Abstract. Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a connected $p$-adic reductive group $G/L$, and a smooth irreducible representation $\pi$ of $G(L)$, Fargues-Scholze <cit.> recently attached a semisimple $L$-parameter to such $\pi$, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of the correspondence after semisimplification. For $G = \GL_{n}$ and its inner forms,  Fargues-Scholze and Hansen-Kaletha-Weinstein <cit.> showed that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart <cit.>. We verify a similar compatibility for $G = \GSp_{4}$ and its unique non-split inner form $G = \GU_{2}(D)$, where $D$ is the quaternion division algebra over $L$, assuming that $L/\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono <cit.>. Analogous to the case of $\GL_{n}$ and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to $\GSp_{4}$, using basic uniformization of abelian type Shimura varieties due to Shen <cit.>, combined with various global results of Kret-Shin <cit.> and Sorensen <cit.> on Galois representations in the cohomology of global Shimura varieties associated to inner forms of $\GSp_{4}$ over a totally real field. After showing the parameters are the same, we apply some ideas from the geometry of the Fargues-Scholze construction explored recently by Hansen <cit.>. This allows us to give a more precise description of the cohomology of this local Shimura variety, verifying a strong form of the Kottwitz conjecture in the process.

§ ACKNOWLEDGEMENTS
It is a pleasure to thank my advisor David Hansen for giving me this project and for numerous suggestions and ideas related to it, as well as Eric Chen, Arthur-César Le-Bras, Yoichi Mieda, Jack Sempliner, and Zhiyu Zhang for some nice conversations related to this work. Special thanks also go to Peter Scholze for sharing with me the draft of <cit.>, pointing out some errors, and generously initiating me in these ideas during my masters thesis, as well as Thomas Haines and Sug-Woo Shin for pointing out some errors in section $3$ and section $5$ respectively. Lastly, I would like to thank the MPIM Bonn for their hospitality during part of the completion of this project and the anonymous referee for pointing out several mistakes and inaccuracies.

§ INTRODUCTION

§.§ Background and Main Theorems

Fix distinct primes $\ell \neq p$, let $\mathbb{Q}_{p}$ denote the $p$-adic numbers, and let $G/\mathbb{Q}_{p}$ be a connected reductive group. Set $\mathbb{C}_{p} := \hat{\overline{\mathbb{Q}}}_{p}$ to be the completion of the algebraic closure of $\mathbb{Q}_{p}$. We fix an isomorphism $i: \overline{\mathbb{Q}}_{\ell} \xrightarrow{\simeq} \mathbb{C}$. Let $W_{\mathbb{Q}_{p}}$ be the Weil group of $\mathbb{Q}_{p}$ and set $\hat{G}$ to be the reductive group over $\overline{\mathbb{Q}}_{\ell}$ with root datum dual to $G$. Let $Q$ be the finite quotient through which $W_{\mathbb{Q}_{p}}$ acts on $\hat{G}$. We define the $L$-group $^LG := Q \ltimes \hat{G}$. We let $\Pi(G)$ denote the set of isomorphism classes of smooth irreducible representations of the $p$-adic group $G(\mathbb{Q}_{p})$, and let $\Phi(G)$ denote the set of $L$-parameters, i.e the set of conjugacy classes of homomorphisms

\[ \phi: W_{\mathbb{Q}_{p}} \times SL_{2}(\mathbb{C}) \rightarrow \phantom{}^{L}G(\mathbb{C}) \]

where $SL_{2}(\mathbb{C})$ acts via an algebraic representation and $W_{\mathbb{Q}_{p}}$ acts via a continuous semisimple homomorphism in a way that commutes with the natural projection $\phantom{}^{L}G(\mathbb{C}) \rightarrow Q$, where $\phantom{}^{L}G(\mathbb{C})$ is endowed with the discrete topology. The local Langlands correspondence is a conjectural map

\[ \LLC_{G}: \Pi(G) \rightarrow \Phi(G) \]

\[ \pi \mapsto \phi_{\pi} \]

that builds a bridge between $L$-parameters and the smooth irreducible representations of $G(\mathbb{Q}_{p})$. Conjecturally (under some additional constraints on $\Phi(G)$ if $G$ is not split), these maps should be surjective with finite fibers called $L$-packets and satisfy various properties such as compatibility with products, maps of $L$-groups, character twists, as well as $L$, $\epsilon$, and $\gamma$-factors. Moreover, one expects that the correspondence is uniquely characterized by some such finite list of properties.

In general, the existence and uniqueness of such a correspondence is completely unknown. However, very recently, Fargues and Scholze <cit.>, using the action of the excursion algebra on the moduli space of $G$-bundles on the Fargues-Fontaine curve, were able to construct a completely general candidate, analogous to the work of V. Lafforgue in the function field setting <cit.>. Namely, they construct a map

\[ \LLC_{G}^{\mathrm{FS}}: \Pi(G) \rightarrow \Phi^{\mathrm{ss}}(G) \]

\[ \pi \mapsto \phi_{\pi}^{\mathrm{FS}} \]

where $\Phi^{\mathrm{ss}}(G)$ denotes the set of conjugacy classes of continuous semisimple maps

\[ \phi: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \]

that commute with the projection $\phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \rightarrow Q$ as above. Fargues and Scholze showed that their map has several good properties such as compatibility with parabolic induction; however, one would also like to check that this correspondence agrees with known instances of the local Langlands correspondence. Precisely, given a candidate for the local Langlands correspondence

\[ \LLC_{G}: \Pi(G) \rightarrow \Phi(G) \]

\[ \pi \mapsto \phi_{\pi} \]

we expect a commutative diagram of the form

\begin{equation*}
\begin{tikzcd}[ampersand replacement=\&]
\Pi(G)  \ar[rr, "\LLC_{G}"] \arrow[drr,"LLC^{\mathrm{FS}}_{G}"] \& \&   \Phi(G) \ar[d,"(-)^{\mathrm{ss}}"] \\
\& \& \Phi^{\mathrm{ss}}(G)
\end{tikzcd}
\end{equation*}

where the semisimplification map $(-)^{\mathrm{ss}}$ precomposes an $L$-parameter $\phi \in \Phi(G)$ with the map
\[g \in W_{\mathbb{Q}_{p}} \mapsto (g,\begin{pmatrix}
|g|^{\frac{1}{2}} & 0 \\
0 & |g|^{\frac{-1}{2}}
\end{pmatrix}) \in W_{\mathbb{Q}_{p}} \times \SL_{2}(\mathbb{C}) \]

and then applies the fixed isomorphism $i^{-1}: \mathbb{C} \xrightarrow{\simeq} \overline{\mathbb{Q}}_{\ell}$, where $|\cdot|: W_{\mathbb{Q}_{p}} \rightarrow W_{\mathbb{Q}_{p}}^{ab} \simeq \mathbb{Q}_{p}^{*} \rightarrow \mathbb{C}^{*}$ is the norm character. We make the following definition.

For $\pi \in \Pi(G)$, we say that a local Langlands correspondence $\LLC_{G}$ is compatible with the Fargues-Scholze local Langlands correspondence if we have an equality: $\phi^{\mathrm{FS}}_{\pi} = \phi_{\pi}^{\mathrm{ss}}$, as conjugacy classes of semi-simple L-parameters.

For $\GL_{n}$, the local Langlands correspondence was constructed by Harris-Taylor/Henniart <cit.> and is uniquely characterized by the preservation of $L$, $\epsilon$, and $\gamma$-factors. In this case, compatibility with the Fargues-Scholze local Langlands correspondence follows from the description of the cohomology of the Lubin-Tate and Drinfeld towers proven in <cit.> and was verified by Fargues and Scholze <cit.>. The main goal of this note is to extend compatibility of the correspondence to $\GSp_{4}$ and its inner form. To this end, we now fix a finite extension $L/\mathbb{Q}_{p}$ and set $G$ to be $\mathrm{Res}_{L/\mathbb{Q}_{p}}\GSp_{4}$ and $J$ to be its unique non-split inner form $\mathrm{Res}_{L/\mathbb{Q}_{p}}\GU_{2}(D)$, where $D/L$ is the quaternion division algebra. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono, respectively <cit.>. It is constructed from the local Langlands correspondence for $\GL_{n}$ and theta lifting and admits a similar unique characterization in terms of the preservation of $L$, $\epsilon$, and $\gamma$ factors. We note that we can and do identify $\Phi(G)$ and $\Phi(J)$ with a subset of homomorphisms:

\[ \phi: W_{L} \times \SL_{2}(\mathbb{C}) \rightarrow \hat{G}(\mathbb{C}) = \GSpin_{5}(\mathbb{C}) \simeq \GSp_{4}(\mathbb{C}) \]

This allows us to introduce a bit of terminology. Namely, we say that a parameter $\phi$ in $\Phi(G)$ or $\Phi(J)$ is supercuspidal if the  $\SL_{2}(\mathbb{C})$-factor acts trivially and $\phi$ does not factor through any proper Levi subgroup of $\GSp_{4}$. This terminology is justified by the fact that this is precisely the case when the $L$-packets over $\phi$ contain only supercuspidal representations. In what follows, we will often abuse notation and drop the superscript $(-)^{\mathrm{ss}}$ when speaking about such parameters, as in this case it merely corresponds to forgetting the trivially acting $\SL_{2}(\mathbb{C})$-factor and applying the isomorphism $i^{-1}$. We now come to our main theorem.

The following is true.

* For any $\pi \in \Pi(G)$ (resp. $\rho \in \Pi(J)$) such that the Gan-Takeda (resp. Gan-Tantono) parameter is not supercuspidal, we have that the Gan-Takeda (resp. Gan-Tantono) correspondence is compatible with the Fargues-Scholze correspondence.

* If $L/\mathbb{Q}_{p}$ is unramified and $p > 2$, we have, for all $\pi \in \Pi(G)$ (resp. $\rho \in \Pi(J)$) such that the Gan-Takeda (resp. Gan-Tantono) parameter is supercuspidal, that the Gan-Takeda (resp. Gan-Tantono) correspondence is compatible with the Fargues-Scholze correspondence.

As will be explained more below, the restrictions in the case where the parameter is supercuspidal are necessary to apply basic uniformization of the generic fiber of abelian type Shimura varieties due to Shen <cit.>. If one were not to impose this assumption, the relevant Shimura varieties would have bad reduction at $p$, which, to the best of our knowledge, prevents the methods of Shen from working. In particular, if one could establish the expected description of basic locus in the sense of the isomorphism (2) of Definition 4.1, for Shimura varieties associated to the group $\mathrm{Res}_{F/\mathbb{Q}}\mathbf{G}$, where $\mathbf{G}$ is an inner form of $\GSp_{4}$ over a totally real field $F$ with an inert prime $p$ such that $F_{p} \simeq L$ for $L$ any extension then our result would hold in complete generality.

As mentioned above, the proof of compatibility for $\GL_{n}$ uses the results of Harris-Taylor <cit.> on the cohomology of the Lubin-Tate/Drinfeld Towers. In particular, if one looks at the rigid generic fiber of the Lubin-Tate tower

\[ \lim_{m \rightarrow \infty} \mathrm{LT}_{n,m,\breve{\mathbb{Q}}_{p}} \]

a tower of $n - 1$-dimensional rigid spaces over $\breve{\mathbb{Q}}_{p}$, for fixed $n \geq 1$ and varying $m \geq 1$, where $\breve{\mathbb{Q}}_{p}$ denotes the completion of the maximal unramified extension of $\mathbb{Q}_{p}$. The cohomology of this tower

\[ R\Gamma_{c}(\mathrm{LT}_{n,\infty},\overline{\mathbb{Q}}_{\ell}) := \colim_{m \rightarrow \infty} R\Gamma_{c}(\mathrm{LT}_{n,m,\mathbb{C}_{p}},\overline{\mathbb{Q}}_{\ell}) \]

based changed to $\mathbb{C}_{p}$ carries commuting actions of $\GL_{n}(\mathbb{Q}_{p})$ and $D^{*}_{\frac{1}{n}}$, the units in the division algebra over $\mathbb{Q}_{p}$ of invariant $\frac{1}{n}$, as well as an action of the Weil group $W_{\mathbb{Q}_{p}}$. In particular, given $\pi \in \Pi(\GL_{n})$ (resp. $\rho \in \Pi(D^{*}_{\frac{1}{n}})$), we can consider the complexes

\[ R\Gamma_{c}(\mathrm{LT}_{n,\infty},\overline{\mathbb{Q}}_{\ell})[\pi] := R\Gamma_{c}(\mathrm{LT}_{n,\infty},\overline{\mathbb{Q}}_{\ell}) \otimes^{\mathbb{L}}_{\mathcal{H}(\GL_{n})} \pi \]

\[ R\Gamma_{c}(\mathrm{LT}_{n,\infty},\overline{\mathbb{Q}}_{\ell})[\rho] := R\Gamma_{c}(\mathrm{LT}_{n,\infty},\overline{\mathbb{Q}}_{\ell}) \otimes^{\mathbb{L}}_{\mathcal{H}(D^*_{\frac{1}{n}})} \rho \]

where $\mathcal{H}(\GL_{n}) := C^{\infty}_{c}(\GL_{n}(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$ (resp. $\mathcal{H}(D^{*}_{\frac{1}{n}})$) is the usual smooth Hecke algebra of $G$ (resp. $D^{*}_{\frac{1}{n}}$). Then the key result of Harris-Taylor and later refined by Boyer and Dat <cit.> is as follows.
Fix $\pi \in \Pi(\GL_{n})$, a supercuspidal representation of $\GL_{n}(\mathbb{Q}_{p})$, let

\[ \mathrm{JL}: \Pi(D^{*}_{\frac{1}{n}}) \rightarrow \Pi(\GL_{n}(\mathbb{Q}_{p})) \]

be the map defined by the Jacquet-Langlands correspondence and $\rho := \mathrm{JL}^{-1}(\pi) \in \Pi(D_{\frac{1}{n}}^{*})$ a Jacquet-Langlands lift of $\pi$. Then the complexes $R\Gamma_{c}(G,b,\mu)[\pi]$ and $R\Gamma_{c}(G,b,\mu)[\rho]$ are concentrated in middle degree $n - 1$. The middle degree cohomology of $R\Gamma_{c}(G,b,\mu)[\pi]$ is isomorphic to

\[ \rho \boxtimes \phi_{\pi}^{\vee} \otimes |\cdot|^{(1 - n)/2} \]

as a $D^{*}_{\frac{1}{n}} \times W_{\mathbb{Q}_{p}}$ representation. Similarly, the middle degree cohomology of $R\Gamma_{c}(G,b,\mu)[\rho]$ is isomorphic to

\[ \pi \boxtimes \phi_{\pi} \otimes |\cdot|^{(1 - n)/2} \]

where $\phi_{\pi} \in \Phi^{\mathrm{ss}}(G)$ is the (semisimplified) L-parameter associated to $\pi$ by Harris-Taylor.

To see why this result is relevant for compatibility, we invoke the observation, due to Scholze-Weinstein <cit.>, that, using Grothendieck-Messing theory, the Lubin-Tate tower at infinite level

\[ \mathrm{LT}_{n,\infty} := \lim_{m \rightarrow \infty} \mathrm{LT}_{n,m,\breve{\mathbb{Q}}_{p}} \]

is representable by a space admitting a moduli interpretation as a space of shtukas over the Fargues-Fontaine curve; namely, the space denoted $\Sht(\GL_{n},b,\mu)_{\infty}$ in the notation of <cit.>, where $b \in B(\GL_{n})$ is an element in the Kottwitz set of $\GL_{n}$ corresponding to a rank $n$ isocrystal of slope $\frac{1}{n}$ and $\mu = (1,0,\ldots,0,0)$ is a dominant cocharacter of $\GL_{n}$. If $X$ denotes the Fargues-Fontaine curve, then this this parametrizes modifications

\[ \mathcal{O}_{X}(-\frac{1}{n}) \rightarrow \mathcal{O}_{X}^{n}  \]

of type $(1,0,\ldots,0,0)$ (i.e this map is an embedding with cokernel a length $1$ torsion sheaf on $X$), where $\mathcal{O}_{X}(-\frac{1}{n})$ is the unique rank $n$ vector bundle on $X$ of slope $-\frac{1}{n}$. This interpretation allows one to relate the complex
$R\Gamma_{c}(\mathrm{LT}_{n,\infty},\overline{\mathbb{Q}}_{\ell})[\pi]$ to the action of a Hecke operator $T_{\mu^{-1}}$ on $\Bun_{G}$, acting on a sheaf $\mathcal{F}_{\pi}$ constructed from the supercuspidal representation $\pi$, where $\mu^{-1} = (0,0,\ldots,0,-1)$ is a dominant inverse of $\mu$. The Fargues-Scholze parameter of $\pi$ is built from the action of the excursion algebra on the sheaf $\mathcal{F}_{\pi}$, which in turn is built from Hecke operators equipped with a factorization structure coming from geometric Satake. It thus is reasonable to expect that the cohomology group  $R\Gamma_{c}(\mathrm{LT}_{n,\infty},\overline{\mathbb{Q}}_{\ell})[\pi]$ should have $W_{\mathbb{Q}_{p}}$-action given by the Fargues-Scholze parameter $\phi_{\pi}^{\mathrm{FS}}: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}\GL_{n}(\overline{\mathbb{Q}}_{\ell}) \simeq \GL_{n}(\overline{\mathbb{Q}}_{\ell})$ composed with the highest weight representation of $\phantom{}^{L}\GL_{n}(\overline{\mathbb{Q}}_{\ell})$ corresponding to the dominant cocharacter $\mu^{-1}$. However, this is just the dual of the standard representation of $\GL_{n}(\overline{\mathbb{Q}}_{\ell})$. Thus, using Theorem 1.2, we can see that

\[ \phi_{\pi}^{\vee} = (\phi_{\pi}^{\mathrm{FS}})^{\vee} \]

where the twist by the norm-character $|\cdot|^{(1 - n)/2}$ is cancelled out by a perverse normalization (also related to the middle degree being the relevant one) in the definition of the Hecke operator $T_{\mu^{-1}}$. This implies compatibility for supercuspidal $\pi$, which, by using compatibility of the Fargues-Scholze correspondence with parabolic induction, is enough to conclude the general case. In a similar fashion, using the description of the $\rho$-isotypic part one can prove compatibility for the inner form $D^{*}_{\frac{1}{n}}$.

