# Emerton–Gee stacks, Serre weights, and Breuil–Mézard conjectures for
$\mathrm{GSp}_{4}$

Heejong Lee  Department of Mathematics, University of Toronto, 40 St. George
Street, Toronto, ON M5S 2E4, Canada<EMAIL_ADDRESS>
###### Abstract.

We construct a moduli stack of rank 4 symplectic projective étale
$(\varphi,\Gamma)$-modules and prove its geometric properties for any prime
$p>2$ and finite extension $K/\mathbf{Q}_{p}$. When $K/\mathbf{Q}_{p}$ is
unramified, we adapt the theory of local models recently developed by Le–Le
Hung–Levin–Morra to study the geometry of potentially crystalline substacks in
this stack. In particular, we prove the unibranch property at torus fixed
points of local models and deduce that tamely potentially crystalline
deformation rings are domain under genericity conditions. As applications, we
prove, under appropriate genericity conditions, an $\mathrm{GSp}_{4}$-analogue
of the Breuil–Mézard conjecture for tamely potentially crystalline deformation
rings, the weight part of Serre’s conjecture formulated by Gee–Herzig–Savitt
for global Galois representations valued in $\mathrm{GSp}_{4}$ satisfying
Taylor–Wiles conditions, and a modularity lifting result for tamely
potentially crystalline representations.

###### Contents

  1. 1 Introduction
  2. 2 Types and weights
    1. 2.1 Preliminaries
    2. 2.2 Serre weights and Deligne–Lusztig representations
    3. 2.3 Tame inertial $L$-parameters
    4. 2.4 Inertial local Langlands for $\mathrm{GSp}_{4}$
    5. 2.5 Combinatorics between types and weights
  3. 3 The theory of local models
    1. 3.1 Global affine Grassmannians
    2. 3.2 Geometry of universal local models
    3. 3.3 Local models in mixed characteristic
    4. 3.4 Products of local models
    5. 3.5 Special fiber of naive local models in mixed characteristic
    6. 3.6 Matching irreducible components and Serre weights
    7. 3.7 Torus fixed points
  4. 4 Moduli stacks in $p$-adic Hodge theory
    1. 4.1 Sympletic étale $(\varphi,\Gamma)$-modules
    2. 4.2 Symplectic Breuil–Kisin modules
    3. 4.3 Symplectic étale $\varphi$-modules
    4. 4.4 Local models for potentially crystalline stacks
    5. 4.5 Potentially crystalline stacks modulo $p$
  5. 5 Global setup
    1. 5.1 Some local Galois deformation rings
    2. 5.2 Congruent patching functors
    3. 5.3 Algebraic automorphic forms
    4. 5.4 Patching argument
  6. 6 Applications
    1. 6.1 The Breuil–Mézard conjecture
    2. 6.2 The weight part of a Serre’s conjecture
    3. 6.3 Modularity lifting in generic tamely potentially crystalline case
  7. A Torus fixed points of certain affine Springer fibers

##  1\. Introduction

Serre’s modularity conjecture predicts that every continuous, irreducible, odd
Galois representations
$\overline{r}:\operatorname{\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p})$
arise from modular forms [Ser87]. Its refined version predicts the minimal
weight of modular forms that give rise to $\overline{r}$ in terms of the
restriction of $\overline{r}$ at $p$. The weight part of Serre’s conjecture,
saying that the original conjecture implies the refined conjecture, was
crucial for the proof of the original conjecture due to Kisin and
Khare–Winterberger [KW09a, KW09b, Kis09].

The weight part of Serre’s conjecture has been generalized, most notably, to
2-dimensional Galois representations over a totally real field [BDJ10], to
Galois representations over a totally real field valued in $\mathrm{GL}_{n}$
[Her09] and in more general groups $G$ [GHS18]. (Both [Her09] and [GHS18]
assume that $p$ is unramified in the totally real field and the Galois
representations are tamely ramified at places above $p$. See [LLHLMa,
Conjecture 1.6.2] for a generalization to wildly ramified representations, and
[LLHLMb, Definition 1.1.1] to a totally real field in which $p$ ramifies.) In
these generalizations, the notion of weight is replaced by _Serre weights_ ,
irreducible $\overline{\mathbf{F}}_{p}$-representations of
$G(\mathbf{F}_{q})$.

Although stated in the global context, the weight part of Serre’s conjecture
is closely related to the mod $p$ Langlands program. Let $K/\mathbf{Q}_{p}$ be
a finite extension. The mod $p$ Langlands program predicts a certain
correspondence between admissible smooth
$\overline{\mathbf{F}}_{p}$-representations of a $p$-adic reductive group
$G(K)$ and continuous representations of
$\operatorname{\mathrm{Gal}}(\overline{K}/K)$ valued in the $C$-group
$\prescript{C}{}{G}(\overline{\mathbf{F}}_{p})$ defined in [BG14]. (In this
article, we only consider $G=\mathrm{GL}_{n}$ or $G=\mathrm{GSp}_{4}$, in
which case $\prescript{C}{}{G}$ is isomorphic to $G\times\mathrm{GL}_{1}$, and
it is equivalent to work with $G$.) Such correspondence has been established
only in the case $G=\mathrm{GL}_{2}$ and $K=\mathbf{Q}_{p}$ [Col10, Paš13] or
$G=\mathrm{GL}_{1}$. Despite many recent advances (for example, see [CEG+16,
BHH+a]) in the program, its intricate nature still remains poorly understood,
and a precise conjecture for $G=\mathrm{GL}_{n}$ was only made recently
[EGH22] in a categorical language. However, using global methods, one can
construct a candidate for a mod $p$ Langlands correspondence, namely, an
admissible smooth $\overline{\mathbf{F}}_{p}$-representation $\Pi$ of
$\mathrm{GL}_{n}(K)$ associated with an $n$-dimensional continuous
$\overline{\mathbf{F}}_{p}$-representation $\overline{\rho}$ of
$\operatorname{\mathrm{Gal}}(\overline{K}/K)$. One major open problem is to
prove that $\Pi$ only depends on $\overline{\rho}$ and not on non-canonical
choices made in the global argument. The weight part of Serre’s conjecture
describes the Serre weights appearing in the
$\mathrm{GL}_{n}({\mathcal{O}}_{K})$-socle of $\Pi$ explicitly in terms of
$\overline{\rho}|_{I_{K}}$. Thus, the weight part of Serre’s conjecture has
been a guiding principle in the mod $p$ Langlands program.

The theory of Galois deformations has played an important role in many
advances in proving the weight part of Serre’s conjecture ([LLHLM18, LLHLM20,
LLHLMa]), in the mod $p$ and $p$-adic Langlands program ([Col10, Paš13,
CEG+16, BHH+b]), and in proving modularity lifting results ([TW95, Wil95,
CHT08, BLGGT14, BCGP21]). Recently, Emerton–Gee constructed an object which
algebraically interpolates deformation spaces of $n$-dimensional $p$-adic
Galois representations (called the Emerton–Gee stack [EG23]). Thus, it allows
us to employ more geometric methods to study $p$-adic Galois representations.
Most importantly, it is an essential ingredient in formulating the categorical
mod $p$ and $p$-adic Langlands conjecture [EGH22]. When $K/\mathbf{Q}_{p}$ is
unramified, [LLHLMa] developed a theory of local models to study the geometry
of potentially crystalline substacks in Emerton–Gee stacks. In particular, it
is applied to analyze singularities of tamely potentially crystalline
deformation rings. This led to several applications, including the weight part
of Serre’s conjecture and the Breuil–Mézard conjecture. The objective of this
article is to generalize Emerton–Gee stacks, local models, and their
applications to Serre weight conjectures and the Breuil–Mézard conjecture to
the group $\mathrm{GSp}_{4}$.

#### Emerton–Gee stacks for $\mathrm{GSp}_{4}$

Let $p$ be a prime and $K/\mathbf{Q}_{p}$ be a finite extension with the ring
of integers ${\mathcal{O}}_{K}$, a uniformizer $\varpi$, and the residue field
$k$. We also fix a sufficiently large finite extension $E/\mathbf{Q}_{p}$ with
the ring of integers ${\mathcal{O}}$, a uniformizer $\varpi$, and the residue
field $\mathbf{F}$.

In [EG23], Emerton–Gee constructed ${\mathcal{X}}_{n,K}$ a moduli stack of
rank $n$ projective étale $(\varphi,\Gamma)$-modules. For a finite
${\mathcal{O}}$-algebra $A$, rank $n$ projective étale $(\varphi,\Gamma)$
modules with $A$-coefficients are equivalent to continuous representations of
$G_{K}:=\operatorname{\mathrm{Gal}}(\overline{K}/K)$ on rank $n$ projective
$A$-modules. However, for general $A$, étale $(\varphi,\Gamma)$-modules behave
better than $G_{K}$-representations in algebraic families. In particular, a
family of reducible étale $(\varphi,\Gamma)$-modules can converge to an
irreducible one, unlike $G_{K}$-representations. Thus, ${\mathcal{X}}_{n,K}$
is considered the correct notion of a moduli stack of “$p$-adic Langlands
parameters”.

To generalize Emerton–Gee stacks to $\mathrm{GSp}_{4}$, we define a
_symplectic_ projective étale $(\varphi,\Gamma)$-module to be a triple
$(M,N,\alpha)$ where $M$ is a rank 4 projective étale $(\varphi,\Gamma)$
module, $N$ is a rank 1 projective étale $(\varphi,\Gamma)$-modules, and
$\alpha:M\simeq M^{\vee}\otimes N$ is an essentially self-dual isomorphism
satisfying a certain sign condition (Definition 4.1.1). These reflect the
underlying 4-dimensional vector space of $\mathrm{GSp}_{4}$, the similitude
character of $\mathrm{GSp}_{4}$, and the non-degenerate skew-symmetric
bilinear form on the 4-dimensional vector space, respectively. Under
Fontaine’s equivalence, symplectic projective étale $(\varphi,\Gamma)$-modules
with $A$-coefficients for finite local ${\mathcal{O}}$-algebra $A$ are
equivalent to continuous representations
$\rho:G_{K}\rightarrow\mathrm{GSp}_{4}(A)$. We let
${\mathcal{X}}_{{\mathrm{Sym}},K}$ be a category fibered in groupoids over
$\operatorname{\mathrm{Spf}}{\mathcal{O}}$ whose groupoid of $A$-points, for a
$p$-adically complete ${\mathcal{O}}$-algebra $A$, is the groupoid of
symplectic projective étale $(\varphi,\Gamma)$-modules with $A$-coefficients.
The following Theorem generalizes main properties of ${\mathcal{X}}_{n,K}$ to
${\mathcal{X}}_{{\mathrm{Sym}},K}$.

######  Theorem 1 (Theorem 4.1.5, Proposition 4.1.6, Theorem 4.1.10).

Suppose that $p>2$.

  1. (1)

The category fibered in groupoids ${\mathcal{X}}_{{\mathrm{Sym}},K}$ is a
Noetherian formal algebraic stack.

  2. (2)

For each Hodge type $\lambda$ and inertial type $\tau$, there exist a closed
substack ${\mathcal{X}}_{{\mathrm{Sym}},K}^{\lambda,\tau}$ (resp.
${\mathcal{X}}_{{\mathrm{Sym}},K}^{{\mathrm{ss}},\lambda,\tau}$) which is
furthermore a $p$-adic formal algebraic stack and flat over ${\mathcal{O}}$.
It is uniquely characterized as ${\mathcal{O}}$-flat closed substack such that
for any finite flat ${\mathcal{O}}$-algebra $A$,
${\mathcal{X}}_{{\mathrm{Sym}},K}^{\lambda,\tau}(A)$ (resp.
${\mathcal{X}}_{{\mathrm{Sym}},K}^{{\mathrm{ss}},\lambda,\tau}(A)$) is
equivalent to the groupoid of representations
$\rho:G_{K}\rightarrow\mathrm{GSp}_{4}(A)$ such that $\rho\otimes_{A}E$ is
potentially crystalline (resp. semistable) representations with Hodge type
$\lambda$ and inertial type $\tau$.

  3. (3)

The underlying reduced substack
${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{red}}}\subset{\mathcal{X}}_{{\mathrm{Sym}},K}$
is an algebraic stack over $\mathbf{F}$ and equidimensional of dimension
$4[K:\mathbf{Q}_{p}]$. Moreover, its irreducible components are naturally
labelled by Serre weights.

The proofs of item (1) and (2) use the corresponding results for the
Emerton–Gee stack for $\mathrm{GL}_{4}\times\mathrm{GL}_{1}$ and can be easily
generalized to $\mathrm{GSp}_{2n}$. To prove item (3), we first construct
irreducible components corresponding to Serre weights and prove that their
union is equal to the stack ${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{red}}}$,
following the strategy for $\mathrm{GL}_{n}$. The construction of irreducible
components requires a generalization of a result of Emerton–Gee constructing a
family of extensions using abelian Galois cohomology. One difference between
$\mathrm{GSp}_{4}$ and $\mathrm{GL}_{n}$ is the presence of a non-abelian
unipotent radical of a maximal parabolic subgroup $Q$ called the Klingen
parabolic subgroup. It is unclear how the construction of Emerton–Gee can be
generalized to the non-abelian case. Instead, we construct a family of
extensions valued in $Q$ inside a family of extensions valued in the minimal
parabolic of $\mathrm{GL}_{4}$ containing $Q$ by only using abelian Galois
cohomology. To prove that their union is equal to the stack
${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{red}}}$, we need to prove the
existence of crystalline lifts for any continuous representation
$G_{K}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$. In this case, it seems
inevitable to use non-abelian Galois cohomology. We prove this by using the
main result in [Linb], which develops an obstruction theory for lifting mod
$p$ representations of $G_{K}$ valued in reductive groups.

While we were writing this article, Zhongyipan Lin posted a preprint that
constructs generalizations of Emerton–Gee stacks to tamely ramified reductive
groups and proves their Noetherian formal algebraicity using a Tannakian
formalism [Lina].

#### Local models for potentially crystalline stacks

From now on, we assume that $K/\mathbf{Q}_{p}$ is unramified of degree $f$.

In [LLHLMa], the authors constructed local models of potentially crystalline
substacks of ${\mathcal{X}}_{n,K}$ using Breuil–Kisin modules. One can
associate a Breuil–Kisin module with descent data to a lattice in a
potentially crystalline representation. Using this, they identify the
potentially crystalline stacks (with bounded $p$-adic Hodge type) with a
certain closed substack of the moduli stack of Breuil–Kisin modules with
descent data constructed in [CL18]. It is proved in _loc. cit._ that the
moduli stack of Breuil–Kisin modules with descent data is smoothly equivalent
to a Pappas–Zhu local model studied in [PZ13]. The innovation in [LLHLMa] is a
construction of certain subvarieties in Pappas–Zhu local models whose open
neighborhoods are, after $p$-adic completion, smoothly equivalent to open
neighborhoods of potentially crystalline stacks.

The Pappas–Zhu local models exist for any connected reductive group which
splits over a tamely ramified base change. Let $M(\leq\lambda)$ be the
Pappas–Zhu local model over $\operatorname{\mathrm{Spec}}{\mathcal{O}}$
associated to the group $\Res_{K/\mathbf{Q}_{p}}\mathrm{GSp}_{4}$, Iwahori
subgroup of $\mathrm{GSp}_{4}(K)$, and a regular Hodge type $\lambda$.
Following the idea of [LLHLMa], we construct a closed subvariety
$M(\lambda,\nabla_{{\mathbf{a}}_{\tau}})\subset M(\leq\lambda)$ for a tame
inertial type $\tau$. The following Theorem generalizes the main result on
local models in [LLHLMa] to $\mathrm{GSp}_{4}$.

######  Theorem 2 (Theorem 4.4.3).

Let ${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{reg}}}^{\leq\lambda,\tau}$
(resp. $M_{\mathrm{reg}}(\leq\lambda,\nabla_{{\mathbf{a}}_{\tau}})$) be the
union of ${\mathcal{X}}_{{\mathrm{Sym}},K}^{\lambda^{\prime},\tau}$ (resp.
$M(\lambda^{\prime},\nabla_{{\mathbf{a}}_{\tau}})$) for all regular Hodge
types $\lambda^{\prime}\leq\lambda$. Suppose that $\lambda$ is regular and
$\tau$ is sufficiently generic (depending on $\lambda$). There exist Zariski
open covers
$\\{{\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{reg}}}^{\leq\lambda,\tau}(\widetilde{z})\\}_{\widetilde{z}}$
of ${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{reg}}}^{\leq\lambda,\tau}$ and
$\\{U_{{\mathrm{reg}}}(\widetilde{z},\leq\lambda,\nabla_{{\mathbf{a}}_{\tau}})\\}_{\widetilde{z}}$
of $M_{\mathrm{reg}}(\leq\lambda,\nabla_{{\mathbf{a}}_{\tau}})$, and for each
$\widetilde{z}$, a local model diagram

${\widetilde{U}_{{\mathrm{reg}}}(\widetilde{z},\leq\lambda,\nabla_{{\mathbf{a}}_{\tau}})^{\wedge_{p}}}$${{\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{reg}}}^{\leq\lambda,\tau}(\widetilde{z})}$${U_{{\mathrm{reg}}}(\widetilde{z},\leq\lambda,\nabla_{{\mathbf{a}}_{\tau}})^{\wedge_{p}}.}$

Here,
$\widetilde{U}_{{\mathrm{reg}}}(\widetilde{z},\leq\lambda,\nabla_{{\mathbf{a}}_{\tau}}):=T^{f}\times
U_{{\mathrm{reg}}}(\widetilde{z},\leq\lambda,\nabla_{{\mathbf{a}}_{\tau}})\rightarrow
U_{{\mathrm{reg}}}(\widetilde{z},\leq\lambda,\nabla_{{\mathbf{a}}_{\tau}})$ is
the trivial $T^{f}$-torsor, and the left diagonal arrow is a $T^{f}$-torsor
under a different $T^{f}$-action, where $T$ is the diagonal maximal torus of
$\mathrm{GSp}_{4}$, and ${\wedge_{p}}$ denotes the $p$-adic completion.

Note that if $\lambda$ is equal to a cocharacter $\eta$ lifting the half sum
of positive coroots, then
${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{reg}}}^{\leq\eta,\tau}={\mathcal{X}}_{{\mathrm{Sym}},K}^{\eta,\tau}$
and
$M_{\mathrm{reg}}(\leq\eta,\nabla_{{\mathbf{a}}_{\tau}})=M(\eta,\nabla_{{\mathbf{a}}_{\tau}})$.

Similar to symplectic étale $(\varphi,\Gamma)$-modules, we define _symplectic_
Breuil–Kisin modules and denote by $Y_{{\mathrm{Sym}}}^{\leq\lambda,\tau}$ the
moduli stack of symplectic Breuil–Kisin modules of height $\leq\lambda$ and
inertial type $\tau$. We prove that the stack
$Y_{{\mathrm{Sym}}}^{\leq\lambda,\tau}$ is $p$-adic formal algebraic stack
over $\operatorname{\mathrm{Spf}}{\mathcal{O}}$ (Proposition 4.2.5) and
locally smoothly equivalent to $M(\leq\lambda)^{\wedge_{p}}$ (Theorem 4.2.16).
There exists a closed substack
$Y_{{\mathrm{Sym}}}^{\leq\lambda,\tau,\nabla_{\infty}}\subset
Y_{{\mathrm{Sym}}}^{\leq\lambda,\tau}$ which is isomorphic to
${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{reg}}}^{\leq\lambda,\tau}$
(Proposition 4.4.2). We note that the locus
$Y_{{\mathrm{Sym}}}^{\leq\lambda,\tau,\nabla_{\infty}}$ is _analytic_ , i.e.
it is given by formal power series, while in contrast, the subvariety
$M(\lambda,\nabla_{{\mathbf{a}}_{\tau}})\subset M(\leq\lambda)$ is algebraic.
However, these conditions defining substack/subvariety are “congruent modulo
power of $p$” (in a suitable sense). Then the left arrow in the above diagram
is induced by applying Elkik’s approximation theorem [Elk73]. In particular,
it is highly non-canonical but canonical modulo $p$.

The local model $M(\lambda,\nabla_{{\mathbf{a}}_{\tau}})$ is a projective
variety equipped with a natural $T^{f}$-action. Its $T^{f}$-fixed points are
given by certain elements $\widetilde{z}$ in the extended affine Weyl group of
$(\mathrm{GSp}_{4})^{f}$. Under the above local model diagram, they correspond
to tame
$\overline{\rho}\in{\mathcal{X}}_{{\mathrm{Sym}},K}^{\leq\lambda,\tau}(\mathbf{F})$.
The main result on the geometry of local models is the following.

######  Theorem 3 (Theorem 3.4.1).

Suppose that $\lambda$ is regular and $\tau$ is sufficiently generic
(depending on $\lambda$). Then $M(\lambda,\nabla_{{\mathbf{a}}_{\tau}})$ is
unibranch at any $T^{f}$-fixed point $\widetilde{z}\in
M(\lambda,\nabla_{{\mathbf{a}}_{\tau}})(\mathbf{F})$.

