With the results of the previous section in place, we can now start making progress towards our goal of proving compatibility. So we again let $G = \mathrm{Res}_{L/\mathbb{Q}_{p}}\GSp_{4}$ and $J = \mathrm{Res}_{L/\mathbb{Q}_{p}}\GU_{2}(D)$, for $L/\mathbb{Q}_{p}$ a finite extension. As mentioned in section $1$, the case where a representation $\pi \in \Pi(G)$ (resp. $\rho \in \Pi(J)$), is a sub-quotient of a parabolic induction easily follows from Theorem 3.6 (3), (4), (5), and compatibility of the (semi-simplified) Gan-Takeda (resp. Gan-Tantono) parameter with parabolic induction. We record this as a corollary now.

Let $\pi \in \Pi(G)$ (resp. $\rho \in \Pi(J)$) be representations occurring as a sub-quotient of a parabolic induction. For such $\pi$ (resp. $\rho$), the Fargues-Scholze and Gan-Takeda (resp. Gan-Tantono) local Langlands correspondences are compatible.

To tackle the remaining cases where the $L$-parameter $\phi$ is mixed supercuspidal or supercuspidal, we note that these are the cases where the $L$-parameter is discrete (i.e the $L$-parameter does not factor through a Levi subgroup). This case is disposable to the results of Hansen-Kaletha-Weinstein <cit.>. We will now let $\mu$ be the Siegel cocharacter of $G$ so that the $\sigma$-centralizer of the  unique basic element $b \in B(G,\mu)$ is $J$. We now state the main result of <cit.> specialized to the case of the Shimura datum $(G,b,\mu)$.
Let $\phi$ be a mixed supercuspidal or supercuspidal parameter and $S_{\phi} := Z_{\hat{G}}(\mathrm{Im}(\phi))$ as before. Let $\Pi_{\phi}(G)$ and $\Pi_{\phi}(J)$ denote the $L$-packets over $\phi$. Set $\pi \in \Pi_{\phi}(G)$ (resp. $\rho \in \Pi_{\phi}(J)$) to be smooth irreducible representations of $G$ (resp. $J$). If $\phi$ is supercuspidal or mixed supercuspidal, we have the following equality in the Grothendieck group $K_{0}(G(\mathbb{Q}_{p}))^{ell}$ of elliptic admissible representations of $G(\mathbb{Q}_{p})$ of finite length

\[ [R\Gamma_{c}^{\flat}(G,b,\mu)[\rho]] = -\sum_{\pi \in \Pi_{\phi}(G)} Hom_{S_{\phi}}(\delta_{\pi,\rho},\mathrm{std}\circ\phi)\pi \]

and the following equality

\[ [R\Gamma_{c}^{\flat}(G,b,\mu)[\pi]] = -\sum_{\rho \in \Pi_{\phi}(J)} Hom_{S_{\phi}}(\delta_{\pi,\rho}^{\vee},(\mathrm{std}\circ\phi)^{\vee})\rho \]

in the Grothendieck group of elliptic admissible $J(\mathbb{Q}_{p})$-representations of finite length, where $\delta_{\pi,\rho}$ is the algebraic representation of $S_{\phi}$ in Definition 2.3. Moreover, if the Fargues-Scholze parameter of $\pi$ (resp. $\rho$) is supercuspidal this is true in the Grothendieck group $K_{0}(G(\mathbb{Q}_{p}))$ (resp. $K_{0}(J_{b}(\mathbb{Q}_{p}))$) of all admissible representations of finite length.

* To deduce the result for the $\pi$-isotypic part, we have implicitly used the two towers isomorphism $\Sht(G,b,\mu)_{\infty} \simeq \Sht(J,\hat{b},\mu^{-1})_{\infty}$, where $\mu^{-1}$ is a dominant inverse to $\mu$ and $\hat{b} = b^{-1} \in B(G,\mu^{-1})$ is the unique basic element <cit.>, and $B(G,\mu^{-1}) \simeq B(J,\mu^{-1})$ under the inner twisting. This inverse explains the appearance of duals in the formula for the $\pi$-isotypic part.

* We see that, via Corollary 3.3, this, in the case that $\phi$ is supercuspidal, should provide us insight into the multiplicity spaces $V_{\pi_{b'}}$ appearing in Corollary 3.11, assuming compatibility of the Fargues-Scholze and Gan-Tantono/Gan-Takeda local Langlands correspondences. Namely, we will see later (Theorem 3.17 and 3.18) that $R\Gamma_{c}(G,b,\mu)[\rho] \simeq R\Gamma_{c}^{\flat}(G,b,\mu)[\rho]$ and is concentrated in middle degree $3$ if $\phi_{\rho}^{\mathrm{FS}}$ is supercuspidal. Assuming compatibility, Corollary 3.11 will therefore tell us that $R\Gamma_{c}(G,b,\mu)[\rho]$ will be a direct sum over representations $\pi \in \Pi_{\phi}(G)$ with $W_{L}$-action given by $\mathrm{std}\circ \phi_{\rho}^{\mathrm{FS}} = \mathrm{std}\circ \phi_{\rho}$ decomposed as a representation of $S_{\phi}$. The summands in the decomposed $S_{\phi}$-representation correspond to the weight spaces appearing in the above description in the Grothendieck group.

We now wish to write out the precise formula for the $\rho$ and $\pi$-isotypic parts, using the refined local Langlands discussed in section $2$.