One may expect, given the above sketch of compatibility for $\GL_{n}$, that, to prove Theorem 1.1, one must similarly provide a description of the cohomology of the $\pi \in \Pi(G)$ (resp. $\rho \in \Pi(J)$)-isotypic part of a local Shimura variety/shtuka space at infinite level associated to $G$. In the case where the associated $L$-parameters are supercuspidal, this is the content of the Kottwitz conjecture <cit.>.  To this end, we consider the cohomology of a Shtuka space associated to the group $G = \mathrm{Res}_{L/\mathbb{Q}_{p}}\GSp_{4}$. Namely, the space denoted $\Sht(G,b,\mu)_{\infty}$, where $\mu$ is the Siegel cocharacter and $b \in B(G)$ is a basic element in the Kottwitz set of $G$, corresponding to a rank $4$ isocrystal with a polarization and automorphism group equal to $J = \mathrm{Res}_{L/\mathbb{Q}_{p}}(\GU_{2}(D))$. This space carries a commuting $G(\mathbb{Q}_{p})$ and $J(\mathbb{Q}_{p})$ action. The quotients

\[ \Sht(G,b,\mu)_{K} := \Sht(G,b,\mu)_{\infty}/\underline{K} \]

for varying compact open $K \subset G(\mathbb{Q}_{p})$ are, as before, representable by rigid analytic varieties over $\Spa(\breve{L})$ of dimension $3$, where $\breve{L} := L\breve{\mathbb{Q}}_{p}$. They define a tower of local Shimura varieties in the sense of Rapoport-Viehmann <cit.>, which uniformize the basic locus of certain global Shimura varieties analogous to the Lubin-Tate case described above. Letting $\Sht(G,b,\mu)_{K,\mathbb{C}_{p}}$ be the base-change of these spaces to $\mathbb{C}_{p}$, we can then consider the analog of the complexes described above

\[ R\Gamma_{c}(G,b,\mu) := \colim_{K \rightarrow 1} R\Gamma_{c}(\Sht(G,b,\mu)_{K,\mathbb{C}_{p}},\overline{\mathbb{Q}}_{\ell}) \]

This complex is concentrated in degrees $0 \leq i \leq 6 = 2\dim(\Sht(G,b,\mu)_{K})$ and admits an action of $G(\mathbb{Q}_{p}) \times J(\mathbb{Q}_{p}) \times W_{L}$. This allows one to consider the $\rho$ and $\pi$-isotypic parts, i.e we set

\[ R\Gamma_{c}(G,b,\mu)[\rho] := R\Gamma_{c}(G,b,\mu) \otimes^{\mathbb{L}}_{\mathcal{H}(J)}   \rho \]

\[ R\Gamma_{c}(G,b,\mu)[\pi] := R\Gamma_{c}(G,b,\mu) \otimes^{\mathbb{L}}_{\mathcal{H}(G)}   \pi \]

where $\mathcal{H}(G)$ (resp. $\mathcal{H}(J)$) are the usual smooth Hecke algebra of $G$ (resp. $J$). To deduce compatibility, one needs to realize the (semi-simplified) L-parameter $\phi_{\pi}$ (resp. $\phi_{\rho}$) of Gan-Takeda (resp. Gan-Tantono) in these two cohomology groups. We will sketch how to do this in the next section using uniformization and global methods. Interestingly,  after knowing compatibility, one can use ideas from the geometry of the Fargues-Scholze construction to provide a more precise of the complexes $R\Gamma_{c}(G,b,\mu)[\rho]$ and $R\Gamma_{c}(G,b,\mu)[\pi]$. Namely, recent work of Hansen <cit.> allows us to deduce that if $\phi_{\rho}$ (resp. $\phi_{\pi}$) is supercuspidal the complexes $R\Gamma_{c}(G,b,\mu)[\rho]$ (resp. $R\Gamma_{c}(G,b,\mu)[\pi]$) are concentrated in middle degree $3$. It then follows from work of Hansen-Kaletha-Weinstein <cit.> on a weakening of the Kottwitz conjecture and work of Fargues-Scholze <cit.> describing the Hecke action on objects with supercuspidal Fargues-Scholze parameter that one can actually deduce a strong form of the Kottwitz conjecture for these representations. To state this result, we first note that, if we are given a supercuspidal parameter $\phi: W_{L} \rightarrow \GSp_{4}(\overline{\mathbb{Q}}_{\ell})$, even though the parameter $\phi$ is irreducible, its composition with the standard embedding $\mathrm{std}: \GSp_{4}(\overline{\mathbb{Q}}_{\ell}) \hookrightarrow \GL_{4}(\overline{\mathbb{Q}}_{\ell})$ may not be. In particular, the size of the $L$-packets $\Pi_{\phi}(G) := \LLC_{G}^{-1}(\phi)$ and $\Pi_{\phi}(J) := \LLC_{J}^{-1}(\phi)$ over $\phi$ are governed by this.

* (stable) $\mathrm{std}\circ \phi$ is irreducible. In this case, the $L$-packets each contain one supercuspidal member.

* (endoscopic) $\mathrm{std}\circ \phi \simeq \phi_{1} \oplus \phi_{2}$, where $\phi_{i}: W_{L} \rightarrow \GL_{2}(\bar{\mathbb{Q}}_{\ell})$ are distinct irreducible $2$-dimensional representations with $\det(\phi_{1}) = \det(\phi_{2})$. In this case, the $L$-packets over $\phi$ each contain two supercuspidal members.

This allows us to state our main consequence of Theorem 1.1, which (almost) verifies the strong form of the Kottwitz conjecture for $\GSp_{4}/L$ and $\GU_{2}(D)/L$.

Let $L/\mathbb{Q}_{p}$ be an unramified extension with $p > 2$. Let $\pi$ (resp. $\rho$) be members of the $L$-packet over a supercuspidal parameter $\phi: W_{L} \rightarrow \GSp_{4}(\overline{\mathbb{Q}}_{\ell})$ as above. Then the complexes

\[ R\Gamma_{c}(G,b,\mu)[\pi] \]

\[ R\Gamma_{c}(G,b,\mu)[\rho] \]

are concentrated in middle degree $3$.

* If $\phi$ is stable supercuspidal, with singleton $L$-packets $\{\pi\} = \Pi_{\phi}(G)$ and $\{\rho\} = \Pi_{\phi}(J)$, then the cohomology of $R\Gamma_{c}(G,b,\mu)[\pi]$ in middle degree is isomorphic to

\[ \rho \boxtimes (\mathrm{std}\circ \phi)^{\vee} \otimes |\cdot|^{-3/2}   \]

as a $J(\mathbb{Q}_{p}) \times W_{L}$-module, and the cohomology of $R\Gamma_{c}(G,b,\mu)[\rho]$ in middle degree is isomorphic to

\[ \pi \boxtimes \mathrm{std}\circ \phi \otimes |\cdot|^{-3/2} \]

as a $G(\mathbb{Q}_{p}) \times W_{L}$-module.

* If $\phi$ is an endoscopic parameter, with $L$-packets $\Pi_{\phi}(G) = \{\pi^{+},\pi^{-}\}$ and $\Pi_{\phi}(J) = \{\rho_{1},\rho_{2}\}$[For an explanation of the notation, see the discussion at the end of section 2.2.], the cohomology of $R\Gamma_{c}(G,b,\mu)[\pi]$ in middle degree is isomorphic to

\[ \rho_{1} \boxtimes \phi_{1}^{\vee} \otimes |\cdot|^{-3/2} \oplus \rho_{2} \boxtimes \phi_{2}^{\vee} \otimes |\cdot|^{-3/2} \]

\[ \rho_{1} \boxtimes \phi_{2}^{\vee} \otimes |\cdot|^{-3/2} \oplus \rho_{2} \boxtimes \phi_{1}^{\vee} \otimes |\cdot|^{-3/2} \]

as a $J(\mathbb{Q}_{p}) \times W_{L}$-module. Similarly, the cohomology of $R\Gamma_{c}(G,b,\mu)[\rho]$ in middle degree is isomorphic to

\[  \pi^{+} \boxtimes \phi_{1} \otimes |\cdot|^{-3/2} \oplus \pi^{-} \boxtimes \phi_{2} \otimes |\cdot|^{-3/2} \]

\[  \pi^{+} \boxtimes \phi_{2} \otimes |\cdot|^{-3/2} \oplus \pi^{-} \boxtimes \phi_{1} \otimes |\cdot|^{-3/2} \]

as a $G(\mathbb{Q}_{p}) \times W_{L}$-module. Here we write $\mathrm{std}\circ \phi \simeq \phi_{1} \oplus \phi_{2}$, with $\phi_{i}$ distinct irreducible $2$-dimensional representations of $W_{L}$ and $\det(\phi_{1}) = \det(\phi_{2})$.

Moreover, both possibilities for the cohomology of $R\Gamma_{c}(G,b,\mu)[\rho]$ (resp. $R\Gamma_{c}(G,b,\mu)[\pi]$) in the endoscopic case occur for some choice of representation $\rho \in \Pi_{\phi}(J)$ (resp. $\pi \in \Pi_{\phi}(G)$). In particular, knowing the precise form of either $R\Gamma_{c}(G,b,\mu)[\rho]$ or $R\Gamma_{c}(G,b,\mu)[\pi]$ for some $\rho \in \Pi_{\phi}(J)$ or $\pi \in \Pi_{\phi}(G)$ determines the precise form of the cohomology in all other cases.

* Results of this form when $L = \mathbb{Q}_{p}$ have also been shown by Ito-Meida <cit.>.

* If one knew Arthur's multiplicity formula for inner forms of $\GSp_{4}$ over totally real fields, one should be able to determine the cohomology in the endoscopic case more precisely, using basic uniformization and the more precise description of the cohomology of the global Shimura variety this multiplicity formula would provide (See for example <cit.> for this kind of analysis in the case of unitary groups.). However, to our knowledge the multiplicity formula is unknown in this case. In the case that $L = \mathbb{Q}_{p}$, one can apply what is known about the multiplicity formula for $\GSp_{4}/\mathbb{Q}$ <cit.>. This is carried out by Ito-Mieda <cit.>. The correct answer, for the $\rho$-isotypic part, should be that, if $\rho = \rho_{1}$, we are in the first case, and if $\rho = \rho_{2}$, we are in the second case. Similarly, for the $\pi$-isotypic part, if $\pi = \pi^{+}$ is the unique generic member of the $L$-packet for a fixed choice of Whittaker datum, we should be in the first case and, if $\pi = \pi^{-}$, we should be in the second case. It might also be possible to show this using a weaker argument. Our analysis reduces us to checking that $R\Gamma_{c}(G,b,\mu)[\rho_{1}]$ admits a sub-quotient isomorphic to $\pi_{+} \boxtimes \phi_{1} \otimes |\cdot|^{-3/2}$, which may be possible to show through basic uniformization and a small global argument.

* We hope that the perspective we take on the Kottwitz conjecture in this paper will help provide further advancements in our knowledge of the cohomology of local Shimura varieties. In particular, by invoking the use of these very general geometric tools from the Fargues-Scholze construction, we require global input only to show compatibility, which, as we will see in the next section, requires substantially less than the input needed to determine the precise form of the cohomology such as a multiplicity formula for the automorphic spectrum.

We will now conclude the introduction by providing a sketch of the proof of Theorem 1.1.

§.§ Proof Sketch of the Main Theorems

As before, we set $G = \mathrm{Res}_{L/\mathbb{Q}_{p}}\GSp_{4}$ and $J = \mathrm{Res}_{L/\mathbb{Q}_{p}}\GU_{2}(D)$.
Similar to the case of $\GL_{n}$, the idea behind proving compatibility for $G$ and $J$ is to use the compatibility of the Fargues-Scholze local Langlands correspondence with parabolic induction to reduce to the case where $\pi$ is a supercuspidal representation. However, this is a little bit more subtle than the case of $\GL_{n}$. Unlike $\GL_{n}$, the local Langlands correspondence for these groups is not a bijection. As seen before, the $L$-packets can be either of size $1$ or $2$. Given an $L$-parameter $\phi: W_{L} \times \SL_{2}(\overline{\mathbb{Q}}_{\ell}) \rightarrow \GSp_{4}(\overline{\mathbb{Q}}_{\ell})$, there are three distinct possibilities.

* The $L$-packets $\Pi(G)_{\phi}$ and $\Pi(J)_{\phi}$ do not contain any supercuspidal representations.

* The $L$-packets $\Pi(G)_{\phi}$ and $\Pi(J)_{\phi}$ contain a mix of supercuspidal and non-supercuspidal representations.

* The $L$-packets $\Pi(G)_{\phi}$ and $\Pi(J)_{\phi}$ contain only supercuspidals.

Case $(1)$ is straight forward. Since compatibility is known for $\GL_{n}$ and its inner forms and any proper Levi subgroup of $G$ (resp. $J$) is a product of such groups, it follows from compatibility of the Fargues-Scholze correspondence with parabolic induction and products that the correspondences are compatible for any representation lying in such an $L$-packet.

Case $(2)$ is a bit more subtle, here $\phi^{\mathrm{ss}}$ factors through a Levi subgroup of $\GSp_{4}(\overline{\mathbb{Q}}_{\ell})$, but $\phi$ itself does not. In particular, the restriction to the $\SL_{2}$ factor of $\phi$ is non-trivial. In this case, we can write $\Pi_{\phi}(G) = \{\pi_{disc},\pi_{sc}\}$ (resp. $\Pi_{\phi}(J) := \{\rho_{disc},\rho_{sc}\}$ or $\Pi_{\phi}(J) = \{\rho^{1}_{disc},\rho^{2}_{disc}\}$, depending on whether the parameter is of Saito-Kurokawa or Howe-Piatetski–Schapiro type), where $\pi_{disc}$ (resp. $\rho^{1}_{disc}$, $\rho^{2}_{disc}$, and $\rho_{disc}$) are non-supercuspidal (essentially) discrete series representation of $G$ (resp. $J$), and $\pi_{sc}$ (resp. $\rho_{sc}$) is a supercuspidal representation of $G(\mathbb{Q}_{p})$ (resp. $J(\mathbb{Q}_{p})$). The key observation is that $\pi_{disc}$ (resp. $\rho_{disc}$) is an irreducible sub-quotient of a parabolic induction, so, in this case, we can apply the same argument as in case (1) to deduce compatibility of the two correspondences. It remains to see that the same is true for $\pi_{sc}$. To do this, we use a description of the $\rho$-isotypic part of the Shtuka space $\Sht(G,b,\mu)_{\infty}$ introduced in section $1.1$. Namely, we consider the complex

\[ R\Gamma_{c}(G,b,\mu)[\rho_{disc}] \simeq R\Gamma_{c}(G,b,\mu) \otimes^{\mathbb{L}}_{\mathcal{H}(G)} \rho_{disc}  \]

of $G(\mathbb{Q}_{p}) \times W_{L}$-modules. Recent work of Hansen-Kaletha-Weinstein <cit.> then tells us the form of this cohomology group (or rather a small variant thereof) as a $J(\mathbb{Q}_{p})$-representation. In particular, if we let $R\Gamma_{c}(G,b,\mu)[\rho_{disc}]_{sc}$ denote the summand of $R\Gamma_{c}(G,b,\mu)[\rho_{disc}] $ where $J(\mathbb{Q}_{p})$ acts via a supercuspidal representation, then in the Grothendieck group of admissible $J(\mathbb{Q}_{p})$-representations of finite length $R\Gamma_{c}(G,b,\mu)[\rho_{disc}]_{sc}$ is equal to  $-2\pi_{sc}$.

Similar to the case of $G = \GL_{n}$, this complex describes the action of the Hecke operator $T_{\mu}$  acting on a sheaf $\mathcal{F}_{\rho_{disc}}$ constructed from $\rho_{disc}$ on $\Bun_{G}$. Moreover, the complex $R\Gamma_{c}(G,b,\mu)[\rho_{disc}]_{sc}$ can be interpreted as a complex of sheaves on the open Harder-Narasimhan(=HN)-strata $\Bun_{G}^{\mathbf{1}} \subset \Bun_{G}$ corresponding to the trivial $G$-bundle on the Fargues-Fontaine curve $X$. It follows from the above description in the Grothendieck group that the excursion algebra will act on this complex via eigenvalues valued in the parameter $\phi_{\pi_{sc}}^{\mathrm{FS}}$. However, since the excursion algebra is built from Hecke operators, it will also commute with the action of Hecke operators on $\mathcal{F}_{\rho_{disc}}$. This allows us to conclude that it also must act via eigenvalues valued in $\phi_{\rho_{disc}}^{\mathrm{FS}}$ giving a chain of equalities

\[ \phi_{\pi_{sc}}^{\mathrm{FS}} = \phi_{\rho_{disc}}^{\mathrm{FS}} = \phi_{\rho_{disc}}^{\mathrm{ss}} = \phi_{\pi_{sc}}^{\mathrm{ss}}\]

where the first equality follows from the previous argument and the second equality follows from the above analysis of induced representations. Similarly, one deduces compatibility for $\rho_{sc}$ by applying a similar argument to the $\pi$-isotypic part.