Our proof of Theorem 3 follows [LLHLMa] closely. The novel idea in _loc. cit._
is comparing the (mixed characteristic) local model
$M(\lambda,\nabla_{{\mathbf{a}}})$ (defined over
$\operatorname{\mathrm{Spec}}{\mathcal{O}}$) with an equal characteristic
local model (defined over $\operatorname{\mathrm{Spec}}\mathbf{F}[v]$) inside
an universal local model (defined over
$\operatorname{\mathrm{Spec}}\mathbf{Z}[v]$). In general, the mixed and equal
characteristic local models are _not_ a base change of the universal local
model. However, this is true under a genericity hypothesis, and they naturally
have the same special fiber. We prove the unibranch property in the equal
characteristic case and use it to prove the mixed characteristic case. Here,
it is also crucial to compare the base change of the normalization of
universal local models and normalizations of mixed and equal characteristic
local models. This requires the base change of the normalization of universal
local models to be normal, which also holds under a genericity hypothesis.
Using these ideas, we first prove a preliminary unibranch property (Theorem
3.3.4). Then using this, we prove the main unibranch property for products of
local models (Theorem 3.4.1).

Let $\overline{\rho}:G_{K}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$ be a
continuous representation. We denote by $R_{\overline{\rho}}^{\lambda,\tau}$
the unique ${\mathcal{O}}$-flat quotient of a framed deformation ring
$R_{\overline{\rho}}^{\square}$ of $\overline{\rho}$ whose
$\overline{\mathbf{Q}}_{p}$-points parameterize potentially crystalline lifts
of type $(\lambda,\tau)$. As a direct consequence of Theorem 2 and 3, we
obtain the following result.

######  Corollary 4.

Suppose that $\lambda$ is regular and $\tau$ is sufficiently generic
(depending on $\lambda$). If
$\overline{\rho}:G_{K}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$ is tame, then
$R_{\overline{\rho}}^{\lambda,\tau}$ is a domain (if is nonzero).

The importance of this Corollary is related to patching argument. Roughly
speaking, patching argument constructs a module
$M_{\infty}(\lambda-\eta,\tau)$ over (a certain modification of)
$R_{\overline{\rho}}^{\lambda,\tau}$. Standard commutative algebra argument
shows that the support of $M_{\infty}(\lambda-\eta,\tau)$ is a union of
irreducible components in
$\operatorname{\mathrm{Spec}}R_{\overline{\rho}}^{\lambda,\tau}$. It is a
folklore conjecture that the support is indeed equal to
$\operatorname{\mathrm{Spec}}R_{\overline{\rho}}^{\lambda,\tau}$, which almost
immediately implies a modularity lifting result in a relevant setup. Moreover,
it has applications to the Serre weight conjectures and the Breuil–Mézard
conjecture (see [GHS18, §4.1] for its relationship to the Breuil–Mézard
conjecture). Unfortunately, this is hard to prove in general. However, if
$M_{\infty}(\lambda-\eta,\tau)\neq 0$ and $R_{\overline{\rho}}^{\lambda,\tau}$
is domain, then the support of $M_{\infty}(\lambda-\eta,\tau)$ has to be equal
to $\operatorname{\mathrm{Spec}}R_{\overline{\rho}}^{\lambda,\tau}$.

######  Remark 5.

As in [LLHLMa], our main results on local models, as well as results in the
remaining introduction, hold under suitable genericity conditions. In
particular, genericity conditions imply that the Hodge type $\lambda$ is small
relative to $p$. We refer readers to §1.3 in _loc. cit._ for discussions on
genericity conditions.

#### The geometric Breuil–Mézard conjecture

The original Breuil–Mézard conjecture [BM02] measures the complexity of the
special fiber of potentially crystalline (or semistable) deformation rings in
terms of the mod $p$ representation theory of $\mathrm{GL}_{n}(k)$. It’s
geometric formulations are stated in [EG14] (for deformation rings) and [EG23]
(for Emerton–Gee stacks) and proven in [LLHLMa] in the tamely potentially
crystalline case under genericity assumptions. Also, Dotto formulated the
Breuil–Mézard conjecture for central division algebras and proved that it
follows from the conjecture for $\mathrm{GL}_{n}$ [Dot].

We formulate the geometric Breuil–Mézard conjecture for $\mathrm{GSp}_{4}$ in
the tamely potentially crystalline case and prove it under genericity
assumptions. We first explain the conjecture for deformation rings. Let
$R^{\square}_{\overline{\rho}}$ be a framed deformation ring of
$\overline{\rho}$. Given a regular Hodge type $\lambda$ and a mildly generic
tame inertial type $\tau$, there exists a $(4f+11)$-dimensional cycle
$Z(R_{\overline{\rho}}^{\lambda,\tau}/\varpi)$ in
$\operatorname{\mathrm{Spec}}R_{\overline{\rho}}^{\square}/\varpi$ which
counts the irreducible components in
$\operatorname{\mathrm{Spec}}R_{\overline{\rho}}^{\lambda,\tau}/\varpi$ with
appropriate multiplicities. On the other hand, one can associate to the pair
$(\lambda,\tau)$ a locally algebraic representation
$\sigma(\tau)\otimes_{\mathcal{O}}V(\lambda-\eta)$ of
$\mathrm{GSp}_{4}({\mathcal{O}}_{K})$. Here, $\sigma(\tau)$ is an irreducible
representation of $\mathrm{GSp}_{4}(k)$ corresponding to $\tau$ under the
inertial local Langlands correspondence, which we discuss below, and
$V(\lambda-\eta)$ is the irreducible algebraic representation of
$\mathrm{GSp}_{4}({\mathcal{O}}_{K})$ of highest weight $\lambda-\eta$.

######  Conjecture 6.

Let $\overline{\rho}:G_{K}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$ be a
continuous representation. For each Serre weight $\sigma$, there exists a
$(4f+11)$-dimensional cycle $Z_{\sigma}$ in
$\operatorname{\mathrm{Spec}}R_{\overline{\rho}}^{\square}/\varpi$ such that
for any regular Hodge type $\lambda$ and any mildly generic tame inertial type
$\tau$,

(1.0.1) $\displaystyle
Z(R_{\overline{\rho}}^{\lambda,\tau}/\varpi)=\sum_{\sigma}[\overline{\sigma(\tau)\otimes
V(\lambda-\eta)}:\sigma]Z_{\sigma}.$

Similar to the case of deformation rings, one can construct a $4f$-dimensional
cycle ${\mathcal{Z}}_{\lambda,\tau}$ in
${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{red}}}$ attached to
${\mathcal{X}}_{{\mathrm{Sym}},K}^{\lambda,\tau}\times_{\operatorname{\mathrm{Spec}}{\mathcal{O}}}\operatorname{\mathrm{Spec}}\mathbf{F}$
.

######  Conjecture 7.

For each Serre weight $\sigma$, there exists a $4f$-dimensional cycle
${\mathcal{Z}}_{\sigma}$ in ${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{red}}}$
such that for any regular Hodge type $\lambda$ and any mildly generic tame
inertial type $\tau$,

(1.0.2)
$\displaystyle{\mathcal{Z}}_{\lambda,\tau}=\sum_{\sigma}[\overline{\sigma(\tau)\otimes
V(\lambda-\eta)}:\sigma]{\mathcal{Z}}_{\sigma}.$

######  Remark 8.

Both Conjecture 6 and 7 should extend to any inertial type $\tau$. We only
state them for mildly generic (more precisely, $1$-generic in the sense of
§2.1.7) tame inertial types because we prove the inertial local Langlands
under this condition. Also, they are expected to generalize to potentially
semistable deformation cases.

Our main result on the Breuil–Mézard conjecture is the following.

######  Theorem 9 (Corollary 6.1.17 and 6.1.18).

Let $\Lambda$ be a finite set of regular Hodge types $\lambda$.

  1. (1)

For each Serre weight $\sigma$, there exists a $4f$-dimensional cycle
${\mathcal{Z}}_{\sigma}$ in ${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{red}}}$
such that (1.0.2) holds for any regular Hodge type $\lambda\in\Lambda$ and any
sufficiently generic (depending on $\Lambda$) regular tame inertial type
$\tau$.

  2. (2)

Let $\overline{\rho}:G_{K}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$ be
sufficiently generic (depending on $\Lambda$). For each Serre weight $\sigma$,
there exists a $(4f+11)$-dimensional cycle $Z_{\sigma}$ in
$\operatorname{\mathrm{Spec}}R_{\overline{\rho}}^{\square}/\varpi$ such that
(1.0.1) holds for any regular Hodge type $\lambda\in\Lambda$ and any regular
tame inertial type $\tau$.

By using patching functors and Corollary 4, we first construct cycles
$Z_{\sigma}$ satisfying (1.0.1) for regular $\lambda$, sufficiently generic
$\tau$, and tame $\overline{\rho}$ (Theorem 6.1.3). Then we algebraically
interpolate $Z_{\sigma}$ for various $\overline{\rho}$ to construct
${\mathcal{Z}}_{\sigma}$ in item (1) following the axiomatic argument in
[LLHLMa, §8.3]. Item (2) (where we do _not_ assume that $\overline{\rho}$ is
tame) essentially follows from item (1) by pulling back the cycles
${\mathcal{Z}}_{\sigma}$ to a versal ring for
${\mathcal{X}}_{{\mathrm{Sym}},K,{\mathrm{red}}}$ at $\overline{\rho}$.

#### The weight part of Serre’s conjecture

Let $F$ be a totally real field of even degree over $\mathbf{Q}$ in which $p$
is unramified. Let ${\mathcal{G}}$ be an inner form of $\mathrm{GSp}_{4}$ over
$F$ which is compact modulo center at infinity and splits at all finite
places. Fix an isomorphism
$\iota:\overline{\mathbf{Q}}_{p}\buildrel\sim\over{\rightarrow}\mathbf{C}$.
Given a Hecke character
$\chi:\mathbb{A}_{F}^{\times}/F^{\times}\rightarrow\mathbf{C}^{\times}$, a
level $U\subset{\mathcal{G}}(\mathbb{A}^{\infty}_{F})$, and a
${\mathcal{O}}[{\mathcal{G}}({\mathcal{O}}_{F}\otimes_{\mathbf{Z}}\mathbf{Z}_{p})]$-module
$W$, we define a space of algebraic automorphic forms $S_{\chi}(U,W)$ to be
the ${\mathcal{O}}$-module of continuous functions
$f:{\mathcal{G}}(F){\backslash}{\mathcal{G}}(\mathbb{A}^{\infty}_{F})\rightarrow
W$ such that $f(zg)=(\iota^{-1}\circ\chi(z))f(g)$ and $f(gu)=u_{p}^{-1}\cdot
f(g)$ for all $z\in
Z(\mathbb{A}_{F}^{\infty}),g\in{\mathcal{G}}(\mathbb{A}^{\infty}_{F})$, and
$u\in U$.

Let $\overline{r}:G_{F}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$ be a
continuous and absolutely irreducible representation which is the mod $p$
reduction of the Galois representation attached to a regular algebraic
cuspidal automorphic representation of ${\mathcal{G}}(\mathbb{A}_{F})$ (or
equivalently, of $\mathrm{GSp}_{4}(\mathbb{A}_{F})$). Then $\overline{r}$
determines a maximal ideal ${\mathfrak{m}}_{\overline{r}}$ of an appropriate
Hecke algebra, and $S_{\chi}(U,\sigma)_{{\mathfrak{m}}_{\overline{r}}}\neq 0$
for some $\chi$, $U$, and Serre weight $\sigma$. In this case, we say $\sigma$
is a _modular_ Serre weight for $\overline{r}$ and write $W(\overline{r})$ for
the set of modular Serre weights for $\overline{r}$. When $\overline{r}$ is
tame at places above $p$, [GHS18] defines a set
$W^{?}(\otimes_{v|p}\overline{r}|_{I_{F_{v}}})$ (see Definition 2.4.7) using
combinatorial recipes, where ${I_{F_{v}}}$ denotes the inertia subgroup at
$v$, and conjectures that
$W(\overline{r})=W^{?}(\otimes_{v|p}\overline{r}|_{I_{F_{v}}})$. We verify
this conjecture under technical genericity assumptions.

######  Theorem 10 (Theorem 6.2.5).

Let $\overline{r}:G_{F}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$ be as above.
Moreover, we assume that $\overline{r}|_{I_{F_{v}}}$ is tame and sufficiently
generic for $v|p$, and $\overline{r}$ satisfies _Taylor–Wiles conditions_
(Definition 5.4.1). Then
$W(\overline{r})=W^{?}(\otimes_{v|p}\overline{r}|_{I_{F_{v}}})$.

Previously, a similar conjecture was made by Herzig–Tilouine [HT13, Conjecture
1] when $F=\mathbf{Q}$ using étale cohomologies of Siegel modular varieties
instead of algebraic automorphic forms on ${\mathcal{G}}$. We expect that our
method can be used to prove the conjecture of Herzig–Tilouine (or its
generalizations to totally real fields) if the conjecture on vanishing of mod
$p$ étale cohomologies of Hilbert–Siegel modular varieties localized at non-
Eisenstein maximal ideal outside middle degree is known (c.f. [Car, Conjecture
4.3]).

Analogous conjectures for a rank $n$ unitary group $U(n)$ over totally real
field were proven under technical assumptions (when $U(n)$ splits at places
above $p$ by [GLS15] for $n=2$, by [LLHLM18, LLHLM20] for $n=3$, and by
[LLHLMa] for general $n$; when $U(n)$ ramifies at places above $p$ by [KM] for
$n=2$). For the group $\mathrm{GSp}_{4}$, [GG12] proved modularity of _obvious
weights_ for $\overline{r}$ ordinary at places above $p$ assuming modularity
of a single obvious weight. Also, see [Yam, Theorem 1.3] for a result obtained
by a different approach. Our result is independent of [GG12, Yam].

#### Congruent Kisin–Taylor–Wiles patching functors

The proofs of Theorem 9 and 10 use patching functors. Patching functors are
first introduced in [EGS15] and have been a central object in the mod $p$ and
$p$-adic Langlands program (see [CEG+16] and [EGH22]). Our innovation in the
global argument is introducing a _congruent family of fixed similitude
patching functors_ (Definition 5.2.4). It is a collection of fixed similitude
patching functors, labelled by an appropriate set of similitude characters,
that are congruent modulo $p$, i.e. they are identified when restricted to
$p$-torsion objects. A single fixed similitude patching functor generalizes a
(weak) patching functor in [CEG+16, LLHLMa] to $\mathrm{GSp}_{4}$ except the
presence of the fixed similitude. The fixed similitude prevents important
applications such as the comparison of the mod $p$ reduction of patched
modules over a crystalline deformation ring and a potentially crystalline
deformation ring. However, a congruent family of fixed similitude patching
functors makes such applications available through the congruence. We remark
that our notion of _congruent_ patching functors can describe Taylor’s “Ihara
avoidance” argument; see §5.4.8.

We construct patching functors for local and global applications. In the local
case, it is a functor from a category of finite ${\mathcal{O}}$-modules with
continuous $\mathrm{GSp}_{4}({\mathcal{O}}_{K})$-action with a fixed central
character to a category of finitely generated modules over a certain
deformation ring. In the global case, we replace the group
$\mathrm{GSp}_{4}({\mathcal{O}}_{K})$ by
$\mathrm{GSp}_{4}({\mathcal{O}}_{F}\otimes_{\mathbf{Z}}\mathbf{Z}_{p})$ where
$F$ is a totally real field. Although the construction of patching functors
depends on non-canonical choices (e.g. Taylor–Wiles primes), these choices can
be made independent of the fixed similitude character. Then the congruence
property of fixed similitude patching functors in the global case follows
naturally. We construct patching functors in the local case using the global
ones, after realizing a given local Galois representation as a restriction of
an automorphic global Galois representation with a minimality assumption,
following [EG14, Appendix A] for $\mathrm{GL}_{n}$. The congruent property is
less obvious in the local case and requires the weight elimination result
(Theorem 5.4.4) and certain potentially crystalline deformation rings with
fixed similitude characters that are formally smooth and congruent modulo $p$
(Theorem 5.1.4). In particular, our congruent patching functors improve a
congruent _pair_ of fixed similitude patching functors introduced in our
previous work with John Enns [EL], where the patching functors in the local
case was not available because the weight elimination result was not
available.

We also incorporate the “Ihara avoidance” argument in our patching functors to
prove the following modularity lifting result.

######  Theorem 11 (Theorem 6.3.1).

Let $r,r^{\prime}:G_{F}\rightarrow\mathrm{GSp}_{4}({\mathcal{O}})$ be
continuous representations that are isomorphic modulo $\varpi$, unramified at
all but finitely many places, and potentially crystalline with regular Hodge
type $\lambda$ and sufficiently generic (depending on $\lambda$) tame inertial
type $\tau$ at places above $p$. We further assume that $r\bmod\varpi$ is tame
at places above $p$ and satisfies Taylor–Wiles assumptions. If $r^{\prime}$ is
automorphic, then $r$ is also automorphic.

#### The inertial local Langlands

For a mildly generic tame inertial type $\tau$, we construct an irreducible
representation $\sigma(\tau)$ of $\mathrm{GSp}_{4}(k)$ in characteristic zero
such that if a smooth representation $\pi$ of $\mathrm{GSp}_{4}(K)$ contains
$\sigma(\tau)$ (viewed as a $\mathrm{GSp}_{4}({\mathcal{O}}_{K})$ via
inflation) as a $\mathrm{GSp}_{4}({\mathcal{O}}_{K})$-subrepresentation, then
$\tau$ is isomorphic to the restriction to inertia of the $L$-parameter of
$\pi$ (Theorem 2.4.1). This result is essential to our global arguments. It is
proven by using the explicit theta correspondence in [GT11b] (in the non-
supercuspidal case) and the depth-zero regular supercuspidal local Langlands
correspondence in [DR09] (in the supercuspidal case).

#### Transfer from $\mathrm{GSp}_{4}$ to $\mathrm{GL}_{4}$

We define natural maps sending Deligne–Lusztig representations and Serre
weights of $\mathrm{GSp}_{4}(k)$ to those of $\mathrm{GL}_{4}(k)$ (Proposition
2.4.10), which might be of independent interests. They are compatible with mod
$p$ reduction under a very mild assumption. In the case of Deligne–Lusztig
representations, it is also compatible with inertial local Langlands in a
suitable sense. Moreover, this also shows that a Serre weight $\sigma$ of
$\mathrm{GSp}_{4}(k)$ is in the set $W^{?}(\overline{\rho}|_{I_{K}})$ of
conjectural Serre weights associated with tame
$\overline{\rho}:G_{K}\rightarrow\mathrm{GSp}_{4}(\mathbf{F})$ if and only if
its transfer to $\mathrm{GL}_{4}(k)$ is in the set of conjectural Serre
weights of $\mathrm{GL}_{4}(k)$ associated with $\overline{\rho}$ viewed as
$\mathrm{GL}_{4}(\mathbf{F})$-valued representation (Corollary 2.4.12).

#### The work of Arthur

In this article, we use Arthur’s multiplicity formula for the discrete
spectrum of $\mathrm{GSp}_{4}$ announced in [Art04] and later proven in [GT19]
using the main results in [Art13]. We also use results from [Mok14, BCGP21]
which rely on Arthur’s formula for $\mathrm{GSp}_{4}$. At the time of writing,
[Art13] is conditional on the twisted weighted fundamental lemma announced in
[CL10], but whose proofs have not appeared, as well as on the references
[A25], [A26], and [A27] in [Art13], which have not appeared publicly.

Specifically, we use [Mok14, Theorem 3.5] and [GT19, Theorem 7.4.1] for the
construction of a Galois representation attached to a regular algebraic
cuspidal automorphic representation of an inner form of $\mathrm{GSp}_{4}$,
and our construction of patching functors relies on it. Among theorems and
corollaries stated above, proofs of Theorem 1 and 3 do not require patching
functors, and all others rely on patching functors (and thus on [Art13]).

#### Organization of the paper

In §2, we first recall basic notions related to the representation theory of
$\mathrm{GSp}_{4}(k)$. Then we prove the inertial local Langlands for
$\mathrm{GSp}_{4}$ in §2.4. In §2.5, we establish several results on mod $p$
reduction of certain representations of $\mathrm{GSp}_{4}(k)$ which will be
used throughout the paper.

In §3, we introduce local models and prove its properties. Our presentation
closely follows [LLHLMa]. When it is possible, we tried to simplify our
argument by deducing results from analogous results for $\mathrm{GL}_{4}$.
After introducing the global affine Grassmannian and universal local models in
§3.1–3.2, we introduce mixed characteristic local models and prove the
unibranch property in §3.3–3.4. In §3.5–3.7, we discuss irreducible components
in the special fiber of mixed characteristic local models. In §3.7, we
classify torus fixed points in each irreducible components under genericity
conditions.

We introduce moduli stacks of symplectic étale $(\varphi,\Gamma)$-modules in
§4.1, of Breuil–Kisin modules in §4.2, and of étale $\varphi$-modules in §4.3.
Then we bring them together to prove our main result on local model diagrams
for potentially crystalline stacks (Theorem 4.4.3) and their for mod $p$
reduction (Theorem 4.5.4). Both results rely on the existence of certain local
lifts whose proofs are postponed to §5.4.