* ($\phi$ stable)
In this case, the $L$-packet $\Pi_{\phi}(G) = \{\pi\}$ is a singleton so the RHS of the above formula for the $\rho$-isotypic part has one term

\[ -\pi Hom_{S_{\phi}}(\delta_{\pi,\rho},\mathrm{std}\circ\phi_{\rho}) \]

In this case, $S_{\phi} = \mathbb{G}_{m}$ and $\delta_{\pi,\rho}$ is simply the identity representation. Thus, this Hom space gets identified with the characters of $\GL_{4}$, so the formula reduces to

\[ -4\pi \]

* ($\phi$ endoscopic)
In this case, the $L$-packet has size $2$ and, as seen in section $2.1$, $\Pi_{\phi}(G) = \{ \pi^{+}, \pi^{-} \}$, where $\pi^{+}$ (resp. $\pi^{-}$) corresponds to the trivial (resp. non-trivial) character of the component group. $\rho$ can be either of the two representations corresponding to the irreducible representation of $S_{\phi}$ given by $\tau_{i}$ for $i = 1,2$ the projection to the two coordinates of $S_{\phi}$. However, the RHS remains the same regardless of which one it corresponds to. So, without loss of generality, we assume that $\rho = \rho_{1}$. Then the RHS of the above formula for the $\rho$-isotypic part has two terms

\[ \pi^{+}Hom_{S_{\phi}}(\tau_{1},\mathrm{std}\circ\phi_{\rho})\]

\[ \pi^{-}Hom_{S_{\phi}}(\tau_{1} \otimes \tau_{\pi^{-}} \simeq \tau_{2}, \mathrm{std}\circ \phi_{\rho}) \]

However, writing $\mathrm{std}\circ \phi_{\rho} \simeq \phi_{1} \oplus \phi_{2}$, these get identified with

\[ -\pi^{+}Hom_{\overline{\mathbb{Q}}_{\ell}^{*}}(\overline{\mathbb{Q}}_{\ell}^{*},\phi_{1})\]

\[ -\pi^{-}Hom_{\overline{\mathbb{Q}}_{\ell}^{*}}(\overline{\mathbb{Q}}_{\ell}^{*}, \phi_{2}) \]

which will both be identified with characters of $\GL_{2}$. Thus, the RHS is equal to

\[ - 2\pi^{+} - 2\pi^{-}\]

Similarly, for the $\pi$-isotypic part, we get that the RHS of the above formula is given by

\[ -4\rho \]

in the stable case and

\[ -2\rho_{1} - 2\rho_{2} \]

in the endoscopic case.

As mentioned in section 1.2, we will now use the previous result to perform a bootstrap to the supercuspidal representations occurring in the $L$-packets $\Pi_{\phi}(G)$ (resp. $\Pi_{\phi}(J)$), for $\phi$ a mixed supercuspidal parameter. For this, we will mention one last result from the Fargues-Scholze local Langlands correspondence.
For $G$ any connected reductive group over $\mathbb{Q}_{p}$, the action of the excursion algebra on $\Dlis(\Bun_{G},\overline{\mathbb{Q}}_{\ell})$ commutes with Hecke operators. Moreover, it is compatible with restriction to the HN-strata $\Bun_{G}^{b}$ for $b \in B(G)$ in the following sense. Given a Schur irreducible object $A \in \Dlis(\Bun_{G},\ol{\mathbb{Q}}_{\ell})$ and $b \in B(G)$, if we let $B$ be a smooth irreducible constituent of $j_{b}^{*}(A)$ then we can view it as a sheaf on the neutral strata of $\Bun_{J_{b}}^{1}$ and let $\phi_{B}^{\mathrm{FS}}$ denote its Fargues-Scholze parameter with respect to the excursion algebra on $\Bun_{J_{b}}$. Then the Fargues-Scholze parameter of $A$ is the composition

\[ W_{\mathbb{Q}_{p}} \xrightarrow{\phi_{B}^{\mathrm{FS}}} \phantom{}^{L}J_{b}(\ol{\mathbb{Q}}_{\ell}) \ra \phantom{}^{L}G(\ol{\mathbb{Q}}_{\ell}) \]

where $\phantom{}^{L}J_{b}(\ol{\mathbb{Q}}_{\ell}) \ra \phantom{}^{L}G(\ol{\mathbb{Q}}_{\ell})$ is the twisted embedding, as defined in <cit.>.

The commutation of Hecke operators and the excursion algebra follows from the interpretation of the excursion algebra in terms of endomorphisms coming from multiplication by the ring of global functions of the stack of $L$-parameters, as discussed in Section 3.2.

From this, we can deduce the following useful corollary.

For $G$ any connected reductive group with $(G,b,\mu)$ a local Shimura datum and $\pi \in \Pi(G)$ and $\rho \in \Pi(J_{b})$ smooth irreducible representations. All smooth irreducible representations occurring in the cohomology of $R\Gamma_{c}(G,b,\mu)[\pi]$ have Fargues-Scholze parameter equal to $\phi_{\pi}^{\mathrm{FS}}$. Similarly, all smooth irreducible representations occurring in the cohomology of $R\Gamma_{c}(G,b,\mu)[\rho]$ have Fargues-Scholze parameter equal to $\phi_{\rho}^{\mathrm{FS}}$. The same is also true for $R\Gamma_{c}^{\flat}(G,b,\mu)[\rho]$ and $R\Gamma_{c}^{\flat}(G,b,\mu)[\pi]$.