Case (3) is by far the most involved and takes up the majority of the paper. This is the case in which the $L$-parameter $\phi$ is supercuspidal. First, one can make a reduction to showing compatibility for just  $\rho \in \Pi(J)$ with supercuspidal Gan-Tantono parameter $\phi$, by using the commutation of Hecke operators and the excursion algebra similar to what was done in case $(2)$. Now, the key point again is that the complex $R\Gamma_{c}(G,b,\mu)[\rho]$ describes the action of the Hecke operator $T_{\mu}$ on a sheaf $\mathcal{F}_{\rho}$ on $\Bun_{G}$. To make further progress towards compatibility, we use that the Hecke operators can be in turn described using the spectral action of the derived category of perfect complexes on the stack of $L$-parameters, as constructed in <cit.>. In particular, a Hecke operator defines a vector bundle on the stack of $L$-parameters, whose action on the sheaf $\mathcal{F}_{\rho}$ via the spectral action is precisely $T_{\mu}$. Using this, we argue using the support of the spectral action of certain averaging operators, considered by <cit.> in the case of $\GL_{n}$, to show that, if $\mathrm{std}\circ \phi \otimes |\cdot|^{-3/2}$ occurs as a $W_{L}$-stable sub-quotient of the complex

\[ \bigoplus_{\rho' \in \Pi_{\phi}(J)} R\Gamma_{c}(G,b,\mu)[\rho'] \]

we have an equality:

\[ \mathrm{std}\circ \phi_{\rho}^{\mathrm{FS}} = \mathrm{std}\circ \phi\]

for all $\rho \in \Pi_{\phi}(J)$. Now a $\GSp_{4}$-valued parameter is in turn determined by its composition with $\mathrm{std}$ and its similitude character, which is precisely the central character of $\rho$. Therefore, since the Fargues-Scholze correspondence is compatible with central characters, this is enough to conclude that $\phi = \phi_{\rho}^{\mathrm{FS}}$. This reduces the question of showing compatibility for $\rho \in \Pi(\GU_{2}(D))$ with supercuspidal Gan-Tantono parameter $\phi$ to the following.

Let $\phi$ be a supercuspidal parameter with associated $L$-packet $\Pi_{\phi}(J)$. Then the direct summand of

\[ \bigoplus_{\rho' \in \Pi_{\phi}(J)} R\Gamma_{c}(G,b,\mu)[\rho'] \]

where $G(\mathbb{Q}_{p})$ acts via a supercuspidal representation

\[ \bigoplus_{\rho' \in \Pi_{\phi}(J)} R\Gamma_{c}(G,b,\mu)[\rho']_{sc} \]

is concentrated in middle degree $3$ and admits a non-zero $W_{L}$-stable sub-quotient with $W_{L}$-action given by $\mathrm{std}\circ \phi \otimes |\cdot|^{-3/2}$.

Just as one does in proving Theorem 1.2, the key idea is to directly relate the complex

\[ \bigoplus_{\rho' \in \Pi_{\phi}(J)} R\Gamma_{c}(G,b,\mu)[\rho']_{sc} \]

to the cohomology of a global Shimura variety using basic uniformization of the generic fiber as proven by Shen <cit.> and an analogue of Boyer's trick <cit.>. This allows us to in turn prove Proposition 1.4 using global results on Galois representations in the cohomology of these Shimura varieties due to Kret-Shin <cit.> and Sorensen <cit.>. More specifically, in this case the relevant Shimura datum is given by $(\mathbf{G},X)$, where $\mathbf{G}$ is a $\mathbb{Q}$-inner form of $\mathbf{G}^{*} := \mathrm{Res}_{F/\mathbb{Q}}(\GSp_{4})$ for $F/\mathbb{Q}$ a totally real extension with $p$ inert and $F_{p} \simeq L$. The relevant uniformization result is then applicable if $L/\mathbb{Q}_{p}$ is an unramified extension and $p > 2$. To state the key consequence of this uniformization result, we introduce some notation. We let $\mathbb{A}$ and $\mathbb{A}_{f}$ denote the adeles and finite adeles of $\mathbb{Q}$, respectively. If $K^{p} \subset \mathbf{G}(\mathbb{A}^{p}_{f})$ denotes the level away from $p$ and $K_{p} \subset G(\mathbb{Q}_{p})$ denotes the level at $p$, we let $\mathcal{S}(\mathbf{G},X)_{K_{p}K^{p}}$ be the rigid analytic Shimura variety over $\mathbb{C}_{p}$ of level $K_{p}K^{p}$. We set $\xi$ be a regular weight of an algebraic representation $\mathcal{V}_{\xi}$ of $\mathbf{G}$ over $\mathbb{Q}$ and let $\mathcal{L}_{\xi}$ denote the associated $\overline{\mathbb{Q}}_{\ell}$ local system on $\mathcal{S}(\mathbf{G},X)_{K_{p}K^{p}}$. We then define

\[ R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi}) := \colim_{K_{p} \rightarrow \{1\}} R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}K_{p}},\mathcal{L}_{\xi})   \]

The basic uniformization result of Shen then furnishes a $\mathbb{Q}$-inner form $\mathbf{G}'$ of $\mathbf{G}$ satisfying that $\mathbf{G}'_{\mathbb{Q}_{p}} \simeq J$ together with a  $G(\mathbb{Q}_{p}) \times W_{L}$-invariant map

\[ \Theta: R\Gamma_{c}(G,b,\mu) \otimes^{\mathbb{L}}_ {\mathcal{H}(J_{b})} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi}) \rightarrow R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi})  \]

functorial in the level $K^{p}$. Here $\mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi})$ denotes the space of algebraic automorphic forms  of level $K^{p}$ valued in the algebraic representation $\mathcal{V}_{\xi}$ in the sense of <cit.>. We want to use this uniformization map to apply global results on the cohomology of $R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi})$ to study the action of $W_{L}$ on $R\Gamma_{c}(G,b,\mu)$. To do this, we show an analogue of Boyer's trick, which says that the non-basic Newton strata of the adic flag variety $\mathcal{F}\ell_{G,\mu^{-1}} := (G/P_{\mu^{-1}})^{ad}$ are parabolically induced as spaces with $G(\mathbb{Q}_{p})$-action. Using the Hodge-Tate period map from the Shimura variety $\mathcal{S}(\mathbf{G},X)_{K^{p}}$ to $\mathcal{F}\ell_{G,\mu^{-1}}$, this implies that, if we pass to the part of the cohomology on both sides where $G(\mathbb{Q}_{p})$ acts via a supercuspidal representation, we get a $W_{L} \times G(\mathbb{Q}_{p})$-equivariant isomorphism:

\[ \Theta_{sc}: R\Gamma_{c}(G,b,\mu)_{sc} \otimes^{\mathbb{L}}_ {\mathcal{H}(J_{b})} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi}) \xrightarrow{\simeq} R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi})_{sc}  \]

After showing this, we fix a $\rho$ having supercuspidal Gan-Tantono parameter $\phi$, and, via an argument using the simple trace formula, choose a globalization of $\rho$ to a cuspidal automorphic representation $\Pi'$ of $\mathbf{G}'$, which occurs as a $J(\mathbb{Q}_{p})$-stable direct summand of $\mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi})$ and is an unramified twist of Steinberg at some non-empty set of places $S_{st}$, for some sufficiently large regular weight $\xi$ and sufficiently small level $K^{p}$. We set $S$ to be a finite set of places outside of which $\Pi'$ is unramified. The Hecke eigenvalues of $\Pi'$ then define a maximal ideal $\mathfrak{m} \subset \mathbb{T}^{S}$ in the abstract commutative Hecke algebra of $\mathbf{G}'$ away from the finite places $S$. Regarding both sides of $\Theta$ as $\mathbb{T}^{S}$-modules, we can localize at $\mathfrak{m}$ to get a map

\[ \Theta_{\mathfrak{m}}: (R\Gamma_{c}(G,b,\mu) \otimes^{\mathbb{L}}_ {\mathcal{H}(J_{b})} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi}))_{\mathfrak{m}} \rightarrow R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi})_{\mathfrak{m}}  \]

We write $K^{p} = K^{S_{st} \cup \{p\}}K^{p}_{S_{st}}$
for $K^{S_{st} \cup \{p\}} \subset \mathbf{G}(\mathbb{A}_{f}^{S_{st} \cup \{p\}})$. Taking colimits on both sides as $K^{p}_{S_{st}} \rightarrow \{1\}$, we see that $\Theta_{\mathfrak{m}}$ induces a map:

\[ (R\Gamma_{c}(G,b,\mu) \otimes^{\mathbb{L}}_ {\mathcal{H}(J_{b})} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{S_{st} \cup \{p\}},\mathcal{L}_{\xi}))_{\mathfrak{m}} \rightarrow R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{S_{st} \cup \{p\}}},\mathcal{L}_{\xi})_{\mathfrak{m}}  \]

For $v \in S_{st}$, we consider the projection of both side to the Bernstein component of $\mathbf{G}(F_{v}) \simeq \mathbf{G}'(F_{v})$ corresponding to the Steinberg rerpresentation. This gives us a map:

\[ \Theta_{\mathfrak{m}}^{st}: (R\Gamma_{c}(G,b,\mu) \otimes^{\mathbb{L}}_ {\mathcal{H}(J_{b})} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{S_{st} \cup \{p\}},\mathcal{L}_{\xi}))^{st}_{\mathfrak{m}} \rightarrow R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{S_{st} \cup \{p\}}},\mathcal{L}_{\xi})^{st}_{\mathfrak{m}}  \]

A priori, this only implies that $\mathbf{G}(F_{v}) \simeq \mathbf{G}'(F_{v})$ acts via an unramified twist of the Steinberg or trivial representation, but the latter case is ruled out since $\xi$ is regular, by a standard strong approximation argument (See the proof of Theorem 5.2). The key point is now, by analyzing the simple twisted trace formula of Kottwitz-Shelstad <cit.> and stable trace formulas of Arthur <cit.>, we can prove a strong multiplicity one type result (Proposition 5.4), for cuspidal automorphic representations that are unramified twists of Steinberg at some sufficiently large non-empty set of places and regular of weight $\xi$ at infinity. This implies that the representations of  $\mathbf{G}'$) occurring on LHS of $\Theta_{\mathfrak{m}}^{st}$ must have local constituent at $p$ with Langlands parameter $\phi$, since we localized at the Hecke eigensystem defined by $\Pi'$ at the unramified places. Moreover, since we know $\Theta_{sc}$ is an isomorphism, this implies that we have an isomorphism

\[ \Theta_{\mathfrak{m},sc}^{st}: (R\Gamma_{c}(G,b,\mu)_{sc} \otimes^{\mathbb{L}}_ {\mathcal{H}(J_{b})} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{S_{st} \cup \{p\}},\mathcal{L}_{\xi}))^{st}_{\mathfrak{m}} \xrightarrow{\simeq} R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{S_{st} \cup \{p\}}},\mathcal{L}_{\xi})^{st}_{\mathfrak{m},sc}  \]

and, since the local constituents of the automorphic representations of $\mathbf{G}'$ at $p$ occurring in the LHS are all in the $L$-packet $\Pi_{\phi}(J)$ by the strong multipicity one result, we can reduce Proposition 1.4 to showing that $R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi})^{st}_{\mathfrak{m},sc}$ is concentrated in degree $3$ and has $W_{L}$-action given (up to multiplicity) by $\mathrm{std}\circ \phi \otimes |\cdot|^{-3/2}$. This follows from the analysis carried out in  Kret-Shin <cit.>. In particular, it follows from their results that this complex will be concentrated in degree $3$ and that the traces of Frobenius in $\Gamma_{F} := \Gal(\overline{F}/F)$ on the étale cohomology of the associated global Shimura variety over $\overline{F}$  are given by $\mathrm{std}\circ \phi_{\tau_{v}}$, where $\tau_{v}$ are the local constituents of some weak transfer $\tau$ of $\Pi'$ to an automorphic representation of $\mathrm{Res}_{F/\mathbb{Q}}\GSp_{4} =: \mathbf{G}^{*}$ and $\phi_{\tau_{v}}$ is the associated Gan-Takeda parameter. This allows one, up to multiplicities, to describe the Galois action on the global Shimura variety in terms of the composition $\mathrm{std}\circ \rho_{\tau}$, where $\rho_{\tau}$ is a global $\GSp_{4}(\overline{\mathbb{Q}}_{\ell})$-valued representation of the absolute Galois group of $F$ constructed by Sorensen <cit.> from $\tau$ characterized by the property that $i\WD(\mathrm{std}\circ \rho_{\tau})|_{W_{F_{v}}}^{F-s.s.} \simeq \phi_{\tau_{v}} \otimes |\cdot|^{-3/2}$ for all but finitely many places $v$ of $F$. This would give one precisely the desired description of the $W_{L}$-action on $R\Gamma_{c}(G,b,\mu)[\rho]_{sc}$ if one knew that $\phi_{\tau_{p}} = \phi_{\rho}$. Since $\Pi'$ is globalization of $\rho$, one needs to choose $\tau$ to be a strong transfer of $\Pi'$ at the prime $p$. This latter goal is accomplished using analysis of the simple trace formula as done in Kret-Shin <cit.> combined with the character identities proven by Chan-Gan <cit.>. These results on strong transfers also aid us in deducing the strong multiplicity one type result mentioned above.

In section $2$, we give an overview of the Gan-Takeda and Gan-Tantono local Langlands correspondence, putting it in the framework of the refined local Langlands correspondence of Kaletha in preparation for applications to the Kottwitz conjecture. In section $3$, we describe the Fargues-Scholze local Langlands correspondence and related ideas, giving the proof of compatibility in cases (1) and (2) and reducing case (3) to Proposition 1.4, via some properties of the spectral action discussed in section $3.2$. In section $4$, we discuss basic uniformization of the relevant Shimura varieties and prove the aforementioned analogue of Boyer's trick, showing that the uniformization map $\Theta_{sc}$ is an isomorphism. In section $5$, we analyze the simple trace formula with fixed central character in a fashion similar to Kret-Shin <cit.> to deduce the existence of the required strong transfers, as well as combine this with analysis of the simple twisted trace formula to deduce the required strong multiplicity one result. In section $6$, we apply the results of section $5$ combined with results of Kret-Shin <cit.> and Sorensen <cit.> to compute the relevant Galois action on the global Shimura variety. Finally, in section $7$, we put the results of the previous sections together to prove Proposition 1.4. We then conclude with the application to the proofs of Theorem 1.1 and 1.3, as well as formally deduce compatibility for the local Langlands correspondence for $\Sp_{4}$ and its non quasi-split inner form $\SU_{2}(D)$, as constructed by Gan-Takeda <cit.> and Choiy <cit.>, respectively. We finish the section with a brief discussion of an application to the cohomology of the related (non-minuscule) local Shtuka spaces.

§ CONVENTIONS AND NOTATIONS

For a diamond or $v$-stack, we freely use the formalism in <cit.> of $\ell$-adic cohomology of diamonds and $v$-stacks. We will fix isomorphisms $i: \overline{\mathbb{Q}}_{\ell} \xrightarrow{\simeq} \mathbb{C}$ and $j: \overline{\mathbb{Q}}_{p} \xrightarrow{\simeq} \mathbb{C}$ and use the (geometric) normalization of local class field theory that sends the Frobenius to the inverse of the uniformizer. For a supercuspidal $L$-parameter, we will often abuse notation and use $\phi$ to denote both the $L$-parameter and the semisimplified parameter $\phi^{\mathrm{ss}}$, as in this case this merely corresponds to forgetting the trivially acting $\SL_{2}(\mathbb{C})$-factor and applying the isomorphism $i$. For a reductive group $H/\mathbb{Q}_{p}$, we will write $R\mathcal{H}om_{H(\mathbb{Q}_{p})}(-,\ol{\mathbb{Q}}_{\ell}): \D(H(\mathbb{Q}_{p}),\ol{\mathbb{Q}}_{\ell})^{\mathrm{op}} \rightarrow  \D(H(\mathbb{Q}_{p}),\ol{\mathbb{Q}}_{\ell})$ for the derived smooth duality functor, where $\D(H(\mathbb{Q}_{p}),\ol{\mathbb{Q}}_{\ell})$ is the unbounded derived category of smooth representations. Namely, it is derived functor induced by the left exact smooth duality functor $V \mapsto (V^{*})^{\mathrm{sm}}$, where $(V^{*})^{\mathrm{sm}}$ is the set of all functions $f: V \ra \ol{\mathbb{Q}}_{\ell}$ such that there exists $K \subset H(\mathbb{Q}_{p})$ a compact open such that for all $v \in V$ and $k \in K$ we have that $f(kv) = f(v)$. Normally, in the literature the space $\Sht(G,b,\mu)_{\infty}$ parametrizes modifications $\mathcal{E}_{0} \dashrightarrow \mathcal{E}_{b}$ with meromorphy $\mu$. For us, it will denote the space parametrizing modifications of type $\mu^{-1}$. This convention limits the appearances of duals (cf. Remark 3.8).

§ LOCAL LANGLANDS FOR $\GSP_{4}$ AND $\GU_{2}(D)$

§.§ Local Langlands for $\GSp_{4}$

In this section, we will describe the local Langlands correspndence of Gan-Takeda for the group $G := \GSp_{4}/L$, where $L/\mathbb{Q}_{p}$ is a finite extension. We fix a choice of Whittaker datum $\mathfrak{m} := (B,\psi)$ throughout section $2$, where $B$ is the Borel and $\psi$ is a generic character of $L$.

As before, we consider the set $\Phi(G)$ of admissible homomorphisms

\[ \phi: W_{L} \times \SL_{2}(\mathbb{C}) \rightarrow \hat{G}(\mathbb{C}) = \GSpin_{5}(\mathbb{C}) \simeq \GSp_{4}(\mathbb{C}) \]

taken up to $\hat{G}$-conjugacy, where $W_{L}$ acts via a continuous semisimple homomorphism with respect to the discrete topology and $\SL_{2}(\mathbb{C})$ acts via an algebraic representation. Similarly, let $\Pi(G)$ denote the isomorphism classes of smooth irreducible representations of the group $G(L)$. We can now state the main theorem of Gan-Takeda.
There is a surjective finite to one map

\[ \LLC_{G}: \Pi(\GSp_{4}) \rightarrow \Phi(\GSp_{4}) \]

\[ \pi \mapsto \phi_{\pi} \]

with the following properties:

* $\pi$ is an (essentially) discrete series representation of $\GSp_{4}(L)$ if and only if its $L$-parameter does not factor through any proper Levi subgroup of $\GSp_{4}(\mathbb{C})$.

* Given an $L$-parameter $\phi$, we set $S_{\phi} := Z_{\hat{G}}(\mathrm{Im}(\phi))$ to be the centralizer of $\phi$. The fiber $\Pi_{\phi}(G)$ can be naturally parametrized by the set of irreducible characters of the component group

\[ A_{\phi} := \pi_{0}(S_{\phi}) \simeq \pi_{0}(S_{\phi}/Z(\GSp_{4}))   \]

which is either trivial or equal to $\mathbb{Z}/2\mathbb{Z}$. When $A_{\phi} = \mathbb{Z}/2\mathbb{Z}$, exactly one of the two representations in $\Pi_{\phi}(G)$ is generic for the fixed choice of Whittaker datum, and is indexed by the trivial character of $A_{\phi}$.

