{p}(p)^{n-r}\cdot\alpha(U_{p,r})=\alpha(U_{p,2n-r}).$

To get the claimed relation for the normalised $U_{p,r}^{\circ}$’s, recall
$U_{p,r}^{\circ}=p^{\lambda_{1}+\cdots+\lambda_{r}}U_{p,r}$. We have
$p^{\lambda_{1}+\cdots+\lambda_{n-r}}=p^{\lambda_{1}+\cdots+\lambda_{r}}\cdot
p^{(n-r){\sf w}}$, and $\eta_{p}(p)=\eta_{0}(p)p^{-{\sf w}}$. ∎

However, this statement is certainly not if-and-only-if in general. When
$r=n$, for example, the statement (5.1) is vacuous, so is satisfied by all
$\tilde{\pi}$. It is desirable to find analogous relations that _exactly_
characterise the $r$-spin (hence $P$-spin) refinements. For this, we will use
the canonical function
$\gamma_{\tilde{\pi}}:\\{1,...,n\\}\hookrightarrow\\{1,...,2n\\}$ attached to
$\tilde{\pi}$, which – by Proposition 3.17 – exactly determines when
$\tilde{\pi}$ is $P$-spin.

For any $p$-refinement $\alpha$, by Proposition 2.10,
$\alpha(U_{p,r}^{\circ})\neq 0$ for all $r$. We will repeatedly use the
following simple observation.

######  Lemma 5.2.

Let $\tilde{\pi}$ be a $p$-refinement and let
$\sigma=\Psi_{\theta}(\tilde{\pi})$. Then

$\displaystyle\theta_{\sigma(r)}(p)$
$\displaystyle=p^{\tfrac{2n-2r+1}{2}}\cdot\frac{\alpha(U_{p,r})}{\alpha(U_{p,r-1})}$
(5.2) $\displaystyle=p^{\tfrac{2n-2r+1}{2}}\cdot
p^{-\lambda_{r}}\cdot\frac{\alpha(U_{p,r}^{\circ})}{\alpha(U_{p,r-1}^{\circ})}.$

Here, by convention,
$\alpha(U_{p,0})=\alpha(U_{p,0}^{\circ})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=1$.

###### Proof.

The first equality follows from Proposition 2.10(i), which says for any $r$,
we have $\alpha(U_{p,r}^{\circ})=\delta_{B}^{-1/2}(t_{p,r})\cdot
p^{\lambda_{1}+\cdots+\lambda_{r}}\cdot\theta_{\sigma(1)}(p)\cdots\theta_{\sigma(r)}(p).$
The second equality follows as
$U_{p,r}^{\circ}=p^{\lambda_{1}+\cdots+\lambda_{r}}U_{p,r}$. ∎

Recall the map
$\gamma_{\tilde{\pi}}:\\{1,...,n\\}\hookrightarrow\\{1,...,2n\\}$ from
Definition 3.14. Crucially, by definition of $\gamma_{\tilde{\pi}}$, (2.4)
tells us
$\theta_{\sigma(i)}\cdot\theta_{\sigma(2n+1-\gamma_{\tilde{\pi}}(i))}=\eta_{p}$.
For ease of notation, let
$\alpha_{r}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\alpha(U_{p,r})$.

######  Lemma 5.3.

For each $1\leqslant s\leqslant n$, we have

$\alpha_{s}\cdot\prod_{i=1}^{s}p^{2\gamma_{\tilde{\pi}}(i)-2n-1}\frac{\alpha_{2n+1-\gamma_{\tilde{\pi}}(i)}}{\alpha_{2n-\gamma_{\tilde{\pi}}(i)}}=\delta_{B}^{-1/2}(t_{p,s})\cdot\eta_{p}(p)^{s}.$
(5.3)

As $\pi_{p}$ is regular, $\gamma_{\tilde{\pi}}$ is the unique map
$\\{1,...,n\\}\hookrightarrow\\{1,...,2n\\}$ with this property.

###### Proof.

We know
$\alpha_{s}=\delta_{B}^{-1/2}(t_{p,s})\theta_{\sigma(1)}(p)\cdots\theta_{\sigma(s)}(p)$.
By Lemma 5.2, the left-hand side is

$\delta_{B}^{-1/2}(t_{p,s})\theta_{\sigma(1)}(p)\cdots\theta_{\sigma(s)}(p)\cdot\prod_{i=1}^{s}\theta_{\sigma(2n+1-\gamma_{\tilde{\pi}}(i))}(p)=\delta_{B}^{-1/2}\prod_{i=1}^{s}\big{[}\theta_{\sigma(i)}\theta_{\sigma(2n+1-\gamma_{\tilde{\pi}}(i))}\big{]}(p).$

We deduce (5.3) since
$\theta_{\sigma(i)}\theta_{\sigma(2n+1-\gamma_{\tilde{\pi}}(i))}=\eta_{p}$ for
each $i$.

It remains to prove uniqueness. Suppose
$\gamma:\\{1,...,n\\}\hookrightarrow\\{1,...,2n\\}$ is another function such
that (5.3) holds (with $\gamma$ in place of $\gamma_{\tilde{\pi}}$) for
$1\leqslant s\leqslant n$. Regularity of $\pi_{p}$ means all the
$\theta_{i}(p)$’s are distinct. Dividing (5.3) for $s$ by (5.3) for $s-1$
gives

$\theta_{\sigma(s)}\cdot\theta_{\sigma(2n+1-\gamma(s))}(p)=\eta_{p}(p)=\theta_{\sigma(s)}\cdot\theta_{\sigma(2n+1-\gamma_{\tilde{\pi}}(s))}(p).$

Regularity implies
$\sigma(2n+1-\gamma(s))=\sigma(2n+1-\gamma_{\tilde{\pi}}(s))$, so
$\gamma(s)=\gamma_{\tilde{\pi}}(s)$, and $\gamma=\gamma_{\tilde{\pi}}$. ∎

######  Proposition 5.4.

For each $1\leqslant s\leqslant n$, we have

$\alpha(U_{p,s}^{\circ})\cdot\prod_{i=1}^{s}p^{2\gamma_{\tilde{\pi}}(i)-2n-1}\cdot
p^{\lambda_{\gamma_{\tilde{\pi}}(i)}-\lambda_{i}}\cdot\frac{\alpha(U_{p,2n+1-\gamma_{\tilde{\pi}}(i)}^{\circ})}{\alpha(U_{p,2n-\gamma_{\tilde{\pi}}(i)}^{\circ})}=\delta_{B}^{-1/2}(t_{p,s})\cdot\eta_{0}(p)^{s}.$
(5.4)

If $\pi_{p}$ is regular, then $\gamma_{\tilde{\pi}}$ is the unique map
$\\{1,...,n\\}\hookrightarrow\\{1,...,2n\\}$ with this property.

###### Proof.

The direct analogue of (5.3) with normalised eigenvalues is

$\alpha(U_{p,s}^{\circ})\cdot\prod_{i=1}^{s}p^{2\gamma_{\tilde{\pi}}(i)-2n-1}\cdot
p^{-\lambda_{2n+1-\gamma_{\tilde{\pi}}(i)}}\cdot\frac{\alpha(U_{p,2n+1-\gamma_{\tilde{\pi}}(i)}^{\circ})}{\alpha(U_{p,2n-\gamma_{\tilde{\pi}}(i)}^{\circ})}=p^{\lambda_{1}+\cdots+\lambda_{s}}\cdot\delta_{B}^{-1/2}(t_{p,s})\cdot\eta_{p}(p)^{s}.$

To get the stated form, we use that
$\lambda_{\gamma_{\tilde{\pi}}(i)}+\lambda_{2n+1-\gamma_{\tilde{\pi}}(i)}={\sf
w}$ and $\eta_{p}(p)=\eta_{0}(p)p^{-{\sf w}}$. ∎

#### 5.2. Zariski-density of $p$-refined spherical points

In our proofs of Theorems 4.9 and 4.10, we will require a Zariski-dense set of
classical points with good properties. This is furnished by the following.
Note we do _not_ require RASCARs here, only RACARs.

######  Proposition 5.5.

Let $\mathscr{C}\subset\mathscr{L}_{K}^{G}$ be a classical family containing a
classical point corresponding to a $p$-refined RACAR that is spherical and
regular at $p$. Then $\mathscr{C}$ contains a Zariski-dense set of classical
points corresponding to $p$-refined RACARs that are spherical and regular at
$p$.

###### Proof.

Any classical point $y\in\mathscr{C}$ corresponds to an eigensystem
$\alpha_{y}$ appearing in a RACAR $\pi_{y}$ such that $\pi_{y,p}$ is Iwahori-
spherical (admits non-zero Iwahori-invariants). By [Cas80, Prop. 2.6], any
such $\pi_{y,p}$ is a $\operatorname{GL}_{2n}(\mathbf{Q}_{p})$-submodule of an
unramified principal series representation
$\operatorname{Ind}_{B}^{G}\theta_{y}$, for an unramified character
$\theta_{y}=(\theta_{y,1},...,\theta_{y,2n})$. First we prove that
$\operatorname{Ind}_{B}^{G}\theta_{y}$ is irreducible for a Zariski-dense set
of $y\in\mathscr{C}$, as then $\pi_{y,p}=\operatorname{Ind}_{B}^{G}\theta_{y}$
is spherical.