The first part follows immediately from Proposition 3.14 and Corollary 3.3. It remains to see the same is true for the complexes $R\Gamma_{c}^{\flat}(G,b,\mu)[\rho]$ and $R\Gamma_{c}^{\flat}(G,b,\mu)[\pi]$. This can be done by writing them as $j_{1}^{*}T_{\mu}Rj_{b*}(\rho)$ and $j_{b}^{*}T_{\mu^{-1}}Rj_{1*}(\pi)$, as in <cit.>, where it again follows from Proposition 3.14 and Corollary 3.3.

We now exploit this corollary to deduce compatibility in the mixed supercuspidal case.

Let $\phi$ be an $L$-parameter of Howe-Piatetski–Schapiro or Saito-Kurokawa type. Then, for any $\pi \in \Pi_{\phi}(G)$ (resp. $\rho \in \Pi_{\phi}(J)$), the Fargues-Scholze and Gan-Takeda (resp. Gan-Tantono) local Langlands correspondences are compatible.

We give the proof for the Gan-Takeda local Langlands correspondence with the proof for the Gan-Tantono correspondence being completely analogous. If $\phi$ is of Saito-Kurokawa type then, as seen in section $2.2$, we can write $\Pi_{\phi}(G) = \{\pi_{sc},\pi_{disc}\}$ and $\Pi_{\phi}(J) = \{\rho_{sc},\rho_{disc}\}$, where $\pi_{sc}$ (resp. $\rho_{sc}$) is a supercuspidal representation of $G$ (resp. $J$) and $\pi_{disc}$ (resp. $\rho_{disc}$) is a non-supercuspidal representaiton. We note that the Gan-Takeda (resp. Gan-Tantono) correspondences are compatible with the Fargues-Scholze correspondence for $\pi_{disc}$ (resp. $\rho_{disc}$), by Corollary 3.12.

If we let $\mu$ be the Siegel cocharacter and $b \in B(G,\mu)$ be the unique basic element. Then the $\sigma$-centralizer $J_{b}$ is isomorphic to  $J$ and we can consider the complex

\[ R\Gamma_{c}^{\flat}(G,b,\mu)[\rho_{disc}] \]

of $J(\mathbb{Q}_{p}) \times W_{L}$-representations. We then let $R\Gamma_{c}^{\flat}(G,b,\mu)[\rho_{disc}]_{sc}$ denote the direct summand of $R\Gamma_{c}^{\flat}(G,b,\mu)[\rho_{disc}]$, where $J(\mathbb{Q}_{p})$ acts via a supercuspidal representation. Theorem 3.13 tells us that we can describe this complex in the Grothendieck group of admissible $G(\mathbb{Q}_{p})$-representations of finite length as

\[ [R\Gamma_{c}^{\flat}(G,b,\mu)[\rho_{disc}]_{sc}] = -2\pi_{sc} \]

which tells us that $\pi_{sc}$, occurs as a non-zero sub-quotient of the complex $R\Gamma_{c}^{\flat}(G,b,\mu)[\rho_{disc}]$. By Corollary 3.15, we know that we have an equality:

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

However, Corollary 3.12 tells us that $\phi_{\rho_{disc}}^{\mathrm{FS}} = \phi_{\rho_{disc}}^{\mathrm{ss}}$, which is equal to $\phi_{\pi_{sc}}^{\mathrm{ss}}$, so we get the desired equality. The analysis in the Howe-Piatetski–Schapiro case is the same, where one can look at the $\rho$-isotypic part for any of the two non-supercuspidals in $\Pi_{\phi}(J)$.

In the remaining part of this section, we address proving compatibility in the case where the parameter $\phi$ is supercuspidal. Before tackling the question of compatibility, we address some geometric properties of the sheaves $\mathcal{F}_{\rho}$ , for $\rho$ with supercuspidal Fargues-Scholze parameter. This will be leveraged in proving the strong form of the Kottwitz Conjecture for the $\rho$ and $\pi$-isotypic parts in section $8$, as mentioned in Remark 3.10 (2).

Now, considering again $G$ a general connected reductive group, $b \in B(G)_{basic}$ a basic element, and a smooth irreducible representation $\rho$ of the $\sigma$-centralizer $J_{b}(\mathbb{Q}_{p})$. We will now address some further consequences of the Fargues-Scholze parameter $\phi^{\mathrm{FS}}_{\rho}$ being supercuspidal. It turns out that the sheaves defined by representations with these parameters have interesting geometric properties, which were leveraged in <cit.> to prove various general results on the cohomology groups.
In particular, Hansen shows the following:
Let $(G,b,\mu)$ be a basic local Shimura datum with $E$ the reflex field of $\mu$ as before, and let $\rho$ be a smooth irreducible representation of $J_{b}(\mathbb{Q}_{p})$. Suppose the following conditions hold:

* The spaces $(\Sht(G,b,\mu)_{K})_{K \subset G(\mathbb{Q}_{p})}$ occur in the basic uniformization at $p$ of a global Shimura variety in the sense of Definition 4.1.

* The Fargues-Scholze parameter $\phi_{\rho}^{\mathrm{FS}}: W_{\mathbb{Q}_{p}} \rightarrow ^{L}G(\overline{\mathbb{Q}}_{\ell})$ is supercuspidal.

Then the complex $R\Gamma_{c}(G,b,\mu)[\rho]$ defined above is concentrated in middle degree $d = \dim(\Sht(G,b,\mu)_{\infty}) = \langle 2\rho_{G},\mu \rangle$.