In §5, we discuss our setup for global arguments and construct a congruent
family of fixed similitude patching functors. As applications, we prove the
weight elimination result and the existence of certain local lifts in §5.4.

Finally, we formulate the geometric Breuil–Mézard conjecture and prove its
restricted versions in §6.1. We prove our result on the weight part of a
Serre’s conjecture in §6.2 and a modularity lifting result in §6.3.

### Acknowledgement

The author would like to thank his advisor Florian Herzig for suggesting to
work on the weight part of Serre’s conjecture for $\mathrm{GSp}_{4}$, his
constant encouragement, and numerous comments. The debt that this article owes
to the works of Daniel Le, Bao V. Le Hung, Brandon Levin, and Stefano Morra
will be obvious to the reader, and it is a pleasure to acknowledge this. The
author would like to thank Daniel Le, Bao V. Le Hung, and Stefano Morra for
the discussions and for answering many questions about their work. He would
like to thank Pablo Boixeda Alvarez, Wee Teck Gan, Tasho Kaletha, and
Zhongyipan Lin for helpful correspondence. Also, he would like to thank Shaun
Stevens for pointing out Jamie Lust’s thesis.

Part of the work was carried out during visits at the Ulsan National Institute
of Science and Technology and the Korea Institute of Advanced Science. The
author would like to heartily thank these institutions and Chol Park and
Myungjun Yu for their support. The author is partially supported by the
Ontario Trillium Scholarship.

## Notation

Let $p>2$ be a prime. We let $K/\mathbf{Q}_{p}$ denote a finite field
extension with the ring of integers ${\mathcal{O}}_{K}$ and the residue field
$k$. Throughout this paper except §4.1, we assume $K$ to be unramified of
degree $f$ over $\mathbf{Q}_{p}$. We take $E$ to be a finite extension of
$\mathbf{Q}_{p}$ with ramification degree $e$, the ring of integers
${\mathcal{O}}$, and the residue field $\mathbf{F}$. We assume that $E$ is
sufficiently large unless mentioned otherwise. We let
$[\cdot]:\mathbf{F}^{\times}\rightarrow W(\mathbf{F})^{\times}$ denote the
multiplicative lift.

Let $F$ be a field. We write
$G_{F}:=\operatorname{\mathrm{Gal}}(\overline{F}/F)$ where $\overline{F}$ is a
separable closure of $F$. If $F$ is a non-archimedean local field, we write
$I_{F}\subset G_{F}$ for the inertia subgroup and $W_{F}$ for the Weil group.

Fix a separable closure $\overline{K}$ of $K$ and suppose that
$K/\mathbf{Q}_{p}$ is unramified of degree $f$. We choose $\pi\in\overline{K}$
such that $\pi^{p^{f}-1}=-p$. We denote by
$\omega_{K}:G_{K}\rightarrow{\mathcal{O}}_{K}^{\times}$ the character defined
by $g(\pi)=\omega_{K}(g)\pi$ for $g\in G_{K}$. For an embedding
$\sigma:K\hookrightarrow E$, we write
$\omega_{K,\sigma}=\sigma\circ\omega_{K}$.

Let $\epsilon$ denote the $p$-adic cyclotomic character. If $V$ is a de Rham
representation of $G_{K}$ over $E$, then for each
$\sigma\in\operatorname{\mathrm{Hom}}_{\mathbf{Q}_{p}}(K,E)$, we let
$\operatorname{{\mathrm{HT}}}_{\sigma}(V)$ denote the multiset of Hodge–Tate
weights labelled by $\sigma$ normalized so that
$\operatorname{{\mathrm{HT}}}_{\sigma}(\epsilon)=\left\\{1\right\\}$.

Let $F$ be a number field. For a place $v$ of $F$, we let $F_{v}$ be the
completion of $F$ at $v$ with the ring of integers ${\mathcal{O}}_{F_{v}}$, a
uniformizer $\varpi_{v}$, and the residue field $k_{v}$ of size $q_{v}$. We
write $\operatorname{\mathrm{Frob}}_{F_{v}}$ for a geometric Frobenius element
in $G_{F_{v}}$. We normalize the Artin map
$\operatorname{\mathrm{Art}}_{F_{v}}:F_{v}^{\times}\buildrel\sim\over{\rightarrow}W^{\mathrm{ab}}_{F_{v}}$
so that uniformizers are mapped to geometric Frobenius elements.

Let $G$ be a split connected reductive group over $\mathbf{Z}$. In the body of
the paper, we take either $G=\mathrm{GSp}_{4}$ or $G=\mathrm{GL}_{n}$. We
write $B\subset G$ for a choice of Borel subgroup, $T\subset B$ for a maximal
torus, and $U\subset B$ for the unipotent radical of $B$. Let
$\Phi^{+}\subset\Phi$ be the subset of positive roots in the set of roots for
$(G,B,T)$. We denote by $\Delta$ the set of simple roots. We write $X^{*}(T)$
for the group of characters of $T$ and $X_{*}(T)$ for the group of
cocharacters of $T$. We write $\Lambda_{R}\subset X^{*}(T)$ and
$\Lambda_{R}^{\vee}\subset X_{*}(T)$ for the root lattice and coroot lattice.
We let $W$ denote the Weyl group, $W_{a}$ denote the affine Weyl group, and
$\widetilde{W}$ denote the extended affine Weyl group for $G$.

We write $G^{\vee}$ for the split reductive group over $\mathbf{Z}$ defined by
the root datum $(X_{*}(T),X^{*}(T),\Phi^{\vee},\Phi)$. We write
$T^{\vee}\subset G^{\vee}$ for the induced maximal split torus. We have
isomorphisms $X^{*}(T^{\vee})\simeq X_{*}(T)$ and $X_{*}(T^{\vee})\simeq
X^{*}(T)$. We let $W^{\vee}$, $W_{a}^{\vee}$, and $\widetilde{W}^{\vee}$
denote the Weyl group, affine Weyl group, and extended affine Weyl group for
$G^{\vee}$.

We denote by ${\mathcal{O}}_{p}$ a finite étale $\mathbf{Z}_{p}$-algebra. In
the body of this paper, we take ${\mathcal{O}}_{p}$ to be either
${\mathcal{O}}_{K}$ or ${\mathcal{O}}_{F}\otimes_{\mathbf{Z}}\mathbf{Z}_{p}$
for some totally real field $F$ in which $p$ is unramified. Let
$F_{p}={\mathcal{O}}_{p}[1/p]$. Then $F_{p}$ is isomorphic to a product
$\prod_{v\in S_{p}}F_{v}$ for a finite set $S_{p}$ and finite unramified
extensions $F_{v}/\mathbf{Q}_{p}$. Also there is an isomorphism
${\mathcal{O}}_{p}\simeq\prod_{v\in S_{p}}{\mathcal{O}}_{F_{v}}$ where
${\mathcal{O}}_{F_{v}}$ is the ring of integer of $F_{v}$.

We define
$G_{0}:=\Res_{{\mathcal{O}}_{p}/\mathbf{Z}_{p}}G_{/{\mathcal{O}}_{p}}$ and
$\underline{G}:=(G_{0})_{/{\mathcal{O}}}$. Let $B$ be a choice of Borel
subgroup and $T\subset B$ be a maximal split torus. We define
$B_{0},\underline{B}$ and $T_{0},\underline{T}$ similarly to
$G_{0},\underline{G}$. Let
${\mathcal{J}}=\operatorname{\mathrm{Hom}}_{\mathbf{Z}_{p}}({\mathcal{O}}_{p},{\mathcal{O}})$.
Then $(\underline{G},\underline{B},\underline{T})$ is naturally identified
with
($G^{\mathcal{J}}_{/{\mathcal{O}}},B^{\mathcal{J}}_{/{\mathcal{O}}},T^{\mathcal{J}}_{/{\mathcal{O}}})$.
The root datum of $(\underline{G},\underline{B},\underline{T})$ is given by

$\displaystyle(X^{*}(\underline{T}),X_{*}(\underline{T}),\underline{\Phi},\underline{\Phi}^{\vee})\simeq(X^{*}(T)^{\mathcal{J}},X_{*}(T)^{\mathcal{J}},\Phi^{\mathcal{J}},\Phi^{\vee,{\mathcal{J}}}).$

We have $\underline{\Lambda}_{R}\simeq\Lambda_{R}^{\mathcal{J}}$,
$\underline{W}\simeq W^{\mathcal{J}}$, $\underline{W}_{a}\simeq
W_{a}^{\mathcal{J}}$,
$\underline{\widetilde{W}}\simeq\widetilde{W}^{\mathcal{J}}$, and similarly
for $\underline{\Lambda}_{R}^{\vee}$, $\underline{W}^{\vee}$,
$\underline{W}_{a}^{\vee}$, $\underline{\widetilde{W}}^{\vee}$.

Let $\varphi$ be the absolute Frobenius on ${\mathcal{O}}_{p}/p$ and its lift
to ${\mathcal{O}}_{p}$. If $S$ is a set and
$s=(s_{\sigma})_{\sigma\in{\mathcal{J}}}\in S^{\mathcal{J}}$, then we define
$\pi(s)$ by $\pi(s)_{\sigma}=s_{\sigma\circ\varphi^{-1}}$. When
${\mathcal{O}}_{p}={\mathcal{O}}_{K}$, we fix an embedding
$\sigma_{0}:K\hookrightarrow E$ and define
$\sigma_{j}:=\sigma_{0}\circ\varphi^{-j}$ for $j\in\mathbf{Z}/f\mathbf{Z}$.
This identifies ${\mathcal{J}}$ with $\mathbf{Z}/f\mathbf{Z}$.

We write $\mathrm{Diag}(a_{1},\dots,a_{n})$ for the diagonal matrix in
$\mathrm{GL}_{n}$ with entries $a_{1},\dots,a_{n}$. If $\mu$ is a cocharacter
and $a$ is a scalar, we write $a^{\mu}$ to denote $\mu(a)$. We write
$\operatorname{\mathrm{Ind}}$ for the unnormalized parabolic induction and
$\operatorname{\mathrm{ind}}$ for the compact induction.

##  2\. Types and weights

In this section, we recall basic notions regarding the representation theory
and the extended Weyl group of $\mathrm{GSp}_{4}$ and prove the inertial local
Langlands correspondence for $\mathrm{GSp}_{4}$ (Theorem 2.4.1). We follow
[LLHLMa, §2] closely throughout this section except §2.4.

###  2.1. Preliminaries

####  2.1.1. The group $\mathrm{GSp}_{4}$

Let $G=\mathrm{GSp}_{4}$ be the split reductive group over $\mathbf{Z}$
defined by

$\displaystyle\mathrm{GSp}_{4}(R)=\left\\{g\in\mathrm{GL}_{4}(R)\mid\prescript{t}{}{g}Jg=\operatorname{\mathrm{sim}}(g)J\text{
for some $\operatorname{\mathrm{sim}}(g)\in R^{\times}$}\right\\}$

for any commutative ring $R$, where

$\displaystyle J={\begin{pmatrix}&&&1\\\ &&1&\\\ &-1&&\\\
-1&&&\end{pmatrix}}.$

Then $\operatorname{\mathrm{sim}}:g\mapsto\operatorname{\mathrm{sim}}(g)$
defines a character of $\mathrm{GSp}_{4}$, called the _similitude character_.
Let $W=N(T)/T$ be the Weyl group of $G$. We identify $W$ with the subgroup of
$N(T)$ generated by two simple reflections

$\displaystyle s_{1}:={\begin{pmatrix}&1&&\\\ 1&&&\\\ &&&1\\\
&&1&\end{pmatrix}},\ s_{2}:={\begin{pmatrix}1&&&\\\ &&1&\\\ &-1&&\\\
&&&1\end{pmatrix}}.$

We write $w_{0}=s_{1}s_{2}s_{1}s_{2}$ for the longest element in $W$. We use
the same notation to denote ${\mathcal{J}}$-fold product of $w_{0}$ in
$\underline{W}$.

Let $B$ be the upper triangular Borel subgroup of $\mathrm{GSp}_{4}$. For any
subset $A\subset\\{s_{1},s_{2}\\}$, we have a standard parabolic subgroup
$P_{A}$ generated by $A$ and $B$. When $A=\\{s_{1}\\}$, we call $S:=P_{A}$ the
_Siegel parabolic subgroup_ , and when $A=\\{s_{2}\\}$, we call $Q:=P_{A}$ the
_Klingen parabolic subgroup_. If $P$ is a standard parabolic subgroup with
unipotent radical $N$, we write $\overline{P}$ and $\overline{N}$ to denote
the opposite parabolic and its unipotent radical.

We identify the character group $X^{*}(T)$ with $\mathbf{Z}^{3}$ by defining
the character corresponding to $(a,b;c)\in\mathbf{Z}^{3}$ by

$\displaystyle(a,b;c):\mathrm{Diag}(x,y,zy^{-1},zx^{-1})\mapsto
x^{a}y^{b}z^{c}.$

Similarly, we identify the cocharacter group $X_{*}(T)$ with $\mathbf{Z}^{3}$
by

$\displaystyle(a,b;c):x\mapsto\mathrm{Diag}(x^{a},x^{b},x^{c-b},x^{c-a}).$

Sets of simple roots and coroots are given by

$\displaystyle\Delta=\left\\{\alpha_{1}=(1,-1;0),\alpha_{2}=(0,2;-1))\right\\}$
$\displaystyle\Delta^{\vee}=\left\\{\alpha_{1}^{\vee}=(1,-1;0),\alpha_{2}^{\vee}=(0,1;0)\right\\}$

Sets of positive roots and coroots are given by

$\displaystyle\Phi^{+}$
$\displaystyle=\left\\{\alpha_{1},\alpha_{2},\alpha_{3}=2\alpha_{1}+\alpha_{2},\alpha_{4}=\alpha_{1}+\alpha_{2}\right\\}$
$\displaystyle\Phi^{\vee,+}$
$\displaystyle=\left\\{\alpha_{1}^{\vee},\alpha_{2}^{\vee},\alpha_{3}^{\vee},\alpha_{4}^{\vee}\right\\}$

where $\alpha_{3}=\alpha_{1}+\alpha_{2}$, $\alpha_{4}=2\alpha_{1}+\alpha_{2}$,
$\alpha_{3}^{\vee}=\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}$, and
$\alpha_{4}^{\vee}=\alpha_{1}^{\vee}+\alpha_{2}^{\vee}$.

The dual group of $\mathrm{GSp}_{4}$ is isomorphic to $\mathrm{GSp}_{4}$ by an
exceptional isomorphism. We often write $\mathrm{GSp}_{4}^{\vee}$ (and
$T^{\vee}$ for its diagonal maximal torus) to emphasize that we are working on
the dual side. We fix the duality isomorphism by

$\displaystyle(X^{*}(T),X_{*}(T),\Phi,\Phi^{\vee})$
$\displaystyle\buildrel\sim\over{\rightarrow}(X_{*}(T^{\vee}),X^{*}(T^{\vee}),\Phi^{\vee},\Phi)$
$\displaystyle\phi:X^{*}(T)$ $\displaystyle\xrightarrow{}X_{*}(T^{\vee})$
$\displaystyle(a,b;c)$ $\displaystyle\mapsto(a+b+c,a+c;a+b+2c).$

Then $\phi$ maps $\alpha_{1}$ to $\alpha_{2}^{\vee}$ and $\alpha_{2}$ to
$\alpha_{1}^{\vee}$. We also write a map
$\phi:W\buildrel\sim\over{\rightarrow}W^{\vee}$ sending $s_{1}\mapsto s_{2}$,
$s_{2}\mapsto s_{1}$, so that

$\displaystyle\phi(w(\lambda))=\phi(w)(\phi(\lambda))$

for any $w\in W$ and $\lambda\in X^{*}(T)$. We fix an element $\eta=(2,1;0)\in
X^{*}(T)$. We often use the same notation to denote the ${\mathcal{J}}$-fold
product of $\eta$ in $X^{*}(\underline{T})$ and its image in
$X_{*}(\underline{T}^{\vee})$ under $\phi$.

####  2.1.2. The group $\mathrm{GL}_{n}$

If $G=\mathrm{GL}_{n}$, we add subscript $n$ to the objects introduced above
when we need to distinguish them from the case $G=\mathrm{GSp}_{4}$. For
example, we write $T=T_{n},B=B_{n},U=U_{n},W=W_{n}$. We identify
$X^{*}(T_{n})\simeq X_{*}(T_{n})\simeq\mathbf{Z}^{n}$ in the standard way. We
fix an element $\eta^{\prime}=(n-1,n-2,\dots,0)\in X^{*}(T_{n})$ and use the
same notation for its ${\mathcal{J}}$-fold product in
$X^{*}(\underline{T}_{n})$.

####  2.1.3. Affine Weyl group

Let $G$ be a split reductive group over $\mathbf{Z}$. Recall that the affine
Weyl group $W_{a}\simeq\Lambda_{R}\rtimes W$ and the extended Weyl group
$\widetilde{W}\simeq X^{*}(T)\rtimes W$ for $G$. Similarly, we write
$W^{\vee}_{a}\simeq\Lambda^{\vee}_{R}\rtimes W^{\vee}$ and
$\widetilde{W}^{\vee}\simeq X_{*}(T^{\vee})\rtimes W^{\vee}$ for the affine
Weyl group and extended Weyl group for $G^{\vee}$.

Let ${\mathcal{A}}$ be the set of alcoves of
$X^{*}(T)\otimes_{\mathbf{Z}}\mathbf{R}$. We denote by $A_{0}$ the dominant
base alcove with respect to our choice of the Borel subgroup $B$. The group
$W_{a}$ acts simply transitively on ${\mathcal{A}}$, and the choice of $A_{0}$
defines a Bruhat order on $W_{a}$ which we denote by $\leq$. We also recall
the upper arrow ordering on the set ${\mathcal{A}}$ (see, [Jan03, §II.6.5]).
This induces an ordering on $W_{a}$ which we denote by $\uparrow$.

Let $\Omega\leq\widetilde{W}$ be the subgroup stabilizing $A_{0}$. Then we
have $\widetilde{W}\simeq W_{a}\rtimes\Omega$. We extend a Bruhat order and
upper arrow order to $\widetilde{W}$ by: for
$\widetilde{w}_{1},\widetilde{w}_{2}\in W_{a}$ and
$\delta_{1},\delta_{2}\in\Omega$, when $\delta_{1}=\delta_{2}$,
$\widetilde{w}_{1}\delta_{1}\leq\widetilde{w}_{2}\delta_{2}$ (resp.
$\widetilde{w}_{1}\delta_{1}\uparrow\widetilde{w}_{2}\delta_{2}$) if and only
if $\widetilde{w}_{1}\leq\widetilde{w}_{2}$ (resp.
$\widetilde{w}_{1}\uparrow\widetilde{w}_{2}$), and when
$\delta_{1}\neq\delta_{2}$, $\widetilde{w}_{1}\delta_{1}$,
$\widetilde{w}_{2}\delta_{2}$ are incomparable.

We define a Bruhat order on $W_{a}^{\vee}$ induced by the antidominant base
alcove and extend to $\widetilde{W}^{\vee}$ as in the previous paragraph. We
define a bijection

$\displaystyle(-)^{*}:\widetilde{W}$
$\displaystyle\rightarrow\widetilde{W}^{\vee}$
$\displaystyle\widetilde{w}=t_{\nu}w$
$\displaystyle\mapsto\widetilde{w}^{*}:=\phi(w)^{-1}t_{\phi(\nu)}.$

This is an antihomomorphism of groups and preserves Bruhat ordering on both
sides. We also write $(-)^{*}$ for its inverse, i.e. if
$\widetilde{w}\in\widetilde{W}$ and $\widetilde{z}=\widetilde{w}^{*}$, then
$\widetilde{z}^{*}=\widetilde{w}$.

######  Definition 2.1.4.

We define the _admissible set_ associated to $\lambda_{0}\in X^{*}(T)$ as

$\displaystyle{\mathrm{Adm}}(\lambda_{0}):=\left\\{\widetilde{w}\in\widetilde{W}\mid\widetilde{w}\leq
t_{w\lambda_{0}}\text{ for some $w\in W$}\right\\}.$

If $\lambda\in X^{*}(\underline{T})$, we define
${\mathrm{Adm}}(\lambda)=\prod_{j\in{\mathcal{J}}}{\mathrm{Adm}}(\lambda_{j})$

We similarly define ${\mathrm{Adm}}^{\vee}(\lambda)$ for $\lambda\in
X_{*}(\underline{T}^{\vee})$. The map $(-)^{*}$ induces a bijection between
${\mathrm{Adm}}(\phi^{-1}(\lambda))$ and ${\mathrm{Adm}}^{\vee}(\lambda)$.

The _$p$ -dot action_ of $t_{\nu}w\in\widetilde{W}$ on $\lambda\in
X^{*}(T)\otimes\mathbf{R}$ is defined as

$\displaystyle t_{\nu}w\cdot\lambda:=w(\lambda+\eta)-\eta+p\nu$

A _$p$ -alcove_ is a connected component of the complement

$\displaystyle\left(X^{*}(T)\otimes\mathbf{R}\right)\backslash\cup_{\alpha\in\Phi,m\in\mathbf{Z}}\left\\{x\mid\langle
x+\eta,\alpha^{\vee}\rangle=pm\right\\}.$

We let $C_{0}$ denote the dominant base $p$-alcove, i.e.