* The similitude character $sim(\phi_{\pi})$ of $\GSp_{4}(L)$ is equal to the central character $\omega_{\pi}$ via the isomorphism given by local class field theory.

* Given a character $\chi$ of $L^{*}$ and letting $\lambda: \GSp_{4} \rightarrow L^{*}$ be the similitude character of $\GSp_{4}(L)$, we have, via local class field theory, that the $L$-parameter of $\pi \otimes (\chi \circ \lambda)$ is equal to $\phi_{\pi} \otimes \chi$.

* If $\pi \in \Pi(\GSp_{4})$ is a representation, for any smooth irreducible representation $\sigma$ of $\GL_{r}(L)$, we have that

\[ \gamma(s, \pi \times \sigma,\psi) = \gamma(s,\phi_{\pi} \otimes \phi_{\sigma},\psi) \]

\[ L(s, \pi \times \sigma,\psi) = L(s,\phi_{\pi} \otimes \phi_{\sigma},\psi) \]

\[ \epsilon(s, \pi \times \sigma,\psi) = \epsilon(s,\phi_{\pi} \otimes \phi_{\sigma},\psi) \]

where the RHS are the Artin local factors associated to the representations of $W_{L} \times \SL_{2}(\mathbb{C})$ and the LHS are the local factors of Shahidi <cit.> with respect to the morphisms of $L$-groups defined in <cit.> in the case that $\pi$ is a generic supercuspidal or non-supercuspidal and are the local factors defined by Townsend <cit.> if $\pi$ is a non-generic supercuspidal representation.

The map $\LLC_{G}$ is uniquely determined by the properties (1),(3), and (5), where one can take $r \leq 2$ in $(5)$.

When the paper of Gan-Takeda was released there was no good theory of $L$,$\epsilon$, and $\gamma$ factors for nongeneric supercuspidal representations satisfying the usual properties. (See the 10-Commandents in <cit.>) Instead, to uniquely characterize the correspondence for these representations, they use an equality between the Plancharel measure on the family of inductions from $\GSpin_{5}(L) \times \GL_{r}(L) \simeq \GSp_{4}(L) \times \GL_{r}(L)$ to $\GSpin_{2r + 5}(L)$ for $r \leq 2$. However, this theory of $L$,$\epsilon$, and $\gamma$ factors was later constructed by Nelson Townsend in his PhD thesis <cit.>.

We now make the following definition.

Write $\mathrm{std}:\GSp_{4} \hookrightarrow \GL_{4}$ for the standard embedding. We say a discrete $L$-parameter is stable if the $L$-packet $\Pi_{\phi}(G)$ has size $1$ and is endoscopic if it has size $2$. Equivalently, by Theorem 2.1 (2), this is equivalent to saying that the character group $A_{\phi}^{\vee}$ of the component group $A_{\phi}$ has cardinality $1$ or $2$, respectively. By <cit.>, this can be characterized as follows.

* (stable) $\mathrm{std}\circ \phi$ is an irreducible representation of $W_{L} \times \SL_{2}(\mathbb{C})$. In this case, $S_{\phi} = Z(\hat{G}) = \mathbb{G}_{m}$, so $A_{\phi}$ is trivial.

* (endoscopic) $\mathrm{std}\circ \phi \simeq \phi_{1} \oplus \phi_{2}$, where the $\phi_{i}: W_{L} \times \SL_{2}(\mathbb{C}) \rightarrow \GL_{2}(\mathbb{C})$ for $i = 1,2$ are distinct irreducible $2$-dimensional representations of $W_{L} \times \SL_{2}(\mathbb{C})$ with $\det(\phi_{1}) = \det(\phi_{2})$. In this case, $A_{\phi} \simeq \mathbb{Z}/2\mathbb{Z}$. We recall that $\GSp_{4}(\mathbb{C})$ has a unique endoscopic group $\GSO_{2,2}$, and the endoscopic parameters lie in the image of the map $\Phi(\GSO_{2,2}) \rightarrow \Phi(\GSp_{4})$. More specifically, the dual group of $\GSO_{2,2}$ is $\GSpin_{4}$ and one has an identification $\GSpin_{4} \simeq (\GL_{2}(\mathbb{C}) \times \GL_{2}(\mathbb{C}))^{0} := \{(g_{1},g_{2}) \in \GL_{2}(\mathbb{C}) \times \GL_{2}(\mathbb{C}) |  \det(g_{1}) = \det(g_{2}) \}$ and the map $\Phi(\GSO_{2,2}) \rightarrow \Phi(\GSp_{4}) \xrightarrow{\mathrm{std} \circ} \Phi(\GL_{4})$ comes from the inclusion $(\GL_{2}(\mathbb{C}) \times \GL_{2}(\mathbb{C}))^{0} \subset \GL_{4}(\mathbb{C})$. Using this, we can compute that one has an identification

\[ S_{\phi} \simeq \{(a,b) \in \mathbb{C}^{*} \times \mathbb{C}^{*}| a^{2} = b^{2} \} \subset (\GL_{2}(\mathbb{C}) \times \GL_{2}(\mathbb{C}))^{0} \]

where the center $Z(\GSp_{4})(\mathbb{C}) \simeq \mathbb{C}^{*}$ embeds diagonally.

For an $L$-parameter $\phi$ we see, by Theorem 2.1 (2), that the size of the $L$-packet $\Pi_{\phi}(G)$ is at most $2$, this allows us to subdivide into three cases:

* The $L$-packet $\Pi_{\phi}(G)$ does not contain any supercuspidal representations.

* The $L$-packet $\Pi_{\phi}(G)$ contains one supercuspidal and one non-supercuspidal.

* The $L$-packet contains only supercuspidals.

In Case $(1)$ the parameter will not be discrete. Case $(2)$ is where the parameter $\phi$ does not factor through a Levi-subgroup so it is discrete, but its semisimplification $\phi^{\mathrm{ss}}$ as defined in section $1$ does. Case $(2)$ does not occur when the parameter is a stable discrete parameter, by definition. The relevant case is when the parameter is discrete endoscopic. To understand this, we let $\nu(n)$ denote the unique $n$-dimension irreducible representation of $\SL_{2}(\mathbb{C})$ then there are two cases:

* (Saito-Kurokawa Type) We have $\mathrm{std}\circ \phi = \phi_{0} \oplus \chi \boxtimes \nu(2)$, where $\phi_{0}$ is a $2$-dimensional irreducible representation of $W_{L}$ and $\chi$ is a character, with $\chi^{2} = \det(\phi_{0})$. Therefore, the semisimplification $\phi^{\mathrm{ss}}$ satisfies: $\mathrm{std}\circ \phi^{\mathrm{ss}} = \phi_{0} \oplus \chi \otimes |\cdot|^{\frac{1}{2}} \oplus \chi \otimes |\cdot|^{-\frac{1}{2}}$.

* (Howe-Piatetski–Shapiro Type) We have $\mathrm{std}\circ \phi = \chi_{1} \boxtimes \nu(2) \oplus \chi_{2} \boxtimes \nu(2)$, where $\chi_{1}$ and $\chi_{2}$ are distinct characters of $W_{L}$ satisfying $\chi_{1}^{2} = \chi_{2}^{2}$. Therefore, the semisimplification $\phi^{\mathrm{ss}}$ satisfies: $\mathrm{std}\circ \phi^{\mathrm{ss}} = \chi_{1} \otimes |\cdot|^{\frac{1}{2}} \oplus \chi_{1} \otimes |\cdot|^{-\frac{1}{2}} \oplus \chi_{2} \otimes |\cdot|^{\frac{1}{2}} \oplus \chi_{2} \otimes |\cdot|^{-\frac{1}{2}}$.

* The terminology here is explained by Arthur's classification <cit.> of the global automorphic representations of $\GSp_{4}$ appearing in the papers <cit.> and  <cit.>, respectively.

* We will mention in the next section how to distinguish these two cases via the number of supercuspidals in the $L$-packet $\Pi_{\phi}(\GU_{2}(D))$ defined by the Gan-Tantono local Langlands correspondence.

Case (3) is the situation where the parameter $\phi$ is supercuspidal as defined in the introduction. In particular, in the supercuspidal case the restriction of the parameter $\phi$ to the $\SL_{2}(\mathbb{C})$ factor is trivial, so the irreducible representations occurring in the decomposition of $\mathrm{std}\circ \phi$ are just representations of $W_{L}$.

For the purposes of applying the weak form of the Kottwitz Conjecture proven in Hansen-Kaletha-Weinstein <cit.>, we formulate this correspondence in terms of the refined local Langlands of Kaletha <cit.> with respect to the fixed choice of Whittaker datum $\mathfrak{m}$. Now, given a parameter $\phi$ that is either mixed supercuspidal or supercuspidal, we have by Theorem 2.1 (2) a correspondence between the $L$-packet $\Pi_{\phi}(G)$ and the set of irreducible characters $A^{\vee}_{\phi}$. This in turn gives rise to an irreducible character of the group $S_{\phi}$ via the composition:

\[ S_{\phi} \rightarrow \pi_{0}(S_{\phi}) = A_{\phi} \]

This allows us to make the following definition.

For $\phi$ a supercuspidal or mixed-supercuspidal parameter $\phi$ as above and $\pi \in \Pi_{\phi}(G)$, we denote the character of $S_{\phi}$ described above by $\tau_{\pi}$.

§.§ Local Langlands for $\GU_{2}(D)$

In this section, we describe the local Langlands correspondence for the unique non-split inner form $J = \GU_{2}(D)$, the group of similitudes of the unique $2$-dimensional Hermitian vector space over the quaternion division algebra $D/L$. As in the previous section, we let $\Pi(J)$ denote the set of irreducible admissible representations of $J$, and $\Phi(J)$ be the set of $L$-parameters of $J$. This is a subset of the previous set $\Phi(\GSp_{4})$ as we will now explain. $J$ has a unique up to conjugacy minimal parabolic whose Levi factor is

\[ D^{*} \times \GL_{1}. \]

This defines a form of the Siegel parabolic of $\GSp_{4}$ and it determines a dual parabolic subgroup $P^{\vee}(\mathbb{C})$ in the dual group $\GSp_{4}(\mathbb{C})$ of $\GU_{2}(D)$. This is the Heisenberg parabolic subgroup of $\GSp_{4}(\mathbb{C})$, its conjugacy class is said to be relevant for $J$ while all other conjugacy classes of proper parabolics are said to be irrelevant. We say $\phi \in \Phi(\GSp_{4})$ is relevant if it does not factor through any irrelevant parabolic subgroups of $\GSp_{4}(\mathbb{C})$. We define $\Phi(J)$ to be the subset of relevant $\phi$ in $\Phi(\GSp_{4})$. We set $B_{\phi} := \pi_{0}(Z_{Sp_{4}}(\mathrm{Im}(\phi)))$. One has an exact sequence:

\[ \langle \pm 1 \rangle \rightarrow B_{\phi} \rightarrow A_{\phi} \rightarrow 0 \]

Implying that one has an injection on the group of irreducible characters $\hat{A}_{\phi} \hookrightarrow \hat{B}_{\phi}$, which identifies $\hat{A}_{\phi}$ as the subgroup of (index at most $2$) of characters trivial on the image of the center $Z(Sp_{4})(\mathbb{C}) $. One can check that $\hat{B}_{\phi} \neq \hat{A}_{\phi}$ if and only if $\phi$ is relevant for $\GU_{2}(D)$. Now we can state the main theorem of Gan and Tantono.
There is a natural surjective finite-to-one map

\[ \LLC_{J}: \Pi(\GU_{2}(D)) \rightarrow \Phi(\GU_{2}(D)) \]

\[ \rho \mapsto \phi_{\rho} \]

with the following properties:

* $\rho$ is an (essentially) discrete series representation of $\GU_{2}(D)$ if and only if its parameter $\phi_{\rho}$ does not factor through any proper Levi subgroup of $\GSp_{4}(\mathbb{C})$.

* For an $L$-parameter $\phi$, the fiber $\Pi_{\phi}(J)$ can be naturally parametrized by the set $\hat{B}_{\phi} \setminus \hat{A}_{\phi}$. This set has size either $1$ or $2$.

* The similitude character $sim(\phi_{\rho})$ of $\phi_{\rho}$ is equal to the central character $\omega_{\rho}$ of $\rho$, via the isomorphism given by local class field theory.

* Given a character $\chi$ of $L^{*}$ and letting $\lambda: \GU_{2}(D) \rightarrow L^{*}$ be the similitude character of $\GU_{2}(D)$, we have, via local class field theory, that the $L$-parameter of $\rho \otimes (\chi \circ \lambda)$ is equal to $\phi_{\rho} \otimes \chi$.

* If $\rho \in \Pi(\GU_{2}(D))$ is a non-supercuspidal representation then, for any smooth irreducible representation $\sigma$ of $\GL_{r}(L)$, we have that

\[ \gamma(s, \rho \times \sigma,\psi) = \gamma(s,\phi_{\rho} \otimes \phi_{\sigma},\psi) \]

\[ L(s, \rho \times \sigma,\psi) = L(s,\phi_{\rho} \otimes \phi_{\sigma},\psi) \]

\[ \epsilon(s, \pi \times \sigma,\psi) = \epsilon(s,\phi_{\rho} \otimes \phi_{\sigma},\psi) \]

where the RHS are the Artin local factors associated to the representations of $W_{L} \times \SL_{2}(\mathbb{C})$ and the LHS are the local factors of Shahidi, as defined in <cit.>.

* Suppose that $\rho$ is a supercuspidal representation. For any irreducible supercuspidal representation $\sigma$ of $\GL_{r}(L)$ with $L$-parameter $\phi_{\sigma}$, if $\mu(s,\rho \boxtimes \sigma, \psi)$ denotes the Plancharel measure associated to the family of induced representations $I_{P}(\pi \boxtimes \sigma,s)$ on $\GSpin_{r+4,r+1}$, where we have regarded $\rho \boxtimes \sigma$ as a representation of the Levi subgroup $\GSpin_{4,1} \times \GL_{r} \simeq \GU_{2}(D) \times \GL_{r}$, then $\mu(s,\rho \boxtimes \sigma)$ is equal to

\[ \gamma(s,\phi_{\rho}^{\vee} \otimes \phi_{\sigma},\psi) \cdot \gamma(-s,\phi_{\rho} \otimes \phi_{\sigma}^{\vee}, \overline{\psi}) \cdot \gamma(2s, Sym^{2}\phi_{\sigma} \otimes sim(\phi_{\rho})^{-1},\psi)) \cdot \gamma(-2s,Sym^{2}\phi_{\sigma}^{\vee} \otimes sim(\phi_{\rho}), \overline{\psi}) \]

The map $\LLC_{J}$ is uniquely determined by the properties (1), (3), (5), and (6), with $r \leq 4$ in (5) and (6).

We now further elaborate on the structure of the $L$-packets $\Pi_{\phi}(J) := \LLC_{J}^{-1}(\phi)$ in the case where the parameter $\phi$ is mixed supercuspidal. If the parameter $\phi$ is of this form, then, it follows from <cit.> and the description of $\LLC_{J}$ provided in <cit.>, that the $L$-packet $\Pi_{\phi}(J)$ has following structure, as alluded to in Remark 2.2 (2).

* (Saito-Kurokawa Type) The $L$-packet $\Pi_{\phi}(J) = \{\rho_{disc},\rho_{sc}\}$ contains one supercuspidal representation $\rho_{sc}$ and one non-supercuspidal representation $\rho_{disc}$.

* (Howe-Piatetski–Shapiro Type) The $L$-packet $\Pi_{\phi}(J) = \{\rho_{disc}^{1},\rho_{disc}^{2}\}$ contains no supercuspidal representations.

We now would also like to briefly comment on the structure of the set $\hat{B}_{\phi} \setminus \hat{A}_{\phi}$, confirming the expectation that the size of the $L$-packets $\Pi_{\phi}(G)$ and $\Pi_{\phi}(J)$ is always the same.

* (stable) In the case that the $L$-parameter $\phi$ is stable, we have that $B_{\phi} = \mathbb{Z}/2\mathbb{Z}$ and, as noted in section 2.1, $A_{\phi} = 1$. This means the set $\hat{B}_{\phi} \setminus \hat{A}_{\phi}$ consists of one element corresponding to the non-trivial character.

* (endoscopic) In the case that the parameter $\phi$ is endoscopic, we have that the decomposition $\mathrm{std}\circ \phi \simeq \phi_{1} \oplus \phi_{2}$ induces an exact sequence

\[ A_{\phi} = \mathbb{Z}/2\mathbb{Z} \xrightarrow{\Delta} B_{\phi} = Z(\SL_{2}) \times Z(\SL_{2}) = \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \]

and so the set $\hat{B}_{\phi} \setminus \hat{A}_{\phi}$ has size $2$ and is indexed by two characters $\eta_{+-}$ and $\eta_{-+}$ each non-trivial on one of the two $\mathbb{C}^{*}$-factors under the isomorphism

\[ S_{\phi} \simeq \{(a,b) \in \mathbb{C}^{*} \times \mathbb{C}^{*}| a^{2} = b^{2} \} \subset (\GL_{2}(\mathbb{C}) \times \GL_{2}(\mathbb{C}))^{0} \]

from section 2.1.