One of the key ideas in the argument is to exploit the behavior of the sheaf $j_{b!}(\mathcal{F}_{\rho})$ under Verdier duality, where $j_{b}: \Bun_{G}^{b} \hookrightarrow \Bun_{G}$ is the inclusion of the open HN-strata corresponding to $b \in B(G)_{basic}$. In particular, by Proposition 3.14 and Theorem 3.6 (4), one can see that the natural map $j_{b!}(\mathcal{F}_{\rho}) \rightarrow Rj_{b*}(\mathcal{F}_{\rho})$ is an isomorphism. Namely, Proposition 3.14 implies that a non-zero restriction of $Rj_{b*}(\mathcal{F}_{\rho})$ to any non-basic HN-strata must be valued in representations having Fargues-Scholze parameter $\phi_{\rho}^{\mathrm{FS}}$ under the relevant twisted embedding, which is impossible since the $\sigma$-centralizers of non-basic elements are extended pure inner forms of proper Levi subgroups of $G$ and, by assumption, the parameter $\phi_{\rho}^{\mathrm{FS}}$ is supercuspidal. This implies that, if we apply Verdier duality to both sides of
the isomorphism

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

supplied by Corollary 3.3, we see that the LHS is isomorphic to

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

On the other hand, on the RHS we act through Verdier duality on the tower $(\Sht(G,b,\mu)_{K})_{K \subset G(\mathbb{Q}_{p})}$, which are smooth rigid spaces of dimension $d$. So, in particular, the dualizing object is isomorphic to $\overline{\mathbb{Q}}_{\ell}[2d](d)$. This allows one to deduce the following consequence for the cohomology groups $R\Gamma_{c}(G,b,\mu)[\rho]$.
Fix a basic local Shimura datum $(G,b,\mu)$ and let $\rho$ be representation of $J_{b}(\mathbb{Q}_p)$ with supercuspidal Fargues-Scholze parameter. Then there is a natural isomorphism

\[ R\mathcal{H}om(R\Gamma_{c}(G,b,\mu)[\rho],\overline{\mathbb{Q}}_{\ell}) \simeq R\Gamma_{c}(G,b,\mu)[\rho^{*}][2d](d) \]

as $W_{E}$-equivariant objects of $\D(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$, where $\rho^{*}$ is the contragredient of $\rho$. In particular, we have a natural $W_{E}$-equivariant isomorphism of admissible $G(\mathbb{Q}_{p})$-representations for all $0 \leq i \leq 2d$

\[ H^{i}(R\Gamma_{c}(G,b,\mu)[\rho])^{*} \simeq H^{2d - i}(R\Gamma_{c}(G,b,\mu)[\rho^{*}])(d) \]

As noted in Remark 3.4, the LHS of the above formula is isomorphic to $R\Gamma_{c}^{\flat}(G,b,\mu)[\rho^{*}][2d](d)$, so it follows by cancelling the shifts and Tate twists and relaxing contragradients that one has an isomorphism

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

as $J_{b}(\mathbb{Q}_{p}) \times W_{E}$-representations for all such $\rho$.

Now we turn our attention to the question of showing compatibility for supercuspidal parameters assuming Proposition 1.4. So again let $L/\mathbb{Q}_{p}$ be a finite extension and let $G := \mathrm{Res}_{L/\mathbb{Q}_{p}}(\GSp_{4})$ be the restriction of scalars of $\GSp_{4}$ and $J := \mathrm{Res}_{L/\mathbb{Q}_{p}}\GU_{2}(D)$ the unique non-split inner form as before. As we will see in section $8$, it essentially follow from Theorem 3.13 and Corollary 3.15 that showing compatibility for $\rho \in \Pi(J)$ with supercuspidal Gan-Tantono parameter implies the corresponding statement for $\pi \in \Pi(G)$ with supercuspidal Gan-Takeda parameter. So we fix such a $\rho$ and assume that the Gan-Tantono parameter $\phi_{\rho}$ is endoscopic supercuspidal with the stable case being strictly easier. We will write $\mathrm{std}\circ \phi_{\rho} \simeq \phi_{1} \oplus \phi_{2}$ for $\phi_{i}$ distinct irreducible $2$-dimensional representations of $W_{L}$ and let $\mu$ be the Siegel cocharacter. The Shtuka space $\Sht(G,b,\mu)_{\infty}$ in this case will have dimension $d := \langle 2\rho_{G},\mu \rangle = 3$. We will assume for the rest of this section that Proposition 1.4 is true.

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 $\mathrm{std}\circ \phi \otimes |\cdot|^{-3/2}$.

First, we combine this with the following lemma.

Let $\phi$ be a supercuspidal parameter then all representations in the $L$-packet $\Pi_{\phi}(J)$ have the same Fargues-Scholze parameter.

We choose a $\pi \in \Pi_{\phi}(G)$ and then apply Corollary 3.15 to deduce that all representations occurring in the cohomology of $R\Gamma_{c}^{\flat}(G,b,\mu)[\pi]$ have Fargues-Scholze parameter equal to $\phi_{\pi}^{\mathrm{FS}}$. However, by Theorem 3.13, we have that all representations in $\rho \in \Pi_{\phi}(J)$ occur in the cohomology of $R\Gamma_{c}^{\flat}(G,b,\mu)[\pi]$, so their Fargues-Scholze parameters are the same as desired.

With this in hand, we are ready to prove the key consequence of Proposition 1.4 using the results on the spectral action obtained in section 3.2.