$\displaystyle C_{0}:=$ $\displaystyle\left\\{x\in
X^{*}(T)\otimes\mathbf{R}\mid 0<\langle
x+\eta,\alpha^{\vee}\rangle<p,\forall\alpha\in\Phi^{+}\right\\}.$

Similarly, we define $p$-alcoves in $X^{*}(\underline{T})\otimes\mathbf{R}$
and denote the dominant base $p$-alcove by $\underline{C}_{0}$.

We say that an alcove $A$ (resp. a $p$-alcove $C$) is _restricted (resp.
$p$-restricted)_ if for all $x\in A$ (resp. $x\in C$) and $\alpha\in\Phi^{+}$,
$0<\langle x,\alpha^{\vee}\rangle<1$ (resp. $0<\langle
x+\eta,\alpha^{\vee}\rangle<p$). We define

$\displaystyle\widetilde{W}^{+}$
$\displaystyle:=\\{\widetilde{w}\in\widetilde{W}\mid\widetilde{w}(A_{0})\text{
is dominant}\\}$ $\displaystyle\widetilde{W}^{+}_{1}$
$\displaystyle:=\\{\widetilde{w}\in\widetilde{W}\mid\widetilde{w}(A_{0})\text{
is restricted}\\}$

and $\underline{\widetilde{W}}^{+}:=\widetilde{W}^{+,{\mathcal{J}}}$ and
$\underline{\widetilde{W}}^{+}_{1}:=\widetilde{W}_{1}^{+,{\mathcal{J}}}$.

We use above notations for $G=\mathrm{GSp}_{4}$. For $G=\mathrm{GL}_{n}$, we
again add subscript $n$ when we need to distinguish them from the case of
$\mathrm{GSp}_{4}$. For example, we write $\widetilde{W}_{n}$,
$\widetilde{W}^{+}_{n}$, and $\widetilde{W}^{+}_{1,n}$ for
$\widetilde{W},\widetilde{W}^{+}$, and $\widetilde{W}^{+}_{1}$ respectively.

####  2.1.5. Transfer map

We write
$\operatorname{{\mathrm{std}}}:\mathrm{GSp}_{4}^{\vee}\xrightarrow{}\mathrm{GL}_{4}^{\vee}$
for the standard representation. It induces a map between the tori
$\underline{T}^{\vee}\xrightarrow{}\underline{T}_{4}^{\vee}$. We also write
the induced map between cocharacter groups by
$\operatorname{{\mathrm{std}}}:X_{*}(\underline{T}^{\vee})\xrightarrow{}X_{*}(\underline{T}_{4}^{\vee})$.
Explicitly, we have

$\displaystyle(a,b;c)\xmapsto{\operatorname{{\mathrm{std}}}}(a,b,c-b,c-a).$

We also define
${\mathcal{T}}:X^{*}(\underline{T})\xrightarrow{}X^{*}(\underline{T}_{4})$ to
be a unique map which makes the following diagram commute

${X^{*}(\underline{T})}$${X^{*}(\underline{T}_{4})}$${X_{*}(\underline{T}^{\vee})}$${X_{*}(\underline{T}_{4}^{\vee})}$$\scriptstyle{{\mathcal{T}}}$$\scriptstyle{\phi}$$\scriptstyle{\operatorname{{\mathrm{std}}}}$

Explicitly, ${\mathcal{T}}$ maps $(a,b;c)\in
X^{*}(\underline{T})\mapsto(a+b+c,a+c,b+c,c)\in X^{*}(\underline{T}_{4})$.

The map $\operatorname{{\mathrm{std}}}$ also induces a map between
$\underline{W}^{\vee}$ and $\underline{W}_{4}^{\vee}$. We again denote by
${\mathcal{T}}:\underline{W}\rightarrow\underline{W}_{4}$ the composition
$\operatorname{{\mathrm{std}}}\circ\phi$ followed by the identification
$\underline{W}_{4}^{\vee}=\underline{W}_{4}$. We have
${\mathcal{T}}(w\lambda)={\mathcal{T}}(w){\mathcal{T}}(\lambda)$ for all
$\lambda\in X^{*}(\underline{T})$, $w\in\underline{W}$. As a result,
${\mathcal{T}}$ extends to a map between $\widetilde{W}$ and
$\widetilde{W}_{4}$. Note that ${\mathcal{T}}$ is equal to the composition

$\displaystyle\widetilde{W}\xrightarrow{(-)^{*}}\widetilde{W}^{\vee}\xrightarrow{\operatorname{{\mathrm{std}}}}\widetilde{W}_{4}^{\vee}\xrightarrow{(-)^{*}}\widetilde{W}_{4}.$

The image of ${\mathcal{T}}$ in $\underline{\widetilde{W}}_{4}$ can be
interpreted as the invariant of a certain involution. Let
$w^{\prime}_{0}\in\underline{W}_{4}$ be the longest element. Let
$\underline{\widetilde{W}}_{4,a,\mathbf{Q}}=(X^{*}(T_{4})\otimes_{\mathbf{Z}}\mathbf{Q})\rtimes
W_{4}$. Define
$\Theta:\underline{\widetilde{W}}_{4,a,\mathbf{Q}}\xrightarrow{}\underline{\widetilde{W}}_{4,a,\mathbf{Q}}$
to be the unique group homomorphism satisfying

$\displaystyle\Theta(s)=w^{\prime}_{0}sw^{\prime}_{0},\ \
\Theta(a,b,c,d)=\frac{a+b+c+d}{2}(1,1,1,1)-w^{\prime}_{0}(a,b,c,d)$

for $s\in\underline{W}_{4}$ and $(a,b,c,d)\in X^{*}(\underline{T}_{4})$. Then
$\widetilde{w}\in\underline{\widetilde{W}}_{4}$ is in the image of
${\mathcal{T}}$ if and only if $\Theta(\widetilde{w})=\widetilde{w}$.

######  Lemma 2.1.6.

The map
${\mathcal{T}}:\underline{\widetilde{W}}\rightarrow\underline{\widetilde{W}}_{4}$
respects Bruhat ordering, i.e. for all
$\widetilde{w}_{1},\widetilde{w}_{2}\in\underline{\widetilde{W}}$,

$\displaystyle\text{$\widetilde{w}_{1}\leq\widetilde{w}_{2}$ in
$\underline{\widetilde{W}}$ if and only if
${\mathcal{T}}(\widetilde{w}_{1})\leq{\mathcal{T}}(\widetilde{w}_{2})$ in
$\underline{\widetilde{W}}_{4}$}.$

Moreover, for $\lambda\in X^{*}(\underline{T})$, we have
${\mathrm{Adm}}(\lambda)\overset{{\mathcal{T}}}{\simeq}{\mathrm{Adm}}({\mathcal{T}}(\lambda))^{\Theta}$.

###### Proof.

The first claim is the result of Kottwitz–Rapoport [KR00, Proposition 2.3].
The second claim is the result of Haines–Châo [HC02, Proposition 5]. Note that
in _loc. cit._ , the set $\mathrm{Perm}(\mu)$ is equal to
${\mathrm{Adm}}(\mu)$ (for the group $\mathrm{GL}_{n}$) by Theorem 1 in _loc.
cit._. ∎

####  2.1.7. Genericity

For a character $\lambda\in X^{*}(T)$ (resp. a cocharacter $\lambda\in
X_{*}(T^{\vee})$), we define
$h_{\lambda}=\max_{\alpha\in\Phi}\\{\langle\lambda,\alpha^{\vee}\rangle\\}$
(resp.
$h_{\lambda}=\max_{\alpha\in\Phi}\\{\langle\phi^{-1}(\lambda),\alpha^{\vee}\rangle\\})$.

######  Definition 2.1.8 (cf. Definition 2.1.10 in [LLHLMa]).

Let $\lambda\in X^{*}(T)$ be a character and $m\in Z_{\geq 0}$.

  1. (1)

We say that $\lambda$ is _$m$ -deep in a $p$-alcove $C$ _ if

$\displaystyle
n_{\alpha}p+m<\langle\lambda+\eta,\alpha^{\vee}\rangle<(n_{\alpha}+1)p-m$

for all $\alpha\in\Phi^{+}$ where $C=\left\\{\lambda\in
X^{*}(T)\otimes\mathbf{R}\mid
n_{\alpha}p<\langle\lambda+\eta,\alpha^{\vee}\rangle<(n_{\alpha}+1)p,\forall\alpha\in\Phi^{+}\right\\}$

  2. (2)

We say $\lambda$ is _$m$ -deep_ if it is $m$-deep in some $p$-alcove $C$.

  3. (3)

For $\widetilde{w}=wt_{\nu}\in\widetilde{W}$, we say that $\widetilde{w}$ is
_$m$ -generic_ if $\nu-\eta$ is $m$-deep.

  4. (4)

For $\widetilde{w}=wt_{\nu}\in\widetilde{W}$, we say that $\widetilde{w}$ is
$m$-small if $h_{\nu}\leq m$.

  5. (5)

For $\widetilde{z}\in\widetilde{W}^{\vee}$, we say that $\widetilde{z}$ is
_$m$ -generic_ (resp.  _$m$ -small_) if $\widetilde{z}^{*}$ is $m$-generic
(resp. $m$-small).

  6. (6)

For
${\mathbf{a}}=({\mathbf{a}}_{1},{\mathbf{a}}_{2},{\mathbf{a}}_{3})\in\mathbf{F}^{3}$,
we say that ${\mathbf{a}}$ is $m$-generic if

$\displaystyle\\{{\mathbf{a}}_{1},{\mathbf{a}}_{2},{\mathbf{a}}_{1}+{\mathbf{a}}_{2},{\mathbf{a}}_{1}-{\mathbf{a}}_{2}\\}\cap\\{-m,-m+1,\dots,m-1,m\\}=\emptyset$

where $\\{-m,-m+1,\dots,m-1,m\\}$ is considered as a subset of $\mathbf{F}$
using $\mathbf{Z}\rightarrow\mathbf{F}$.

  7. (7)

Let $P(X_{1},X_{2},X_{3})\in\mathbf{Z}[X_{1},X_{2},X_{3}]$ be a polynomial and
let $R$ be a commutative ring. We say that ${\mathbf{a}}\in R^{3}$ (resp.
${\mathbf{a}}=({\mathbf{a}}_{j})_{{j\in{\mathcal{J}}}}\in(R^{3})^{\mathcal{J}}$)
is _$P$ -generic_ if $P({\mathbf{a}})\bmod p$ (resp. $P({\mathbf{a}}_{j})\bmod
p$) is in $(R/p)^{\times}$ (resp. for all ${j\in{\mathcal{J}}}$).

  8. (8)

We define $P_{m}(X_{1},X_{2},X_{3})\in\mathbf{Z}[X_{1},X_{2},X_{3}]$ to be the
polynomial

$\displaystyle
P_{m}(X_{1},X_{2},X_{3})=\prod_{a=1}^{m}(X_{1}-X_{2}-a)(X_{2}-a)(X_{1}+X_{2}+a).$

Note that $\lambda-\eta\in C_{0}$ is $m$-deep if and only if $\lambda$ (viewed
as an element in $\mathbf{Z}^{3}$) is $P_{m}$-generic.

######  Lemma 2.1.9.

If $\lambda\in X^{*}(T)$ is $m$-deep, then ${\mathcal{T}}(\lambda)\in
X^{*}(T_{4})$ is $m$-deep in the sense of [LLHLMa, Definition 2.1.10].
Similarly, if $\widetilde{w}\in\widetilde{W}$ is $m$-generic (resp.
$m$-small), then ${\mathcal{T}}(\widetilde{w})\in\widetilde{W}_{4}$ is
$m$-generic (resp. $m$-small).

###### Proof.

This follows from a direct computation. ∎

###  2.2. Serre weights and Deligne–Lusztig representations

Let $G$ be either $\mathrm{GSp}_{4}$ or $\mathrm{GL}_{n}$. Recall that we have
a finite étale $\mathbf{Z}_{p}$-algebra ${\mathcal{O}}_{p}$ and
$G_{0}=\Res_{{\mathcal{O}}_{p}/\mathbf{Z}_{p}}G_{/{\mathcal{O}}_{p}}$. We
remark that $G_{0}(\mathbf{F}_{p})=G(k)$ when
${\mathcal{O}}_{p}={\mathcal{O}}_{K}$, and
$G_{0}(\mathbf{F}_{p})=G({\mathcal{O}}_{F}/p)$ when
${\mathcal{O}}_{p}={\mathcal{O}}_{F}\otimes_{\mathbf{Z}}\mathbf{Z}_{p}$ with
$F$ a totally real field in which $p$ is unramified.

####  2.2.1. Serre weights

A _Serre weight_ of $G_{0}(\mathbf{F}_{p})$ is an irreducible
$\mathbf{F}$-representation of $G_{0}(\mathbf{F}_{p})$. Recall the set of
$p$-restricted dominant weights

$\displaystyle X^{*}_{1}(\underline{T})=\left\\{\lambda\in
X^{*}(\underline{T})\mid\forall\alpha^{\vee}\in\underline{\Delta}^{\vee},0\leq\langle\lambda,\alpha^{\vee}\rangle\leq
p-1\right\\}.$

For $\lambda\in X^{*}_{1}(\underline{T})$, we write $L(\lambda)$ for the
unique up to isomorphism irreducible
$\underline{G}_{/\mathbf{F}}$-representation of highest weight $\lambda$. Then
$F(\lambda):=L(\lambda)|_{\underline{G}(\mathbf{F}_{p})}$ is a Serre weight.
Moreover, we have the following bijection ([GHS18, Lemma 9.2.4])

$\displaystyle\frac{X_{1}^{*}(\underline{T})}{(p-\pi)X^{0}(T)}$
$\displaystyle\xrightarrow{\sim}\left\\{\text{Serre weights of
$G_{0}(\mathbf{F}_{p})$}\right\\}/\simeq$ $\displaystyle\lambda$
$\displaystyle\mapsto F(\lambda)$

where $X^{0}(\underline{T}):=\left\\{\lambda\in
X^{*}(\underline{T})\mid\forall\alpha\in\underline{\Phi},\langle\lambda,\alpha^{\vee}\rangle=0\right\\}$.
For an integer $m\geq 0$, we say a Serre weight $\sigma$ is _$m$ -deep_ if
$\sigma\simeq F(\lambda)$ for some $m$-deep $\lambda$.

Let $X_{\mathrm{reg}}(\underline{T})\subset X^{*}_{1}(\underline{T})$ be the
subset of $\lambda$ such that $0\leq\langle\lambda,\alpha^{\vee}\rangle<p-1$
for all $\alpha^{\vee}\in\underline{\Delta}^{\vee}$. We say a Serre weight
$\sigma$ is _regular_ if $\sigma\simeq F(\lambda)$ for some $\lambda\in
X_{\mathrm{reg}}(\underline{T})$.

Let $\widetilde{w}_{h}:=w_{0}t_{-\eta}$. We denote by ${\mathcal{R}}$ an
endomorphism on $X_{\mathrm{reg}}(\underline{T})$ given by
$\lambda\mapsto\widetilde{w}_{h}\cdot\lambda$. It preserves $X^{0}(T)$ and
induces a map on the set of regular Serre weights by
${\mathcal{R}}(F(\lambda)):=F({\mathcal{R}}(\lambda))$.

######  Definition 2.2.2.

Let $\omega-\eta\in\underline{C}_{0}\cap X^{*}(\underline{T})$ and
$\widetilde{w}_{1}\in\underline{\widetilde{W}}_{1}^{+}$. We define

$\displaystyle
F_{(\widetilde{w}_{1},\omega)}:=F(\pi^{-1}(\widetilde{w}_{1})\cdot(\omega-\eta)).$

Consider the equivalence relation
$(\widetilde{w}_{1},\omega)\sim(t_{\nu}\widetilde{w}_{1},\omega-\nu)$ for all
$\nu\in X^{0}(\underline{T})$. The map $(\widetilde{w}_{1},\omega)\mapsto
F_{(\widetilde{w}_{1},\omega)}$ sends equivalent pairs to the same Serre
weights. We call the equivalence class of $(\widetilde{w}_{1},\omega)$ as a
_lowest alcove presentation_ of $F_{(\widetilde{w}_{1},\omega)}$. Let
$m\in\mathbf{Z}_{\geq 0}$. If $\omega-\eta$ is $m$-deep in
$\underline{C}_{0}$, we say that $(\widetilde{w}_{1},\omega)$ is _$m$
-generic_ lowest alcove presentation of $F_{(\widetilde{w}_{1},\omega)}$.

####  2.2.3. Deligne–Lusztig representations

Let $(s,\mu)\in\underline{W}\times X^{*}(\underline{T})$. By [GHS18,
Proposition 9.2.1 and 9.2.2], we can attach a Deligne–Lusztig representation
$R_{s}(\mu)$ of $G_{0}(\mathbf{F}_{p})$. By [DL76, Proposition 10.10], it is a
genuine representation of $G_{0}(\mathbf{F}_{p})$ if $(s,\mu)$ is a good pair
(see [LLHL19, §2.2]).

######  Definition 2.2.4.

Let $R$ be a Deligne–Luszitg representation and $m\in\mathbf{Z}_{\geq 0}$.

  1. (1)

We say that $(s,\mu-\eta)$ is a _$m$ -generic lowest alcove presentation_ of
$R$ if $R\simeq R_{s}(\mu)$ and $\mu-\eta$ is $m$-deep in $\underline{C}_{0}$.

  2. (2)

We say that $R$ is _$m$ -generic_ if there exists a $m$-generic lowest alcove
presentation of $R$.

######  Remark 2.2.5.

If $\mu-\eta$ is $0$-deep in $\underline{C}_{0}$, then $(s,\mu)$ is a good
pair. Moreover if $\mu-\eta$ is $1$-deep in $\underline{C}_{0}$, then
$R_{s}(\mu)$ is irreducible by [DL76, Theorme 6.8].

The following Proposition gives the Jordan–Hölder factors of the reduction of
Deligne–Lusztig representations in generic cases.

######  Proposition 2.2.6 (Proposition 2.3.6 in [LLHLMa]).

Let $(s,\mu-\eta)$ be a lowest alcove presentation of $R_{s}(\mu)$ that is
6-generic if $G=\mathrm{GSp}_{4}$ and $2(n-1)$-generic if $G=\mathrm{GL}_{n}$.
For $\lambda\in X^{*}_{1}(\underline{T})$,
$F(\lambda)\in\mathrm{JH}(\overline{R_{s}(\mu)})$ if and only if there exists
$\widetilde{w}=wt_{-\nu}\in\underline{\widetilde{W}}$ such that

$\displaystyle\widetilde{w}\cdot(\mu-\eta+s\pi(\nu)))\uparrow\widetilde{w}_{h}\cdot\lambda$

and
$\widetilde{w}\cdot\underline{C}_{0}\uparrow\widetilde{w}_{h}\cdot\underline{C}_{0}$.

######  Definition 2.2.7.

Let $w\in\underline{W}$. Let $\nu\in X^{*}(\underline{T})$ be an element up-
to-$X^{0}(\underline{T})$ uniquely determined by the condition
$wt_{-\nu}\in\underline{\widetilde{W}}_{1}$. We define
$\widehat{w}:=wt_{-\nu}\bmod X^{0}(\underline{T})$.

By abuse of notation, we also let $\widehat{w}$ denote a representative in its
class.

######  Definition 2.2.8.

Let $R\simeq R_{s}(\mu)$ be a Deligne–Lusztig representation that is 6-generic
if $G=\mathrm{GSp}_{4}$ and $2(n-1)$-generic if $G=\mathrm{GL}_{n}$. By
Proposition 2.2.6, we can define a function

$\displaystyle F_{R}:\underline{W}$
$\displaystyle\rightarrow\mathrm{JH}(\overline{R_{s}(\mu)})$ $\displaystyle w$
$\displaystyle\mapsto
F(\widetilde{w}_{h}^{-1}\pi^{-1}(\widehat{w})\cdot(\mu-\eta+s(\widehat{w}^{-1}(0)))).$

Note that this does not depend on the choice of $\widehat{w}$. The map $F_{R}$
depends on the choice of the lowest alcove presentation, which will be clear
in the context. So we suppress the dependence on it in the notation.

Serre weights in the image of $F_{R}$ are called outer weights in [LLHLM20].

###  2.3. Tame inertial $L$-parameters

Recall that we take ${\mathcal{O}}_{p}$ to be a finite étale
$\mathbf{Z}_{p}$-algebra. It is isomorphism to $\prod_{v\in
S_{p}}{\mathcal{O}}_{v}$ where $S_{p}$ is a finite set and ${\mathcal{O}}_{v}$
is the ring of intergers in some finite unramified extension
$F_{v}/\mathbf{Q}_{p}$. Following [LLHLMa, §1.8.2], we have the following
definitions.

######  Definition 2.3.1.

Let $A$ be a topolocigal ${\mathcal{O}}$-algebra.