We now wish to put this local Langlands correspondence for the inner form in the context of the refined local Langlands correspondence of Kaletha <cit.>. We consider the Kottwitz set $B(G)$ <cit.> and let $b \in B(G)$ be the basic element whose associated $\sigma$-centralizer $J_{b} = J$. We take this to be the basic element whose slope homorphism is the dominant rational cocharacter of $G$ given by $(1/2,1/2,1/2,1/2)$. Let $Z(\GSp_{4}) \simeq \mathbb{G}_{m}$ be the center. We recall that we have an isomorphism $\pi_{1}(G) \simeq X_*(Z(\hat{G})) \simeq \mathbb{Z}$ and that the $\kappa$-invariant of $b$ is sent to the element $1 \in \mathbb{Z}$ under this isomorphism. This indexes the identity representation of $\mathbb{G}_{m}$, denoted  $id_{\mathbb{G}_{m}}$. Thus, given a discrete parameter (= supercuspidal or mixed supercuspidal) $\phi: W_{L} \times \SL_{2}(\mathbb{C}) \rightarrow \GSp_{4}(\mathbb{C})$, the refined local Langlands correspondence asserts bijections

\[ \Pi_{\phi}(G) \longleftrightarrow \{ \text{irreducible algebraic representations $\tau$ of $S_{\phi}$ s.t }   \tau|_{Z(\hat{G})} = \mathbf{1} \} \]

\[ \pi \mapsto \tau_{\pi} \]

\[ \Pi_{\phi}(J) \longleftrightarrow \{\text{irreducible algebraic representations $\tau$ of $S_{\phi}$ s.t }   \tau|_{Z(\hat{G})} = id_{\mathbb{G}_{m}} \}\]

\[ \rho \mapsto \tau_{\rho} \]

where $\mathbf{1}$ is the trivial representation. In section $2.1$, we saw how for $\pi \in \Pi_{\phi}(G)$ to construct the desired $\tau_{\pi}$. Here it is uniquely pinned down by the property that the trivial representation corresponds to the unique $\mf{m}$-generic representation. In the case of the inner form, the situation is a bit more tricky. Consider $\rho \in \Pi(J)$ with associated $L$-parameter $\phi_{\rho}$. If $\phi_{\rho}$ is stable then $S_{\phi} = \mathbb{G}_{m}$ and $\tau_{\rho}$ is simply $id_{\mathbb{C}^{*}}$. If $\phi$ is endoscopic, then, as noted in section 2.1, we have an inclusion:

\[ Z(\GSp_{4})(\mathbb{C}) = \mathbb{C}^{*} \xrightarrow{\Delta} S_{\phi} \simeq \{(a,b) \in \mathbb{C}^{*} \times \mathbb{C}^{*}| a^{2} = b^{2} \} \]

We consider the characters $\tau_{i}: S_{\phi} \rightarrow \mathbb{C}^{*}$ for $i = 1,2$ given by projecting to the first and second coordinate. These satisfy the property that $\tau_{i}|_{\mathbb{C}^{*}} = id_{\mathbb{C}^{*}}$ on the diagonally embedded center as desired. Similarly, under the parametrization of Gan-Tantono $\Pi_{\phi}(J) = \{\rho_{+-},\rho_{-+}\}$, where $\rho_{+-}$ and $\rho_{-+}$ correspond to the characters $\eta_{+-}$ and $\eta_{-+}$ described above. Specifically, if $\pi_{1}$ and $\pi_{2}$ are the unique discrete series representations of $\GL_{2}(L)$ in the $L$-packet over $\phi_{1}$ and $\phi_{2}$ then

\[ \rho_{+-} := \theta(\mathrm{JL}^{-1}(\tau_{2}) \boxtimes \tau_{1}) \text{ and } \rho_{-+} := \theta(\mathrm{JL}^{-1}(\tau_{1}) \boxtimes \tau_{2}) \]

where $\theta$ denotes the non-zero local theta lift from $D^{*} \times \GL_{2}(F)$ to $\GU_{2}(D)$, as in <cit.>, and $\mathrm{JL}: \Pi(D^{*}) \ra \Pi(\GL_{2})$ is the Jacquet-Langlands correspondence. Now we would like to match these two representations with $\tau_{1}$ and $\tau_{2}$ under the refined local Langlands of Kaletha. Suppose we fix such a matching. Now, we consider a refined endoscopic datum $\mf{c}$ for the quasi-split reductive group $G$, which we recall is a tuple $(H,s,\mc{H},\eta)$ which consists of

* a quasi-split group $H$ over $F$,

* an extension $\mc{H}$ of $W_F$ by $\widehat{H}$ such that the map $W_F \to \mathrm{Out}(\widehat{H})$ coincides with the map $\rho_H: W_F \to \mathrm{Out}(\widehat{H})$ induced by the action of $W_F$ on $\widehat{H} \subset {}^LH$,

* an element $s \in Z(\widehat{H})^{\Gamma}$,

* an $L$-homomorphism $\eta: \mc{H} \to {}^LG$,

satisfying the condition:

* we have $\eta(\widehat{H}) = Z_{\widehat{G}}(s)^{\circ}$.

Considering $J$ as the $\sigma$-centralizer of the unique basic element $b_{1} \in B(G)$ of the Kottwitz invariant $1$, this defines for us an extended pure inner twisting of $(\xi,b): G \ra J_{b}$ in the sense of <cit.>, and we can attach a canonical transfer factor $\Delta[\mf{m},\mf{c},b_{1}]$ to this datum, as defined in <cit.>. Given a test function $f \in C^{\infty}_{c}(J(\mathbb{Q}_{p}),\ol{\mathbb{Q}}_{\ell})$, we can use these transfer factors to say what it means for $f^{\mf{c}} \in C^{\infty}_{c}(H(\mathbb{Q}_{p}),\ol{\mathbb{Q}}_{\ell})$ to be matching in the sense that their stable orbital integrals normalized with respect to these transfer factors match up.

Now, suppose we have a discrete parameter $\phi \in \Phi(J)$ and a refined endoscopic datum $\mf{c}$, such that $\phi = \eta \circ \phi^{\mf{c}}$ as conjugacy classes of parameters for an $L$-parameter $\phi^{\mf{c}}: W_{L} \times \SL(2,\ol{\mathbb{Q}}_{\ell}) \ra \mathcal{H}$. Then the matching is uniquely described using the endoscopic character identities. This assert an equality

\[ \Theta^{1}_{\phi^{\mf{c}}}(f^{\mf{c}}) = \sum_{\pi \in \Pi_{\phi^{\mf{c}}}(H)} \tr(1|\tau_{\pi})\theta_{\pi}(f^{\mf{c}}) = e(J)\sum_{\rho \in \Pi_{\phi}(J)} \mathrm{tr}(s|\tau_{\rho}) \theta_{\pi}(f) = \Theta_{\phi^{\mf{c}}}^{s}(f) \]

where $e(J) = -1$ is the Kottwitz sign of $J$, as defined in <cit.> and $\theta_{\pi}$ denotes the Harish-Chandra character of $\pi$. Using the linear independence of the distributions $\Theta_{\pi}$ and the fact that the packets $\Pi_{\phi}(J)$ are disjoint, we can see that the matching between $\rho \mapsto \tau_{\rho}$ is uniquely characterized by these relations. To show that there exists a matching between the representations $\tau_{1}$ and $\tau_{2}$ and $\rho_{+-}$ and $\rho_{-+}$, we need to show that these identities are satisfied under the parametrization of Gan-Takeda-Tantono. This will follow from the endoscopic character identities verified by Chan-Gan <cit.>. Namely, in the case that the parameter $\phi$ is stable, one only needs to consider the trivial endoscopic datum $\mf{c}_{\mathrm{triv}} = (G,1,\phantom{}^{L}G,\mathrm{id})$, and these identities follow from <cit.>. In the case that the parameter $\phi$ is endoscopic, the case $\mf{c}_{\mathrm{triv}} = (G,1,\phantom{}^{L}G,\mathrm{id})$ follows from <cit.>, but one also needs to consider the refined endoscopic datum given by $\mf{c} = (\GSO_{2,2},(1,-1),\GSpin_{4},i: \GSpin_{4} \ra \GSpin_{5} \simeq \GSp_{4})$, where $i$ is the map from before and $(1,-1) \in S_{\phi} \subset (\GL_{2}(\mathbb{C}) \times \GL_{2}(\mathbb{C})$. In this case, the identities (up to a sign) follow from combining <cit.> as before and <cit.>, where we note that since $\tau_{1}$ and $\tau_{2}$ are by definition two projections their traces against $(-1,1)$ have the opposite sign, which is consistent with <cit.>. This shows the refined local Langlands correspondence of Kaletha holds for the group $G$; however, in order to describe the exact matching between $\rho_{+-}$ and $\rho_{-+}$ with $\tau_{1}$ and $\tau_{2}$ one needs to exactly compare the signs of the ad-hoc transfer factors fixed in the statement of <cit.> with the canonical ones $\Delta(\mf{m},\mf{c},b_{1})$ constructed by Kaletha. The above argument only shows that there is some matching; nonetheless, for our purposes the choice ends up being irrelevant, so we denote the representations in the $L$-packet $\Pi_{\phi}(J)$ corresponding to the projections $\tau_{1}$ and $\tau_{2}$ by $\rho_{1}$ and $\rho_{2}$, respectively. Similarly, for the representations obtained by pre-composing a character with the composition

\[ S_{\phi} \rightarrow A_{\phi} \]

we denote the elements of the $L$-packet $\Pi_{\phi}(G)$ corresponding to the trivial (non-trivial) character of $A_{\phi}$ by $\pi^{+}$ (resp. $\pi^{-}$). We note that, by Theorem 2.1 (2), $\pi^{+}$ can be characterized by the unique $\mathfrak{m}$-generic representation of this $L$-packet.

Given a supercuspidal or mixed supercuspidal $L$-parameter $\phi$ as above and $\rho \in \Pi_{\phi}(J)$, we let $\tau_{\rho}$ be the irreducible representation of $S_{\phi}$ associated to it via the matching described above. Given $\pi \in \Pi_{\phi}(G)$ and $\rho \in \Pi_{\phi}(J)$, we set

\[ \delta_{\pi,\rho} := \tau_{\pi}^{\vee} \otimes \tau_{\rho}\]

where $\tau_{\pi}^{\vee}$ denotes the contragredient.

Changing the choice of Whittaker datum scales the representations by a $1$-dimensional character of $S_{\phi}$ that is trivial when restricted to the center, so in particular this pairing is independent of the choice of Whittaker datum (See <cit.>).

§ THE FARGUES-SCHOLZE LOCAL LANGLANDS CORRESPONDENCE

We will now discuss the Fargues-Scholze local Langlands correspondence and deduce compatibility in the cases where the Gan-Takeda/Gan-Tantono parameter is not supercuspidal. We will then conclude by reducing the question of compatibility in the supercuspidal case to Proposition 1.4.

§.§ Overview of the Fargues-Scholze Local Langlands Correspondence

For now, let $G$ be any connected reductive group over $\mathbb{Q}_{p}$. Since we are going to be using geometric Satake, we fix a choice of the square root of $p$ in $\overline{\mathbb{Q}}_{\ell}$, so that half Tate-twists are well-defined. For us, we will always take $i^{-1}(\sqrt{p})$, where $i: \ol{\mathbb{Q}}_{\ell} \xrightarrow{\simeq} \mathbb{C}$ is the fixed isomorphism. Fargues-Scholze <cit.> consider the moduli space of $G$-bundles on the Fargues-Fontaine curve $X$, denoted $\Bun_{G}$. This moduli space is an Artin $v$-stack (in the sense of <cit.>) and has the structure that the underlying points of its topological space $|\Bun_{G}|$ are in natural bijection with elements of the Kottwitz set $B(G)$, where the slopes of the $G$-isocrystal associated to $b \in B(G)$ are the negatives of the slopes of the associated vector bundle $\mathcal{E}_{b}$ and the specializations between points of $|\Bun_{G}|$ is dictated by the partial ordering on $B(G)$ induced by the kappa invariant and the slope homomorphism <cit.>. In particular, the connected components of $\Bun_{G}$ are in bijection with $B(G)_{basic} \xrightarrow{\simeq} \pi_{1}(G)_{\Gamma}$. Specifically, for any $b \in B(G)_{basic}$, there is a unique open Harder-Narasimhan strata $\Bun_{G}^{b} \subset \Bun_{G}$ dense inside the associated connected component. We recall that the elements of $B(G)_{basic}$ parametrize extended pure inner forms of $G$, via sending an element $b \in B(G)_{basic}$ to its $\sigma$-centralizer $J_{b}/\mathbb{Q}_{p}$. For such a basic $b$, we have an identification $\Bun_{G}^{b} \simeq  [\ast/\underline{J_{b}(\mathbb{Q}_{p})}] =: B\underline{J_{b}(\mathbb{Q}_{p})}$ of the HN-strata defined by $b$ and the classifying stack of $J_{b}(\mathbb{Q}_{p})$. For any Artin $v$-stack $Z$, Fargues-Scholze define a triangulated category $\D_{\blacksquare}(Z,\overline{\mathbb{Q}}_{\ell})$ of solid $\overline{\mathbb{Q}}_{\ell}$-sheaves <cit.> and isolate a nice full subcategory $\Dlis(Z,\overline{\mathbb{Q}}_{\ell}) \subset \D_{\blacksquare}(Z,\overline{\mathbb{Q}}_{\ell})$ of étale-lisse $\overline{\mathbb{Q}}_{\ell}$-sheaves <cit.>, which may be roughly thought of as the unbounded derived category of étale $\overline{\mathbb{Q}}_{\ell}$ sheaves on $Z$, where one has made an enlargement to capture information about the topology of $p$-adic groups. In any case, the key point for us is that we have the following basic result.
There is an equivalence of categories

\[ \Dlis(B\underline{J_{b}(\mathbb{Q}_{p})}),\overline{\mathbb{Q}}_{\ell})\simeq \D(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell}) \]

where the RHS denotes unbounded derived category of smooth $J_{b}(\mathbb{Q}_{p})$-representations with coefficients in $\overline{\mathbb{Q}}_{\ell}$. Under this equivalence, Verdier duality corresponds to smooth duality.

The main reason for constructing this category $\Dlis$ is that, if one were to take the usual definition for the category of étale $\overline{\mathbb{Q}}_{\ell}$-sheaves on $B\underline{J_{b}(\mathbb{Q}_{p})}$, this equivalence would no longer be true. In particular, one would obtain the bounded derived category of representations of $J_{b}(\mathbb{Q}_{p})$ admitting a $J_{b}(\mathbb{Q}_{p})$-stable $\overline{\mathbb{Z}}_{\ell}$-lattice, where the representation is continuous with respect to the $\ell$-adic topology on the target. This would limit the scope of the Fargues-Scholze LLC as, in general, one wants to consider smooth $\overline{\mathbb{Q}}_{\ell}$-representations of $J_{b}(\mathbb{Q}_{p})$, and hence the need for the enlargement of the derived category to $\Dlis$.

Lemma 3.1 tells us that, given an irreducible smooth representation $\pi$ of $G(\mathbb{Q}_{p})$, we can consider the associated sheaf, denoted $\mathcal{F}_{\pi}$, on $\Bun_{G}^{\mathbf{1}}$ the open HN-strata corresponding to the trivial element $\mathbf{1} \in B(G)$, and take the extension by zero along the open inclusion $j_{!}(\mathcal{F}_{\pi})$[The shriek push-forward is not in general well-defined in the context of solid $\ol{\mathbb{Q}}_{\ell}$-sheaves. However, for the inclusion of HN-strata into $\Bun_{G}$, its existence follows from <cit.>, using <cit.>.]. This realizes the representation $\pi$ in terms of a sheaf on the moduli space $\Bun_{G}$ in an analogous way to how the function-sheaf dictionary realizes cuspidal automorphic forms as functions associated to sheaves in the context of curves over finite fields. Following V. Lafforgue <cit.>, Fargues and Scholze construct a semisimple L-parameter associated to this sheaf by looking at the action of the excursion algebra on this category $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$. This relies on a form of the geometric Satake correspondence for the $B_{dR}^{+}$-affine Grassmannians. For any finite set $I$, let $X^{I}$ be the product of $I$-copies of the diamond $X = \Spd(\Breve{\mathbb{Q}}_{p})/\Frob^{\mathbb{Z}}$. We then have the Hecke stack

\[ \begin{tikzcd}
& & \arrow[dl,"h^{\leftarrow}"] \Hck \arrow[dr,"h^{\rightarrow} \times supp"] & & \\
& \Bun_{G} & & \Bun_{G} \times X^{I}  &
\end{tikzcd} \]

defined as the functor that parametrizes, for $S$ a perfectoid space in characteristic $p$ together with a map $S \rightarrow X^{I}$ defining a tuple of Cartier divisors in the relative Fargues-Fontaine $X_{S}$ over $S$, corresponding to characteristic $0$ untilts $S_{i}^{\sharp}$ for $i \in I$ of $S$, a pair of $G$-torsors $\mathcal{E}_{1}$, $\mathcal{E}_{2}$ together with an isomorphism

\[ \beta:\mathcal{E}_{1}|_{X_{S} \setminus \bigcup_{i \in I} S_{i}^{\sharp}} \xrightarrow{\simeq} \mathcal{E}_{2}|_{X_{S} \setminus \bigcup_{i \in I} S_{i}^{\sharp}}\]

where $h^{\leftarrow}((\mathcal{E}_{1},\mathcal{E}_{2},i,(S_{i}^{\sharp})_{i \in I})) = \mathcal{E}_{1}$ and $h^{\rightarrow} \times supp((\mathcal{E}_{1},\mathcal{E}_{2},\beta,(S_{i}^{\sharp})_{i \in I})) = (\mathcal{E}_{2},(S_{i}^{\sharp})_{i \in I})$. We set $^{L}G^{I}$ to be $I$-copies of the Langlands dual group of $G$, i.e $^{L}G = Q \ltimes \hat{G}(\overline{\mathbb{Q}}_{\ell})$, where $\hat{G}$ is the reductive group having dual root datum to $G$ and is viewed as a reductive group over $\overline{\mathbb{Q}}_{\ell}$. The Weil group acts on $\hat{G}$ via the induced action on root datum through some finite quotient $Q$, which we now fix. Let $\Rep_{\overline{\mathbb{Q}}_{\ell}}(^{L}G^{I})$ denote the category of algebraic $\overline{\mathbb{Q}}_{\ell}$-representations of $I$-copies of $^{L}G$. For each element $W \in \Rep_{\overline{\mathbb{Q}}_{\ell}}(^{L}G^{I})$, the geometric Satake correspondence of Fargues-Scholze <cit.> furnishes a solid $\overline{\mathbb{Q}}_{\ell}$-sheaf $\mathcal{S}_{W}$ on $\Hck$. This allows us to define Hecke operators.