For convenience, drop the subscript $y$. Let
$\sigma=\Psi_{\theta}(\tilde{\pi})$; without loss of generality, replace
$\theta$ with $\theta^{\sigma}$ and assume $\sigma=\mathrm{id}$. By [BZ77,
Thm. 4.2], $\operatorname{Ind}_{B}^{G}\theta$ is reducible if and only if
there exist $r,s$ such that $\theta_{r}=\theta_{s}|\cdot|$. As the
$\theta_{i}$ are unramified, this happens if and only if
$p\cdot\theta_{r}(p)=\theta_{s}(p).$ Using Lemma 5.2 with $\sigma=1$, this is
equivalent to

$p\cdot p^{s-r}\cdot
p^{\lambda_{s}-\lambda_{r}}\cdot\alpha(U_{p,r}^{\circ})\cdot\alpha(U_{p,s-1}^{\circ})=\alpha(U_{p,s}^{\circ})\cdot\alpha(U_{p,r-1}^{\circ}).$
(5.5)

Since the $\alpha(U_{p,i}^{\circ})$ are all analytic and non-zero on
$\mathscr{C}$, the locus $\mathscr{C}_{r,s}$ in $\mathscr{C}$ where (5.5) is
satisfied is a Zariski-closed subspace (with weight support only over subsets
where $\lambda_{r}-\lambda_{s}$ is constant). However, by assumption
$\mathscr{C}$ contains a $p$-refined spherical point, so
$\mathscr{C}_{r,s}\neq\mathscr{C}$, whence
$\mathscr{C}_{r,s}\subset\mathscr{C}$ is a proper subspace of smaller
dimension.

Any classical point $y$ where $\operatorname{Ind}_{B}^{G}\theta_{y}$ is
reducible must live in $\bigcup_{r\neq s}\mathscr{C}_{r,s}$. Since there are
only finitely many possible pairs $(r,s)$, this union is a proper subspace of
$\mathscr{C}$ of smaller dimension. It follows that
$\operatorname{Ind}_{B}^{G}\theta_{y}$ is irreducible for a Zariski-dense set
of $y$, and each of these $y$ corresponds to a $p$-refined $p$-spherical
RACAR.

It remains to check a Zariski-dense subset of these $y$ are regular. Note such
a $y$ is not regular, then there exist $r\neq s$ such that
$\theta_{r}(p)=\theta_{s}(p)$. As above, this happens if and only if

$p^{s-r}\cdot
p^{\lambda_{s}-\lambda_{r}}\cdot\alpha(U_{p,r}^{\circ})\cdot\alpha(U_{p,s-1}^{\circ})=\alpha(U_{p,s}^{\circ})\cdot\alpha(U_{p,r-1}^{\circ}),$

again cutting out a closed subspace in $\mathscr{C}$. We conclude that there
are a Zariski-dense set of $p$-regular points as before. ∎

######  Remark 5.6.

In any positive-dimensional component of $\mathscr{C}_{r,s}$ we must have
$\lambda_{r}-\lambda_{s}$ constant. It follows that any everywhere-ramified
family must vary over some parabolic weight space
$\mathscr{W}_{0,\lambda}^{P}$ for some non-minimal $B\subsetneq P\subset G$.
In particular, we recover that any classical family over the full pure weight
space $\mathscr{W}_{0}$ contains a Zariski-dense set of spherical points.

#### 5.3. Proof of Theorem 4.9

Let $\tilde{\pi}$ be an optimally $P$-spin $p$-refined RASCAR such that
$\pi_{p}$ is spherical and regular, and let $\mathscr{C}$ be a symplectic
family though $\tilde{\pi}$. To prove Theorem 4.9, we must show that
$w(\mathscr{C})\subset\mathscr{W}_{0,\lambda_{\pi}}^{P}$.

Let $\mathfrak{X}$ be the set of classical points in $\mathscr{C}$ that
correspond to $p$-refined RASCARs $\tilde{\pi}_{y}$ such that $\pi_{y,p}$ is
spherical and regular. By Proposition 5.5, the set $\mathfrak{X}$ is Zariski-
dense in $\mathscr{C}$.

For each $y\in\mathfrak{X}$, let
$\gamma_{y}:\\{1,...,n\\}\hookrightarrow\\{1,...,2n\\}$ be the function for
$\tilde{\pi}_{y}$ from Definition 3.14.

######  Lemma 5.7.

The function $\gamma_{y}$ is constant as $y$ varies in $\mathfrak{X}$.

###### Proof.

There are only finitely many functions
$\gamma:\\{1,...,n\\}\hookrightarrow\\{1,...,2n\\}$, so there must exist such
a function $\gamma$ and a Zariski dense subset
$\mathfrak{Y}\subset\mathfrak{X}\subset\mathscr{C}$ such that
$\gamma_{z}=\gamma$ for all $z\in\mathfrak{Y}$.

By Proposition 5.4, at every $y$ in $\mathfrak{Y}$, the Hecke relations

$\alpha_{y}(U_{p,s}^{\circ})\cdot\prod_{i=1}^{s}p^{2\gamma(i)-2n-1}\cdot
p^{\lambda_{y,\gamma(i)}-\lambda_{y,i}}\cdot\frac{\alpha_{y}(U_{p,2n+1-\gamma(i)}^{\circ})}{\alpha_{y}(U_{p,2n-\gamma(i)}^{\circ})}=\delta_{B}^{-1/2}(t_{p,s})\cdot\eta_{0}(p)^{s}$
(5.6)

are satisfied for all $1\leqslant s\leqslant n$, where $w(y)=\lambda_{y}$.
Since $U_{p,r}^{\circ}$ defines an analytic function on $\mathscr{C}$, and
these relations hold for the Zariski-dense $\mathfrak{Y}$, they hold over all
of $\mathscr{C}$. In particular, they hold at every point $y\in\mathfrak{X}$.
Since the points in $\mathfrak{X}$ are regular, the unicity statement in
Proposition 5.4 says $\gamma_{y}=\gamma$ for all $y\in\mathfrak{X}$. ∎

######  Lemma 5.8.

Every point $y\in\mathfrak{X}$ is optimally $P$-spin.

###### Proof.

Let $y\in\mathfrak{X}$, and let $P_{y}$ be the unique spin parabolic such that
$\tilde{\pi}_{y}$ is optimally $P_{y}$-spin. By Proposition 3.17, $P_{y}$ is
determined by the function $\gamma_{y}$. By Lemma 5.7, the function
$\gamma_{y}$ is constant over $\mathfrak{X}$; thus $P_{y}$ is also constant
over $\mathfrak{X}$. But $\mathfrak{X}$ contains $\tilde{\pi}$, which by
assumption is optimally $P$-spin. Thus $P_{y}=P$ for all $y\in\mathfrak{X}$. ∎

######  Lemma 5.9.

For $1\leqslant i\leqslant n$, if $a_{i}\in\Delta_{P}$, then
$\lambda_{y,i}-\lambda_{y,i+1}$ is constant as $y$ varies in $\mathfrak{X}$.

###### Proof.

Let $\gamma$ be the function from the proof of Lemma 5.7. We showed that the
relation (5.6) holds over all of $\mathscr{C}$, and for all $1\leqslant
s\leqslant n$. As the $\alpha_{y}(U_{p,r}^{\circ})$ vary analytically with
$y$, for this to be true for all $s$, the term
$p^{\lambda_{y,\gamma(i)}-\lambda_{i}}$ must be constant for all $1\leqslant
i\leqslant n$. This forces $\lambda_{y,\gamma(i)}-\lambda_{y,i}$ to be
constant.

Now, suppose $a_{i}\in\Delta_{P}$. Then $i\not\in X_{P}$. Now, since the
points of $\mathfrak{X}$ are optimally $P$-spin, by Proposition 3.17 we know
that $\gamma$ does not preserve $\\{1,...,i\\}$. In particular, there exists
some $m\in\\{1,...,i\\}$ such that $\gamma(m)>i$. Also, by dominance, we have
$\lambda_{m}\geqslant\lambda_{i}\geqslant\lambda_{i+1}\geqslant\lambda_{\gamma(m)}$.
Thus if $\lambda_{y,\gamma(m)}-\lambda_{y,m}$ is constant, as $y$ varies over
$\mathfrak{X}$, then so is $\lambda_{y,i}-\lambda_{y,i+1}$. ∎

Finally we prove Theorem 4.9. If $a_{i}\in\Delta_{P}$, then either:

  * (1)

$1\leqslant i\leqslant n$. Lemma 5.9, and Zariski-density of $\mathfrak{X}$,
imply $\lambda_{i}-\lambda_{i+1}$ is constant over $w(\mathscr{C})$.

  * (2)

or $n+1\leqslant i\leqslant 2n-1$; then $1\leqslant 2n-i\leqslant n$. As $P$
is a spin parabolic $a_{i}\in\Delta_{P}$ if and only if
$a_{2n-i}\in\Delta_{P}$, so by (1) $\lambda_{2n-i}-\lambda_{2n-i+1}$ is
constant. As $w(\mathscr{C})$ is in the pure weight space, this implies
$\lambda_{i}-\lambda_{i+1}$ is constant.

By Lemma 4.7, this means that
$w(\mathscr{C})\subset\mathscr{W}_{0,\lambda_{\pi}}^{P}$, as claimed. ∎

### 6\. Existence of $P$-spin families

We have obtained an upper bound on the dimension of symplectic families. We
now prove Theorem 4.10, constructing families realising this bound through
non-critical slope refinements.

#### 6.1. $B$-spin families

Let $\pi$ be a RASCAR of regular weight that is spherical and regular at $p$.
Let $K_{1}(\tilde{\pi})$ be as before Theorem 4.10. In [BDW] and [BDG+] we
proved:

######  Theorem 6.1.

Let $\tilde{\pi}$ be a non-critical $B$-spin refinement. There is a unique
family $\mathscr{C}$ through $\tilde{\pi}$ in
$\mathscr{E}^{G}_{K_{1}(\tilde{\pi})}$ that varies over the pure weight space
$\mathscr{W}_{0}$. Moreover $\mathscr{C}$ is an $(n+1)$-dimensional classical
symplectic family étale over $\mathscr{W}_{0}$ at $\tilde{\pi}$.

###### Proof.

When $K_{1}(\pi)=G(\widehat{\mathbf{Z}})$, this is exactly [BDG+, Thm. 13.6].
One can treat general $K_{1}(\pi)$ following exactly the strategy of [BDW,
§7.5,7.6]. ∎

######  Lemma 6.2.

We may shrink $\mathscr{C}$ so that every classical point $y\in V$ corresponds
to a $B$-spin $p$-refined RASCAR $\tilde{\pi}_{y}$ such that
$\pi_{y,p}=\operatorname{Ind}_{B}^{G}\theta_{y}$ is a regular and spherical,
with $\Psi_{\theta_{y}}(\tilde{\pi}_{y})=\Psi_{\theta}(\tilde{\pi})$.