  1. (1)

An _$L$ -homomorphism over $A$_ is a continuous homomorphism
$W_{\mathbf{Q}_{p}}\rightarrow\prescript{L}{}{\underline{G}}(A)$. An _$L$
-parameter over $A$_ is a $\underline{G}^{\vee}(A)$-conjugacy class of
$L$-homomorphisms. When $A$ is finite, any $L$-homomorphism extends to
$G_{\mathbf{Q}_{p}}$.

  2. (2)

An _inertial $L$-homomorphism over $A$_ is a continuous homomorphism
$I_{\mathbf{Q}_{p}}\rightarrow\underline{G}^{\vee}(A)$ which has open kernel
and extends to an $L$-homomorphism over $A$. An _inertial $L$-parameter over
$A$_ is a $\underline{G}^{\vee}(A)$-conjugacy class of inertial
$L$-homomorphism.

  3. (3)

An _inertial type for $K$ over $A$_ is a $G^{\vee}(A)$-conjugacy class of
homomorphisms $I_{K}\rightarrow G^{\vee}(A)$ which has open kernel and extends
to homomorphisms $W_{K}\rightarrow G^{\vee}(A)$.

Note that a choice of an isomorphism
$\overline{F}_{v}\simeq\overline{\mathbf{Q}}_{p}$ for each $v\in S_{p}$
induces an embedding $G_{F_{v}}\hookrightarrow G_{\mathbf{Q}_{p}}$. This
induces a bijection between $L$-homomorphisms over $A$ and collections of
continuous homomorphisms $W_{F_{v}}\rightarrow G^{\vee}(A)$ indexed by
$S_{p}$. Once we take $\underline{G}^{\vee}(A)$-conjugacy classes and
$G^{\vee}(A)$-conjugacy classes respectively, the bijection is independent of
the choice of isomorphisms. The same bijection holds for inertial
$L$-parameters and collections of inertial types for $F_{v}$ indexed by
$S_{p}$. In particular, when ${\mathcal{O}}_{p}={\mathcal{O}}_{K}$, the
inertial $L$-parameter over $A$ is equivalent to an inertial type for $K$ over
$A$. When $A$ is finite, one can replace Weil groups in the definition by
Galois groups.

Let $(s,\mu)\in\underline{W}\times X^{*}(\underline{T})$ and
$s^{\prime}={\mathcal{T}}(s)$, $\mu^{\prime}={\mathcal{T}}(\mu)$. Then
[LLHL19, Definition 2.2.1] defines a tame inertial $L$-parameter

$\displaystyle\tau(s^{\prime},\mu^{\prime}+\eta^{\prime}):I_{\mathbf{Q}_{p}}\rightarrow\underline{T}_{4}^{\vee}({\mathcal{O}}).$

By our choice of $(s^{\prime},\mu^{\prime})$, this factors through
$\underline{T}^{\vee}({\mathcal{O}})\subset\underline{T}_{4}^{\vee}({\mathcal{O}})$,
and we let $\tau(s,\mu+\eta)$ denote the induced tame inertial $L$-parameter
valued in $\underline{T}^{\vee}({\mathcal{O}})$.

Let $F^{*}$ denote the endomorphism $p\pi^{-1}$ on
$X_{*}(\underline{T}^{\vee})$, and $d\geq 1$ be an integer such that
$(F^{*}\circ s^{-1})^{d}=p^{d}$. By [LLHLMa, §2.4], we have the following
explicit description:

$\displaystyle\tau(s,\mu+\eta):=\left(\sum_{i=0}^{d-1}(F^{*}\circ\phi(s)^{-1})^{i}(\phi(\mu+\eta))\right)(\omega_{d}):I_{\mathbf{Q}_{p}}\xrightarrow{}\underline{T}^{\vee}({\mathcal{O}}).$

When ${\mathcal{O}}_{p}={\mathcal{O}}_{K}$, we also write $\tau(s,\mu+\eta)$
to denote the corresponding tame inertial type for $K$. By following Example
2.4.1 of _loc. cit._ , we write $s_{\tau}:=s_{0}s_{1}\dots s_{f-1}\in W$,
$r:=\absolutevalue{s_{\tau}}$, and
$\textbf{a}^{(0)}:=(\sum^{f-1}_{j=0}(F^{*}\circ s^{-1})^{j}(\mu+\eta))_{0}\in
X^{*}(T)$. Then we have

$\displaystyle\tau(s,\mu+\eta)=\left(\sum_{k=0}^{r-1}p^{fk}\phi(s_{\tau})^{-k}\left(\phi\left(\textbf{a}^{0}\right)\right)\right)(\omega_{fr}):I_{K}\rightarrow
T^{\vee}({\mathcal{O}}).$

Note that base change $-\otimes_{\mathcal{O}}E$ and
$-\otimes_{\mathcal{O}}\mathbf{F}$ induce bijections between tame inertial
$L$-parameters over ${\mathcal{O}}$, $E$, and $\mathbf{F}$. For a tame
inertial $L$-parameter $\tau$ over ${\mathcal{O}}$ or $E$, we write
$\overline{\tau}$ for the corresponding tame inertial $L$-parameter over
$\mathbf{F}$.

######  Definition 2.3.2.

Let $\tau$ (resp. $\overline{\tau}$) be a tame inertial $L$-parameter over $E$
(resp. $\mathbf{F}$). Let $m\in\mathbf{Z}_{\geq 0}$.

  1. (1)

A pair $(s,\mu)\in\underline{W}\times X^{*}(\underline{T})$ is a _$m$ -generic
lowest alcove presentation_ of $\tau$ (resp $\overline{\tau}$) if $\mu$ is
$m$-deep in $\underline{C}_{0}$ and $\tau\simeq\tau(s,\mu+\eta)$ (resp
$\overline{\tau}\simeq\overline{\tau}(s,\mu+\eta)$).

  2. (2)

We say $\tau$ (resp. $\overline{\tau}$) is _$m$ -generic_ if there is a
$m$-generic lowest alcove presentation $(s,\mu)$ of $\tau$ (resp.
$\overline{\tau}$).

###  2.4. Inertial local Langlands for $\mathrm{GSp}_{4}$

In this subsection, we take $G=\mathrm{GSp}_{4}$ and
${\mathcal{O}}_{p}={\mathcal{O}}_{K}$.

In [GT11a], Gan and Takeda established the local Langlands correspondence for
$\mathrm{GSp}_{4}$, which we denote by ${\mathrm{rec_{GT}}}$. It is a
surjective finite-to-one map that takes equivalence classes of smooth
irreducible representation of $\mathrm{GSp}_{4}(K)$ to
$\mathrm{GSp}_{4}$-conjugacy classes of Weil–Deligne representations of
$W_{K}$ valued in $\mathrm{GSp}_{4}(\mathbf{C})$. Fix once and for all an
isomorphism $\iota:\mathbf{C}\simeq\overline{\mathbf{Q}}_{p}$. This induces a
correspondence $\mathrm{rec}_{\mathrm{GT},\iota}$ over
$\overline{\mathbf{Q}}_{p}$. We define a normalized local Langlands
correspondence by

$\displaystyle\mathrm{rec}_{\mathrm{GT},p}(\pi):=\mathrm{rec}_{\mathrm{GT},\iota}(\pi\otimes\absolutevalue{\operatorname{\mathrm{sim}}}^{-3/2})$

for any smooth irreducible $\overline{\mathbf{Q}}_{p}$-representation $\pi$ of
$\mathrm{GSp}_{4}(K)$.

Let $\tau$ be a tame inertial $L$-parameter with 1-generic lowest alcove
presentation $(s,\mu)$. We attach a tame type $\sigma(\tau):=R_{s}(\mu+\eta)$
to $\tau$. We often view $\sigma(\tau)$ as
$\mathrm{GSp}_{4}({\mathcal{O}}_{K})$-representation by inflation.

######  Theorem 2.4.1.

Let $\pi$ be a smooth irreducible $\overline{\mathbf{Q}}_{p}$-representation
of $\mathrm{GSp}_{4}(K)$ and $\tau$ be a tame inertial type for $K$ with
1-generic lowest alcove presentation $(s,\mu)$. If
$\sigma(\tau)\subset\pi|_{\mathrm{GSp}_{4}({\mathcal{O}}_{K})}$, then
${\mathrm{rec_{GT}}}(\pi)|_{I_{K}}\simeq\tau$. Moreover,
$\operatorname{\mathrm{Hom}}_{\overline{\mathbf{Q}}_{p}[\mathrm{GSp}_{4}({\mathcal{O}}_{K})]}(\sigma(\tau),\pi|_{\mathrm{GSp}_{4}({\mathcal{O}}_{K})})$
is 1-dimensional.

Recall that given a pair $(s,\mu)\in\underline{W}\times X^{*}(\underline{T})$,
we defined $(s_{\tau},\textbf{a}^{(0)})\in W\times X^{*}(T)$ by

$\displaystyle s_{\tau}=s_{0}s_{1}\dots s_{f-1},\
\textbf{a}^{(0)}=\sum_{j=0}^{f-1}((F^{*}\circ s^{-1})^{j}(\mu+\eta))_{0}.$

For such $(s_{\tau},\textbf{a}^{(0)})$, we can associate a pair
$(\mathbb{T},\theta):=(T_{s_{\tau}},\theta_{s_{\tau},{\mathbf{a}}^{(0)}})$ and
the Deligne–Lusztig representation
$\epsilon_{\mathrm{GSp}_{4}}\epsilon_{\mathbb{T}}R_{\mathbb{T}}^{\theta}$
following [Her09, Lemma 4.2]. Using [DM20, Corollary 10.5 and Proposition
12.2] and [CGP15, Proposition A.5.15(1)], one can see that
$\epsilon_{\mathrm{GSp}_{4}}\epsilon_{\mathbb{T}}R_{\mathbb{T}}^{\theta}$ is
isomorphic to $R_{s}(\mu+\eta)$ as $\mathrm{GSp}_{4}(k)\simeq
G_{0}(\mathbf{F}_{p})$-representation. We say that $R_{s}(\mu+\eta)$ is
_cuspidal_ if the torus $\mathbb{T}$ is _not_ contained in any proper Levi
subgroup of $\mathrm{GSp}_{4}$. Otherwise, it is called _non-cuspidal_.

Let $P$ be the minimal standard parabolic subgroup of $\mathrm{GSp}_{4}$
containing $\mathbb{T}$ with Levi factor $M$ and the unipotent radical $N$. By
[DL76, Proposition 8.2] we have

$\displaystyle
R_{\mathbb{T}}^{\theta}=\operatorname{\mathrm{Ind}}_{P(\mathbf{F}_{q})}^{G(\mathbf{F}_{q})}(R_{\mathbb{T},P}^{\theta}).$

Note that $M$ decomposes into a product $\prod_{i=1}^{r}M_{i}$ (see [RS07,
§2.1]). We also write $\mathbb{T}=\prod_{i=1}^{r}\mathbb{T}_{i}$ where
$\mathbb{T}_{i}\subset M_{i}$ is a maximal torus and
$\theta_{i}=\theta|_{\mathbb{T}_{i}}$. By Künneth Theorem, we have an
isomorphism between $M(k)$-representations

$\displaystyle
R^{\theta}_{\mathbb{T},P}=\bigotimes_{i=1}^{r}R^{\theta_{i}}_{\mathbb{T}_{i},M_{i}}.$

We give explicit descriptions of $\mathbb{T}_{i}$ and $\theta_{i}$. Let us
write ${\mathbf{a}}^{(0)}=(a_{1},a_{2};a_{3})$. Recall that we write $T_{2}$
for the diagonal torus of $\mathrm{GL}_{2}$ and $W_{2}$ for the Weyl group for
$\mathrm{GL}_{2}$. Only in this subsection, we write
$W_{2}=\left\\{1,s\right\\}$.

  1. (1)

When $s_{\tau}=1$, we have $P=B$, $r=3$, $M_{i}=\mathbb{T}_{i}=\mathbf{G}_{m}$
for $1\leq i\leq 3$. Then $\theta_{i}$ is given by $a_{i}\in
X^{*}(\mathbf{G}_{m})$.

  2. (2)

When $s_{\tau}\in\left\\{s_{1},s_{2}s_{1}s_{2}\right\\}$, we have $P=S$,
$r=2$, $M_{1}\simeq\mathrm{GL}_{2}$ and $M_{2}\simeq\mathbf{G}_{m}$. Then
$(\mathbb{T}_{1},\theta_{1})=(T_{s},\theta_{s,\mu})$ where $\mu=(a_{1},a_{2})$
(resp. $(a_{1},-a_{2})$) in $X^{*}(T_{2})$ when $s_{\tau}=s_{1}$ (resp.
$s_{2}s_{1}s_{2}$), and $\theta_{2}$ is given by $a_{3}\in
X^{*}(\mathbf{G}_{m})$.

  3. (3)

When $s_{\tau}\in\left\\{s_{2},s_{1}s_{2}s_{1}\right\\}$, we have $P=Q$,
$r=2$, $M_{1}\simeq\mathbf{G}_{m}$, and $M_{2}\simeq\mathrm{GL}_{2}$. Then
$\theta_{1}$ is given by $a_{1}$ (resp. $a_{2}$) in $X^{*}(\mathbf{G}_{m})$,
and $(\mathbb{T}_{2},\theta_{2})=(T_{s},\theta_{s,\mu})$ where
$\mu=(a_{2}+a_{3},a_{3})$ (resp. $(a_{1}+a_{3},a_{3})$) in $X^{*}(T_{2})$ when
$s_{\tau}=s_{2}$ (resp. $s_{1}s_{2}s_{1}$).

  4. (4)

When $s_{\tau}\in\left\\{s_{1}s_{2},s_{2}s_{1},w_{0}\right\\}$, we have
$P=\mathrm{GSp}_{4}$. This is the only case that $R_{\mathbb{T}}^{\theta}$ is
cuspidal.

Let $\pi$ and $\sigma(\tau)$ be as in Theorem 2.4.1. When $\sigma(\tau)$ is
non-cuspidal, we can use the above data to describe $\pi$ as a parabolically
induced representation.

######  Proposition 2.4.2.

Let $\tau$ be a 1-generic tame inertial $L$-parameter such that $\sigma(\tau)$
is non-cuspidal. Let $\pi$ be a smooth irreducible
$\overline{\mathbf{Q}}_{p}$-representation of $G(K)$ such that
$\sigma(\tau)\subset\pi|_{\mathrm{GSp}_{4}({\mathcal{O}}_{K})}$. Following the
above notations, there exists a
$M_{i}({\mathcal{O}}_{K})K^{\times}$-representation
$\widetilde{R}_{\mathbb{T}_{i}}^{\theta_{i}}$ extending
$\epsilon_{M_{i}}\epsilon_{\mathbb{T}_{i}}R_{\mathbb{T}_{i}}^{\theta_{i}}$
such that

$\displaystyle\pi\simeq\operatorname{\mathrm{Ind}}_{P(K)}^{\mathrm{GSp}_{4}(K)}\left(\bigotimes_{i=1}^{r}\operatorname{\mathrm{ind}}_{M_{i}({\mathcal{O}}_{K})K^{\times}}^{M_{i}(K)}\widetilde{R}_{\mathbb{T}_{i}}^{\theta_{i}}\right).$

Moreover, $\pi$ contains $\sigma(\tau)$ with multiplicity one.

###### Proof.

Let ${\mathfrak{P}}\subset\mathrm{GSp}_{4}({\mathcal{O}}_{K})$ be the
parahoric subgroup defined as the inverse image of
$P(k)\subset\mathrm{GSp}_{4}(k)$. We let $\sigma_{\mathfrak{P}}$ and
$\sigma_{M}$ denote the inflation of
$\epsilon_{P}\epsilon_{\mathbb{T}}R^{\theta}_{\mathbb{T},P}$ to
${\mathfrak{P}}$ and $M({\mathcal{O}}_{K})$ respectively. By [Mor99, Lemma
3.6], we have an isomorphism
$\pi^{\sigma_{\mathfrak{P}}}\simeq(\pi_{N})^{\sigma_{M}}$ where $\pi_{N}$ is
the unnormalized Jacquet module. Since
$\sigma(\tau)\subset\pi|_{\mathrm{GSp}_{4}({\mathcal{O}}_{K})}$, we get
$\sigma_{M}\hookrightarrow\pi_{N}|_{M({\mathcal{O}}_{K})}$. The pair
$(M({\mathcal{O}}_{K}),\sigma_{M})$ is $M(K)$-type. This implies that there is
an isomorphism of $M(K)$-representations
$\pi_{N}\buildrel\sim\over{\rightarrow}\tau_{\sigma_{M}}$ for a supercuspidal
representation of $M(K)$

$\displaystyle\tau_{\sigma_{M}}=\bigotimes_{i=1}^{r}\operatorname{\mathrm{ind}}_{M_{i}({\mathcal{O}}_{K})K^{\times}}^{M_{i}(K)}\widetilde{R}_{\mathbb{T}_{i},M_{i}}^{\theta_{i}}$

where $\widetilde{R}_{\mathbb{T}_{i}}^{\theta_{i}}$ is a
$M_{i}({\mathcal{O}}_{K})K^{\times}$-representation extending
$\epsilon_{M_{i}}\epsilon_{\mathbb{T}_{i}}R_{\mathbb{T}_{i}}^{\theta_{i}}$
(see [Mor99, Proposition 4.1]). By Frobenius reciprocity, we get a non-zero
map

$\displaystyle\pi\rightarrow{}\operatorname{\mathrm{Ind}}_{P(K)}^{\mathrm{GSp}_{4}(K)}\tau_{\sigma_{M}}.$

This is an isomorphism by [GT11b, Lemma 5.1.(a), 5.2.(b)] (for $P=S$ or $Q$)
and [BCGP21, Proposition 2.4.6] (for $P=B$).

Let
$\mathrm{GSp}_{4}({\mathcal{O}}_{K})_{1}:=\ker(\mathrm{GSp}_{4}({\mathcal{O}}_{K})\twoheadrightarrow\mathrm{GSp}_{4}(k))$.
Also we let $\widetilde{\sigma}_{M}$ be the
$M({\mathcal{O}}_{K})K^{\times}$-representation extending $\sigma_{M}$ by
letting $K^{\times}$ act by the central character of $\pi$. Then the
multiplicity one assertion follows from

$\displaystyle\pi^{\sigma(\tau)}\simeq\operatorname{\mathrm{Hom}}_{\mathrm{GSp}_{4}(k)}(\sigma(\tau),\pi^{\mathrm{GSp}_{4}({\mathcal{O}}_{K})_{1}})\simeq\pi^{\sigma_{\mathfrak{P}}}\simeq(\pi_{N})^{\sigma_{M}}\simeq(\tau_{\sigma_{M}})^{\widetilde{\sigma}_{M}}\simeq\operatorname{\mathrm{Hom}}_{M(K)}(\tau_{\sigma_{M}},\tau_{\sigma_{M}})$

where the second isomorphism is induced by Frobenius reciprocity for finite
groups, the fourth isomorphism exists because $\pi_{N}\simeq\tau_{\sigma_{M}}$
admits a central character, and the last isomorphism is induced by the
Frobenius reciprocity for compact inductions. ∎

Now suppose that $\sigma(\tau)$ is cuspidal. We write
$\widetilde{\sigma}(\tau)$ for the
$G({\mathcal{O}}_{K})K^{\times}$-representation extending $\sigma(\tau)$ by
letting $K^{\times}$ act by central character of $\pi$. By [DR09, Lemma
4.5.1], we have an isomorphism of $G(K)$-representations

$\displaystyle\pi\simeq\operatorname{\mathrm{ind}}_{G({\mathcal{O}}_{K})K^{\times}}^{G(K)}(\widetilde{\sigma}(\tau)).$

We prove Theorem 2.4.1 using an explicit description of the local Langlands
correspondence. In the non-cuspidal case, we use the explicit theta
correspondence in [GT11b]. In the cuspidal case, we use the explicit
construction of the local Langlands correspondence for tame regular semisimple
elliptic Langlands parameters in [DR09]. Note that the compatibility between
the local Langlands correspondence for $\mathrm{GSp}_{4}$ of DeBacker–Reeder
and Gan–Takeda is proven by Lust ([Lus13, Theorem 1.1]).

####  2.4.3. Explicit theta correspondences

In [GT11b], Gan–Takeda established theta correspondences between the group
$\mathrm{GSp}_{4}$ and various similitude orthogonal groups
$\mathrm{GSO}_{2,2},\mathrm{GSO}_{3,3}$, and $\mathrm{GSO}_{4,0}$. This was
used to prove the local Langlands conjecture for $\mathrm{GSp}_{4}$ ([GT11a]).
The parabolically induced representations of $\mathrm{GSp}_{4}$ has a non-zero
theta lift to $\mathrm{GSO}_{3,3}$. The group $\mathrm{GSO}_{3,3}$ admits an
accidental isomorphism

$\displaystyle\mathrm{GSO}_{3,3}\simeq(\mathrm{GL}_{4}\times\mathrm{GL}_{1})/\left\\{(z,z^{-2})\mid
z\in\mathrm{GL}_{1})\right\\}.$

Using this isomorphism, we view a representation of $\mathrm{GSO}_{3,3}$ as a
representation of $\mathrm{GL}_{4}\times\mathrm{GL}_{1}$.