For each $W \in \Rep_{\overline{\mathbb{Q}}_{\ell}}(^{L}G^{I})$, we define the Hecke operator

\[ T_{W}: \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell}) \rightarrow \D_{\blacksquare}(\Bun_{G} \times X^{I}) \]

\[ A \mapsto R(h^{\rightarrow} \times supp)_{\natural}(h^{\leftarrow *}(A) \otimes^{\mathbb{L}} \mathcal{S}_{W})\]

where $\mathcal{S}_{W}$ is a solid $\overline{\mathbb{Q}}_{\ell}$-sheaf and the functor $R(h^{\rightarrow} \times supp)_{\natural}$ is the natural push-forward. I.e the left adjoint to the restriction functor in the category of solid $\overline{\mathbb{Q}}_{\ell}$-sheaves <cit.>.

These satisfy various compatibilities with respect to composition and restriction to the diagonal. In particular, given two representations $V,W \in \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G)$, we have that

\[ (T_{V} \times id)(T_{W})(\cdot)|_{\Delta} \simeq T_{V \otimes W}(\cdot) \]

where $\Delta: X \rightarrow X^{2}$ is the diagonal map.

We then consider $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{BW_{\mathbb{Q}_{p}}^{I}}$, the category of objects in $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ with continuous action by $W_{\mathbb{Q}_{p}}^{I}$. Examples of objects in this category are objects of $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ tensored by a continuous representation of $W_{\mathbb{Q}_{p}}^{I}$, for a more precise description see <cit.>.  With this in hand, we then have the following theorem of Fargues-Scholze.
The Hecke operator $T_{W}$ for $W \in \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G^{I})$

\[ T_{W}: \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell}) \rightarrow \D_{\blacksquare}(\Bun_{G} \times X^{I}) \]

induces a functor

\[ \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell}) \rightarrow \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{BW_{\mathbb{Q}_{p}}^{I}} \]

and the induced endofunctors of $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ given by forgetting the Weil group action preserve compact and ULA objects.

This should be thought of as a manifestation of Drinfeld's Lemma, where (roughly) the étale fundamental group of $\Spd(\breve{\mathbb{Q}_{p}})/\Frob^{\mathbb{Z}} = X$ should be the same as $W_{\mathbb{Q}_{p}}$.

From now on, when talking about Hecke operators we shall always refer to this induced functor, which we will also abusively denote by $T_{W}$. Theorem 3.2 has direct implications for the cohomology of local Shimura varieties. To study this, consider a minuscule  cocharacter $\mu$ with field of definition $E$, and let $b \in B(G,\mu)$ be the unique basic element in the $\mu$-admissible locus (See <cit.>). We say that the triple $(G,b,\mu)$ defines a local Shimura datum in the sense of Rapoport-Viehmann <cit.>. Attached to such a data, Scholze-Weinstein <cit.> construct a tower of diamonds

\[ p_{K}: (\Sht(G,b,\mu)_{K})_{K \subset G(\mathbb{Q}_{p})} \rightarrow \Spd(\Breve{E})\]

for varying open compact $K \subset G(\mathbb{Q}_{p})$. This is obtained by considering the space $\Sht(G,b,\mu)_{\infty}$ which parametrizes modifications $ \mathcal{E}_{b} \rightarrow \mathcal{E}_{0}$ with meromorphy bounded by $\mu$, where $\mathcal{E}_{b}$ (resp. $\mathcal{E}_{0}$) is the bundle corresponding to $b \in B(G)$ (resp. the trivial bundle) on the Fargues-Fontaine curve. It has commuting actions by $G(\mathbb{Q}_{p})$ and $J_{b}(\mathbb{Q}_{p})$ given by acting via automorphisms on $\mathcal{E}_{0}$ and $\mathcal{E}_{b}$, respectively. The tower is then given by considering the quotients of this space for varying open compact $K \subset G(\mathbb{Q}_{p})$ under the action of $G(\mathbb{Q}_{p})$.

Let $\Sht(G,b,\mu)_{K,\mathbb{C}_{p}}$ be the base-change of the above tower to $\mathbb{C}_{p}$. We define the complex

\[ R\Gamma_{c}(G,b,\mu) := \colim_{K \rightarrow 1} R\Gamma_{c}(\Sht(G,b,\mu)_{K,\mathbb{C}_{p}},\overline{\mathbb{Q}}_{\ell}) \]

a colimit of smooth $W_{E} \times J_{b}(\mathbb{Q}_{p})$-modules with a $G(\mathbb{Q}_{p})$-action, where $W_{E}$ is the Weil group of $E$. A priori it only has an action by the inertia group, but this space admits a non-effective Frobenius descent datum. We then define, for $\rho$ a smooth admissible $J_{b}(\mathbb{Q}_{p})$-representation, the complex

\[ R\Gamma_{c}(G,b,\mu)[\rho] := R\Gamma_{c}(G,b,\mu) \otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})} \rho \]

where $\mathcal{H}(J_{b})$ is the usual smooth Hecke algebra. We also define

\[ R\Gamma_{c}^{\flat}(G,b,\mu)[\rho] := \colim_{K \ra \{1\}} R\mathcal{H}om_{J_{b}(\mathbb{Q}_{p})}(R\Gamma_{c}(G,b,\mu)_{K},\rho)[-2d](-d) \]

Similarly, for $\pi$ a smooth admissible $G(\mathbb{Q}_{p})$-representation, we define $R\Gamma_{c}(G,b,\mu)[\pi]$ and $R\Gamma_{c}^{\flat}(G,b,\mu)[\pi]$.

We note that, by Hom-Tensor duality, $R\mathcal{H}om(R\Gamma_{c}(G,b,\mu)[\rho],\overline{\mathbb{Q}}_{\ell})[-2d](-d)$ is isomorphic to $R\Gamma_{c}^{\flat}(G,b,\mu)[\rho^{*}]$, where $\rho^{*}$ is the contragredient. We will end up using both of these cohomology groups throughout this manuscript. The former is more natural from the point of view of basic uniformization, while the latter is disposable to the results of Hansen-Kaletha-Weinstein <cit.> on the Kottwitz conjecture.

To study these complexes, we specialize the above discussion of Hecke operators to the case where $W = V_{\mu^{-1}}$ is specified by the highest weight representation of highest weight $\mu^{-1}$ a dominant inverse of $\mu$ and $I = \{\ast\}$ is a singleton. The sheaf $\mathcal{S}_{W}$ will then be supported on the closed subspace $\Hck_{\leq \mu^{-1}} = \Hck_{\mu^{-1}}$ of $\Hck$, parametrizing modifications with meromorphy bounded by or equal to $\mu^{-1}$, where the equality follows by the minuscule assumption. The space $\Hck_{\mu^{-1}}$ is cohomologically smooth of dimension $d := \langle 2\rho_{G},\mu \rangle$ , and the sheaf $\mathcal{S}_{W}$, as in the geometric Satake correspondence of <cit.>, behaves like the intersection cohomology of this space, so we have $\mathcal{S}_{W} \simeq \overline{\mathbb{Q}}_{\ell}[d](\frac{d}{2})$. This implies that, to study the action of the Hecke operator $T_{W}$ on $\Bun_{G}$, we can look at the restriction of the diagram defining the Hecke correspondence to this subspace

\[ \begin{tikzcd}
& & \arrow[dl,"h_{\mu^{-1}}^{\leftarrow}"] \Hck_{\mu^{-1}} \arrow[dr,"h_{\mu^{-1}}^{\rightarrow} \times supp"] & & \\
& \Bun_{G} & & \Bun_{G} \times \Spd(\Breve{E})/\Frob^{\mathbb{Z}}  &
\end{tikzcd} \]

In particular, we have an isomorphism:

\[ T_{\mu^{-1}}(A) := T_{W}(A) \simeq R(h_{\mu^{-1}}^{\rightarrow} \times supp)_{\natural}(h_{\mu^{-1}}^{\leftarrow *}(A))[d](\frac{d}{2}) \]

Now consider a smooth admissible representation $\pi$ of $G(\mathbb{Q}_{p})$ and apply the Hecke operator to the sheaf:

\[ j_{!}(\mathcal{F}_{\pi}) \]

Then the fiber of $\Hck_{\mu^{-1}}$ of $h_{\mu^{-1}}^{\leftarrow}$ over $\Bun_{G}^{\mathbf{1}}$ is identified with

\[ [Gr_{G,\mu^{-1}}/\underline{G(\mathbb{Q}_{p})}] \]

the Schubert cell/variety associated to $\mu^{-1}$ in the $B_{dR}^{+}$-affine Grassmannian, quotiented out by $G(\mathbb{Q}_{p})$ acting on the trivial bundle via automorphisms. The sheaf

\[ T_{\mu^{-1}}j_{!}(\mathcal{F}_{\pi}) \]

is then supported on the HN-strata given by the Kottwitz elements in $B(G,\mu)$ since, by <cit.>, any $G$-bundle occurring as a modification of type $\mu^{-1}$ of the trivial bundle has associated Kottwitz element lying in this set. We then consider the restriction

\[ j_{b}^{*}T_{\mu^{-1}}j_{!}(\mathcal{F}_{\pi}) \in \D(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})^{BW_{E}} \]

where $j_{b}: \Bun_{G}^{b} \hookrightarrow \Bun_{G}$ is the inclusion of the open HN-strata defined by $b$. The complex $j_{b}^{*}T_{\mu^{-1}}j_{!}(\mathcal{F}_{\pi})$ will be computed in terms of the cohomology of sheaves supported on the Newton strata

\[ [Gr_{G,\mu^{-1}}^{b}/\underline{G(\mathbb{Q}_{p})}] \]

parametrizing modifications of type $\mu^{-1}$ of the trivial bundle such that the resulting bundle has associated Kottwitz element of type $b$ after pulling back to each geometric point, modulo automorphisms of the trivial bundle. The space $\Sht(G,b,\mu)_{\infty}$ defined above is a pro-étale $\underline{J_{b}(\mathbb{Q}_{p})}$-torsor with respect to the $J_{b}(\mathbb{Q}_{p})$-action by automorphisms of $\mathcal{E}_{b}$

\[ \Sht(G,b,\mu)_{\infty} \rightarrow Gr_{G,\mu^{-1}}^{b} \]

over this Newton strata. Using this description of the infinite level Shimura variety, it then follows from base change and the fact that the sheaves $\mathcal{S}_{W}$ are ULA over $X$ (See <cit.> for details) that we have an isomorphism

\[ R\Gamma_{c}(G,b,\mu)[\pi][d](\frac{d}{2}) \simeq  j_{b} ^{*}T_{\mu^{-1}}j_{!}(\mathcal{F}_{\pi}) \in \D(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})^{BW_{E}}  \]

of $J_{b}(\mathbb{Q}_{p}) \times W_{E}$-modules. For our purposes, it will be also useful to have a description of the $\rho$-isotypic part of this cohomology in terms of Hecke operators, for $\rho$ a smooth irreducible representation of $J_{b}(\mathbb{Q}_{p})$. In particular, analysis similar to the above gives us an isomorphism

\[ R\Gamma_{c}(G,b,\mu)[\rho][d](\frac{d}{2}) \simeq j_{\mathbf{1}}^{*}T_{\mu}j_{b!}(\mathcal{F}_{\rho}) \]

as $G(\mathbb{Q}_{p}) \times W_{E}$-modules. We record these two isomorphisms as a corollary of the above discussion.

Given a local Shimura datum $(G,b,\mu)$ as above and $\pi$ (resp. $\rho$) a smooth irreducible representation of $G(\mathbb{Q}_{p})$ (resp. $J_{b}(\mathbb{Q}_{p})$). There exists an isomorphism

\[ R\Gamma_{c}(G,b,\mu)[\rho][d](\frac{d}{2}) \simeq  j_{\mathbf{1}}^{*}T_{\mu}j_{b!}(\mathcal{F}_{\rho}) \]

of complexes of $G(\mathbb{Q}_{p}) \times W_{E}$-modules and an isomorphism

\[ R\Gamma_{c}(G,b,\mu)[\pi][d](\frac{d}{2}) \simeq j_{b}^{*}T_{\mu^{-1}}j_{\mathbf{1}!}(\mathcal{F}_{\pi}) \]

of complexes of $J_{b}(\mathbb{Q}_{p}) \times W_{E}$-modules.

We have the following basic structural result which, in more generality, follows from the analysis in Fargues-Scholze, but, in the case of a local Shimura datum, also partially follows from standard finiteness results for rigid spaces (See <cit.>). In particular, one can show the following.
For a local Shimura datum $(G,b,\mu)$ as above, the cohomology groups of $R\Gamma_{c}^{\flat}(G,b,\mu)[\rho]$ and $R\Gamma_{c}(G,b,\mu)[\rho]$ are valued in smooth admissible $G(\mathbb{Q}_{p})$-representations of finite length with an action of $W_{E}$. Moreover, they are concentrated in degrees $0 \leq i \leq 2d$.

A sheaf $\mathcal{F} \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ being ULA is equivalent to its stalks at different HN-strata being valued in complexes of smooth admissible representations <cit.>, so indeed the admissibility of the above complex is a consequence of Theorem 3.2 and Corollary 3.3.

Fargues-Scholze use the endofunctors defined by the Hecke algebra on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$  to define the excursion algebra.

For a finite set $I$, a representation $W \in \Rep_{\overline{\mathbb{Q}}_{\ell}}(^LG^{I})$, maps $\alpha: \overline{\mathbb{Q}}_{\ell} \rightarrow \Delta^{*}W$ and $\beta: \Delta^{*}W \rightarrow \overline{\mathbb{Q}}_{\ell}$, and elements $\gamma_{i} \in W_{\mathbb{Q}_{p}}$ for $i \in  I$, one defines the excursion operator on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ to be the composition:

\[ id = T_{\overline{\mathbb{Q}}_{\ell}} \xrightarrow{\alpha} T_{\Delta^{*}W} = T_{W} \xrightarrow{(\gamma_{i})_{i \in I}} T_{W} = T_{\Delta^{*}W}  \xrightarrow{\beta} T_{\overline{\mathbb{Q}}_{\ell}} = id \]

where $\Delta^{*}(W)$ is the precomposition of $W$ with the diagonal embedding $\phantom{}^{L}G \ra \phantom{}^{L}G^{I}$.

This defines a natural endomorphism of the identity functor on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$. If one looks at the induced endofunctor given by the inclusion $\D(G(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell}) \subset \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ induced by the open immersion $j: \Bun_{G}^{\mathbf{1}} \hookrightarrow \Bun_{G}$
then one obtains a natural endomorphism of the identity functor on $\D(G(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$. In other words, we get a family of compatible endomorphisms for all complexes of smooth representations of $G(\mathbb{Q}_{p})$; namely, an element of the Bernstein center. One can verify that this excursion algebra satisfies similar properties to that considered by V. Lafforgue, so, using Lafforgue's reconstruction theorem <cit.>, one can show the following.

To an irreducible smooth $\overline{\mathbb{Q}}_{\ell}$-representation $\pi$ of $G(\mathbb{Q}_{p})$ (or more generally $A \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ any Schur-irreducible object (i.e $\End(A) = \overline{\mathbb{Q}}_{\ell}$)), there is a unique continuous semisimple map

\[ \phi^{\mathrm{FS}}_{\pi}: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \]

characterized by the property that for all $I,W,\alpha,\beta$, and $\gamma_{i} \in W_{\mathbb{Q}_{p}}$ for $i \in I$, the corresponding endomorphism of $\pi$ defined above is given by multiplication by the scalar that results from the composite

\[ \overline{\mathbb{Q}}_{\ell} \xrightarrow{\alpha} \Delta^{*}W = W \xrightarrow{(\phi_{\pi}(\gamma_{i}))_{i \in I}} W = \Delta^{*}W \xrightarrow{\beta} \overline{\mathbb{Q}}_{\ell} \]

By further studying the geometry of $\Bun_{G}$ and the Hecke stacks, one can deduce various good properties of this correspondence.
The mapping defined above

\[ \pi \mapsto \phi^{\mathrm{FS}}_{\pi} \]

enjoys the following properties:

* (Compatibility with Local Class Field Theory) If $G = T$ is a torus, then $\pi \mapsto \phi_{\pi}$ is the usual local Langlands correspondence

* The correspondence is compatible with character twists, passage to contragredients, and central characters.

* (Compatibility with products) Given two irreducible representations $\pi_{1}$ and $\pi_{2}$ of two connected reductive groups $G_{1}$ and $G_{2}$ over $\mathbb{Q}_{p}$, respectively. We have

\[ \pi_{1} \boxtimes \pi_{2} \mapsto \phi^{\mathrm{FS}}_{\pi_{1}} \times \phi^{\mathrm{FS}}_{\pi_{2}}\]

under the Fargues-Scholze local Langlands correspondence for $G_{1} \times G_{2}$.

* (Compatibility with parabolic induction) Given a parabolic subgroup $P \subset G$ with Levi factor $M$ and a representation $\pi_{M}$ of $M$, then the semisimple L-parameter corresponding to any sub-quotient of $ind_{P}^{G}(\pi_{M})$ the (normalized) parabolic induction is the composition

\[ W_{\mathbb{Q}_{p}}\xrightarrow{\phi^{\mathrm{FS}}_{\pi_{M}}} \\  ^{L}M(\overline{\mathbb{Q}}_{\ell}) \rightarrow ^{L}G(\overline{\mathbb{Q}}_{\ell}) \]

where the map $\phantom{}^{L}M(\overline{\mathbb{Q}}_{\ell}) \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$ is the natural embedding.

* (Compatibility with Harris-Taylor/Henniart LLC)
For $G = \GL_{n}$ or an inner form of $G$ the semisimple L-parameter associated to $\pi$ is the (semi-simplified) parameter $\phi_{\pi}^{\mathrm{ss}}$. associated to $\pi$ by Harris-Taylor/Henniart.

* (Compatibility with Restriction of Scalars) The above story works the same for $G'$ a connected reductive group over any finite extension $E'/\mathbb{Q}_{p}$, where one then gets a semisimple L-parameter valued on $W_{E'}$. If $G = \mathrm{Res}_{E'/\mathbb{Q}_{p}}G'$ is the Weil restriction of some $G'/E'$ then $L$-parameters for $G/\mathbb{Q}_{p}$ agree with $L$-parameters for $G'/E'$ in the usual sense.