In other words: ‘each classical point is a $p$-refined $p$-spherical RASCAR,
and for each such point, and all the refinements are in the same position in
the Weyl group.’

###### Proof.

By Proposition 5.5 and its proof, all the classical points corresponding to
RACARs that are ramified at $p$ live inside a proper closed subspace of the
eigenvariety, and since $x$ is not in this closed subspace, we can shrink the
neighbourhood $\mathscr{C}$ to avoid it completely. Then every classical $y$
is unramified principal series at $p$.

In this $\mathscr{C}$, every $y$ is (optimally) $B$-spin by Lemmas 5.7 and
5.8; so $\Psi_{\theta_{y}}(\tilde{\pi}_{y})\in\mathcal{W}_{G}^{0}$. By Remarks
2.11 and 3.3, we can thus conjugate $\theta_{y}$ so that
$\Psi_{\theta_{y}}(\tilde{\pi}_{y})=\Psi_{\theta}(\tilde{\pi})$. ∎

#### 6.2. Refinement-switching

To produce $P$-spin families, we take the part of the $B$-spin family
supported over the $P$-parahoric weight space, and systematically switch
between refinements for each classical point in the family. For
$\operatorname{GL}_{4}$, this is pictorially represented in the figure right,
and we shall now explain the notation.

To enact this strategy, we need to able to pass between optimally $P$-spin and
optimally $B$-spin refinements, and to relate eigenvalues as we do so. Recall
the notion of being $r$-spin from Definition 3.10, and $X$-spin from
Definition 3.11. The following lemma shows you can always ‘improve’ the spin-
ness with a controlled transposition.

######  Lemma 6.3.

Suppose $\tilde{\pi}$ is optimally $X$-spin, for $X\subset\\{1,...,n\\}$.

  1. (i)

Let $1\leqslant i\leqslant n-1$, and suppose: (a) $(i-1)\in X$ or $i=1$, and
(b) $i\notin X$. Let

$k\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left\\{\begin{array}[]{cl}2n-i&:i-1\text{
is maximal in }X,\\\ \mathrm{min}\\{i^{\prime}\in
X:i^{\prime}>i-1\\}&:\text{else}.\end{array}\right.$

Then there exists $i+1\leqslant j\leqslant k$ such that the $p$-refinement
$\tilde{\pi}^{\prime}$ with

$\Psi_{\theta}(\tilde{\pi}^{\prime})=\Psi_{\theta}(\tilde{\pi})\cdot(i,j)$

is $X\cup\\{i\\}$-spin.

  2. (ii)

If $\tilde{\pi}$ is $(n-1)$-spin, then it is $n$-spin (i.e. if $n-1\in X$,
then $n\in X$).

###### Proof.

(i) Let $\sigma=\Psi_{\theta}(\tilde{\pi})$, and let $j$ be the unique integer
such that $\sigma(j)+\sigma(2n+1-i)=2n+1$.

Step 1: Inequalities on $j$. For any $r\in X$, since $\tilde{\pi}$ is
$r$-spin, the sets $\\{\sigma(1),...,\sigma(r)\\}$ and
$\\{\sigma(2n+1-r),...,\sigma(2n)\\}$ pair off so that the sum of each pair is
$2n+1$. In particular, $\sigma(j)$ is in one of these two sets $\iff$
$\sigma(2n+1-i)$ is in the other. Then:

  * •

Apply $(\dagger)$ with $r=i-1$. As
$\sigma(2n+1-i)\not\in\\{\sigma(2n+2-i),...,\sigma(2n)\\}$, we know
$\sigma(j)\notin\\{\sigma(1),...,\sigma(i-1)\\}$. So
$j\not\in\\{1,...,i-1\\}$, i.e. $i\leqslant j$.

  * •

As $\tilde{\pi}$ is $(i-1)$-spin but _not_ $i$-spin,
$\sigma(i)+\sigma(2n+1-i)\neq 2n+1$, so $j\neq i$; hence $i+1\leqslant j$.

  * •

As $i\leqslant n-1$, we have
$\sigma(2n+1-i)\not\in\\{\sigma(1),...,\sigma(i-1)\\}$, so $j\leqslant
2n+1-i$. But $j\neq 2n+1-i$ clearly, so $j\leqslant 2n-i$ (always).

  * •

If $i-1$ is maximal in $X$, then $k=2n-i$ and we are done. Otherwise $k$ is
the next smallest element of $X$; as $i<k$ and $\sigma$ is $k$-spin, we have
$\sigma(2n+1-i)\in\\{\sigma(2n+1-k),...,\sigma(2n)\\}$, so $(\dagger)$ implies
$j\leqslant k$.

Step 2: $\tilde{\pi}^{\prime}$ is $X$-spin. Now, let $\zeta=(i,j)$. If $r\in
X$, then either we have

$r<i\text{ and }j\leqslant r<2n+1-r,\qquad\text{ or }\qquad\text{both
}i,j\leqslant r.$

Either way, $\zeta$ preserves $\\{1,...,r\\}$ and $\\{2n+1-r,...,2n\\}$. In
particular, we have

$\displaystyle\\{\sigma(1),...,\sigma(r)\\}$
$\displaystyle=\\{\sigma\zeta(1),...,\sigma\zeta(r)\\},$
$\displaystyle\\{\sigma(2n+1-r),...,\sigma(2n)\\}$
$\displaystyle=\\{\sigma\zeta(2n+1-r),...,\sigma\zeta(2n)\\},$

so $\sigma\zeta$ is $r$-spin since $\sigma$ is. Since this is true of all
$r\in X$, we conclude $\sigma\zeta=\Psi_{\theta}(\tilde{\pi}^{\prime})$ is
$X$-spin.

Step 3: $\tilde{\pi}^{\prime}$ is $X\cup\\{i\\}$-spin. By above, $\sigma\zeta$
is $(i-1)$-spin. Moreover, by construction
$\sigma\zeta(i)+\sigma\zeta(2n+1-i)=2n+1$, so additionally $\sigma\zeta$ is
$i$-spin. As it is $X$-spin and $i$-spin,
$\sigma\zeta=\Psi_{\theta}(\tilde{\pi}^{\prime})$ is $X\cup\\{i\\}$-spin, as
claimed.

(ii) If $\tilde{\pi}$ is $(n-1)$-spin, then by definition, for each
$r\leqslant n-1$, there is $s\geqslant n+2$ such that
$\sigma(r)+\sigma(s)=2n+1$. This accounts for $n-1$ of the $n$ pairs with this
property, and forces $\sigma(n)+\sigma(n+1)=2n+1$ to be the $n$th and last.
Thus $\tilde{\pi}$ is also $n$-spin. ∎

We now relate the Hecke eigenvalues of $\tilde{\pi}$ and
$\tilde{\pi}^{\prime}$ from the previous lemma. Recall that by Proposition
2.10, since $\theta_{i}(p)\neq 0$ for all $i$, $\alpha$ is finite slope, i.e.
$\alpha(U_{p,i}^{\circ})\neq 0$ for all $i$.

######  Lemma 6.4.

Let $\tilde{\pi}=(\pi,\alpha)$ and
$\tilde{\pi}^{\prime}=(\pi,\alpha^{\prime})$ be two $p$-refinements, with

$\Psi_{\theta}(\tilde{\pi}^{\prime})=\Psi_{\theta}(\tilde{\pi})\cdot(i,j),$

where $(i,j)\in\mathrm{S}_{2n}$ is a transposition with $i<j$. Then for all
$r$,

$\alpha^{\prime}(U_{p,r}^{\circ})=\left\\{\begin{array}[]{cc}p^{i-j}p^{\lambda_{i}-\lambda_{j}}\frac{\alpha(U_{p,j}^{\circ})}{\alpha(U_{p,j-1}^{\circ})}\cdot\frac{\alpha(U_{p,i-1}^{\circ})}{\alpha(U_{p,i}^{\circ})}\cdot\alpha(U_{p,r}^{\circ})&:i\leqslant
r<j\\\ \alpha(U_{p,r}^{\circ})&:\text{otherwise},\end{array}\right.$

where $\pi$ has weight $\lambda=(\lambda_{1},...,\lambda_{2n})$ and we use the
shorthand that “$\alpha(U_{p,0}^{\circ})$” $:=1$.

###### Proof.

Let $\sigma=\Psi_{\theta}(\tilde{\pi})$. By Proposition 2.10 the definition of
$U_{p,r}^{\circ}$ we have

$\alpha(U_{p,r}^{\circ})=\delta_{B}^{-1/2}(t_{p,r})\cdot
p^{\lambda_{1}+\cdots+\lambda_{r}}\cdot\theta_{\sigma(1)}(p)\cdots\theta_{\sigma(r)}(p).$

Now $\alpha^{\prime}(U_{p,r}^{\circ})$ can be described in the same way,
except with $\sigma$ replaced with $\sigma(i,j)$. When $r<i$ or $r\geqslant
j$, this is identical to $\alpha(U_{p,r}^{\circ})$; when $i\leqslant r<j$,
this means $\theta_{\sigma(i)}(p)$ is replaced by
$\theta_{[\sigma(i,j)](i)}(p)=\theta_{\sigma(j)}(p)$ in the product. Via Lemma
5.2, in this case

$\displaystyle\alpha^{\prime}(U_{p,r}^{\circ})$
$\displaystyle=\alpha(U_{p,r}^{\circ})\cdot\theta_{\sigma(j)}(p)\cdot\theta_{\sigma(i)}(p)^{-1}$
$\displaystyle=\alpha(U_{p,r}^{\circ})\cdot\left[p^{-\lambda_{j}}p^{(2n+1-2j)/2}\frac{\alpha(U_{p,j}^{\circ})}{\alpha(U_{p,j-1}^{\circ})}\right]\cdot\left[p^{-\lambda_{i}}p^{(2n+1-2i)/2}\frac{\alpha(U_{p,i}^{\circ})}{\alpha(U_{p,i-1}^{\circ})}\right]^{-1},$

which simplifies to the claimed expression. ∎

We will use Lemma 6.4 to define maps between families on the eigenvariety.
This requires adding inverses to the Hecke algebra.