Let $\pi$ be a irreducible smooth $\mathbf{C}$-representation of
$\mathrm{GSp}_{4}(K)$. If the theta lifting of $\pi$ to $\mathrm{GSO}_{3,3}$
is given by (non-zero) $\Pi\boxtimes\chi$, the $L$-parameter of
$\Pi\boxtimes\chi$ (which is valued in
$\mathrm{GL}_{4}(\mathbf{C})\times\mathrm{GL}_{1}(\mathbf{C})$) factors
through the map

$\displaystyle\mathrm{GSp}_{4}(\mathbf{C})\xrightarrow{\operatorname{{\mathrm{std}}}\times\operatorname{\mathrm{sim}}}\mathrm{GL}_{4}(\mathbf{C})\times\mathrm{GL}_{1}(\mathbf{C})$

which provides the $L$-parameter ${\mathrm{rec_{GT}}}(\pi)$ of $\pi$.

In the following Theorem, we write $\phi_{\tau}$ for the $L$-parameter
attached to a smooth irreducible $\overline{\mathbf{Q}}_{p}$-representation
$\tau$ of $\mathrm{GL}_{2}(K)$ by [HT01] conjugated by $\iota$. We write
$\omega_{\tau}$ for the central character of $\tau$.

######  Proposition 2.4.4.

Let $\pi$ be a smooth irreducible $\overline{\mathbf{Q}}_{p}$-representation
of $\mathrm{GSp}_{4}(K)$.

  1. (1)

($P=B$ the Borel subgroup) If
$\pi\simeq\operatorname{\mathrm{Ind}}_{B(K)}^{\mathrm{GSp}_{4}(K)}(\chi_{1},\chi_{2};\chi)$,
then

$\displaystyle\mathrm{rec}_{\mathrm{GT},p}(\pi)=\chi_{1}\chi_{2}\chi\absolutevalue{\cdot}^{-3}\oplus\chi_{1}\chi\absolutevalue{\cdot}^{-2}\oplus\chi_{2}\chi\absolutevalue{\cdot}^{-1}\oplus\chi:W_{K}\xrightarrow{}T(E)\subset\mathrm{GSp}_{4}(E).$

  2. (2)

($P=Q$ the Klingen parabolic) If
$\pi\simeq\operatorname{\mathrm{Ind}}_{Q(K)}^{\mathrm{GSp}_{4}(K)}(\chi\otimes\tau)$,
then

$\displaystyle\mathrm{rec}_{\mathrm{GT},p}(\pi)=\phi_{\tau}\absolutevalue{\cdot}^{-1/2}\oplus\phi_{\tau}\chi\absolutevalue{\cdot}^{-5/2}:W_{K}\xrightarrow{}M_{Q}(E)\subset\mathrm{GSp}_{4}(E).$

  3. (3)

($P=S$ the Siegel parabolic) If
$\pi\simeq\operatorname{\mathrm{Ind}}_{P(K)}^{\mathrm{GSp}_{4}(K)}(\tau\otimes\chi)$,
then

$\displaystyle\mathrm{rec}_{\mathrm{GT},p}(\pi)=\chi\oplus\phi_{\tau}\chi\absolutevalue{\cdot}^{-3/2}\oplus\chi\omega_{\tau}\absolutevalue{\cdot}^{-3}:W_{K}\xrightarrow{}M_{S}(E)\subset\mathrm{GSp}_{4}(E).$

###### Proof.

This is a special case of [GT11b, Proposition 13.1] (vi), (iv), (v) (for (1),
(2), (3) respectively). Note that the induction in _loc. cit._ is normalized.
∎

###### Proof of Theorem 2.4.1 in the non-cuspidal case.

This follows from Proposition 2.4.2 and 2.4.4, and the inertial local
Langlands correspondence for $\mathrm{GL}_{2}$ and $\mathrm{GL}_{1}$ (e.g.
[LLHLMa, Proposition 2.5.5]). ∎

####  2.4.5. Depth zero regular supercuspidal local Langlands

Let $\tau\simeq\tau(s,\mu+\eta)$ be a 1-generic tame inertial type for $K$
over $E$ and $\psi:G_{K}\rightarrow\mathrm{GSp}_{4}(E)$ be a continuous
representation extending $\tau$. We also assume $\sigma(\tau)$ cuspidal. After
taking conjugation, we can assume that the image of $I_{K}$ is contained in
$T(E)$ and $\psi(\operatorname{\mathrm{Frob}}_{K})\in N_{G}(T)$. We can write
$\psi(\operatorname{\mathrm{Frob}}_{K})=wt$ for a unique $w\in W$ and some
$t\in T(E)$. Note that $t$ gives a well-defined class in $T/(1-w)T$, and thus
$\operatorname{\mathrm{sim}}(t)$ is independent of the choice of $t$. By the
construction of $\tau$, we must have $w=\phi(s_{\tau})$. The cuspidality of
$\sigma(\tau)$ implies $w\in\left\\{s_{1}s_{2},s_{2}s_{1},w_{0}\right\\}$.
Then $\psi$ is TRSELP (tame regular semisimple elliptic $L$-parameter) in the
sense of [DR09, §4.1].

For any TRSELP $\psi$, DeBacker–Reeder constructed the $L$-packet of depth-
zero supercuspidal representations associated with it. These representations
are distributed among the pure inner forms of $\mathrm{GSp}_{4}$. The pure
inner forms are parameterized by Galois cohomology
$H^{1}(K,\mathrm{GSp}_{4})$. Since $H^{1}(K,\mathrm{Sp}_{4})=1$ (because
$\mathrm{Sp}_{4}$ is simply-connected) and $H^{1}(K,\mathbf{G}_{m})=1$, we
have $H^{1}(K,\mathrm{GSp}_{4})=1$. Thus, all these representations are of
$\mathrm{GSp}_{4}$.

Now we explain the construction in [DR09, §4] in a special case. The
$L$-packet of $\psi$, denoted by $\Pi(\psi)$, is parameterized by
$\mathrm{Irr}(C_{\psi})$ the (finite) set of irreducible representations of
$C_{\psi}=\pi_{0}(Z_{G}(\imaginary\psi))$. One can check that $C_{\psi}$ is
trivial when $s_{\tau}=s_{1}s_{2}$ or $s_{2}s_{1}$ and is isomorphic to
$\mathbf{Z}/2\mathbf{Z}$ when $s_{\tau}=w_{0}$. In both cases, we simply take
the trivial representation of $C_{\psi}$. Then the corresponding element in
$\Pi(\psi)$ is given by

$\displaystyle\operatorname{\mathrm{ind}}^{G(K)}_{G({\mathcal{O}}_{K})K^{\times}}\widetilde{R}_{\mathbb{T}}^{\theta}$

where
$(\mathbb{T},\theta)=(T_{s_{\tau}},\theta_{s_{\tau},{\mathbf{a}}^{(0)}})$,
$\widetilde{R}_{\mathbb{T}}^{\theta}|_{G({\mathcal{O}}_{K})}\simeq\epsilon_{G}\epsilon_{\mathbb{T}}{R}_{\mathbb{T}}^{\theta}$,
and $\widetilde{R}_{\mathbb{T}}^{\theta}|_{K^{\times}}$ sends $p$ to
$\operatorname{\mathrm{sim}}(t)$.

###### Proof of Theorem 2.4.1 in the cuspidal case.

We know that $\sigma(\tau)\subset\pi|_{G({\mathcal{O}}_{K})}$ implies that
$\pi\simeq\operatorname{\mathrm{ind}}_{G({\mathcal{O}}_{K})K^{\times}}^{G(K)}(\widetilde{\sigma}(\tau))$
for some $\widetilde{\sigma}(\tau)$ extending $\sigma(\tau)$. Let
$\omega_{\pi}$ be the central character of $\pi$. By the above construction,
$\pi$ is contained in the $L$-packet of $\psi$ such that
$\psi|_{I_{K}}\simeq\tau$ and
$\operatorname{\mathrm{sim}}(\psi)(\operatorname{\mathrm{Frob}}_{K})=\omega_{\pi}(p)$.
The multiplicity one assertion follows from

$\displaystyle\operatorname{\mathrm{Hom}}_{G({\mathcal{O}}_{K})}(\sigma(\tau),\pi|_{G({\mathcal{O}}_{K})})\simeq\operatorname{\mathrm{Hom}}_{G({\mathcal{O}}_{K})K^{\times}}(\widetilde{\sigma}(\tau),\pi|_{G({\mathcal{O}}_{K})K^{\times}})\simeq\operatorname{\mathrm{Hom}}_{G(K)}(\pi,\pi)$

where the first isomorphism exists because $\pi$ admits a central character,
and the second isomorphism follows from the Frobenius reciprocity for compact
inductions. ∎

####  2.4.6. Serre weights of a tame inertial $L$-parameter

We define the conjectural set of Serre weights associated to a tame inertial
$L$-parameter following [GHS18, Definition 9.2.5].

######  Definition 2.4.7.

Let $\overline{\rho}$ be a tame inertial $L$-parameter over $\mathbf{F}$. We
define $W^{?}(\overline{\rho})$ to be the set
${\mathcal{R}}(\mathrm{JH}(\overline{\sigma}([\overline{\rho}])))$.

######  Definition 2.4.8.

Let $\overline{\rho}\simeq\overline{\tau}(s,\mu)$ be a 6-generic tame inertial
$L$-parameter over $\mathbf{F}$. We define a function

$\displaystyle F_{\overline{\rho}}:\underline{W}$ $\displaystyle\rightarrow
W^{?}(\overline{\rho})$ $\displaystyle w$
$\displaystyle\mapsto{\mathcal{R}}(F_{\sigma([\overline{\rho}])}(w)).$

We define $W_{\mathrm{obv}}(\overline{\rho})$ to be the image of
$F_{\overline{\rho}}$ and call its elements as _obvious weights_ of
$\overline{\rho}$.

Note that $W_{\mathrm{obv}}(\overline{\rho})$ does not depend on the choice of
the lowest alcove presentation of $\overline{\rho}$ and coincides with
$W_{\mathrm{obv}}(\overline{\rho})$ defined in [LLHLMa, Definition 2.6.3]. In
_loc. cit._ , $F_{\overline{\rho}}(w)$ is called as the obvious weight of
$\overline{\rho}$ corresponding to $w$.

####  2.4.9. Transfer to $\mathrm{GL}_{4}$

Recall that we have a map ${\mathcal{T}}$ defined in §2.1 which maps
$\underline{W}$ to $\underline{W}_{4}$ and $X^{*}(\underline{T})$ to
$X^{*}(\underline{T}_{4})$. Using this, we define the _transfer_ of
Deligne–Lusztig representations and Serre weights of $G_{0}(\mathbf{F}_{p})$
to $(\mathrm{GL}_{4})_{0}(\mathbf{F}_{p})$.

######  Proposition 2.4.10.

The map ${\mathcal{T}}$ induces a well-defined assignment from Deligne–Lusztig
representations (resp. Serre weights) of $G_{0}(\mathbf{F}_{p})$ to the
Deligne–Lusztig representations (resp. Serre weights) of
$(\mathrm{GL}_{4})_{0}(\mathbf{F}_{p})$ given by

$\displaystyle R_{s}(\mu)$
$\displaystyle\mapsto{\mathcal{T}}(R_{s}(\mu)):=R_{{\mathcal{T}}(s)}({\mathcal{T}}(\mu))$
$\displaystyle F(\lambda)$
$\displaystyle\mapsto{\mathcal{T}}(F(\lambda)):=F({\mathcal{T}}(\lambda)).$

Suppose that $\mu-\eta\in\underline{C}_{0}$ is $6$-deep. Then
$F(\lambda)\in\mathrm{JH}(\overline{R_{s}(\mu)})$ implies
${\mathcal{T}}(F(\lambda))\in\mathrm{JH}(\overline{{\mathcal{T}}(R_{s}(\mu))})$.
The converse is true if furthermore
$\mu-\eta=(a_{j},b_{j};c_{j})_{j\in{\mathcal{J}}}$ and
$\absolutevalue{a_{j}-p/2}>3/2$ for each ${j\in{\mathcal{J}}}$.

###### Proof.

The first claim for Deligne–Lusztig representations follows from the fact that
the map ${\mathcal{T}}$ respects the Weyl group action on the character
lattice and $p$-dot actions. For Serre weights, it follows from that
${\mathcal{T}}$ maps $X^{*}_{1}(\underline{T})$ and $X^{0}(\underline{T})$
into $X^{*}_{1}(\underline{T}_{4})$ and $X^{0}(\underline{T}_{4})$
respectively.

Suppose that $F(\lambda)\in\mathrm{JH}(\overline{R_{s}(\mu)})$. By Proposition
2.2.6, there exists $w\in\underline{W}$ and
$\widetilde{w}\in\underline{\widetilde{W}}_{1}$ such that
$\widehat{w}\uparrow\widetilde{w}$ and

$\displaystyle F(\lambda)\simeq
F\left(\widetilde{w}_{h}^{-1}\widetilde{w}\cdot(\mu-\eta+s\pi(\widehat{w}^{-1}(0)))\right)\in\mathrm{JH}(\overline{R_{s}(\mu)}).$

Let ${\mathcal{T}}(\widetilde{w}_{h})=\widetilde{w}_{h}^{\prime}$. By applying
${\mathcal{T}}$ to the above equation, we have

$\displaystyle{\mathcal{T}}(F(\lambda))\simeq
F\left(\widetilde{w}_{h}^{\prime{-1}}{\mathcal{T}}(\widetilde{w})\cdot({\mathcal{T}}(\mu)-\eta^{\prime}+{\mathcal{T}}(s\pi(\widehat{w}^{-1}(0))))\right).$

Since two ordering $\leq$ and $\uparrow$ coincide on
$\underline{\widetilde{W}}^{+}$, we have
${\mathcal{T}}(\widehat{w})\uparrow{\mathcal{T}}(\widetilde{w})$ by Lemma
2.1.6. Then the above equality implies
${\mathcal{T}}(F(\lambda))\in\mathrm{JH}(\overline{{\mathcal{T}}(R_{s}(\mu))})$
by Proposition 2.2.6.

For the converse, suppose
${\mathcal{T}}(F(\lambda))\in\mathrm{JH}(\overline{{\mathcal{T}}(R_{s}(\mu))})$.
Then there exists $w^{\prime}\in\underline{W}_{4}$ and
$\widetilde{w}^{\prime}\in\underline{\widetilde{W}}_{1}$ such that
$\widehat{w}^{\prime}\uparrow\widetilde{w}^{\prime}$ and

$\displaystyle{\mathcal{T}}(F(\lambda))\simeq
F\left(\widetilde{w}_{h}^{\prime{-1}}\widetilde{w}^{\prime}\cdot({\mathcal{T}}(\mu)-\eta^{\prime}+{\mathcal{T}}(s)\pi(\widehat{w}^{\prime{-1}}(0)))\right).$

This shows that
$\widetilde{w}_{h}^{\prime{-1}}\widetilde{w}^{\prime}\cdot({\mathcal{T}}(\mu)-\eta^{\prime}+{\mathcal{T}}(s)\pi(\widehat{w}^{\prime{-1}}(0)))$
is fixed by $\Theta$ (defined in §2.1). Suppose that $\widetilde{w}^{\prime}$
is fixed by $\Theta$. Then $\widehat{w}^{\prime{-1}}(0)$ is fixed by $\Theta$,
and simple computation shows that $\widehat{w}^{\prime}$ is fixed by $\Theta$
as well. Thus $\widetilde{w}^{\prime}={\mathcal{T}}(\widetilde{w})$ and
$\widehat{w}^{\prime}={\mathcal{T}}(\widehat{w})$ for some
$\widetilde{w}\in\underline{\widetilde{W}}_{1}$ and $w\in\underline{W}$. As in
the previous paragraph, this shows that
$F(\lambda)\in\mathrm{JH}(\overline{R_{s}(\mu)})$.

We finish the proof by showing that $\widetilde{w}^{\prime}$ is fixed by
$\Theta$. Let $\widetilde{w}_{0}\in\underline{\widetilde{W}}$ be an element
such that $\lambda\in\widetilde{w}_{0}\cdot\underline{C}_{0}$. Then we can
write $\widetilde{w}^{\prime}={\mathcal{T}}(\widetilde{w}_{0})\delta^{\prime}$
for some $\delta^{\prime}\in\underline{\Omega}_{4}$. Suppose that
$\delta^{\prime}$ is not fixed by $\Theta$. Since
$\Omega_{4}/{\mathcal{T}}(\Omega)$ is cyclic group of order 2, we can assume
that $\delta^{\prime}_{j}$ is the generator of $\Omega_{4}$ sending
$(a,b,c,d)$ to $(b,c,d,a-p)$ for at least one $j\in{\mathcal{J}}$. Let us
write $\mu-\eta=(a_{j},b_{j};c_{j})_{j\in{\mathcal{J}}}$ and
${\mathcal{T}}(s)\pi(\widehat{w}^{\prime{-1}})=(x_{j},y_{j},z_{j},w_{j})_{j\in{\mathcal{J}}}$.
Then

$\displaystyle\left(\delta^{\prime}\cdot({\mathcal{T}}(\mu-\eta)+{\mathcal{T}}(s)\pi(\widehat{w}^{\prime{-1}}))\right)_{j}$
$\displaystyle=\delta^{\prime}_{j}\cdot(a_{j}+b_{j}+c_{j}+x_{j},a_{j}+c_{j}+y_{j},b_{j}+c_{j}+z_{j},c_{j}+w_{j})$
$\displaystyle=(a_{j}+c_{j}+y_{j},b_{j}+c_{j}+z_{j},c_{j}+w_{j},a_{j}+b_{j}+c_{j}+x_{j}-p).$

Since
$\delta^{\prime}\cdot({\mathcal{T}}(\mu-\eta)+{\mathcal{T}}(s)\pi(\widehat{w}^{\prime{-1}}))$
is fixed by $\Theta$, we have $2a_{j}-p=z_{j}+w_{j}-y_{j}-x_{j}$. However,
$\absolutevalue{z_{j}+w_{j}-y_{j}-x_{j}}\leq 3$ by [LLHL19, Remark 4.1.4].
This leads to a contradiction and $\delta^{\prime}$ is fixed by $\Theta$. ∎

######  Example 2.4.11.

The condition $\absolutevalue{a_{j}-p/2}>3/2$ in the above Lemma may seem
dubious but it is necessary. For example, let $K=\mathbf{Q}_{p}$ and
$(s,\mu-\eta)\in W\times X^{*}(T)$ be a lowest alcove presentation of
$R_{s}(\mu)$ where $s=e$, $\mu-\eta=(a,b;c)$. We take $a=(p-1)/2$. Let
$w^{\prime}\in W_{4}$ be the element sending $(x,y,z,w)$ to $(y,z,w,x)$. Then
$\widehat{w}^{\prime}=w^{\prime}t_{-\nu}$ where $\nu=(1,0,0,0)$. Let

$\displaystyle\lambda^{\prime}$
$\displaystyle:=\widetilde{w}_{h}^{\prime{-1}}\widehat{w}\cdot({\mathcal{T}}(\mu)-\eta^{\prime}+\nu)$
$\displaystyle=\widetilde{w}_{h}^{\prime{-1}}\cdot{\mathcal{T}}(a,a-b;b+c-a)$

so that
$F(\lambda^{\prime})\in\mathrm{JH}(\overline{{\mathcal{T}}(R_{e}(\mu))})$.
However, one can easily check that

$\displaystyle
F(\widetilde{w}_{h}^{-1}\cdot(a,a-b;b+c-a))\notin\mathrm{JH}(\overline{R_{e}(\mu)}).$

######  Corollary 2.4.12.

Let $\overline{\rho}\simeq\tau(s,\mu)$ be a 6-generic tame inertial
$L$-parameter. Recall the set
$W^{?}(\operatorname{{\mathrm{std}}}(\overline{\rho}))$ defined in [LLHLMa,
Definition 2.6.1] Suppose that
$\mu-\eta=(a_{j},b_{j};c_{j})_{j\in{\mathcal{J}}}$ and
$\absolutevalue{a_{j}-p/2}>3/2$ for each ${j\in{\mathcal{J}}}$. Then

$\displaystyle{\mathcal{T}}(W^{?}(\overline{\rho}))=W^{?}(\operatorname{{\mathrm{std}}}(\overline{\rho}))\cap\\{{\mathcal{T}}(F(\lambda))\mid\lambda\in
X_{1}^{*}(\underline{T})\\}.$

The following Corollary shows the compatibility between the inertial local
Langlands correspondence and the transfer of Deligne–Lusztig representations.

######  Corollary 2.4.13.

Let $\pi$ be a smooth irreducible $\overline{\mathbf{Q}}_{p}$-representation
of $\mathrm{GSp}_{4}(K)$ and $\psi=\mathrm{rec}_{\mathrm{GT},p}(\pi)$. Let
$\Pi$ be a smooth irreducible $\overline{\mathbf{Q}}_{p}$-representation of
$\mathrm{GL}_{4}(K)$ corresponding to $\operatorname{{\mathrm{std}}}(\psi)$
under the local Langlands correspondence of [HT01] (conjugated by $\iota$).
Let $\tau$ be a 1-generic tame inertial $L$-parameter. If
$\sigma(\tau)\subset\pi|_{\mathrm{GSp}_{4}({\mathcal{O}}_{K})}$, then
${\mathcal{T}}(\sigma(\tau))\subset\Pi|_{\mathrm{GL}_{4}({\mathcal{O}}_{K})}$.