Assume that $L/\mathbb{Q}_{p}$ is an unramified extension and that $p > 2$ and that Proposition 1.4 is true. Then, for $\rho$ a smooth irreducible representation of $J(\mathbb{Q}_{p})$ with supercuspidal Gan-Tantono parameter $\phi$, the Gan-Tantono and Fargues-Scholze correspondences coincide.

As mentioned in the introduction, the key will be the isomorphism

\[ \bigoplus_{\rho' \in \Pi_{\phi}(J)} j_{\mathbf{1}}^{*}T_{\mu}j_{b!}(\mathcal{F}_{\rho'}) \simeq \bigoplus_{\rho' \in \Pi_{\phi}(J)} R\Gamma_{c}(G,b,\mu)[\rho'][3](\frac{3}{2}) \]

supplied by Corollary 3.3. Now Proposition 1.4 tells us that one of the summands on the RHS admits a sub-quotient with $W_{L}$-action given by $\phi_{1}$ and one of them admits a sub-quotient with $W_{L}$-action given by $\phi_{2}$. Applying Lemma 3.20 and 3.10 therefore tells us that $\mathrm{std}\circ \phi_{\rho}^{\mathrm{FS}}$ admits a sub-quotient isomorphic to $\phi_{1}$ or $\phi_{1}(1)$ and a sub-quotient isomorphic to $\phi_{2}$ or $\phi_{2}(1)$. This gives four possibilities for what the parameter $\mathrm{std}\circ \phi_{\rho}^{\mathrm{FS}}$ is. Since $\phi_{\rho}^{\mathrm{FS}}$ is a $\GSp_{4}$-valued parameter only two of these are possible; namely, $\phi_{1} \oplus \phi_{2}$ or $(\phi_{1} \oplus \phi_{2})(1)$. However, the second possibility can be ruled out since the similitude character of $\mathrm{std}\circ \phi_{\rho}^{\mathrm{FS}}$ must coincide with the central character of $\rho$ by Theorem 3.6 (2), which agrees with the similitude character of $\mathrm{std}\circ \phi_{\rho} = \phi_{1} \oplus \phi_{2}$. Therefore, we conclude that $\mathrm{std}\circ \phi = \mathrm{std}\circ \phi_{\rho}^{\mathrm{FS}}$ and, by the aforementioned equality of similitude characters of these two parameters and <cit.>, this is enough to conclude that $\phi_{\rho} = \phi_{\rho}^{\mathrm{FS}}$, as conjugacy classes of $\GSp_{4}$-valued parameters.

§ BASIC UNIFORMIZATION

In this section, we will briefly review what basic uniformization of the generic fiber of a global Shimura variety means, following <cit.>. Then we will apply it to our particular case and derive an analogue of Boyer's trick, providing useful consequences for the proof of Proposition 1.4.

§.§ A Review of Basic Uniformization

We now recall briefly what basic uniformization means. Let $\mathbf{G}/\mathbb{Q}$ be a connected reductive group over $\mathbb{Q}$ and let $(\mathbf{G},X)$ be a Shimura datum, with associate conjugacy class of Hodge cocharacters $\mu: \mathbb{G}_{m,\mathbb{C}} \rightarrow \mathbf{G}_{\mathbb{C}}$. Fix a prime $p$, and set $G := \mathbf{G}_{\mathbb{Q}_{p}}$. Using our fixed isomorphism $\mathbb{C} \simeq \overline{\mathbb{Q}}_{p}$, we can and do regard $\mu$ as a conjugacy class of cocharacters $\mu: \mathbb{G}_{m,\overline{\mathbb{Q}}_{p}} \rightarrow G_{\overline{\mathbb{Q}}_{p}}$. This allows us to consider the $\mu$-admissible locus $B(G,\mu)$ in the Kottwitz set of $G$. Let $\mathbb{A}$ (resp. $\mathbb{A}_{f}$) denote the adeles (resp. finite adeles) of $\mathbb{Q}$ and $\mathbb{A}_{f}^{p}$ denote the finite adeles away from $p$. For any compact open subgroup $K \subset \mathbf{G}(\mathbb{A}_{f})$, let $\mathcal{S}(\mathbf{G},X)_{K}$ be the associated rigid analytic Shimura variety over $\mathbb{C}_{p}$ of level $K$. We let $K = K^{p}K_{p}$, where $K^{p} \subset \mathbf{G}(\mathbb{A}_{f}^{p})$ and $K_{p} \subset G(\mathbb{Q}_{p})$ are open compact subgroups. We set

\[ \mathcal{S}(\mathbf{G},X)_{K^{p}} = \lim_{K_{p} \rightarrow \{1\}} \mathcal{S}(\mathbf{G},X)_{K^{p}K_{p}} \]

If $(\mathbf{G},X)$ is of pre-abelian type, this is (up to completing the structure sheaf) representable by a perfectoid space and in general it is a diamond. By the results of <cit.>, there exists a canonical $G(\mathbb{Q}_{p})$-equivariant Hodge-Tate period map

\[ \pi_{HT}: \mathcal{S}(\mathbf{G},X)_{K^{p}} \rightarrow \mathcal{F}\ell_{G,\mu^{-1}} \]

where $\mathcal{F}\ell_{G,\mu^{-1}} := (G_{\mathbb{C}_{p}}/P_{\mu^{-1}})^{ad}$ is the adic space associated to the flag variety defined by the parabolic $P_{\mu^{-1}} \subset G_{\mathbb{C}_{p}}$ given by $\mu^{-1}$ via the dynamical method. By the $G(\mathbb{Q}_{p})$-equivariance, $\pi_{HT}$ descends to a map:

\[ \pi_{HT,K_{p}}: \mathcal{S}(\mathbf{G},X)_{K_{p}K_{p}} \rightarrow [\mathcal{F}\ell_{G,\mu^{-1}}/\underline{K_{p}}]\]