######  Definition 6.5.

Let
$\mathcal{H}^{\mathrm{frac}}=\mathcal{H}_{p}^{\mathrm{frac}}\cdot\mathcal{H}^{p}$,
where

$\mathcal{H}_{p}^{\mathrm{frac}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbf{Q}_{p}[U_{p,r}^{\circ},(U_{p,r}^{\circ})^{-1}:1\leqslant
r\leqslant 2n].$

Now fix $K=K_{1}(\tilde{\pi})$ from before Theorem 4.10. Let
$\mathscr{E}=\mathscr{E}_{K}^{G}$ from Theorem 4.1, defined by the action of
$\mathcal{H}$ on overconvergent cohomology. Let also
$\mathscr{E}^{\prime}=\mathscr{E}^{\prime}_{K}$ be the eigenvariety defined by
the same eigenvariety datum, but using instead the action of
$\mathcal{H}^{\mathrm{frac}}$ on the _finite-slope_ overconvergent cohomology.

######  Lemma 6.6.

We have $\mathscr{E}=\mathscr{E}^{\prime}$.

###### Proof.

Both eigenvarieties are defined by writing down local pieces
$\mathscr{E}_{\Omega,h}=\mathrm{Sp}(\mathbf{T}_{\Omega,h})$ and
$\mathscr{E}^{\prime}_{\Omega,h}=\mathrm{Sp}(\mathbf{T}_{\Omega,h}^{\prime})$,
where $\mathbf{T}_{\Omega,h}$ (resp. $\mathbf{T}_{\Omega,h}^{\prime}$) is the
image of $\mathcal{H}\otimes\mathcal{O}_{\Omega}$ (resp
$\mathcal{H}^{\mathrm{frac}}\otimes\mathcal{O}_{\Omega}$) in the
$\mathrm{End}_{\mathcal{O}_{\Omega}}(\mathrm{H}^{\bullet}_{\mathrm{c}}(S_{K},\mathscr{D}_{\Omega})^{\leqslant
h})$. As each $U_{p,r}^{\circ}$ acts invertibly on the slope $\leqslant h$
cohomology (see e.g. [Urb11, §2.3.1]), the image of $U_{p,r}^{\circ}$ in
$\mathbf{T}_{\Omega,h}$ is invertible; and hence
$\mathbf{T}_{\Omega,h}=\mathbf{T}_{\Omega,h}^{\prime}$, so
$\mathscr{E}_{\Omega,h}=\mathscr{E}_{\Omega,h^{\prime}}$.

Both $\mathscr{E}$ and $\mathscr{E}^{\prime}$ are defined by the same gluing
of the same local pieces, so they are equal. ∎

######  Definition 6.7.

For $\lambda=(\lambda_{1},...,\lambda_{2n})\in X^{*}(T)$, and $i<j$, define a
map

$\phi_{ij}^{\lambda}:\mathcal{H}\longrightarrow\mathcal{H}^{\mathrm{frac}}$

to be the identity map on all operators away from $p$, and at $p$ by

$\phi_{ij}^{\lambda}(U_{p,r}^{\circ})=\left\\{\begin{array}[]{cc}p^{i-j}p^{\lambda_{i}-\lambda_{j}}\frac{U_{p,j}^{\circ}}{U_{p,j-1}^{\circ}}\cdot\frac{U_{p,i-1}^{\circ}}{U_{p,i}^{\circ}}\cdot
U_{p,r}^{\circ}&:i\leqslant r<j\\\
U_{p,r}^{\circ}&:\text{otherwise},\end{array}\right.$

######  Lemma 6.8.

Let $\pi$ have weight $\lambda_{\pi}$, and let $\tilde{\pi}=(\pi,\alpha)$ and
$\tilde{\pi}^{\prime}=(\pi,\alpha^{\prime})$ be $p$-refinements with

$\Psi_{\theta}(\tilde{\pi}^{\prime})=\Psi_{\theta}(\tilde{\pi})\cdot(i,j)$

as elements of $\mathcal{W}_{G}$. Then
$\alpha^{\prime}=\alpha\circ\phi_{ij}^{\lambda_{\pi}}$ and
$\alpha^{\prime}\circ\phi_{ij}^{\lambda_{\pi}}=\alpha$.

###### Proof.

Note also
$\Psi_{\theta}(\tilde{\pi}^{\prime})\cdot(i,j)=\Psi_{\theta}(\tilde{\pi})$.
Both statements are then direct from Lemma 6.4. ∎

#### 6.3. From $P$-spin to $B$-spin

Let $\tilde{\pi}=(\pi,\alpha)$ be an optimally $P$-spin non-critical slope
refinement.

######  Proposition 6.9.

  * (i)

There exists an element
$\tau=(i_{1},j_{1})\cdots(i_{k},j_{k})\in\mathcal{W}_{G}$, where $k\leqslant
n-\\#X_{P}$, and a $B$-spin $p$-refinement
$\tilde{\pi}^{\prime}=(\pi,\alpha^{\prime})$ with

$\Psi_{\theta}(\tilde{\pi}^{\prime})=\Psi_{\theta}(\tilde{\pi})\cdot\tau.$

  * (ii)

The refinement $\tilde{\pi}^{\prime}$ from (i) has non-critical slope.

  * (iii)

We have $\alpha^{\prime}\circ\phi_{\tau}^{\lambda_{\pi}}=\alpha$, where for
any classical $\lambda$ we let

$\phi_{\tau}^{\lambda}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\phi_{i_{k},j_{k}}^{\lambda}\circ\cdots\circ\phi_{i_{1},j_{1}}^{\lambda}:\mathcal{H}\longrightarrow\mathcal{H}^{\mathrm{frac}}.$

  * (iv)

We have $\phi_{\tau}^{\lambda}=\phi_{\tau}^{\lambda_{\pi}}$ for any classical
$\lambda\in\mathcal{W}_{\lambda_{\pi}}^{P}$.

###### Proof.

(i) We iterate Lemma 6.3. Let $X_{P}=\\{I_{1},...,I_{\\#X_{P}}\\}$. Let
$1\leqslant i_{1}\leqslant n$ be minimal with $i_{1}\not\in X_{P}$. Then there
exists some $r$ such that $I_{r}<i_{1}<I_{r+1}$ (where
$I_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=0$ and
$I_{\\#X_{P}+1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=2n-I_{\\#X_{P}}$).
By Lemma 6.3, there exists $I_{r}<i_{1}<j_{1}\leqslant I_{r+1}$ and an
$(X_{P}\cup\\{i_{1}\\})$-spin $\tilde{\pi}^{(1)}$ satisfying

$\Psi_{\theta}(\tilde{\pi}^{(1)})=\Psi_{\theta}(\tilde{\pi})\cdot(i_{1},j_{1}).$

Iterating this process $k\leqslant n-\\#X_{P}$ times, we obtain a
$p$-refinement $\tilde{\pi}^{\prime}=\tilde{\pi}^{(k)}$ which is
$\\{1,...,n\\}$-spin with
$\Psi_{\theta}(\tilde{\pi}^{\prime})=\Psi_{\theta}(\tilde{\pi})\cdot(i_{1},j_{i})\cdots(i_{k},j_{k})$.
By Proposition 3.12 $\tilde{\pi}^{\prime}$ is $B$-spin.

(ii) From Definition 2.13, $\tilde{\pi}^{\prime}$ has non-critical slope if

$v_{p}(\alpha^{\prime}(U_{p,i}^{\circ}))<\lambda_{i}-\lambda_{i+1}+1,\qquad
1\leqslant i\leqslant 2n-1$ (6.1)

By assumption this is true for $\alpha$. To see it for $\alpha^{\prime}$:

  * (ii-1)

If $i\geqslant n$: by the proof of [Roc, Thm. 4] (more precisely, the sentence
following the second displayed equation), as $\tilde{\pi}$ has non-critical
slope, it is $n$-spin. In particular, $n\in X_{P}$. By construction this
forces $1\leqslant i_{r},j_{r}\leqslant n$ for all $n$. By Proposition 2.10,
we see $\alpha^{\prime}(U_{p,i}^{\circ})=\alpha(U_{p,i}^{\circ})$. So
$\alpha^{\prime}(U_{p,i}^{\circ})$ is non-critical slope as
$\alpha(U_{p,i}^{\circ})$ is.

  * (ii-2)

If $i<n$: as $\tilde{\pi}^{\prime}$ is $i$-spin, we have
$v_{p}(\alpha^{\prime}(U_{p,i}^{\circ}))=v_{p}(\alpha^{\prime}(U_{p,2n-i}^{\circ}))$
by Lemma 5.1. This is non-critical slope by (ii-1).

(iii) Follows from iterating Lemma 6.8.