###### Proof.

By Theorem 2.4.1, we have $\psi|_{I_{K}}\simeq\tau$. Then the claim follows
from the construction of ${\mathcal{T}}(\sigma(\tau))$ and [Sho18, Theorem 3.7
(2)]. (Note that although the local Langlands correspondence used in _loc.
cit._ is normalized, it only differs by unramified twist.) ∎

######  Remark 2.4.14.

The converse of the above Corollary is not true, as the $L$-packet of $\psi$
can have two elements (e.g. if $\psi|_{I_{K}}\simeq\tau(s,\mu)$ and
$s_{\tau}=w_{0}$).

###  2.5. Combinatorics between types and weights

Let $G$ be either $\mathrm{GSp}_{4}$ or $\mathrm{GL}_{n}$. We first introduce
some notations and definitions.

######  Notation 2.5.1.

Let $(s,\mu)$ be a lowest alcove presentation of a Deligne–Lusztig
representation $R$ (resp. a tame inerital $L$-parameter $\tau$). We let
$\widetilde{w}(R)$ (resp. $\widetilde{w}(\tau)$) denote
$t_{\mu+\eta}s\in\underline{\widetilde{W}}$. If $\overline{\rho}$ is another
tame inertial $L$-parameter with a lowest alcove presentation
$(s^{\prime},\mu^{\prime})$, we write

$\displaystyle\widetilde{w}(\overline{\rho},\tau):=\widetilde{w}(\tau)^{-1}\widetilde{w}(\overline{\rho})\in\underline{\widetilde{W}}.$

######  Definition 2.5.2.

We call an element of $X^{*}(\underline{Z})$ as an _algebraic central
character_ and an element of
$X^{*}(\underline{Z})/(p-\pi)X^{*}(\underline{Z})$ as a _central character_.

Note that
$X^{*}(\underline{Z})/(p-\pi)X^{*}(\underline{Z})\simeq\operatorname{\mathrm{Hom}}(Z({\mathcal{O}}_{p}/p),\mathbf{F}^{\times})$
which justifies the term central character.

Recall that we have an isomorphism

(2.5.3)
$\displaystyle\underline{\Omega}\simeq\underline{\widetilde{W}}/\underline{W}_{a}\simeq
X^{*}(\underline{Z}).$

If $\sigma$ is a Serre weight with a lowest alcove presentation
$(\widetilde{w}_{1},\omega)$, then the central character of $\sigma$ can be
described as the image of $t_{\omega-\eta}\widetilde{w}_{1}$ under (2.5.3)
composed with $X^{*}(\underline{Z})\twoheadrightarrow
X^{*}(\underline{Z})/(p-\pi)X^{*}(\underline{Z})$. Similarly, if $R$ is a
Deligne–Lusztig representation with a lowest alcove presentation $(s,\mu)$,
then the central character of $R$ is given by the lift of the image of
$\widetilde{w}(R)$ under (2.5.3) composed with
$X^{*}(\underline{Z})\twoheadrightarrow
X^{*}(\underline{Z})/(p-\pi)X^{*}(\underline{Z})$.

For a dominant character $\lambda\in X^{*}(\underline{T})$, let
$W(\lambda)_{/{\mathcal{O}}}$ be the unique up to isomorphism irreducible
algebraic $\underline{G}_{/{\mathcal{O}}}$-representation of highest weight
$\lambda$. Let $V(\lambda)$ be the restriction of
$W(\lambda)_{/{\mathcal{O}}}$ to $G_{0}(\mathbf{Z}_{p})$. We define a _type_
(of $G_{0}(\mathbf{Z}_{p})$) to be a pair $(\lambda+\eta,\tau)$ where
$\lambda\in X_{*}(\underline{T}^{\vee})$ is a dominant cocharacter and $\tau$
is a 1-generic tame inertial $L$-parameter. To a type $(\lambda+\eta,\tau)$,
we associate a locally algebraic representations of $G_{0}(\mathbf{Z}_{p})$

$\displaystyle\sigma(\lambda,\tau):=\sigma(\tau)\otimes_{{\mathcal{O}}}V(\phi^{-1}(\lambda)).$

Let $W(\lambda)_{/\mathbf{F}}$ be the dual Weyl module of highest weight
$\lambda$ for the algebraic group $\underline{G}_{/\mathbf{F}}$ and
$W(\lambda)$ be the restriction of $W(\lambda)_{/\mathbf{F}}$ to
$G_{0}(\mathbf{F}_{p})$. Note that
$V(\lambda)\otimes_{\mathcal{O}}\mathbf{F}\simeq W(\lambda)$.

######  Definition 2.5.4.

  1. (1)

Let $(\widetilde{w}_{1},\omega)$ be a lowest alcove presentation of a Serre
weight $\sigma$. We say $(\widetilde{w}_{1},\omega)$ is _compatible with
$\zeta\in X^{*}(\underline{Z})$_ if the image of
$t_{\omega-\eta}\widetilde{w}_{1}$ in $X^{*}(\underline{Z})$ under (2.5.3) is
equal to $\zeta$. We also say that $\sigma$ _has algebraic central character
$\zeta$ (with respect to $(\widetilde{w}_{1},\omega)$)_.

  2. (2)

Let $(s,\mu)$ be a lowest alcove presentation of a Deligne–Lusztig
representation $R$ (resp. a tame inertial $L$-parameter $\tau$ over $E$). Let
$\lambda\in X^{*}(\underline{T})$. We say $(s,\mu)$ is _$\lambda$ -compatible
with $\zeta\in X^{*}(\underline{Z})$_ if the image of
$t_{\lambda}t_{\mu+\eta}s$ in $X^{*}(\underline{Z})$ under (2.5.3) is equal to
$\zeta$. When $\lambda=0$, we also say that $R$ (resp. $\tau$) _has algebraic
central character $\zeta$ (with respect to $(s,\mu)$)_.

  3. (3)

Let $(s,\mu)$ be a lowest alcove presentation of a tame inertial $L$-parameter
$\overline{\tau}$ over $\mathbf{F}$. Let $\lambda\in X^{*}(\underline{T})$. We
say $(s,\mu)$ is _$\lambda$ -compatible with $\zeta\in X^{*}(\underline{Z})$_
if the image of $t_{\lambda}t_{\mu}s$ in $X^{*}(\underline{Z})$ under (2.5.3)
is equal to $\zeta$. (Note that this differs by $t_{\eta}$ from item (2); see
[LLHLMa, Remark 2.4.2].) When $\lambda=0$, we also say that $\tau$ _has
algebraic central character $\zeta$ (with respect to $(s,\mu)$)_.

  4. (4)

We say that a lowest alcove presentation of tame inertial type $\tau$ over $E$
(or $\overline{\tau}$ over $\mathbf{F}$) is _( $\lambda$-)compatible with_ a
lowest alcove presentation of a Serre weight (or Deligne–Lusztig
representation) if the former is ($\lambda$-)compatible with $\zeta\in
X^{*}(\underline{Z})$ and the latter is compatible with $\zeta$.

######  Remark 2.5.5.

A choice of algebraic central character lifting the central character of Serre
weights or Deligne–Lusztig representations corresponds to a choice of lowest
alcove presentations (see [LLHLMa, Lemma 2.2.4 and 2.3.2]). Later, we will
study objects whose constructions depend on choices of lowest alcove
presentations and their connections. The ($\lambda$-)compatibility is a notion
to make such choices consistent.

Let $\lambda\in X^{*}(\underline{T})$ be a dominant character. We have the set
of _admissible pairs_ defined in [LLHLMa, §2.1]

$\displaystyle\mathrm{AP}(\lambda+\eta):=\left\\{(\widetilde{w}_{1},\widetilde{w}_{2})\in(\underline{\widetilde{W}}_{1}^{+}\times\underline{\widetilde{W}}^{+})/X^{0}(\underline{T})\mid\widetilde{w}_{1}\uparrow
t_{\lambda}\widetilde{w}_{h}^{-1}\widetilde{w}_{2}\right\\}.$

######  Proposition 2.5.6 (Proposition 2.3.7 in [LLHLMa]).

Let $\lambda\in X^{*}(\underline{T})$ be a dominant character. Let $m$ be an
integer such that $m\geq\max\\{h_{\lambda}+3,6\\}$ if $G=\mathrm{GSp}_{4}$ and
$m\geq\max\\{h_{\lambda}+n-1,2(n-1)\\}$ if $G=\mathrm{GL}_{n}$. For
$m$-generic Deligne–Lusztig representation $R$, we have the following
bijection

$\displaystyle\begin{split}\mathrm{AP}(\lambda+\eta)&\simeq\mathrm{JH}(\overline{R}\otimes_{\mathbf{F}}W(\lambda))\\\
(\widetilde{w}_{1},\widetilde{w}_{2})&\mapsto
F_{(\widetilde{w}_{1},\widetilde{w}(R)\widetilde{w}_{2}^{-1}(0))}.\end{split}$

Moreover, every Jordan–Hölder factor are $(m-h_{\lambda}-3)$-deep and the
lowest alcove presentations
$(\widetilde{w}_{1},\widetilde{w}(R)\widetilde{w}_{2}^{-1}(0))$ of these Serre
weights are $\lambda$-compatible with the lowest alcove presentation of $R$.

######  Remark 2.5.7.

Suppose that we have $\lambda=0$ in Proposition 2.5.6. By [LLHLMa, Proposition
2.1.6], the condition
$\widetilde{w}_{1}\uparrow\widetilde{w}_{h}^{-1}\widetilde{w}_{2}$ is
equivalent to $\widetilde{w}_{2}\uparrow\widetilde{w}_{h}\widetilde{w}_{1}$.
Let us write

$\displaystyle\nu=\pi^{-1}(\widetilde{w}_{1})\cdot(\mu-\eta+s\widetilde{w}_{2}^{-1}(0))$

for $(\widetilde{w}_{1},\widetilde{w}_{2})\in\mathrm{AP}(\eta)$ so that
$F_{(\widetilde{w}_{1},\widetilde{w}(R)\widetilde{w}_{2}^{-1}(0))}=F(\nu)$.
Then

$\displaystyle\pi^{-1}(\widetilde{w}_{2})\cdot(\mu-\eta+s\widetilde{w}_{2}^{-1}(0))\uparrow\widetilde{w}_{h}\cdot\nu$

as in Proposition 2.2.6.

######  Proposition 2.5.8 (Proposition 2.6.2 in [LLHLMa]).

Let $m$ be an integer such that $m\geq 6$ if $G=\mathrm{GSp}_{4}$ and $m\geq
2(n-1)$ if $G=\mathrm{GL}_{n}$. Let $\overline{\rho}$ be a tame inertial
$L$-parameter over $\mathbf{F}$ with a $m$-generic lowest alcove presentation.
Then there is a bijection

$\displaystyle\\{(\widetilde{w},\widetilde{w}_{2})\in(\underline{\widetilde{W}}^{+}_{1}\times\underline{\widetilde{W}}^{+})/X^{0}(\underline{T})\mid\widetilde{w}_{2}\uparrow\widetilde{w}\\}$
$\displaystyle\buildrel\sim\over{\rightarrow}W^{?}(\overline{\rho})$
$\displaystyle(\widetilde{w},\widetilde{w}_{2})$ $\displaystyle\mapsto
F_{(\widetilde{w},\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0))}.$

Moreover, every Jordan–Hölder factor are $(m-3)$-deep and the lowest alcove
presentations
$(\widetilde{w},\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0))$ of
these Serre weights are compatible with the lowest alcove presentation of
$\widetilde{w}(\overline{\rho})$.

######  Definition 2.5.9.

Let $\sigma$ and $\kappa$ be Serre weights. We write $\sigma\uparrow\kappa$ if
there exist $\lambda,\lambda^{\prime}\in X^{*}_{1}(\underline{T})$ such that
$\sigma\simeq F(\lambda)$, $\kappa\simeq F(\lambda^{\prime})$, and
$\lambda\uparrow\lambda^{\prime}$.

######  Lemma 2.5.10.

Let $\overline{\rho}$ be a tame inertial $L$-parameter over $\mathbf{F}$ that
is 6-generic if $G=\mathrm{GSp}_{4}$ and $2(n-1)$-generic if
$G=\mathrm{GL}_{n}$. Let $\sigma,\kappa\in W^{?}(\overline{\rho})$ be Serre
weights. Then $\sigma\uparrow\kappa$ if and only if $\sigma\simeq
F_{(\widetilde{w},\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0))}$
and $\kappa\simeq
F_{(\widetilde{w}^{\prime},\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0))}$
for $\widetilde{w},\widetilde{w}^{\prime}\in\underline{\widetilde{W}}^{+}_{1}$
and $\widetilde{w}_{2}\in\underline{\widetilde{W}}^{+}$ such that
$\widetilde{w}_{2}\uparrow\widetilde{w}\uparrow\widetilde{w}^{\prime}$.

###### Proof.

Suppose that $\sigma\simeq
F_{(\widetilde{w},\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0))}$
and $\kappa\simeq
F_{(\widetilde{w}^{\prime},\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{\prime-1}(0))}$
for $\widetilde{w},\widetilde{w}^{\prime}\in\underline{\widetilde{W}}^{+}_{1}$
and
$\widetilde{w}_{2},\widetilde{w}_{2}^{\prime}\in\underline{\widetilde{W}}^{+}$
such that $\widetilde{w}_{2}\uparrow\widetilde{w}$ and
$\widetilde{w}_{2}^{\prime}\uparrow\widetilde{w}^{\prime}$. By changing
$\widetilde{w}_{2}$ up to $X^{0}(\underline{T})$ (and $\widetilde{w}$
accordingly), we can guarantee that
$\pi^{-1}(\widetilde{w}_{2})\cdot(\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0)-\eta)$
and
$\pi^{-1}(\widetilde{w}_{2}^{\prime})\cdot(\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{\prime-1}(0)-\eta)$
are in the same $\underline{W}_{a}$-orbit (under the $p$-dot action). Let
$\widetilde{w}_{2}=wt_{\nu}$ and
$\widetilde{w}_{2}^{\prime}=w^{\prime}t_{\nu^{\prime}}$. Then
$(p\pi^{-1}-1)\nu\equiv(p\pi^{-1}-1)\nu^{\prime}\bmod\underline{\Lambda}_{R}$.
By applying $(1+p\pi^{-1}+\dots(p\pi^{-1})^{r-1})$ to this equation where $r$
is an integer such that $\pi^{r}=\mathrm{id}$, we deduce that
$\nu\equiv\nu^{\prime}\bmod\underline{\Lambda}_{R}$. Then
$p\pi^{-1}(\nu)-\widetilde{w}(\overline{\rho})\nu-\eta$ and
$p\pi^{-1}(\nu)-\widetilde{w}(\overline{\rho})\nu^{\prime}-\eta$ are in the
same $\underline{W}_{a}$-orbit. Since they are in the same $p$-alcove
$\underline{C}_{0}+\pi^{-1}(p\nu)$, they have to be equal. Thus, we conclude
that $\nu=\nu^{\prime}$, which implies that
$\widetilde{w}_{2}=\widetilde{w}_{2}^{\prime}$. Then
$\widetilde{w}\uparrow\widetilde{w}^{\prime}$ follows immediately from
$\sigma\uparrow\kappa$. ∎

For the remaining of this subsection, we let $G=\mathrm{GSp}_{4}$. The
following lemmas will be useful in proving main results in §6.

######  Lemma 2.5.11 (Corollary 2.6.5 in [LLHLMa]).

Let $\lambda\in X^{*}(\underline{T})$ be a dominant character. Let
$\overline{\rho}$ and $\tau$ be tame inertial $L$-parameters over $\mathbf{F}$
and $E$, with $\lambda$-compatible 6-generic and
$\max\\{6,h_{\lambda}+3\\}$-generic lowest alcove presentation respectively,
such that $\widetilde{w}(\overline{\rho},\tau)=t_{w^{-1}(\lambda+\eta)}$ for
some $w\in\underline{W}$. Then we have

$\displaystyle\mathrm{JH}(\overline{\sigma}(\lambda,\tau))\cap
W^{?}(\overline{\rho})=\left\\{(F_{\overline{\rho}}(w))\right\\}.$

######  Lemma 2.5.12.

Let $\lambda\in X^{*}_{1}(\underline{T})$ be a $12$-deep dominant character.
Let $\overline{\rho}$ be a tame inertial $L$-parameter over $\mathbf{F}$ with
a 12-generic lowest alcove presentation
$(s_{\overline{\rho}},\mu_{\overline{\rho}})$. For each $s\in\underline{W}$,
we consider a tame inertial type $\tau(s,\widetilde{w}_{h}\cdot\lambda+\eta)$.
Suppose that for each $s\in\underline{W}$,
$\tau(s,\widetilde{w}_{h}\cdot\lambda+\eta)$ admits a lowest alcove
presentation such that
$\widetilde{w}(\overline{\rho},\tau(s,\widetilde{w}_{h}\cdot\lambda+\eta))\in{\mathrm{Adm}}(\eta)$.
Then $F(\lambda)\in W^{?}(\overline{\rho})$.

###### Proof.

This can be proven as [LLHL19, Lemma 4.1.10]. ∎

######  Remark 2.5.13.

Let $(s,\mu+\eta)$ be the lowest alcove presentation of
$\tau(s,\widetilde{w}_{h}\cdot\lambda+\eta)$ as in the previous Lemma. We
remark that the condition
$\widetilde{w}(\overline{\rho},\tau(s,\widetilde{w}_{h}\cdot\lambda+\eta))\in{\mathrm{Adm}}(\eta)$
implies that the images of $st_{\mu+\eta}$ and
$s_{\overline{\rho}}t_{\mu_{\overline{\rho}}}$ under (2.5.3) are equal, or
equivalently, $\mu_{\overline{\rho}}-\mu\in\underline{\Lambda}_{R}$ (cf. the
condition (P3) in [LLHL19, §4.1]). This condition uniquely determines the
lowest alcove presentation $(s,\mu+\eta)$ of
$\tau(s,\widetilde{w}_{h}\cdot\lambda+\eta)$ (if it exists).

######  Lemma 2.5.14.

Let
$\overline{\rho}:I_{\mathbf{Q}_{p}}\rightarrow\prescript{L}{}{\underline{G}}(\mathbf{F})$
be a 6-generic tame inertial $L$-parameter. Let $\sigma\in
W^{?}(\overline{\rho})$ be a Serre weight with a lowest alcove presentation
$(\widetilde{w}_{h}^{-1}\widetilde{w},\widetilde{w}_{2})$ for some
$(\widetilde{w},\widetilde{w}_{2})\in\underline{\widetilde{W}}_{1}^{+}\times\underline{\widetilde{W}}^{+}$
such that $\widetilde{w}_{2}\uparrow\widetilde{w}_{h}^{-1}\widetilde{w}$.
Suppose that $\tau$ is a tame type with a $6$-generic lowest alcove
representation such that
$\widetilde{w}(\overline{\rho},\tau)=\widetilde{w}^{-1}w_{0}\widetilde{w}_{2}$.
Then,

  1. (1)

$\sigma\in\mathrm{JH}(\overline{\sigma}(\tau))$; and

  2. (2)

if $\kappa\in W^{?}(\overline{\rho})\cap\mathrm{JH}(\overline{\sigma}(\tau))$
and $\sigma\uparrow\kappa$, then $\sigma=\kappa$.

###### Proof.

Since
$\widetilde{w}(\overline{\rho})\widetilde{w}_{2}^{-1}(0)=\widetilde{w}(\tau)\widetilde{w}^{-1}(0)$,
we have
$\sigma=F_{(\widetilde{w}_{h}^{-1}\widetilde{w},\widetilde{w}(\tau)\widetilde{w}^{-1}(0))}$
and
$F_{(\widetilde{w}_{h}^{-1}\widetilde{w},\widetilde{w}(\tau)\widetilde{w}^{-1}(0))}\in\mathrm{JH}(\overline{\sigma}(\tau))$
by Proposition 2.5.6. Suppose that $\kappa\in
W^{?}(\overline{\rho})\cap\mathrm{JH}(\overline{\sigma}(\tau))$ and
$\sigma\uparrow\kappa$. Let $(\widetilde{s},\omega)$ be a lowest alcove
presentation of $\kappa$ compatible with $\overline{\rho}$. By Proposition
2.5.6, there exists $\widetilde{s}_{1}\in\widetilde{W}^{+}$ such that
$\widetilde{s}\uparrow\widetilde{w}_{h}^{-1}\widetilde{s}_{1}$ and
$\omega=\widetilde{w}(\tau)\widetilde{s}_{1}^{-1}(0)$. By Lemma 2.5.10,
$\sigma\uparrow\kappa$ implies that $\widetilde{s}_{1}=\widetilde{w}$ and
$\widetilde{w}_{h}^{-1}\widetilde{w}\uparrow\widetilde{s}$. By [LLHLMa,
Proposition 2.1.6], $\widetilde{w}_{h}^{-1}\widetilde{w}\uparrow\widetilde{s}$
is equivalent to $\widetilde{w}_{h}\widetilde{s}\uparrow\widetilde{w}$ and
$\widetilde{s}\uparrow\widetilde{w}_{h}^{-1}\widetilde{s}_{1}$ is equivalent
to $\widetilde{s}_{1}=\widetilde{w}\uparrow\widetilde{w}_{h}\widetilde{s}$.
Thus $\widetilde{s}=\widetilde{w}_{h}^{-1}\widetilde{w}$ and $\sigma=\kappa$.
∎

##  3\. The theory of local models

In this section, we generalize the theory of local models in [LLHLMa] to the
group $\mathrm{GSp}_{4}$. Note that when we write $\mathrm{GSp}_{4}$ in this
section, we mean the dual group $\mathrm{GSp}_{4}^{\vee}$. In particular, the
set of coroots $\Phi^{\vee}$ of $\mathrm{GSp}_{4}$ is identified with a set of
roots of $\mathrm{GSp}_{4}^{\vee}$ by the duality isomorphism $\phi$. If
$\alpha^{\vee}\in\Phi^{\vee}$, we let
$U_{\alpha^{\vee}}\subset\mathrm{GSp}_{4}^{\vee}$ denote the root subgroup
associated to the root $\phi(\alpha^{\vee})$ of $\mathrm{GSp}_{4}^{\vee}$. In
other words, $U_{\alpha}^{\vee}\subset\mathrm{GSp}_{4}^{\vee}$ is a subgroup
such that $U_{\alpha^{\vee}}\simeq\mathbf{G}_{a}$ and
$tut^{-1}=\phi(\alpha^{\vee})(t)u$ for any $t\in T^{\vee}$ and $u\in
U_{\alpha^{\vee}}$.