We let $b \in B(G,\mu)$ be the unique basic element, and let $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ be the basic Newton stratum. This parametrizes, for $S$ a perfectoid space in characterstic $p$, modifications $\mathcal{E}_{0} \dashrightarrow \mathcal{E}$ of type $\mu^{-1}$ between the trivial $G$-bundle $\mathcal{E}_{0}$ on the relative Fargues-Fontaine curve $X_{S}$ and $\mathcal{E}$ a bundle isomorphic to the $G$-bundle $\mathcal{E}_{b}$ corresponding to $b \in B(G)$ after pulling back to a geometric point of $S$. $(G,b,\mu)$ defines a local Shimura datum, as in section $3.1$, so we may consider the infinite level Shimura variety/Shtuka space $\Sht(G,b,\mu)_{\infty}$ and its base change $\Sht(G,b,\mu)_{\infty,\mathbb{C}_{p}}$. By pulling back along $\pi_{HT}$, we get an open subspace $\mathcal{S}(\mathbf{G},X)^{b}_{K^{p}} \subset \mathcal{S}(\mathbf{G},X)_{K^{p}}$, which descends to an open subspace $\mathcal{S}(\mathbf{G},X)^{b}_{K}$, for $K \subset \mathbf{G}(\mathbb{A}_{f})$ an open compact. We now have the key definition.

We say a global Shimura datum $(\mathbf{G},X)$ satisfies basic uniformization at $p$ if there exists (in fact this is always true see <cit.>) a unique up to isomorphism $\mathbb{Q}$-inner form $\mathbf{G}'$ of $\mathbf{G}$ satisfying

* $\mathbf{G}'_{\mathbb{A}^{p}_{f}} \simeq \mathbf{G}_{\mathbb{A}^{p}_{f}}$ as algebraic groups over $\mathbb{A}^{p}_{f}$,

* $\mathbf{G}'_{\mathbb{Q}_{p}} \simeq J_b$, where $J_{b}$ is the inner form of $G$ given by the $\sigma$-centralizer of the basic element $b \in B(G,\mu)$,

* $\mathbf{G}'(\mathbb{R})$ is compact modulo center,

and a $\mathbf{G}(\mathbb{A}_{f})$-equivariant isomorphism of diamonds over $\mathbb{C}_{p}$

\begin{equation}
\lim_{K^{p} \rightarrow \{1\}} \mathcal{S}(\mathbf{G},X)^{b}_{K^{p}} \simeq (\underline{\mathbf{G}'(\mathbb{Q})}\backslash\underline{\mathbf{G}'(\mathbb{A}_{f})} \times_{\Spd(\mathbb{C}_{p})} \Sht(G,b,\mu)_{\infty,\mathbb{C}_{p}})/\underline{J_{b}(\mathbb{Q}_{p})}
\end{equation}

where $J_{b}(\mathbb{Q}_{p})$ acts diagonally, such that, under the identification $\mathcal{F}\ell_{G,\mu^{-1}}^{b} \simeq \Sht(G,b,\mu)_{\infty,\mathbb{C}_{p}}/\underline{J_{b}(\mathbb{Q}_{p})}$, the morphism

\[ \pi_{HT}: \lim_{K^{p} \rightarrow \{1\}} \mathcal{S}(\mathbf{G},X)^{b}_{K^{p}} \rightarrow \mathcal{F}\ell_{G,\mu^{-1}}^{b} \]

identifies with the projection

\[ (\underline{\mathbf{G}'(\mathbb{Q})}\backslash\underline{\mathbf{G}'(\mathbb{A}_{f})} \times_{\Spd(\mathbb{C}_{p})} \Sht(G,b,\mu)_{\infty,\mathbb{C}_{p}})/\underline{J_{b}(\mathbb{Q}_{p})} \rightarrow \Sht(G,b,\mu)_{\infty,\mathbb{C}_{p}}/\underline{J_{b}(\mathbb{Q}_{p})} \]

where $\mathbf{G}(\mathbb{A}_{f}) \simeq \mathbf{G}'(\mathbb{A}_{f}^{p}) \times G(\mathbb{Q}_{p})$ acts on the RHS via the natural action of $\mathbf{G}'(\mathbb{A}_{f}^{p})$ on $\mathbf{G}'(\mathbb{Q})\backslash\mathbf{G}'(\mathbb{A}_{f})$ and $G(\mathbb{Q}_{p})$ acts on $\Sht(G,b,\mu)_{\infty,\mathbb{C}_{p}}$. Moreover, if the reflex field of the cocharacter $\mu: \mathbb{G}_{m,\overline{\mathbb{Q}}_{p}} \rightarrow G_{\overline{\mathbb{Q}}_{p}}$ is $E/\mathbb{Q}_{p}$ then this isomorphism descends to an isomorphism of diamonds over $\breve{E} := E\breve{\mathbb{Q}}_{p}$.