(iv) By Lemma 4.7, we know $\lambda_{i}-\lambda_{i+1}$ is constant in
$\mathcal{W}_{\lambda_{\pi}}^{P}$ whenever $i\not\in X_{P}$. In the map
$\phi_{i_{r},j_{r}}^{\lambda}$, the only dependence on $\lambda$ is in the
term

$p^{\lambda_{i_{r}}-\lambda_{j_{r}}}=p^{\lambda_{i_{r}}-\lambda_{i_{r}+1}}\cdots
p^{\lambda_{j_{r}-1}-\lambda_{j_{r}}}.$ (6.2)

By construction, we know that $I_{s}<i_{r}\leqslant j_{r}\leqslant I_{s+1}$
fall between two adjacent elements of $X_{P}$, so that
$i_{r},i_{r}+1,...,j_{r}-1\not\in X_{P}$. Thus all of the terms in the product
(6.2) are constant as $\lambda$ varies in $\mathcal{W}_{\lambda_{\pi}}^{P}$.
The result follows. ∎

#### 6.4. From $B$-spin to $P$-spin

Let $\tilde{\pi}$ and $\tilde{\pi}^{\prime}$ be as in Proposition 6.9. By
Theorem 6.1 and Lemma 6.6, there exists a unique $n+1$-dimensional symplectic
family $\mathscr{C}^{\prime}\subset\mathscr{E}^{\prime}$ through
$\tilde{\pi}^{\prime}$. Assume $\mathscr{C}^{\prime}$ is as in Lemma 6.2, and
let

$\mathscr{C}^{\prime}_{P}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathscr{C}^{\prime}\times_{\mathscr{W}}\mathscr{W}_{0,\lambda_{\pi}}^{P}$

be the $(\\#X_{P}+1)$-dimensional subspace varying only over
$\mathscr{W}_{0,\lambda_{\pi}}^{P}$. By Lemma 6.2, every classical point
$y^{\prime}\in\mathscr{C}^{\prime}_{P}$ corresponds to a $p$-refined
$\tilde{\pi}_{y}^{\prime}=(\pi_{y},\alpha_{y}^{\prime})$ with
$\pi_{y,p}=\operatorname{Ind}_{B}^{G}\theta_{y}$ spherical and regular. Let
$\tilde{\pi}_{y}=(\pi_{y},\alpha_{y})$ be the unique $p$-refinement with

$\Psi_{\theta_{y}}(\tilde{\pi}_{y}^{\prime})=\Psi_{\theta_{y}}(\tilde{\pi}_{y})\cdot\tau,$

for $\tau$ as in Proposition 6.9.

######  Lemma 6.10.

The refinement $\tilde{\pi}_{y}$ is optimally $P$-spin and we have

$\alpha_{y}^{\prime}\circ\phi_{\tau}^{\lambda_{\pi}}=\alpha_{y}.$ (6.3)

###### Proof.

By Lemma 6.2, we know
$\Psi_{\theta_{y}}(\tilde{\pi}_{y}^{\prime})=\Psi_{\theta}(\tilde{\pi}^{\prime})$.
In particular, we have

$\Psi_{\theta_{y}}(\tilde{\pi}_{y})=\Psi_{\theta_{y}}(\tilde{\pi}_{y}^{\prime})\cdot\tau^{-1}=\Psi_{\theta}(\tilde{\pi}^{\prime})\cdot\tau^{-1}=\Psi_{\theta}(\tilde{\pi}),$

so that $\tilde{\pi}_{y}$ is optimally $P$-spin. The identity (6.3) follows by
iterating Lemma 6.8 as in Proposition 6.9(iii). Here we use (iv) of that
result to see $\phi_{\tau}^{\lambda}=\phi_{\tau}^{\lambda_{\pi}}$. ∎

######  Lemma 6.11.

For a Zariski-dense set of classical $y^{\prime}\in\mathscr{C}^{\prime}_{P}$,
the $p$-refinement $\tilde{\pi}_{y}$ is non-critical slope, and thus
corresponds to a classical $P$-spin point $y\in\mathscr{E}$.

###### Proof.

Up to shrinking $\mathscr{C}^{\prime}_{P}$, we may assume that the slope of
each $U_{p,i}^{\circ}$ is constant along $\mathscr{C}^{\prime}_{P}$. As
$\phi_{\tau}^{\lambda_{\pi}}(U_{p,i}^{\circ})$ is a product of
$U_{p,i}^{\circ}$’s and terms constant over
$\mathscr{W}_{0,\lambda_{\pi}}^{P}$, the slope of
$\alpha_{y}(U_{p,i}^{\circ})=\alpha_{y}^{\prime}\circ\phi_{\tau}^{\lambda_{\pi}}(U_{p,i}^{\circ})$
is constant, equal to $v_{p}(\alpha(U_{p,i}^{\circ}))$, for all $i$ and for
any classical $y^{\prime}\in\mathscr{C}^{\prime}_{P}$.

By assumption $\tilde{\pi}$ is non-critical slope (for $\lambda_{\pi}$). For a
Zariski-dense set of classical weights $\lambda_{y}\in
w(\mathscr{C}^{\prime}_{P})$, the non-critical slope condition (6.1) for
$\lambda_{y}$ is strictly weaker than for $\lambda_{\pi}$; so above all such
weights, the points $\tilde{\pi}_{y}$ are non-critical slope. ∎

#### 6.5. Proof of Theorem 4.10

Let us take stock. We started with a non-critical slope $P$-spin refinement
$\tilde{\pi}$, and via an element $\tau$ in the Weyl group, associated to it a
non-critical slope $B$-spin refinement $\tilde{\pi}^{\prime}$. This varies in
a unique $(n+1)$-dimensional family
$\mathscr{C}^{\prime}\subset\mathscr{E}^{\prime}=\mathscr{E}$. Applying
$\tau^{-1}$ to each $p$-refined classical point $y^{\prime}$ in
$\mathscr{C}_{P}^{\prime}$ gives another $P$-spin point $y\in\mathscr{E}$. We
now show this association can be interpolated over
$\mathscr{W}_{0,\lambda_{\pi}}^{P}$.

######  Proposition 6.12.

There exists a finite map $t:\mathscr{C}^{\prime}_{P}\to\mathscr{E}$ over
$\mathscr{W}_{0,\lambda_{\pi}}^{P}$ which interpolates the association
$y^{\prime}\mapsto y$. Thus there exists an $(\\#X_{P}+1)$-dimensional
symplectic family through $\tilde{\pi}$.

###### Proof.

We use an interpolation idea that originally dates back to Chenevier [Che05].
The precise version we use is [JN19, Thm. 3.2.1], which says: suppose we have
eigenvariety data $\mathcal{D}_{1},\mathcal{D}_{2}$, using Hecke algebras
$\mathcal{H}_{1},\mathcal{H}_{2}$, giving eigenvarieties
$\mathscr{E}_{1},\mathscr{E}_{2}$. Suppose there is a map
$\phi:\mathcal{H}_{2}\to\mathcal{H}_{1}$ and a Zariski-dense set of points
$y_{1}\in\mathscr{E}_{1}$ with $\alpha_{y_{1}}\circ\phi$ appearing as a point
$y_{2}\in\mathscr{E}_{2}$. Then there is a finite map
$\mathscr{E}_{1}\to\mathscr{E}_{2}$ interpolating the transfer $y_{1}\mapsto
y_{2}$. We need only explain why our situation fits this.

Let
$\Omega_{P}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=w(\mathscr{C}^{\prime}_{P})$.
The part of the eigenvariety $\mathscr{E}$ over $\Omega_{P}$ is constructed
from an eigenvariety datum

$\mathcal{D}_{2}=(\Omega_{P},\mathscr{Z},\mathscr{H},\mathcal{H},\psi)$

in the sense of [JN19, Def. 3.1.1]. Also [JN19, Cor. 3.1.5] allows us to
realise $\mathscr{C}_{P}^{\prime}$ inside the eigenvariety attached to an
eigenvariety datum

$\mathcal{D}_{1}=(\Omega_{P},\mathscr{Z}_{\mathscr{C}_{P}^{\prime}},\mathscr{H}^{\prime},\mathcal{H}^{\mathrm{frac}},\psi),$

where we shrink the weight space to be $P$-parabolic, and the Fredholm
hypersurface to isolate the component containing $\mathscr{C}_{P}^{\prime}$.

The map of Hecke algebras is
$\phi_{\tau}^{\lambda_{\pi}}:\mathcal{H}\to\mathcal{H}^{\mathrm{frac}}.$ For a
Zariski-dense set of classical $y^{\prime}\in\mathscr{C}_{P}^{\prime}$,
corresponding to eigensystems $\alpha^{\prime}$, by Lemma 6.11 the eigensystem
$\alpha^{\prime}\circ\phi_{\tau}^{\lambda_{\pi}}$ appears in $\mathscr{E}$,
and we deduce existence of $t$ by [JN19].

Now $t(\mathscr{C}_{P}^{\prime})\subset\mathscr{E}$ is the required symplectic
family through $\tilde{\pi}$. ∎

We have now proved existence of an $\\#X_{P}+1$-dimensional symplectic family
$\mathscr{C}$ through any non-critical slope $P$-spin point in
$\mathscr{E}_{K_{1}(\tilde{\pi})}^{G}$. Theorem 4.10 claims that this family
is unique and étale over its image in weight space, an affinoid $\Omega_{P}$
in $\mathscr{W}_{0,\lambda_{\pi}}^{P}$ (noting $P=P_{\tilde{\pi}}$).

To complete the proof, key is the observation that at level
$K_{1}(\tilde{\pi})$, with appropriate signs, the $\tilde{\pi}$-isotypic part
of the top-degree compactly-supported cohomology is 1-dimensional (as in e.g.
[BDW, Prop. 7.19]). Then as in Proposition 7.20 _op. cit_., there exists an
ideal $I\subset\mathcal{O}_{\Omega_{P}}$ such that we have a relation

$\mathcal{O}_{\mathscr{C},\tilde{\pi}}=\mathcal{O}_{\Omega_{P},\lambda_{\pi}}/I$

between the local rings. It suffices to prove $I=0$. If it is not, then every
component through $\tilde{\pi}$ has dimension $<\dim(\Omega_{P})=\\#X_{P}+1$.
But this contradicts the existence of $\mathscr{C}$, so $I=0$ and
$w:\mathscr{C}\to\Omega_{P}$ is étale at $\tilde{\pi}$. This completes the
proof of Theorem 4.10. ∎

######  Remark 6.13.

For $\operatorname{GL}_{2}$, the _infinite fern_ (see [GM98]) is the image of
the Coleman–Mazur eigenvariety in an unobstructed deformation space of
residual Galois representations. If $\pi$ is a $p$-spherical RACAR of
$\operatorname{GL}_{2}$, then there are two $p$-refinements
$\pi_{\alpha},\pi_{\beta}$, each varying in Coleman families; but both
$\pi_{\alpha},\pi_{\beta}$ have the same underlying Galois representation, so
have the same image in the infinite fern, and the images of their families in
the infinite fern cross at this point.