We let $G=\mathrm{GSp}_{4}$ for the remainder of this paper.

Let $X=\mathbb{A}^{1}_{\mathbf{Z}}$ be an affine line with coordinate function
$v$. We denote by $X_{0}=\operatorname{\mathrm{Spec}}\mathbf{Z}$ the zero
section of $X$ and $X^{0}=\operatorname{\mathrm{Spec}}\mathbf{Z}[v,v^{-1}]$.
We often write $t:\operatorname{\mathrm{Spec}}R\rightarrow X$ to denote
$\mathbf{Z}[v]$-algebra $R$ such that $v$ is mapped to $t\in R$.

###  3.1. Global affine Grassmannians

Let ${\mathcal{G}}$ be the Neron blowup of $\mathrm{GSp}_{4/X}$ in $B_{/X}$
along $X_{0}$ defined in [MRR, Definition 3.1]. By Theorem 3.2 of _loc. cit._
, it is a smooth affine group scheme over $X$ with connected fibers. For
$t:\operatorname{\mathrm{Spec}}R\rightarrow X$ with $t$ regular in $R$, the
set of $R$-points is given by

$\displaystyle{\mathcal{G}}(R)=\\{g\in\mathrm{GSp}_{4}(R)\mid\ g\bmod t\in
B(R/t)\\}.$

There is a morphism of $X$-group schemes
${\mathcal{G}}\rightarrow\mathrm{GSp}_{4/X}$. If $g\in{\mathcal{G}}(R)$, we
denote by $\overline{g}$ its image in $\mathrm{GSp}_{4}(R)$. We also have a
similitude character
$\operatorname{\mathrm{sim}}:{\mathcal{G}}\rightarrow\mathbf{G}_{m}$ sending
$g$ to $\operatorname{\mathrm{sim}}(\overline{g})$.

The base change ${\mathcal{G}}\times_{X}X^{0}$ is isomorphic to
$\mathrm{GSp}_{4/X^{0}}$ and the base change along
$\varpi:\operatorname{\mathrm{Spec}}{\mathcal{O}}\rightarrow X$ is isomorphic
to standard Iwahori group scheme ${\mathcal{I}}$ whose $R$-points for any
${\mathcal{O}}$-algebra $R$ are given by

$\displaystyle{\mathcal{I}}(R)=\left\\{g\in\mathrm{GSp}_{4}(R\llbracket
v\rrbracket)\mid g\bmod v\in B(R)\right\\}.$

In particular,
${\mathcal{G}}\times_{X,\varpi}\operatorname{\mathrm{Spec}}{\mathcal{O}}$
coincides with the group scheme constructed in [PZ13, Corollary 4.2] for
$\mathrm{GSp}_{4}$ and ${\mathcal{I}}$ (see also [MRR, Example 3.3]).

We also define a functor $L^{+}{\mathcal{M}}$ whose $R$-points, for
$t:\operatorname{\mathrm{Spec}}R\rightarrow X$, are given by

$\displaystyle
L^{+}{\mathcal{M}}(R):=\\{g\in\operatorname{\mathrm{Lie}}\mathrm{GSp}_{4}(R\llbracket
v-t\rrbracket)\mid\text{$g$ is upper triangular modulo $v$}\\}.$

Let ${\mathcal{G}}_{4}$ be the Bruhat–Tits group for $\mathrm{GL}_{4}$ defined
as in [LLHLMa, §3.1]. Note that the map
$\operatorname{{\mathrm{std}}}:\mathrm{GSp}_{4}\rightarrow\mathrm{GL}_{4}$
induces a morphism $X$-group schemes
${\mathcal{G}}\rightarrow{\mathcal{G}}_{4}$ (see [MRR, §2.4]) which we denote
by $\operatorname{{\mathrm{std}}}$ as well. It is easy to see that
$\operatorname{{\mathrm{std}}}:{\mathcal{G}}\rightarrow{\mathcal{G}}_{4}$ is a
closed immersion.

We write $L{\mathcal{G}}$ for the loop group and $L^{+}{\mathcal{G}}$ for the
positive loop group of ${\mathcal{G}}$. For
$t:\operatorname{\mathrm{Spec}}R\rightarrow X$, their $R$-points are given by

$\displaystyle L{\mathcal{G}}(R)$
$\displaystyle={\mathcal{G}}(R(\\!(v-t)\\!))$ $\displaystyle
L^{+}{\mathcal{G}}(R)$ $\displaystyle={\mathcal{G}}(R\llbracket
v-t\rrbracket)$

where we consider $R\llbracket v-t\rrbracket$ and $R(\\!(v-t)\\!)$ as
$\mathbf{Z}[v]$-algebra by sending $v$ to $v$. It is known that
$L^{+}{\mathcal{G}}$ is representable by a (not finite type) group scheme and
$L{\mathcal{G}}$ is representable by an ind-group scheme.

######  Remark 3.1.1.

When $R$ is Noetherian, $v$ is regular in both $R\llbracket v-t\rrbracket$ and
$R(\\!(v-t)\\!)$. Thus we have the following description:

$\displaystyle L{\mathcal{G}}(R)$
$\displaystyle=\left\\{g\in\mathrm{GSp}_{4}(R(\\!(v-t)\\!))\mid g\bmod v\in
B(R(\\!(v-t)\\!)/v)\right\\}$ $\displaystyle L^{+}{\mathcal{G}}(R)$
$\displaystyle=\left\\{g\in\mathrm{GSp}_{4}(R\llbracket v-t\rrbracket)\mid
g\bmod v\in B(R\llbracket v-t\rrbracket/v)\right\\}.$

We define $\mathrm{Gr}_{{\mathcal{G}},X}$ to be the fpqc quotient sheaf
$L^{+}{\mathcal{G}}{\backslash}L{\mathcal{G}}$. By [PZ13, Proposition 6.5],
$\mathrm{Gr}_{{\mathcal{G}},X}$ is representable by an ind-projective ind-
scheme. By the properties of ${\mathcal{G}}$, the generic fiber
$\mathrm{Gr}_{{\mathcal{G}},X}\times_{X}X^{0}$ is isomorphic to
$\mathrm{Gr}_{\mathrm{GSp}_{4}}\times_{\mathbf{Z}}X^{0}$, a constant familiy
of affine Grassmannian for the group $\mathrm{GSp}_{4}$ over $X^{0}$, and its
special fiber $\mathrm{Gr}_{{\mathcal{G}},X}\times_{X}X_{0}$ is the affine
flag variety $\mathrm{Fl}:={\mathcal{I}}{\backslash}\mathrm{GSp}_{4}$.

Let $d\in\mathbf{Z}$ and $h\in\mathbf{Z}_{\geq 0}$. We define subfunctors
$L{\mathcal{G}}^{\operatorname{\mathrm{sim}}=d},L{\mathcal{G}}^{\operatorname{\mathrm{sim}}=d,\leq
h}\subset L{\mathcal{G}}$ by

$\displaystyle L{\mathcal{G}}^{\operatorname{\mathrm{sim}}=d}(R)$
$\displaystyle=\left\\{g\in
L{\mathcal{G}}(R)\mid\operatorname{\mathrm{sim}}(\overline{g})\in(v-t)^{d}(R\llbracket
v-t\rrbracket)^{\times}\right\\}$ $\displaystyle
L{\mathcal{G}}^{\operatorname{\mathrm{sim}}=d,\leq h}(R)$
$\displaystyle=\left\\{g\in
L{\mathcal{G}}^{\operatorname{\mathrm{sim}}=d}(R)\mid\overline{g}\in\frac{1}{(v-t)^{h}}M_{4}(R\llbracket
v-t\rrbracket)\right\\}.$

Both of them are stable under left multiplication by $L^{+}{\mathcal{G}}$ and
induce fpqc quotient subsheafs

$\displaystyle\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}:=L^{+}{\mathcal{G}}{\backslash}L{\mathcal{G}}^{\operatorname{\mathrm{sim}}=d,\leq
h}\subset\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d}:=L^{+}{\mathcal{G}}{\backslash}L{\mathcal{G}}^{\operatorname{\mathrm{sim}}=d}\subset\mathrm{Gr}_{{\mathcal{G}},X}.$

The sheaf $\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}$ is representable by a projective scheme over $X$ and
$\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d}=\varinjlim_{h}\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}$.

Our next goal is describing affine open charts of the projective scheme
$\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq h}$. We
define the negative loop group to be a subfunctor $L^{--}{\mathcal{G}}\subset
L{\mathcal{G}}$ such that for $t:\operatorname{\mathrm{Spec}}R\rightarrow X$,

$\displaystyle
L^{--}{\mathcal{G}}(R)=\left\\{g\in{\mathcal{G}}(R(\\!(v-t)\\!))\middle|\begin{array}[]{c}\overline{g}\in\mathrm{GSp}_{4}\left(R\left[\frac{1}{v-t}\right]\right)\\\
\overline{g}\bmod{\frac{1}{v-t}}\in\overline{U}(R)\\\
\overline{g}\bmod\frac{v}{v-t}\in
B\left(R\left[\frac{1}{v-t}\right]/\left(\frac{v}{v-t}\right)\right)\end{array}\right\\}.$

######  Lemma 3.1.2.

The multiplication map

$\displaystyle L^{+}{\mathcal{G}}\times_{X}L^{--}{\mathcal{G}}\rightarrow
L{\mathcal{G}}$

is formally étale in the sense of [LLHLMa, Definition 3.2.4], and so is the
map $L^{--}{\mathcal{G}}\rightarrow\mathrm{Gr}_{{\mathcal{G}},X}$.

###### Proof.

Since this map is a restriction of a monomorphism
$L^{+}{\mathcal{G}}_{4}\times_{X}L^{--}{\mathcal{G}}_{4}\rightarrow
L{\mathcal{G}}_{4}$ ([LLHLMa, Lemma 3.2.2]), it is a monomorphism. Then we can
show that it is formally étale following the argument in [LLHLMa, Lemma 3.2.6]
using a version of [LLHLMa, Lemma 3.2.3] for our setup and the formal
smoothness of $L^{+}{\mathcal{G}}$ and $L^{--}{\mathcal{G}}$. Note that formal
smoothness easily follows from the smoothness of ${\mathcal{G}}$. ∎

Let $\widetilde{z}=wt_{\nu}\in\widetilde{W}^{\vee}$. We define
${\mathcal{U}}(\widetilde{z})$ to be the subfunctor of $L{\mathcal{G}}$ given
by

$\displaystyle{\mathcal{U}}(\widetilde{z})(R)=\left\\{g\in
L{\mathcal{G}}(R)\middle|\begin{array}[]{cl}\overline{g}(v-t)^{-\nu}\in\mathrm{GSp}_{4}(R[\frac{1}{v-t}]),\\\
\overline{g}(v-t)^{-\nu}w^{-1}\bmod\frac{1}{v-t}\in\overline{U}(R),\\\
\overline{g}(v-t)^{-\nu}\bmod\frac{v}{v-t}\in
B(R[\frac{1}{v-t}]/(\frac{v}{v-t}))\end{array}\right\\}.$

######  Lemma 3.1.3.

The subfunctor ${\mathcal{U}}(\widetilde{z})\subset L{\mathcal{G}}$ is stable
under left multiplication by $L^{--}{\mathcal{G}}$ and is left
$L^{--}{\mathcal{G}}$-torsor. The natural map
${\mathcal{U}}(\widetilde{z})\rightarrow\mathrm{Gr}_{{\mathcal{G}},X}$ is
monomorphism.

###### Proof.

The first claim follows from the definition of ${\mathcal{U}}(\widetilde{z})$
and the second claim follows from Lemma 3.1.2. ∎

For $d=\operatorname{\mathrm{sim}}(\nu)$ and $h\in\mathbf{Z}_{\geq 0}$, we
define ${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq h}$ to
be the base change
${\mathcal{U}}(\widetilde{z})\times_{L{\mathcal{G}}}L{\mathcal{G}}^{\operatorname{\mathrm{sim}}=d,\leq
h}$. The following Proposition shows that
${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq h}$ is
represented by a finite type affine scheme over $X$.

######  Proposition 3.1.4.

For a Noetherian $\mathbf{Z}[v]$-algebra $R$,
${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq h}(R)$ is the
set of matrices $A\in\mathrm{GSp}_{4}(R[(v-t)^{\pm 1}])$ satisfying:

  * •

For $1\leq i,j\leq 4$,

$\displaystyle
A_{ij}=v^{\delta_{i>j}}\left(\sum_{k=-h}^{\nu^{\prime}_{j}-\delta_{i>j}-\delta_{i<w^{\prime}(j)}}c_{ij,k}(v-t)^{k}\right)$

and $c_{w^{\prime}(j)j,\nu^{\prime}_{j}-\delta_{w^{\prime}(j)>j}}=1$ where
$(\nu^{\prime}_{1},\nu^{\prime}_{2},\nu^{\prime}_{3},\nu^{\prime}_{4})=\operatorname{{\mathrm{std}}}(\nu)$
and $w^{\prime}=\operatorname{{\mathrm{std}}}(w)$,

  * •

$\operatorname{\mathrm{sim}}(A)=\operatorname{\mathrm{sim}}(w)(v-t)^{d}$.

###### Proof.

This follows from the fact that
${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq
h}={\mathcal{U}}_{4}(\operatorname{{\mathrm{std}}}(\widetilde{z}))^{\det,\leq
h}\times_{L{\mathcal{G}}_{4}}L{\mathcal{G}}$ and Proposition 3.2.8 in [LLHLMa]
(where
${\mathcal{U}}_{4}(\operatorname{{\mathrm{std}}}(\widetilde{z}))^{\det,\leq
h}$ denotes the affine chart defined in §3.2 of _loc. cit._). ∎

######  Proposition 3.1.5.

The map ${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq
h}\rightarrow\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}$ is an open immersion.

###### Proof.

We claim that ${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq
h}\rightarrow\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}$ is formally étale. Since a formally étale monomorphism between finite type
schemes is an open immersion ([LLHLMa, Remark 3.2.5] and [Sta18, Tag 025G]),
this completes the proof.

Let $A$ be an Artinian local ring with residue field $k$. Suppose we have the
following commutative diagram

${\operatorname{\mathrm{Spec}}k}$${{\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq
h}}$${\operatorname{\mathrm{Spec}}A}$${\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}.}$

We write $\overline{t}\in k$ and $t\in A$ for the image of the coordinate $v$
of $X$. Since $k$ and $A$ are Noetherian, we can interpret the image of
$\operatorname{\mathrm{Spec}}k$ (and $\operatorname{\mathrm{Spec}}A$) in
${\mathcal{U}}(\widetilde{z})$ as a matrix
$g_{k}\in\mathrm{GSp}(k(\\!(v-\overline{t})\\!))$ satisfying certain
conditions. We claim that there exists
$g_{A}\in{\mathcal{U}}(\widetilde{z})(A)$ lifting $g_{k}$. Since
$g_{k}(v-\overline{t})^{-\nu}\in\mathrm{GSp}_{4}(k[{\frac{1}{v-\overline{t}}}])$,
we can find $g_{A}\in L{\mathcal{G}}(A)$ lifting $g_{k}$ and
$g_{A}(v-t)^{-\nu}\in\mathrm{GSp}_{4}(A[{\frac{1}{v-t}}])$ by the smoothness
of $\mathrm{GSp}_{4}$. The set of such $g_{A}$ is
$\operatorname{\mathrm{Lie}}\mathrm{GSp}_{4}({\mathfrak{m}}_{A}[\frac{1}{v-t}])(v-t)^{\nu}$-coset.
To show that there exists $g^{\prime}_{A}\in{\mathcal{U}}(\widetilde{z})(A)$
lifting $g_{k}$, we need to find $g^{\prime}_{A}$ satisfying two conditions:

$\displaystyle
g^{\prime}_{A}(v-t)^{-\nu}w^{-1}\bmod\frac{1}{v-t}\in\overline{U}(A),\ \
g^{\prime}_{A}(v-t)^{-\nu}\bmod\frac{v}{v-t}\in
B(A[\frac{1}{v-t}]/(\frac{v}{v-t})).$

Thus, we need to find
$N\in\operatorname{\mathrm{Lie}}\mathrm{GSp}_{4}({\mathfrak{m}}_{A}[\frac{1}{v-t}])$
such that

$\displaystyle
g_{A}(v-t)^{-\nu}w^{-1}+Nw^{-1}\bmod\frac{1}{v-t}\in\overline{U}(A),\ \
g_{A}(v-t)^{-\nu}+N\bmod\frac{v}{v-t}\in B(A[\frac{1}{v-t}]/(\frac{v}{v-t})).$

The existence of such $N$ follows from the surjectivity of the quotient map

$\displaystyle\operatorname{\mathrm{Lie}}\mathrm{GSp}_{4}({\mathfrak{m}}_{A}[\frac{1}{v-t}])\rightarrow\operatorname{\mathrm{Lie}}\mathrm{GSp}_{4}({\mathfrak{m}}_{A})\times\operatorname{\mathrm{Lie}}\mathrm{GSp}_{4}({\mathfrak{m}}_{A}[\frac{1}{v-t}]/(\frac{v}{v-t})).$

Once we know that ${\mathcal{U}}(\widetilde{z})(A)\neq\emptyset$, we can use
Lemma 3.1.2 to find
$\operatorname{\mathrm{Spec}}A\rightarrow{\mathcal{U}}(\widetilde{z})$ which
lifts $g_{k}$ and when composed with
${\mathcal{U}}(\widetilde{z})\rightarrow\mathrm{Gr}_{{\mathcal{G}},X}$
provides the $A$-point of
$\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq h}$ given
in the above diagram. Then it has to factor through
${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq h}$. Thus
${\mathcal{U}}(\widetilde{z})^{\operatorname{\mathrm{sim}},\leq
h}\rightarrow\mathrm{Gr}_{{\mathcal{G}},X}^{\operatorname{\mathrm{sim}}=d,\leq
h}$ is formally étale. ∎

###  3.2. Geometry of universal local models

We introduce universal local models and discuss their basic properties.

####  3.2.1. Schubert varieties

Given a dominant cocharacter $\lambda\in X_{*}(T^{\vee})$, we denote by
$s_{\lambda}$ a section $X\rightarrow\mathrm{Gr}_{{\mathcal{G}},X}$ induced by
the element $(v-t)^{\lambda}\in L{\mathcal{G}}(R)$ for any
$\mathbf{Z}[v]$-algebra $R$. A _global Schubert variety_
${\mathcal{S}}_{X}(\lambda)$ is defined as the minimal irreducible closed
subscheme of $\mathrm{Gr}_{{\mathcal{G}},X}$ containing the section
$s_{\lambda}$ and stable under the right multiplication of
$L^{+}{\mathcal{G}}$ (cf. [Zhu14, Definition 3.1]). The map
$S_{X}(\lambda)\rightarrow X$ is proper. We also write
${\mathcal{S}}_{X^{0}}(\lambda)={\mathcal{S}}_{X}(\lambda)\times_{X}X^{0}$. As
$\mathrm{Gr}_{{\mathcal{G}},X}\times_{X}X^{0}\simeq\mathrm{Gr}_{\mathrm{GSp}_{4}}\times_{\mathbf{Z}}X^{0}$,
${\mathcal{S}}_{X^{0}}(\lambda)$ is the constant family of the Schubert
variety in $\mathrm{Gr}_{\mathrm{GSp}_{4}}$ for $\lambda$ over $X^{0}$.

Let ${\mathrm{Conv}}(\lambda)$ be the convex hull of the subset
$W\lambda\subset X_{*}(T^{\vee})$. A _open Schubert cell_