* (Compatibility with Isogenies) If $G' \rightarrow G$ is a map of reductive groups inducing an isomorphism of adjoint groups, $\pi$ is an irreducible smooth representation of $G(E)$ and $\pi'$ is an irreducible constituent of $\pi|_{G'(E)}$ then $\phi_{\pi'}$ is the image of $\phi_{\pi}$ under the induced map $\hat{G} \rightarrow \hat{G'}$.

In (5), the compatibility of the Fargues-Scholze local Langlands correspondence with the Harris-Taylor/Henniart local Langlands correspondence for an arbitrary inner form of $\GL_{n}$ is not included in the paper of Fargues-Scholze <cit.>. However, it follows from the work of Hansen-Kaletha-Weinstein on the Kottwitz conjecture <cit.>.

§.§ The Spectral Action

With these basic structural properties out of the way, we turn our attention to the "spectral action" on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$, which will be very important to proving compatibility of the two correspondences in the case where the parameter is supercuspidal, as well as deducing applications to the Kottwitz conjecture. We recall that an $L$-parameter over $\overline{\mathbb{Q}}_{\ell}$ can be thought of as a continuous (not necessarily semisimple) homomorphism

\[ \phi: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \]

commuting with the natural projection to $Q$. One can use the classical construction of Grothendieck-Deligne to see that this coincides with the definition given in section $2$ for $\GSp_{4}$ after applying the isomorphism $i$, where the monodromy operation is recovered through the exponential of the action of $W_{\mathbb{Q}_{p}}$ on the $\ell$-power roots of unity, and assuming Frobenius semi-simplicity of $\phi$. Such a continuous map can be thought of as a continuous $1$-cocycle $W_{\mathbb{Q}_{p}} \rightarrow \hat{G}(\overline{\mathbb{Q}}_{\ell})$, with respect to the action of $W_{\mathbb{Q}_{p}}$ on $\hat{G}(\overline{\mathbb{Q}}_{\ell})$. If we let $A/\mathbb{Z}_{\ell}$ be any $\mathbb{Z}_{\ell}$-algebra endowed with a topology given by writing $A = \colim_{A' \subset A} A'$, where $A'$ is a finitely generated $\mathbb{Z}_{\ell}$-module with its $\ell$-adic topology, then we can defined a moduli space, denoted $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})$, over $\mathbb{Z}_{\ell}$, whose $A$-points are the continuous $1$-cocycles $W_{\mathbb{Q}_{p}} \rightarrow \hat{G}(A)$ with respect to the natural action of $W_{\mathbb{Q}_{p}}$ on $\hat{G}(A)$. This defines a scheme considered in <cit.> and <cit.> which, by <cit.>, can be written as a union of open and closed affine subschemes $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}}/P,\hat{G})$ as $P$ runs through subgroups of wild inertia of $W_{E}$, where each $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}}/P,\hat{G})$ is a flat local complete intersection over $\mathbb{Z}_{\ell}$ of dimension $\dim(G)$. This allows us to consider the Artin stack quotient $[\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})/\hat{G}]$, where $\hat{G}$ acts via conjugation. We then consider the base change to $\overline{\mathbb{Q}}_{\ell}$, denoted $[\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]$ and referred to as the stack of Langlands parameters, as well as the category $\Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])$ of perfect complexes of coherent sheaves on this space. We let $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ denote the triangulated sub-category of compact objects in $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ (which are precisely the objects with quasi-compact support on $\Bun_{G}$ and which restrict to compact objects in $\D(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$ for all $b \in B(G)$ by <cit.>). We then have the key theorem of Fargues-Scholze.
There exists a natural compactly supported $\overline{\mathbb{Q}}_{\ell}$-linear action of $\Perf(\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G})$ on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ satisfying the property that the restriction along the map

\[ \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G) \rightarrow \Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])^{BW_{\mathbb{Q}_{p}}} \]

induces the action of Hecke operators

\[ \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G^{I}) \rightarrow \End(\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega})^{BW_{\mathbb{Q}_{p}}^{I}} \]

for a varying finite index set $I$.

* Here the map

\[ \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G) \rightarrow \Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])^{BW_{\mathbb{Q}_{p}}} \]

associates to a representation $V$, with associated map $r_{V}: \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \rightarrow GL(V)(\overline{\mathbb{Q}}_{\ell})$ a vector bundle on $[\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G})]$ of rank equal to $\dim(V)$ with $W_{\mathbb{Q}_{p}}$-action. This bundle, denoted $C_{V}$, has the property that its evaluation at a $\overline{\mathbb{Q}}_{\ell}$-point corresponding to a parameter $\phi: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$ is precisely $r_{V} \circ \phi$.

* The compactly supported condition means that, for all $\mathcal{F} \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$, the functor $\Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]) \rightarrow \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ induced by acting on $\mathcal{F}$ factors through an action of  $\Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}}/P,\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G})])$, where $P$ is a subgroup of wild inertia. Fargues and Scholze state this action in terms of a $(\infty,1)$-category acting on an $(\infty,1)$-category, we suppress this technicality for simplicity. However, we note that this enhanced action has the concrete implication that if one has a morphism, cone, or homotopy limit/colimit in the derived category $\Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])$ that acting on an object $A \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ will produce a corresponding morphism, cone, or homotopy limit/colimit.
* In fact, Fargues-Scholze show that giving such a compactly supported $\overline{\mathbb{Q}}_{\ell}$-linear action is equivalent (when properly formulated) to giving a $\Rep_{\overline{\mathbb{Q}}_{\ell}}(Q^{I})$-linear monoidal functor

\[ \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G^{I}) \rightarrow \End_{\overline{\mathbb{Q}}_{\ell}}(\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega})^{BW^{I}_{\mathbb{Q}_{p}}}\]

In the case that $I = \{*\}$, the fact that the Hecke action satisfies this monoidal property is precisely Remark 3.2.

To study this spectral action, we consider, as in <cit.>, the coarse quotient in the category of schemes

\[ \mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}} / / \hat{G}\]

$\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}$ by the action of $\hat{G}$ via conjugation. Given an $L$-parameter $\phi:W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$, it follows by <cit.> or <cit.>, that the $\hat{G}$-orbit of $\phi$ defines a closed $\overline{\mathbb{Q}}_{\ell}$-point of $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}} / / \hat{G}$ if and only if $\phi$ is a semisimple parameter. Moreover, the natural map

\[ \pi: [\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}] \rightarrow \mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}} / / \hat{G}\]

evaluated on a $\overline{\mathbb{Q}}_{\ell}$-point in the stack quotient defined by an $L$-parameter $\phi$ defines a closed $\overline{\mathbb{Q}}_{\ell}$-point in the coarse moduli space given by its semisimplification $\phi^{\mathrm{ss}}$. We can fit excursion operators into this picture as follows. We let $\mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell}) := \mathcal{O}(\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}})^{\hat{G}}$ be the ring of functions on the stack of $L$-parameters/the coarse moduli space, which we refer to as the spectral Bernstein center. As noted above, the excursion operators define a family of commuting endomorphisms of the identity functor on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$. We let $\mathcal{Z}^{geom}(G,\overline{\mathbb{Q}}_{\ell})$ be the ring of such endomorphisms as in <cit.>, which we refer to as the geometric Bernstein center. In <cit.>, Fargues and Scholze construct a canonical map of rings

\[ \mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathcal{Z}^{geom}(G,\overline{\mathbb{Q}}_{\ell}) \]

which is given by excursion operators in the following sense. By <cit.>, there is an identification between $\mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell})$ and the algebra of excursion operators. In particular, an excursion operator, as in Definition 3.3, associated to the datum $I$, $W$, $\alpha$, $\beta$, and $\gamma_{i} \in W_{\mathbb{Q}_{p}}$ for $i \in I$ defines a function $f_{I,W,\alpha,\beta,(\gamma_{i})_{i \in I}} \in \mathcal{O}_{[Z^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]} = \mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell})$ on  $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}} / / \hat{G}$, whose evaluation on the closed point of the coarse moduli space associated to a semisimple parameter $\phi: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$ is precisely the scalar that results from the endomorphism:

\[ \overline{\mathbb{Q}}_{\ell} \xrightarrow{\alpha} \Delta^{*}W = W \xrightarrow{(\phi(\gamma_{i}))_{i \in I}} W = \Delta^{*}W \xrightarrow{\beta} \overline{\mathbb{Q}}_{\ell} \]

We note that multiplication by $f_{I,W,\alpha,\beta,(\gamma_{i})_{i \in I}}$ defines an endomorphism

\[ \mathcal{O}_{[Z^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]} \rightarrow \mathcal{O}_{[Z^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]} \]

of the structure sheaf on the Artin stack $[Z^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]$. If we act on a Schur-irreducible object $A \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ then we obtain an endomorphism

\[ \{ \mathcal{O}_{[Z^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]} \star A = A \rightarrow \mathcal{O}_{[Z^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]} \star A = A \} \in \End(A) = \overline{\mathbb{Q}}_{\ell} \]

which will be precisely the scalar given by evaluating $\phi_{A}^{\mathrm{FS}}$ on the excursion datum (See also <cit.>). In this way, we see that the action of excursion operators can be obtained through the spectral action. We will leverage this interpretation of excursion operators to prove the following key lemma.

Let $A \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ be any Schur-irreducible object with Fargues-Scholze parameter $\phi_{A}^{\mathrm{FS}}$. Set $x$ to be the closed point defined by the parameter $\phi_{A}^{\mathrm{FS}}$ in the coarse moduli space $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}} / / \hat{G}$ and let $\pi^{-1}(x)$ denote the closed subset defined by the preimage in $[\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]$.  Suppose we have $C \in \Perf([\mathcal{Z}^{1}(W_{E},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])$ with support disjoint from $\pi^{-1}(x)$ then $C$ acts by zero on $A$ via the spectral action.

If we look at the action on $A \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ via the map

\[ \mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathcal{Z}^{geom}(G,\overline{\mathbb{Q}}_{\ell}) \]

given by excursion operators this factors through the maximal ideal $\mathfrak{m}_{A} \subset \mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell}) = \mathcal{O}(Z^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}})^{\hat{G}}$ defined by the closed point $\phi_{A}^{\mathrm{FS}}$ in the coarse moduli space. By the conditions on the support of $C$, this implies that we can write the identity element as $1 = 1_{C} + 1_{A} \in \mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell})$, where $1_{C}$ is a function that annihilates $C$ and $1_{A}$ is in the annihilator of $\mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell})/\mathfrak{m}_{A}$. We consider the spectral action of $C$ on $A$

\[ C \star A \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell}) \]

and look at the endomorphism induced by multiplication by $1$ on $C$

\[ C \star A \rightarrow C \star A \]

which is just the identity. However, since $1_{C}$ annihilates $C$, this is the same as the action of $1_{A}$ on $C \star A$, but, it follows by the above discussion that acting via multiplication by $1_{A}$ is the same as acting via the map

\[ \mathcal{Z}^{spec}(G,\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathcal{Z}^{geom}(G,\overline{\mathbb{Q}}_{\ell}) \]

given by excursion operators, and the action of $1_{A}$ after applying this map is zero. This would lead to a contradiction unless $C \star A$ is also zero.

To take advantage of this lemma, we now introduce the following endofunctors of $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$, which are analogues of the averaging operators considered by <cit.> in the Fargues-Scholze geometric Langlands correspondence for $\GL_{n}$ and by <cit.> in the classical geometric Langlands correspondence over function fields.

Let $\phi$ be a representation of $W_{\mathbb{Q}_{p}}$ and $V$ a representation of $\phantom{}^{L}G$ with $T_{V}$ the associated Hecke operator. We consider the endofunctor of $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$

\[ A \mapsto R\Gamma(W_{\mathbb{Q}_{p}},T_{V}(A) \otimes \phi^{\vee})\]

where $R\Gamma(W_{\mathbb{Q}_{p}},-): \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{BW_{\mathbb{Q}_{p}}} \rightarrow \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ is the derived functor given by continuous group cohomology with respect to $W_{\mathbb{Q}_{p}}$. We denote this endofunctor by $Av_{V,\phi}: \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell}) \rightarrow \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$.

We now would like to realize the functor $Av_{V,\phi}$ as the spectral action of an object in $\Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G})])$ similar to <cit.>. An obvious guess would be that one should take the vector bundle $C_{V}$ corresponding to the Hecke operator $T_{V}$, as in Remark 3.7 (1), and then twist this by the constant sheaf defined by $\phi^{\vee}$, which we denote by

\[ C_{V} \otimes \phi^{\vee} \in \Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])^{BW_{\mathbb{Q}_{p}}}\]

More precisely, this is the vector bundle with $W_{\mathbb{Q}_{p}}$-action whose evaluation at a $\overline{\mathbb{Q}}_{\ell}$-point corresponding to a $L$-parameter $\tilde{\phi}: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$ is the vector space with $W_{\mathbb{Q}_{p}}$-action given by tensoring the representation

\[ \tilde{\phi}: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \xrightarrow{r_{V}} GL(V)\]

with $\phi^{\vee}$. To obtain the desired perfect complex, it is natural to apply $R\Gamma(W_{\mathbb{Q}_{p}},-)$ to the vector bundle
to $C_{V} \otimes \phi^{\vee}$, which we denote by $\mathcal{A}v_{V,\phi}$. We note that $\mathcal{A}v_{V,\phi}$ is a perfect complex. Indeed, as $p$ is invertible in $\overline{\mathbb{Q}}_{\ell}$, the wild inertia $P \subset W_{\mathbb{Q}_{p}}$ will always act through a finite quotient on $C_{V} \otimes \phi^{\vee}$ and has no higher cohomology which implies that the invariants

\[ (C_{V} \otimes \phi^{\vee})^{P} \]

are a direct summand of the vector bundle $C_{V} \otimes \phi^{\vee}$. If we choose a generator $\tau \in I/P$ in the tame quotient of the inertia subgroup $I \subset W_{\mathbb{Q}_{p}}$ together with a Frobenius lift $\sigma \in W_{\mathbb{Q}_{p}}/P$ then the complex $\mathcal{A}v_{V,\phi}$ is computed as the homotopy limit of the diagram:

\begin{equation}
\begin{tikzcd}
& (C_{V} \otimes \phi^{\vee})^{P} \arrow[r,"\tau - 1"] \arrow[d,"\sigma - 1"] & (C_{V} \otimes \phi^{\vee})^{P} \arrow[d,"\sigma(1 + \tau + \ldots + \tau^{p - 1}) - 1"] \\
& (C_{V} \otimes \phi^{\vee})^{P} \arrow[r,"\tau - 1"] & (C_{V} \otimes \phi^{\vee})^{P}
\end{tikzcd}
\end{equation}

This gives a presentation of $\mathcal{A}v_{\phi,V}$ as a perfect complex, as in <cit.>. To see this, note that $\sigma$ and $\tau$ generate a dense discrete subgroup of $W_{\mathbb{Q}_{p}}/P$ subject to the relationship that $\sigma^{-1}\tau\sigma = \tau^{p}$, and it follows that the limit of the diagram is just $((C_{V} \otimes \phi^{\vee})^{P})^{W_{\mathbb{Q}_{p}}/P}$ and that the homotopy limit is in turn the derived functor $R\Gamma(W_{\mathbb{Q}_{p}},C_{V} \otimes \phi^{\vee}) \simeq R\Gamma(W_{\mathbb{Q}_{p}}/P,(C_{V} \otimes \phi^{\vee})^{P})$, where we have used the vanishing of the higher Galois cohomology with respect to $P$ for the last isomorphism.

Then we have the following Lemma which is a verbatim generalization of <cit.>.

There exists a canonical identification

\[ Av_{V,\phi}(-) \simeq \mathcal{A}v_{V,\phi} \star (-)\]

of endofunctors of $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$

Equation (1) gives a diagram of perfect complexes on $[\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]$. Acting via the spectral action on an object $\mathcal{F} \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ then gives a diagram

\[
\begin{tikzcd}
& (T_{V}(\mathcal{F}) \otimes \phi^{\vee})^{P} \arrow[r,"\tau - 1"] \arrow[d,"\sigma - 1"] & (T_{V}(\mathcal{F}) \otimes \phi^{\vee})^{P} \arrow[d,"\sigma(1 + \tau + \ldots + \tau^{p - 1}) - 1"] \\
& (T_{V}(\mathcal{F}) \otimes \phi^{\vee})^{P} \arrow[r,"\tau - 1"] & (T_{V} \otimes \phi^{\vee})^{P}
\end{tikzcd}\]

However, if we take the homotopy limit of the diagram in $(1)$, the claim follows from the fact that the spectral action commutes with homotopy limits, as noted in Remark 3.7 (3).

With this identification in hand, we can apply Lemma 3.8 to prove the following key consequence.

Let $A \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ be an ULA Schur-irreducible object with Fargues-Scholze parameter $\phi_{A}^{\mathrm{FS}}$, $V$ a representation of $\phantom{}^{L}G$, and $\phi$ an irreducible representation of $W_{\mathbb{Q}_{p}}$. If the cohomology sheaves of $T_{V}(A) \in \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{BW_{\mathbb{Q}_{p}}}$ with respect to the standard $t$-structure on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{BW_{\mathbb{Q}_{p}}}$  have a non-zero sub-quotient as $W_{\mathbb{Q}_{p}}$-modules with $W_{\mathbb{Q}_{p}}$-action given by $\phi$ or $\phi(1)$  then $r_{V} \circ \phi_{A}^{\mathrm{FS}}$ also has such a sub-quotient.

During the creation of this manuscript, a similar result was obtained by Koshikawa through a similar but simpler proof <cit.>. As we will similarly conclude in Section 3.3, he shows that if the cohomology $R\Gamma_{c}(G,b,\mu)[\rho](\frac{d}{2})$ admits a sub-quotient with $W_{E}$-action given by an irreducible representation $\phi$ then the Fargues-Scholze parameter $\phi_{\rho}^{\mathrm{FS}}$ admits a sub-quotient given by $\phi^{\vee}$. However, in his paper, the Shtuka space $\Sht(G,b,\mu)_{\infty}$ parametrizes modifications of the form $\mathcal{E}_{0} \dashrightarrow \mathcal{E}_{b}$ of type $\mu$, whereas for us it parametrizes modifications of type $\mu^{-1}$. This explains why, under our conventions, there is no dual appearing.