We now mention some consequences of uniformization which will be key to us in what follows. Let $\mathcal{H}(J_{b}) := C^{\infty}_{c}(J_b(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$ be the usual smooth Hecke algebra. We fix an algebraic representation of  $\mathbf{G}/\mathbb{Q}$, denoted $\mathcal{V}_{\xi}$, of some regular highest weight $\xi$. The isomorphism $i: \overline{\mathbb{Q}}_{\ell} \xrightarrow{\simeq} \mathbb{C}$ then determines a $\overline{\mathbb{Q}}_{\ell}$-local system   $\mathcal{L}_{\xi}$ on the Shimura variety $\mathcal{S}(G,X)_{K^{p}}$. We now consider the space of algebraic automorphic forms valued in $\mathcal{V}_{\xi}$:
\[\mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi}) := \colim_{K_{p} \rightarrow \{1\}} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p}K_{p},\mathcal{L}_{\xi})\]

in the sense of Gross <cit.>. Namely, it is the space of all continuous functions $\phi: \mathbf{G}'(\mathbb{A}_{f}) \ra \mathcal{V}_{\xi}(\ol{\mathbb{Q}}_{\ell})$ with respect to the pro-finite topology on the target and the discrete topology on the source such that, for all $k \in K^{p}$, $\gamma \in G(\mathbb{Q})$, and $g \in \mathbf{G}'(\mathbb{A}_{f})$, we have that $\phi(gk) = \phi(g)$ and $\phi(\gamma g) = \gamma\phi(g)$, where $\gamma$ acts via $\mathcal{V}_{\xi}$. The isomorphism (2) then allows us to deduce an isomorphism

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

of $G(\mathbb{Q}_{p}) \times W_{E}$-modules, which when composed with the morphism

\[ R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)^{b}_{K^{p}},\mathcal{L}_{\xi}) \rightarrow R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi}) \]

coming from excision with respect to the open strata $\mathcal{S}(\mathbf{G},X)^{b}_{K^{p}} \hookrightarrow \mathcal{S}(\mathbf{G},X)_{K^{p}}$, gives rise to the uniformization map mentioned in the introduction. We show this now.

Assume that $(\mathbf{G},X)$ satisfies basic uniformization at $p$, then there exists a $G(\mathbb{Q}_{p}) \times W_{E}$-equivariant 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 $K^{p}$.

We fix a level at $K_{p} \subset \mathbf{G}(\mathbb{A}_{f})$ and write $\pi_{HT,K_{p}}: \mathcal{S}(\mathbf{G},X)_{K^{p}K_{p}} \ra [\mathcal{F}\ell_{G,\mu^{-1}}/\underline{K_{p}}]$ for the induced Hodge-Tate period map. Then we have an isomorphism:

\[ R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi}) \simeq \colim_{K_{p} \ra \{1\}}  R\Gamma([\mathcal{F}\ell_{G,\mu^{-1}}/\underline{K_{p}}], R\pi_{HT,K_{p}!}(\mathcal{L}_{\xi})) \]

Write $j_{K_{p}}: [\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{K_{p}}] \hookrightarrow [\mathcal{F}\ell_{G,\mu^{-1}}/\underline{K_{p}}]$ for the open inclusion of the basic locus quotiented out by $K_{p}$.  Now, we claim that, under this identification, the desired map is given by

\[ R\Gamma_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{K_{p}}], j_{K_{p}}^{*}R\pi_{HT,K_{p}!}(\mathcal{L}_{\xi})) \ra   R\Gamma([\mathcal{F}\ell_{G,\mu^{-1}}/\underline{K_{p}}], R\pi_{HT,K_{p}!}(\mathcal{L}_{\xi}))  \]

coming from applying excision with respect to $j_{K_{p}}$ for varying $K_{p}$. Here we stress that, since we are finite level, this is happening in the usual (not solid) category of étale $\ol{\mathbb{Q}}_{\ell}$-sheaves, where excision is well-defined. The claim would now follow from showing a natural identification
\[ R\Gamma_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{K_{p}}], j_{K_{p}}^{*}R\pi_{HT,K_{p}!}(\mathcal{L}_{\xi})) \simeq R\Gamma_{c}(G,b,\mu)_{K_{p}} \otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi}) \]

for all $K_{p}$. There is a natural map

\[ q_{K_{p}}^{b}: [\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\underline{K_{p}}] \ra [\Spd(\mathbb{C}_{p})/\underline{J_{b}(\mathbb{Q}_{p})}] \]

which is cohomologically smooth as in <cit.>. Since we are working with the usual category of étale $\ol{\mathbb{Q}}_{\ell}$-sheaves, the derived category of $\ol{\mathbb{Q}}_{\ell}$-sheaves on the target is not identified with the unbounded derived category of smooth representations of $J_{b}(\mathbb{Q}_{p})$, but rather the unbounded derived category of smooth $\ell$-complete representations (See Remark 3.1). However, by fixing a $K^{p}$ stable lattice in the $\ol{\mathbb{Z}}_{\ell}$-lattice in the $\ol{\mathbb{Q}}_{\ell}$ realization of the algebraic representation $\mathcal{V}_{\xi}$, we can endow $\Pi := \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi})$ with the structure of such a $J_{b}(\mathbb{Q}_{p})$-representation, and we write $\mathcal{F}_{\Pi}$ for this sheaf. By <cit.>, the fact that we know basic uniformization at $p$ implies we have a natural in $K^{p}$ isomorphism:

\[ (q_{K_{p}}^{b*})(\mathcal{F}_{\Pi}) \simeq j_{K_{p}}^{*}R\pi_{HT,K_{p}!}(\mathcal{L}_{\xi}) \]

Applying $R\Gamma_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{b}/K_{p}],-)$, we obtain

\[  R\Gamma_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{b}/K_{p}],(q_{K_{p}}^{b*})(\mathcal{F}_{\Pi})) \simeq  R\Gamma_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{b}/K_{p}],j_{K_{p}}^{*}R\pi_{HT,K_{p}!}(\mathcal{L}_{\xi})) \]

but now, by <cit.>, we have an isomorphism

\[ R\Gamma_{c}(G,b,\mu)_{K_{p}} \otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p},\mathcal{L}_{\xi}) \simeq R\Gamma_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{b}/K_{p}],(q_{K_{p}}^{b*})(\mathcal{F}_{\Pi})) \]

and this gives the desired result.