The proof here suggest that, given a hypothetical ‘infinite fern’
$\mathscr{I}$ for $\operatorname{GL}_{2n}$, there would be a picture with
higher-dimensional intersections. Consider e.g. $\operatorname{GL}_{4}$; then
the image of the $\operatorname{GL}_{4}$-eigenvariety in $\mathscr{I}$ through
$\pi$ should comprise 24 surfaces (the Iwahori families), intersecting at 6
lines (the $Q$-parahoric families), which all intersect at a single point
(corresponding to $\pi$). Our expectation is that 8 of the surfaces (through
the $B$-spin refinements) comprise classical points, and these intersect at 4
lines (corresponding to 4 classical families at $Q$-parahoric level).

A higher-dimensional ‘infinite fern’ for _polarised_ Galois representations of
$\operatorname{GL}_{n}$ over CM fields is the main focus of [HS].

### 7\. Explicit examples for $\operatorname{GL}_{4}$

We now illustrate the theory concretely for $\operatorname{GL}_{4}$, and give
an explicit example. There are 4 spin parabolics in $G$: $B$, the
(2,2)-parabolic $Q$, the (1,2,1)-parabolic $Q^{\prime}$, and $G$ itself.
Suppose $\pi$ is a RASCAR of $\operatorname{GL}_{4}$ with $\pi_{p}$ spherical,
the transfer of a RACAR $\Pi$ on $\mathrm{GSp}_{4}$, and let
$\mathcal{F}\in\Pi$ be a Siegel newform of level prime to $p$.

There are 6 $Q$-refinements of $\pi_{p}$ (Hecke eigensystems in the
$Q$-parahoric invariants of $\pi_{p}$), corresponding to elements of
$\mathcal{W}_{G}/\mathcal{W}_{L_{Q}}$. These are combinatorially represented
by decomposing $\\{1,2,3,4\\}$ into an ordered disjoint union $A_{1}\sqcup
A_{2}$, where $\\#A_{1}=\\#A_{2}=2$ (cf. [DJR20, §3.3]). Exactly four of these
are ‘$Q$-spin’, factoring through Klingen refinements of $\mathcal{F}$:

$\\{1,2\\}\sqcup\\{3,4\\},\ \ \ \\{1,3\\}\sqcup\\{2,4\\},\ \ \
\\{24\\}\sqcup\\{13\\},\ \ \ \\{34\\}\sqcup\\{12\\},$ (7.1)

whilst $\\{14\\}\sqcup\\{23\\}$ and $\\{23\\}\sqcup\\{14\\}$ do not factor.
These four are the refinements satisfying the combinatorial criterion [DJR20,
Def. 3.5(ii)].

There are 24 Iwahori $p$-refinements, each lying above a unique
$Q$-refinement. Each $Q$-refinement $A_{1}\sqcup A_{2}$ has 4 further Iwahori
refinements, corresponding to orderings on $A_{1}$ and $A_{2}$; e.g. above
$\\{1,2\\}\sqcup\\{3,4\\}$ are {1234}, {2134}, {1243}, {2143}. The table below
lists all the Iwahori $p$-refinements $\tilde{\pi}$, together with the
smallest parabolic $P\subset G$ such that $\tilde{\pi}$ is $P$-spin.

$\tilde{\pi}$ optimally: | $\Psi_{\theta}(\tilde{\pi})$
---|---
$B$-spin | {1234}, {1324}, {2143}, {2413}, {3142}, {3412}, {4231}, {4321}
$Q$-spin | {2134}, {3124}, {1243}, {4213}, {1342}, {4312}, {2431}, {3421}
$G$-spin | {2314}, {3214}, {1423}, {4123}, {1432}, {4132}, {2341}, {3241}

(Any $Q^{\prime}$-spin refinement is automatically a $B$-spin refinement by
Lemma 6.3(ii)). We conjecture that the dimension of the symplectic locus
through the optimally $B$-spin, $Q$-spin and $G$-spin refinements is 3, 2 and
1 respectively; we have proved this for non-critical slope $\tilde{\pi}$.

######  Example.

From the tables at www.smf.compositio.nl, there is a unique non-endoscopic
Siegel modular form $\mathcal{F}$ on $\mathrm{GSp}_{4}$ of level 1 that
transfers to a RASCAR $\pi$ on $\operatorname{GL}_{4}$ of weight
$\lambda=(12,1,-1,-12)$; and $\pi$ is everywhere spherical.

At $p=11$, by examining the Newton polygon, one sees this $\pi$ admits a
parahoric-ordinary $Q$-refinement $\tilde{\pi}^{Q}$, corresponding to an
ordinary Klingen refinement of $\mathcal{F}$. We can normalise $\theta$ so
that this $Q$-refinement is $\\{1,2\\}\sqcup\\{3,4\\}$.

The 4 Iwahori refinements above $\tilde{\pi}^{Q}$ are
$\\{1234\\},\\{2134\\},\\{1243\\},\\{2143\\}$. For $\lambda=(12,1,-1,-12)$,
the non-critical slope bounds (6.1) are $v_{p}(U_{p,1})<12$,
$v_{p}(U_{p,2})<3$, $v_{p}(U_{p,3})<12$. We see:

  * •

$\\{1234\\}$ is $B$-spin. Its $U_{p,i}$-eigenvalues have slopes
$v_{p}(U_{p,1})=v_{p}(U_{p,3})=11$ and $v_{p}(U_{p,2})=0$. This is non-
critical slope, varying in a unique 3-dimensional symplectic family.

  * •

$\\{2134\\}$ is optimally $Q$-spin. The slopes are $v_{p}(U_{p,1})=11$,
$v_{p}(U_{p,2})=0$, $v_{p}(U_{p,3})=1$. This is non-critical slope, varying in
a 2-dimensional symplectic family, inside a 3-dimensional component of the
eigenvariety.

Similarly $\\{1243\\}$ and $\\{2143\\}$ are non-critical slope, optimally
$Q$-spin and $B$-spin respectively.

## Part III. $p$-refined Friedberg–Jacquet Integrals

In Part III, we focus on parahoric $P$-refinements $\tilde{\pi}^{P}$. We give
a conjectural classification of the $P$-spin $P$-refinements in terms of non-
vanishing of twisted global period integrals, and prove various results
towards this by using the results of Part II. Our conjecture generalises
[BDG+, Expectation 7.2], which we prove in some cases.

### 8\. $p$-refined Friedberg–Jacquet integrals: Statements

#### 8.1. Statement of the conjecture

Let $\pi$ be a RACAR of $G(\mathbf{A})$. For $\varphi\in\pi$ and Hecke
characters $\chi,\eta$, let

$Z_{H}(\varphi,\chi,s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\mathbf{A}^{\times}H(\mathbf{Q})\backslash
H(\mathbf{A})}\varphi\left[\begin{pmatrix}h_{1}&\\\
&h_{2}\end{pmatrix}\right]\chi|\cdot|^{s-\tfrac{1}{2}}\left(\frac{\det(h_{1})}{\det(h_{2})}\right)\eta^{-1}\big{(}\det(h_{2})\big{)}dh,$
(8.1)

where $H=\operatorname{GL}_{n}\times\operatorname{GL}_{n}$. In [FJ93, Prop.
2.2] (with [AS14]) Friedberg–Jacquet proved:

######  Theorem 8.1.

Let $\pi$ be a RACAR of $G(\mathbf{A})$. Let $\chi,\eta$ be algebraic Hecke
characters, with $\chi$ finite order. Then for any $s\in\mathbf{C}$, the
following are equivalent:

  1. (1)

There exists $\varphi\in\pi$ such that $Z_{H}(\varphi,\chi,s+1/2)\neq 0$.

  2. (2)

All of the following hold:

     * –

$\pi$ is a functorial transfer of some $\Pi$ on
$\mathrm{GSpin}_{2n+1}(\mathbf{A})$ with central character $\eta$,

     * –

$L(\pi\times\chi,s+1/2)\neq 0$.

For our ‘$p$-refined’ version of Theorem 8.1, we need some notation. Let
$P\subset G$ be a spin parabolic. We define $P$-refinements of RACARs (not
just RASCARs) to be exactly as in Definition 2.8.

######  Notation 8.2.

  * –

Let $\tilde{\pi}^{P}=(\pi,\alpha^{P})$ be a $P$-refinement. We say
$\varphi\in\tilde{\pi}^{P}$ (resp. $\varphi_{p}\in\tilde{\pi}_{p}^{P}$) if
$\varphi\in\pi^{J_{P}}$ (resp. $\varphi_{p}\in\pi_{p}^{J_{P}}$) is an
$\alpha^{P}$-eigenvector for $\mathcal{H}_{p}^{P}$.

  * –

Let $u=\left(\begin{smallmatrix}1&-w_{n}\\\
0&1\end{smallmatrix}\right)\in\operatorname{GL}_{2n}(\mathbf{Q}_{p})$, where
$w_{n}$ is the longest Weyl element in $\operatorname{GL}_{n}(\mathbf{Q}_{p})$
(i.e. antidiagonal 1s). If $P$ is the $(m_{1},...,m_{r})$-parabolic (see
Notation 2.6), let

$t_{P}=\mathrm{diag}(p^{r-1}\mathrm{I}_{m_{1}},...,p\mathrm{I}_{m_{r-1}},\mathrm{I}_{m_{r}})\in
T(\mathbf{Q}_{p}).$

For any $\beta\geqslant 1$, we view $ut_{P}^{\beta}\in
G(\mathbf{Q}_{p})\subset G(\mathbf{A})$ in the obvious way.

######  Definition 8.3.

Let $\tilde{\pi}^{P}$ be a $P$-refined RACAR, and $P\subset G$ a spin
parabolic, with associated $\mathcal{P}\subset\mathcal{G}$. We say
$\tilde{\pi}^{P}$ is a _functorial transfer of a $\mathcal{P}$-refined
$\tilde{\Pi}^{\mathcal{P}}$ on $\mathrm{GSpin}_{2n+1}(\mathbf{A})$_ if $\pi$
is the functorial transfer of $\Pi$, and $\tilde{\pi}^{P}$ is the functorial
transfer of $\tilde{\Pi}^{\mathcal{P}}$ in the sense of Definition 3.6.