We first show that the assumption on the cohomology sheaves of $T_{V}(A)$ implies that averaging operator

\[ Av_{\phi,V}(A) = R\Gamma(W_{\mathbb{Q}_{p}},T_{V}(A) \otimes \phi^{\vee})\]

is non-trivial. Since $A$ is compact we know by Theorem 3.2 that $T_{V}(A)$ is also comapct and therefore supported on a finite number of HN-strata. By applying excision to the HN-strata of $\Bun_{G}$ <cit.>, we can assume, without loss of generality, that $T_{V}(A)$ is supported on a single stalk, where it is given by a complex of smooth irreducible representations of $J_{b}(\mathbb{Q}_{p})$ for $b \in B(G)$. Moreover, since $A$ is ULA, by Theorem 3.2 the sheaf $T_{V}(A)$ is also ULA and therefore it follows that, for all open compact $K \subset J_{b}(\mathbb{Q}_{p})$, $T_{V}(A)^{K}$ is a perfect complex of $\ol{\mathbb{Q}}_{\ell}$-vector spaces. Writing $T_{V}(A) := \colim_{K \ra \{1\}} T_{V}(A)^{K}$ and using that cohomology commutes with colimits, this allows us to apply results from the Galois cohomology of $W_{\mathbb{Q}_{p}}$ on finite-dimensional $\ol{\mathbb{Q}}_{\ell}$ vector spaces to $T_{V}(A)$. To do this, we consider the spectral sequence

\[ E_{2}^{p,q} = H^{p}(W_{\mathbb{Q}_{p}},H^{q}(T_{V}(A) \otimes \phi^{\vee})) \Longrightarrow H^{p + q}( R\Gamma(W_{\mathbb{Q}_{p}},T_{V}(A) \otimes \phi^{\vee})) \]

where cohomology is being taken with respect to the standard $t$-structure on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$. Now recall that the cohomological dimension of $W_{\mathbb{Q}_{p}}$ acting on finite dimensional $\overline{\mathbb{Q}}_{\ell}$-vector spaces is $2$. Therefore, this sequence degenerates at the $E_{3}$ page. Moreover, the only non-zero degeneracy maps are given by

\[ E_{2}^{0, q + 1} = H^{0}(W_{\mathbb{Q}_{p}},H^{q + 1}(T_{V}(A) \otimes \phi^{\vee})) \rightarrow  E_{2}^{2,q} = H^{2}(W_{\mathbb{Q}_{p}},H^{q}(T_{V}(A) \otimes \phi^{\vee})) \]

However, using local Tate-duality on the RHS, we can rewrite this differential as a map:

\[ (H^{q + 1}(T_{V}(A)) \otimes \phi^{\vee})^{W_{\mathbb{Q}_{p}}} \rightarrow H^{0}(W_{\mathbb{Q}_{p}},H^{q}(T_{V}(A) \otimes \phi^{\vee})^{\vee}(1))^{\vee} \simeq ((H^{q}(T_{V}(A))^{\vee} \otimes \phi(1))^{W_{\mathbb{Q}_{p}}})^{\vee}   \]

Now the term on the RHS (resp. LHS) will only be non-zero if $\phi(1)$ (resp. $\phi$) occurs as a sub-quotient of $H^{q}(T_{V}(A))$ (resp. $H^{q + 1 }(T_{V}(A))$). By assumption, this will be true for some value of $q$, but now, since the Euler-Poincaré characteristic of $W_{\mathbb{Q}_{p}}$ acting on $\overline{\mathbb{Q}}_{\ell}$-vector spaces is $0$, one of these values being non-zero implies the $H^{1}(W_{\mathbb{Q}_{p}},-)$ of some cohomology sheaf of $T_{V}$ must also be non-zero. This will then give rise to a non-zero contribution to the cohomology of $R\Gamma(W_{\mathbb{Q}_{p}},T_{V}(A) \otimes \phi^{\vee})$ so the averaging operator $Av_{V,\phi}(A)$ applied to $A$ is non-zero. Lemma 3.9 therefore tells us that the spectral action of the perfect complex $\mathcal{A}v_{V,\phi}$ on $A$ is non-trivial. If $x$ denotes the closed $\overline{\mathbb{Q}}_{\ell}$-point in the coarse moduli space of Langlands parameters defined by $\phi_{A}^{\mathrm{FS}}$ with preimage $\pi^{-1}(x)$ in the stack of Langlands parameters, Lemma 3.8 tells us that $\mathcal{A}v_{V,\phi}$ must have non-zero support on $\pi^{-1}(x)$. The $\overline{\mathbb{Q}}_{\ell}$-points of $\pi^{-1}(x)$ correspond to the set of Langlands parameters whose semisimplification is precisely $\phi_{A}^{\mathrm{FS}}$. The previous analysis tells us that the evaluation of the perfect complex $R\Gamma(W_{\mathbb{Q}_{p}},C_{V} \otimes \phi^{\vee})$ at some such point, corresponding to an $L$-parameter $\tilde{\phi}: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$, must be non-zero. This evaluation is precisely the complex

\[ R\Gamma(W_{\mathbb{Q}_{p}}, r_{V} \circ \tilde{\phi} \otimes \phi^{\vee}) \]

However, by again applying local Tate-duality, this can only be the case if $r_{V} \circ \tilde{\phi}$ has a sub-quotient isomorphic to $\phi$ or $\phi(1)$. Since $\phi$ is irreducible, this can only happen if $r_{V} \circ \tilde{\phi}^{\mathrm{ss}} = r_{V} \circ \phi_{A}^{\mathrm{FS}}$ has this property.

We conclude this section by reviewing how the spectral action behaves on objects with supercuspidal Fargues-Scholze parameter along the lines of <cit.>.  These results will be used to deduce a strong form of the Kottwitz conjecture from compatibility. We recall that a supercuspidal parameter, viewed as a continuous $1$-cocyle with respect to the $W_{\mathbb{Q}_{p}}$-action on $\hat{G}(\overline{\mathbb{Q}}_{\ell})$, denoted $\phi: W_{\mathbb{Q}_{p}} \rightarrow \hat{G}(\overline{\mathbb{Q}}_{\ell})$, satisfies the property that it doesn't factor through any $\hat{P}(\overline{\mathbb{Q}}_{\ell})$ for $P$ a parabolic subgroup of $G$. Equivalently, this is the same as insisting that, if $S_{\phi} := Z_{\hat{G}}(\mathrm{Im}(\phi))$ as before, then the quotient $S_{\phi}/Z(\hat{G})^{\Gamma}$ is finite, where $\Gamma := \Gal(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$. From this, it follows by deformation theory (See the proof of <cit.> for more details) that the unramified twists of the parameter $\phi$ define a connected component

\[ C_{\phi} \hookrightarrow [\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]\]

giving rise to a direct summand

\[ \Perf(C_{\phi}) \hookrightarrow \Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]) \]

Therefore, the spectral action gives rise to a corresponding direct summand

\[ \Dlis^{C_{\phi}}(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega} \subset  \Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega} \]

such that the Schur-irreducible objects in this subcategory all have Fargues-Scholze parameter given by an unramified twist of $\phi$. Now, it follows by Proposition 3.14 in the next section and Theorem 3.6 (4), that, since $\phi$ is supercuspidal, the restriction of any object in $\D^{C_{\phi}}_{lis}(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ to any non-basic HN-strata of $\Bun_{G}$ must be zero. Therefore, one obtains a decomposition

\[ \Dlis^{C_{\phi}}(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega} \simeq \bigoplus_{b \in B(G)_{basic}} \D^{C_{\phi}}(J_{b}(E),\overline{\mathbb{Q}}_{\ell})^{\omega}\]

where $J_{b}$ is the $\sigma$-centralizer of $b$ and  $\D^{C_{\phi}}(J_{b}(E),\overline{\mathbb{Q}}_{\ell})^{\omega} \subset \D(J_{b}(E),\overline{\mathbb{Q}}_{\ell})^{\omega}$ is a full subcategory of the derived category of compact objects in smooth representations of $J_{b}(E)$. It also follows again by Theorem 3.6 (4) that the Schur-irreducible objects of any $\D^{C_{\phi}}(J_{b}(E),\overline{\mathbb{Q}}_{\ell})^{\omega}$ must lie only in the supercuspidal components of the Bernstein center. Now to further analyze this we fix a character $\chi$ of $Z(G)(\mathbb{Q}_{p})$ and consider the subcategory

\[ \D_{\mathrm{lis},\chi}^{C_{\phi}}(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega} \simeq \bigoplus_{b \in B(G)_{basic}} \D_{\chi}^{C_{\phi}}(J_{b}(E),\overline{\mathbb{Q}}_{\ell})^{\omega} \]

where $\D_{\chi}^{C_{\phi}}(J_{b}(E),\overline{\mathbb{Q}}_{\ell})^{\omega}$ is the derived subcategory of $\D^{C_{\phi}}(J_{b}(E),\overline{\mathbb{Q}}_{\ell})^{\omega}$ generated by compact objects with fixed central character $\chi$, via the natural isomorphism $Z(J_{b})(\mathbb{Q}_{p}) \simeq Z(G)(\mathbb{Q}_{p})$ (where we recall that $J_{b}$ is an extended pure inner form of $G$). One can see that the spectral action of $\Perf(C_{\phi})$ preserves this subcategory. Indeed, it follows by <cit.> that $\Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])$ and in turn $\Perf(C_{\phi})$ is generated under cones and retracts by the image of $\Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G)$ in $\Perf([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])$. This reduces us to checking that Hecke operators preserve this subcategory, which, in turn reduces to the observation that, if one looks at the simultaneous action of $J_{b}(\mathbb{Q}_{p}) \times J_{b'}(\mathbb{Q}_{p})$ on the space parametrizing modifications $\mathcal{E}_{b} \rightarrow \mathcal{E}_{b'}$, for $b$ and $b'$ in $B(G)_{basic}$ that, under the canonical identification $Z(J_{b'})(\mathbb{Q}_{p}) \simeq Z(J_{b})(\mathbb{Q}_{p})$, the diagonally embedded center acts trivially. This follows since an element in the center of $J_{b}(\mathbb{Q}_{p})$ acts on the modification by the inverse of an element in the corresponding center of $J_{b'}(\mathbb{Q}_{p})$, where the inverse appears from the fact that $J_{b}(\mathbb{Q}_{p})$ is acting on the left and $J_{b'}(\mathbb{Q}_{p})$ is acting on the right.

Now, via local class field theory, we take $\chi$ to be the central character determined by $\phi$ and local class field theory (as in <cit.>). Since all Schur-irreducible objects in $\D_{\chi}^{C_{\phi}}(J_{b}(E),\overline{\mathbb{Q}}_{\ell})$ lie in the supercuspidal component of the Bernstein-center and have fixed central character $\chi$ and, by <cit.>, supercuspidal representations are injective/projective in the category of smooth representations with fixed central character, we can write $\D_{\chi}^{C_{\phi}}(J_{b}(E),\overline{\mathbb{Q}}_{\ell})^{\omega} = \bigoplus_{\pi} \Perf(\overline{\mathbb{Q}}_{\ell}) \otimes \pi$, where $\pi$ runs over all supercuspidal representations of $J_{b}(E)$ with central character $\chi$ which, a priori, have Fargues-Scholze parameter given by an unramified twist of $\phi$, but, by Theorem 3.6 (2), must indeed be equal to $\phi$.

Now, the closed point of $C_{\phi}$ determined by the parameter $\phi$ gives rise to a closed embedding

\[ [\overline{\mathbb{Q}}_{\ell}/S_{\phi}] \hookrightarrow C_{\phi} \]

and in turn a fully faithful embedding

\[ \Perf([\overline{\mathbb{Q}}_{\ell}/S_{\phi}]) \hookrightarrow \Perf(C_{\phi}) \]

The above discussion and Lemma 3.8 imply that the action of $\Perf(C_{\phi})$ factors over this subcategory in the sense that everything not in the image of this must act by zero. All in all, we conclude that we have a decomposition

\[ \D_{\mathrm{lis},\chi}^{C_{\phi}}(\Bun_{G},\overline{\mathbb{Q}}_{\ell})^{\omega} = \bigoplus_{b \in B(G)_{basic}} \bigoplus_{\pi_{b}} \Perf(\overline{\mathbb{Q}}_{\ell}) \otimes \pi_{b} \]

where the $\pi_{b}$ runs over all supercuspidal representations of $J_{b}(E)$ with Fargues-Scholze parameter $\phi_{\pi_{b}}^{\mathrm{FS}} = \phi$. Moreover, the RHS carries an action of $\Rep_{\overline{\mathbb{Q}}_{\ell}}(S_{\phi})$, the category of finite-dimensional $\overline{\mathbb{Q}}_{\ell}$-representations of $S_{\phi}$. Therefore, given $W \in \Rep_{\overline{\mathbb{Q}}_{\ell}}(S_{\phi})$, we get an object:

\[ \Act_{W}(\pi_{b}) \in \bigoplus_{b \in B(G)_{basic}} \bigoplus_{\pi_{b}} \Perf(\overline{\mathbb{Q}}_{\ell}) \otimes \pi_{b} \]

Assume that $W|_{Z(\hat{G})^{\Gamma}}$ is isotypic, given by some character $\eta: Z(\hat{G})^{\Gamma} \rightarrow \overline{\mathbb{Q}}_{\ell}^{*}$. As $Z(\hat{G})^{\Gamma}$ is diagonalizable with characters given by $B(G)_{basic} \xrightarrow{\simeq} \pi_{1}(G)_{\Gamma}$ via the $\kappa$ map, we obtain an element $b_{\eta} \in B(G)_{basic}$. Then $\Act_{W}(\pi_{b})$ is concentrated on the basic HN-strata given by $b' = b + b_{\eta}$. Therefore, we get an isomorphism

\[ \Act_{W}(\pi_{b}) \simeq \bigoplus_{\pi_{b'}} V_{\pi_{b'}} \otimes \pi_{b'} \]

where $V_{\pi_{b'}} \in \Perf(\overline{\mathbb{Q}}_{\ell})$ and $\pi_{b'}$ runs over all supercuspidals of $J_{b'}$ with Fargues-Scholze parameter  $\phi^{\mathrm{FS}}_{\pi_{b'}} = \phi$. With this in hand, we can elucidate the $W_{\mathbb{Q}_{p}}$-action on the Hecke operator applied to a smooth irreducible object with supercuspidal Fargues-Scholze parameter, similar to what was done in Lemma 3.10 in the general case. Namely, given $V \in \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G)$, we obtain a vector bundle on $[\overline{\mathbb{Q}}_{\ell}/S_{\phi}]$ with $W_{\mathbb{Q}_{p}}$-action given by $\phi$. In other words, we have a functor:

\[ \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G) \rightarrow \Rep_{\overline{\mathbb{Q}}_{\ell}}(S_{\phi})^{BW_{\mathbb{Q}_{p}}} \]

Theorem 3.7 and the above discussion imply that the action of the image of $V$ in $\Rep_{\overline{\mathbb{Q}}_{\ell}}(S_{\phi})^{BW_{\mathbb{Q}_{p}}}$ acting via the spectral action on $\D_{\mathrm{lis},\chi}^{C_{\phi}}(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ is precisely the Hecke operator $T_{V}$. This tells us that, if we decompose $r_{V} \circ \phi$ viewed as a representation of $S_{\phi}$ as a direct sum $\bigoplus_{i \in I} W_{i} \otimes \sigma_{i}$ where $W_{i} \in Rep(S_{\phi})$ is irreducible and $\sigma_{i}$ is a continuous finite-dimensional representation of $W_{\mathbb{Q}_{p}}$, then we have an isomorphism

\[ T_{V}(\pi) \simeq \bigoplus_{i \in I} \Act_{W_{i}}(\pi) \otimes \sigma_{i} \]

as $J_{b'}(\mathbb{Q}_{p}) \times W_{\mathbb{Q}_{p}}$-modules.
We now summarize the above discussion as a corollary for future use.

Let $\phi$ be a supercuspidal parameter of $G$, $b \in B(G)_{basic}$ a basic elment, $V \in \Rep_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G)$ an irreducible representation of some highest weight $\mu$ with dominant inverse $\mu^{-1}$, and $\pi_{b}$ a representation of $J_{b}(E)$ with Fargues-Scholze parameter equal to $\phi$. We set $b_{\mu} \in B(G,\mu^{-1})$ to be the unique basic element and $b' = b + b_{\mu}$.  If we decompose $r_{V} \circ \phi$ viewed as representation of $S_{\phi}$ as a direct sum $\bigoplus_{i \in I} W_{i} \otimes \sigma_{i}$, where $W_{i} \in Rep(S_{\phi})$ is irreducible and $\sigma_{i}$ is a continuous finite-dimensional representation of $W_{\mathbb{Q}_{p}}$, then there exists an isomorphism of $W_{\mathbb{Q}_{p}} \times J_{b'}(\mathbb{Q}_{p})$-modules

\[ T_{\mu}(\pi_{b}) \simeq \bigoplus_{i \in I} \Act_{W_{i}}(\pi) \otimes \sigma_{i} \]

where $\Act_{W_{i}}(\pi) \simeq \bigoplus_{\pi_{b'}} V_{\pi_{b'}} \otimes \pi_{b'}$ with $V_{\pi_{b'}} \in \Perf(\overline{\mathbb{Q}}_{\ell})$ and $\pi_{b'}$ ranging over supercuspidal representation of $J_{b'}(\mathbb{Q}_{p})$ with Fargues-Scholze parameter equal to $\phi$.

As we will start to see in the next section, the work of Hansen <cit.>, Hansen-Kaletha-Weinstein <cit.>, and compatibility of the Fargues-Scholze and Gan-Takeda/Gan-Tantono local Langlands correspondence will allow us to use this Corollary to prove Theorem 1.3. This is suggested already by Corollary 3.3, which shows us that $T_{\mu}(\pi_{b})$ can be computed explicitly using the cohomology of local Shimura varieties.

§.§ Compatibility with the Local Langlands for $\GSp_{4}$ and $\GU_{2}(D)$