######  Conjecture 8.4.

Let $P\subsetneq G$ be a proper spin parabolic, with associated
$\mathcal{P}\subset\mathrm{GSpin}_{2n+1}$. Let
$\tilde{\pi}^{P}=(\pi,\alpha^{P})$ be a $P$-refined RACAR of $G(\mathbf{A})$.
Let $\chi,\eta$ be algebraic Hecke characters, with $\chi$ finite order of
conductor $p^{\beta}>1$. For any $s\in\mathbf{C}$, the following are
equivalent:

  1. (1)

There exists $\varphi\in\tilde{\pi}^{P}$ such that
$Z_{H}(ut_{P}^{\beta}\cdot\varphi,\chi,s+1/2)\neq 0$.

  2. (2)

All of the following hold:

     * –

$\tilde{\pi}^{P}$ is a functorial transfer of some $\tilde{\Pi}^{\mathcal{P}}$
on $\mathrm{GSpin}_{2n+1}(\mathbf{A})$ with central character $\eta$,

     * –

$L(\pi\times\chi,s+1/2)\neq 0$,

     * –

$P$ is contained in the $(n,n)$-parabolic.

We have stated this in a global form close to Theorem 8.1. We shall see a
purely local-at-$p$ reformulation in §8.3. In particular, in condition (1),
the important choice is of a local eigenvector
$\varphi_{p}\in\tilde{\pi}_{p}^{P}$, which is unique up to scalar. In this
sense, locally either the integral vanishes or does not vanish on the entire
$p$-refinement.

#### 8.2. Results towards the conjecture

In the remainder of the paper, we prove a number of results towards this
conjecture. In particular, we prove:

######  Theorem 8.5.

Implication (2) $\Rightarrow$ (1) holds in Conjecture 8.4.

We can also use Theorem 4.9 to prove results towards (1) $\Rightarrow$ (2); as
an example, as in Theorem D of the introduction, we show it when
$\tilde{\pi}^{P}$ has non-$P$-critical slope. To state our (stronger) precise
result, we need some additional terminology.

  * •

Fix a prime-to-$p$ level
$K^{p}\subset\operatorname{GL}_{2n}(\mathbf{A}_{f}^{(p)})$. For a parabolic
$P$, we let
$K_{P}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=K^{p}J_{P}\subset\operatorname{GL}_{2n}(\mathbf{A}_{f})$,
where $J_{P}$ is the $P$-parahoric subgroup.

  * •

For any open compact $K\subset\operatorname{GL}_{2n}(\mathbf{A}_{f})$, let
$S_{K}$ denote the $\operatorname{GL}_{2n}$-locally symmetric space of level
$K$ (see [BDW, §2.3]).

  * •

For any parabolic $P$, let $\mathcal{D}_{\lambda}^{P}$ be the module of weight
$\lambda$ $P$-parahoric distributions for $G$, defined in [BW21, §3.2]. We
have $\mathcal{D}_{\lambda}^{G}=V_{\lambda}^{\vee}$ is the dual of the
algebraic induction, and $\mathcal{D}_{\lambda}^{B}=\mathcal{D}_{\lambda}$ is
the usual module of (Iwahori) locally analytic distributions. We have attached
$p$-adic local systems $\mathscr{V}_{\lambda}/\mathscr{D}_{\lambda}^{P}$ on
$S_{K_{P}}$ (e.g. [BDW, §2.3.2]).

  * •

The _top degree eigenvariety_ was constructed in [BW21, §5], following
[Han17]. It is built from modules
$\mathrm{H}^{t}_{\mathrm{c}}(S_{K_{B}},\mathscr{D}_{\Omega})$, where
$\Omega\subset\mathscr{W}$ is a weight affinoid and $\mathscr{D}_{\Omega}$ is
a local system of locally analytic distributions over $\Omega$ (as in [BDW,
Def. 3.11]; see [BW21, §3.2]). Here $t=n^{2}+n-1$ is the top degree for
cuspidal cohomology.

  * •

We say _$\tilde{\pi}^{P}$ appears in the top degree eigenvariety_ if there
exists an Iwahori refinement $\tilde{\pi}$ above $\tilde{\pi}^{P}$, and a
neighbourhood $\Omega\subset\mathscr{W}_{0,\lambda_{\pi}}^{P}$ of
$\lambda_{\pi}$, such that the natural specialisation map

$r_{\tilde{\pi}}:\mathrm{H}^{t}_{\mathrm{c}}(S_{K_{B}},\mathscr{D}_{\Omega})\xrightarrow{\
r_{\lambda_{\pi}}\
}\mathrm{H}^{t}_{\mathrm{c}}(S_{K_{B}},\mathscr{V}_{\lambda_{\pi}}^{\vee})\twoheadrightarrow\mathrm{H}^{t}_{\mathrm{c}}(S_{K_{B}},\mathscr{V}_{\lambda_{\pi}}^{\vee})_{\tilde{\pi}}$
(8.2)

(induced by
$r_{\lambda_{\pi}}:\mathscr{D}_{\Omega}\xrightarrow{\mod\mathfrak{m}_{\lambda_{\pi}}}\mathscr{D}_{\lambda_{\pi}}\twoheadrightarrow\mathscr{V}_{\lambda_{\pi}}^{\vee}$,
and then projection onto the $\tilde{\pi}$-eigenspace) is surjective. This
implies that the $\tilde{\pi}$-localisation in
$\mathrm{H}^{t}_{\mathrm{c}}(S_{K_{B}},\mathscr{D}_{\Omega})$ is non-zero, and
hence there is a point in the top degree eigenvariety corresponding to
$\tilde{\pi}$.

  * •

We say $\tilde{\pi}^{P}$ is _$P$ -strongly-interior_ if the $P$-parahoric
boundary overconvergent cohomology vanishes
$\mathrm{H}^{\bullet}_{\partial}(S_{K_{P}},\mathscr{D}_{\lambda_{\pi}}^{P})_{\tilde{\pi}^{P}}=0$
(see Def. 5.13 _op. cit_.).

######  Theorem 8.6.

Suppose $\pi$ has regular weight, that $\tilde{\pi}^{P}$ appears in the top
degree eigenvariety, and that $\tilde{\pi}^{P}$ is $P$-strongly-interior. Then
(1) $\Rightarrow$ (2) holds in Conjecture 8.4.

######  Remarks 8.7.

  1. (i)

We cautiously suggest the conditions on $\tilde{\pi}^{P}$ should hold for all
$\tilde{\pi}^{P}$ (whence Conjecture 8.4 would hold in full). Unconditionally,
$\tilde{\pi}^{P}$ is $P$-strongly-interior if it is non-$P$-critical slope;
see [BW21, Lem. 5.14]. It appears in the top-degree eigenvariety if there
exists a non-$B$-critical Iwahori-refinement $\tilde{\pi}$ above
$\tilde{\pi}^{P}$ (see [BW21, Def. 4.1] and [BDW, Prop. 7.8]). Hence Theorem
8.6 implies Theorem D(ii) from the introduction.

  2. (ii)

When $P=B$, this proves [BDG+, Expectation 7.2] for $\tilde{\pi}$ satisfying
the conditions of Theorem 8.6, thus for non-critical slope $\tilde{\pi}$ of
regular weight (see Remark 7.3 _op. cit_.).

#### 8.3. Local reinterpretation

Proofs of all of the above crucially use local zeta integrals. If (1) or (2)
holds in Conjecture 8.4, then Theorem 8.1 implies $\pi$ is symplectic, so that
$\pi$ and $\pi_{v}$ admit Shalika models for all places $v$. Via [FJ93, Prop.
2.3], (8.1) can then be rewritten as a product of local integrals, the
_Friedberg–Jacquet integrals_

$\zeta_{v}(\varphi_{v},\chi_{v},s)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\operatorname{GL}_{n}(\mathbf{Q}_{v})}\mathcal{S}_{\psi_{v}}^{\eta_{v}}(\varphi_{v})\left[\begin{pmatrix}x&\\\
&1\end{pmatrix}\right]\chi_{v}|\cdot|^{s-\tfrac{1}{2}}\Big{(}\det x\Big{)}dx.$

where $\varphi_{v}\in\pi_{v}$ and $\mathcal{S}_{\psi_{v}}^{\eta_{v}}$ is an
intertwining of $\pi_{v}$ into its Shalika model (see e.g. [BDW, §2.6]).

Let $\ell\neq p$ be a finite prime, and $\varphi_{\ell}\in\pi_{\ell}$. By
[FJ93, Prop. 3.1], for each unramified quasi-character
$\chi_{\ell}:F_{\ell}^{\times}\to\mathbf{C}^{\times}$, there exists a
holomorphic function $r_{\ell}(\varphi_{\ell},\chi_{\ell},s)$ such that
$\zeta_{\ell}(\varphi_{\ell}^{\mathrm{FJ}},\chi_{\ell},s)=r_{\ell}(\varphi_{\ell},\chi_{\ell},s)\cdot
L(\pi_{\ell}\times\chi_{\ell},s)$. Moreover there exists
$\varphi_{\ell}^{\mathrm{FJ}}\in\pi_{\ell}$ such that
$r_{\ell}(\varphi_{\ell}^{\mathrm{FJ}},\chi_{\ell},s)=1$. If $\pi_{\ell}$ is
spherical, we may take $\varphi_{\ell}^{\mathrm{FJ}}$ spherical [FJ93, Prop.
3.2].

At infinity, by [Sun19] there exists a vector
$\varphi_{\infty}^{\mathrm{FJ}}\in\pi_{\infty}$ such that
$\zeta_{\infty}(\varphi_{\infty},\chi_{\infty},s)\neq 0$.

######  Lemma 8.8.

The following are equivalent:

  * (1)

There exists $\varphi\in\tilde{\pi}^{P}$ such that
$Z_{H}(ut_{P}^{\beta}\cdot\varphi,\chi,s+1/2)\neq 0$,

  * (1′)

$L(\pi\times\chi,s+1/2)\neq 0$, $\pi$ is symplectic, and there exists
$\varphi_{p}\in\tilde{\pi}_{p}^{P}$ such that
$\zeta_{p}(ut_{P}^{\beta}\cdot\varphi_{p},\chi_{p},s+1/2)\neq 0$.

###### Proof.

Suppose (1) holds. Then $\pi$ is symplectic by Theorem 8.1. Write
$\varphi=\otimes_{v}\varphi_{v}$. Decomposing $Z_{H}$ into a product of local
integrals gives

$\displaystyle 0$ $\displaystyle\neq
Z_{H}(ut_{P}^{\beta}\cdot\varphi,\chi,s+1/2)$
$\displaystyle=\zeta_{\infty}(\varphi_{\infty},\chi_{\infty},s)\cdot
r(\varphi^{(p)},\chi^{(p)},s)\cdot
L(\pi\times\chi,s+1/2)\cdot\zeta_{p}(ut_{P}^{\beta}\cdot\varphi_{p},\chi_{p},s+1/2),$

where $r(\varphi^{(p)},\chi^{(p)},s)=\prod_{\ell\neq
p}r_{\ell}(\varphi_{\ell},\chi_{\ell},s)$. This is a holomorphic function, as
all but finitely many of the $\varphi_{\ell}$ are spherical, meaning all but
finitely many of the $r_{\ell}(\varphi_{\ell},\chi_{\ell},s)$ are 1. Here we
have also used that $L(\pi_{p}\times\chi_{p},s+1/2)=1$, as $\chi_{p}$ is
ramified. We deduce the non-vanishing statements in $(1^{\prime})$. If
conversely $(1^{\prime})$ holds, the same arguments show (1) holds with
$\varphi\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varphi_{p}\otimes(\otimes_{v\neq
p}\varphi_{v}^{\mathrm{FJ}})$. ∎

Proposition 3.7 also gives a local analogue of property (2).

######  Lemma 8.9.

Property (2) in Conjecture 8.4 is equivalent to:

  * (2′)

$L(\pi\times\chi,s+1/2)\neq 0$, $\pi$ is symplectic, $\tilde{\pi}^{P}$ is
$P$-spin, and $P$ is contained in the $(n,n)$-parabolic.

Conjecture 8.4 is then equivalent to the following local statement:

######  Conjecture 8.10.

If $\pi$ is symplectic and $L(\pi\times\chi,s+1/2)\neq 0$, the following are
equivalent:

  * (1′)

There exists $\varphi_{p}\in\tilde{\pi}_{p}^{P}$ such that
$\zeta_{p}(ut_{P}^{\beta}\cdot\varphi_{p},\chi_{p},s+1/2)\neq 0.$

  * (2′)

$P$ is contained in the $(n,n)$-parabolic, and $\tilde{\pi}^{P}$ is $P$-spin.

For the rest of the paper, we assume our prime-to-$p$ level
$K^{p}\subset\operatorname{GL}_{2n}(\mathbf{A}_{f}^{(p)})$ fixes
$\otimes_{\ell\neq p}\varphi_{\ell}^{\mathrm{FJ}}$, which is possible by
[FJ93, Prop. 3.2].

### 9\. Proof of Theorem 8.5

In this section, we give the proof of Theorem 8.5 (that $(2)\Rightarrow(1)$ in
Conjecture 8.4). More specifically, we prove
$(2^{\prime})\Rightarrow(1^{\prime})$ in Conjecture 8.10, which implies the
theorem by §8.3. Our proof is constructive; given a $P$-spin
$\tilde{\pi}^{P}$, we describe explicitly an eigenvector with non-vanishing
local zeta integral. If $P=B$ or the $(n,n)$-parabolic $Q$, then Theorem 8.5
was proved in [BDG+, Cor. 7.15] and [DJR20, Prop. 3.4, Lem. 3.6] respectively.

Our proof for general $P$ is closely modelled on the approach in [BDG+], and
we refer to specific places _op. cit_. for more detail. Recall
$\mathcal{S}_{\psi_{p}}^{\eta_{p}}$ is an intertwining of $\pi_{p}$ into its
Shalika model, and for any $\varphi_{p}\in\pi_{p}$, we let
$W_{\varphi_{p}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{S}_{\psi_{p}}^{\eta_{p}}(\varphi_{p})$.
Then we:

  * (i)

Express $\zeta_{p}(ut_{P}^{\beta}\cdot\varphi_{p},\chi_{p},s+1/2)$ as a non-
zero multiple of a value of $W_{\varphi_{p}}$;

  * (ii)

Show if $P\subset Q$ and $\tilde{\pi}^{P}$ is $P$-spin, there exists
$\varphi_{p}\in\tilde{\pi}_{p}^{P}$ where this specific value of
$W_{\varphi_{p}}$ is non-zero.

#### 9.1. The local zeta integral

######  Proposition 9.1.

Let $\varphi_{p}\in\pi_{p}^{\mathrm{Iw}_{G}}$, and let
$W_{\varphi_{p}}=\mathcal{S}_{\psi_{p}}^{\eta_{p}}(\varphi_{p})$. Let
$\chi_{p}$ be a character of conductor $p^{\beta}>1$. Let
$t=\left(\begin{smallmatrix}z_{1}&\\\ &z_{2}\end{smallmatrix}\right)\in
T(\mathbf{Q}_{p})$, and

$\nu_{\beta}(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=p^{-\beta}z_{2}^{-1}w_{n}z_{1}.$

Then for all $s$,

$\Big{[}\zeta_{p}(ut\cdot\varphi_{p},\chi_{p},s)\neq
0\Big{]}\iff\Big{[}W_{\varphi_{p}}\left(\begin{smallmatrix}\nu_{\beta}(t)&\\\
&1\end{smallmatrix}\right)\neq 0\Big{]}.$ (9.1)

###### Proof.

Recalling $u=\left(\begin{smallmatrix}1&-w_{n}\\\
0&1\end{smallmatrix}\right)$, we have

$\left(\begin{array}[]{cc}x&\\\
&1\end{array}\right)ut=\left(\begin{array}[]{cc}z_{2}&\\\
&z_{2}\end{array}\right)\left(\begin{array}[]{cc}1&-z_{2}^{-1}xw_{n}z_{2}\\\
&1\end{array}\right)\left(\begin{array}[]{cc}z_{2}^{-1}xz_{1}&\\\
&1\end{array}\right).$ (9.2)

Let $\varphi_{p}\in\pi_{p}$ be any $\mathrm{Iw}_{G}$-invariant vector. By
(9.2), as $W_{\varphi_{p}}$ is in the Shalika model, we see

$\zeta_{p}(ut\cdot\varphi_{p},\chi_{p},s)=\eta(\operatorname{det}z_{2})\int_{\operatorname{GL}_{n}(\mathbb{Q}_{p})}\psi_{p}\Big{(}\operatorname{tr}(-z_{2}^{-1}xw_{n}z_{2})\Big{)}W_{\varphi_{p}}\left(\begin{array}[]{cc}z_{2}^{-1}xz_{1}&\\\
&1\end{array}\right)\chi_{p}|\cdot|^{s-\tfrac{1}{2}}\Big{(}\operatorname{det}x\Big{)}dx.$

Let $y=-z_{1}^{-1}w_{n}xz_{1}$, and let
$\omega=-z_{2}^{-1}w_{n}z_{1}=-p^{\beta}\nu_{\beta}(t)$. As
$\mathrm{tr}(-z_{2}^{-1}xw_{n}z_{2})=\mathrm{tr}(y)$, changing variables and
noting $dx=dy$, we see

$\zeta_{p}(ut\cdot\varphi_{p},\chi_{p},s)=(\star)\cdot\mathcal{Q},\qquad(\star)\neq
0,$ (9.3)

where

$\mathcal{Q}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{\operatorname{GL}_{n}(\mathbb{Q}_{p})}\psi_{p}(\operatorname{tr}(y))I(\omega
y)dy,\qquad I(y)=W_{\varphi_{p}}\begin{pmatrix}y&\\\
&1\end{pmatrix}\chi_{p}|\cdot|^{s-\tfrac{1}{2}}(\operatorname{det}y).$

By (9.3), it suffices to prove $\mathcal{Q}\neq 0\iff
W_{\varphi_{p}}\left(\begin{smallmatrix}\nu_{\beta}(t)&\\\
&1\end{smallmatrix}\right)\neq 0$.

We want to reduce the support of the integral $\mathcal{Q}$. Let
$M=\operatorname{GL}_{n}(\mathbb{Q}_{p})\cap M_{n}(\mathbb{Z}_{p})$. By [BDG+,
Lem. 5.1], the support of $I(\omega y)$ (hence $\mathcal{Q}$) is contained in
$\omega^{-1}M$.

As in [BDG+, Not. 5.3], let $A$ denote the set of all diagonal $n\times
n$-matrices of the form

$\gamma=\operatorname{diag}(c_{11},\dots,c_{nn}),\qquad
c_{ii}\in\mathbf{Z}_{p}^{\times}.$

Let $B_{\beta}$ denote the additive group of all $n\times n$-matrices $\delta$
with

$\delta_{i,j}=\left\\{\begin{array}[]{cc}c_{i,j}&\text{ if }i<j\\\ 0&\text{ if
}i=j\\\ p^{\beta}c_{i,j}&\text{ if }i>j\end{array}\right.,\qquad
c_{ij}\in\mathbf{Z}_{p}.$

Let $\alpha=\gamma+\delta$, with $\gamma\in A$, $\delta\in B_{\beta}$. Note
$\alpha\in\mathrm{Iw}_{n}(p^{\beta})$ is in the depth $p^{\beta}$ Iwahori
subgroup of $\operatorname{GL}_{n}(\mathbf{Z}_{p})$. Let
$\varepsilon=\left(\begin{smallmatrix}\alpha^{-1}&\\\
&1\end{smallmatrix}\right)\in\mathrm{Iw}_{G}$. Then
$I(y\alpha^{-1})=\chi_{p}(\operatorname{det}\gamma)^{-1}I(y)$, so