§.§ Boyer's Trick

We will now be interested in applying uniformization to the situation we are interested in, proving an analogue of Boyer's trick <cit.> and deducing some relevant consequences. The relevant results are due to Shen. <cit.>
If $(\mathbf{G},X)$ is a Shimura datum of abelian type and $p > 2$ is a prime where $G$ is unramified then $(\mathbf{G},X)$ satisfies basic uniformization at $p$.

To see a full proof of this exact statement for the full integral model at hyperspecial level, one can look at the proof of <cit.>.

Now, consider a Shimura datum $(\mathbf{G},X)$, where $\mathbf{G}$ is a $\mathbb{Q}$-inner form of $\mathbf{G}^{*} := \mathrm{Res}_{F/\mathbb{Q}}\GSp_{4}$, with $F/\mathbb{Q}$ a totally real field such that $p$ is totally inert and $F_{p} \simeq L$, a fixed unramified extension of $\mathbb{Q}_{p}$ and assume that $\mathbf{G}_{\mathbb{Q}_{p}} \simeq \mathrm{Res}_{L/\mathbb{Q}_{p}}\GSp_{4} = G$. We fix a level $K = K_{p}K^{p} \subset \mathbf{G}(\mathbb{A}_{f})$ as before, and assume from now on that $(\mathbf{G},X)$ is such that the corresponding cocharacter $\mu$ is the Siegel cocharacter. Therefore, the unique basic $b \in B(G,\mu)$ will have $\sigma$-centralizer given by $\mathrm{Res}_{L/\mathbb{Q}_{p}}\GU_{2}(D)$, with $D/L$ the quaternionic division algebra. Since $L/\mathbb{Q}_{p}$ is unramified and $p > 2$, we can apply Theorem $4.2$ to deduce basic uniformization at $p$. Let $\mathbf{G}'$ be the $\mathbb{Q}$-inner form defined above. Now we prove the following result, which plays a similar role to Boyer's trick <cit.> in the study of the cohomology of the Lubin-Tate/Drinfeld towers.

For $b \in B(G,\mu)$ non-basic, the adic Newton strata $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ is parabolically induced as a space with $G(\mathbb{Q}_{p})$-action from a proper parabolic subgroup $P = LU$ of $G$ and a diamond $\mathcal{S}_{P}$ with $P(\mathbb{Q}_{p})$-action. Moreover, the action of $U(\mathbb{Q}_{p})$ on the $\ell$-adic cohomology of $R\Gamma_{c}(\mathcal{S}_{P},\ol{\mathbb{Q}}_{\ell})$ is trivial.

We recall <cit.> that we say $b \in B(G,\mu)$ is Hodge-Newton reducible if there exists a proper Levi subgroup $L$ together with a basic element $b_{L} \in B(L,\mu_{L})$ mapping to $b \in B(G,\mu)$ under the natural map $B(L) \ra B(G)$, where $\mu_{L}$ is a choice of representative for $\mu$ as a geometric dominant cocharacter of $L$. Given such a $b$, we let $P$ be the standard parabolic of $G$ with respect to a choice of Borel such that its Levi factor is $L$, and fix $\mu_{L}$ to be the conjugacy class of dominant cocharacters of $L$ that is dominant respect to $B$. It now follows by <cit.> that $P(\mathbb{Q}_{p}) \subset G(\mathbb{Q}_{p})$ stabilizes a subdiamond $\mathcal{C}^{\mu}_{b}$ and that we have a $L(\mathbb{Q}_{p})$-equivariant isomorphism:

\[ \mathcal{C}_{b_{L}}^{\mu_{L}} \times^{P(\mathbb{Q}_{p})} G(\mathbb{Q}_{p}) \simeq \mathcal{F}\ell_{G,\mu^{-1}}^{b} \]

We recall that, if $\mathcal{J}_{b}$ denotes the group diamond parametrizing automorphisms of the bundle corresponding to $b \in B(G)$, this has a semi-direct product decomposition

\[ \mathcal{J}_{b} := \ul{J_{b}(\mathbb{Q}_{p})} \ltimes \mathcal{J}_{b}^{U} \]

given by the splitting of the HN-filtration of $\mathcal{E}_{b}$ <cit.>. By <cit.>, the space $\mathcal{C}_{b}^{\mu_{L}}$ is isomorphic to

\[ \mathcal{F}\ell_{L,\mu_{L}^{-1}}^{b_{L}} \times \mathcal{J}_{b}^{U} \]

and, by <cit.>, it follows that the action of $U_{b}(\mathbb{Q}_{p})$ on $R\Gamma_{c}(\mathcal{C}_{b_{L}}^{\mu_{L}},\ol{\mathbb{Q}}_{\ell})$ is trivial. Therefore, the claim will follow from checking that the non-basic elements in $B(G,\mu)$ are Hodge-Newton reducible. This follows immediately from the classification of this condition in <cit.>. For clarity, we recall how this works in our case. There are two non-basic elements $b \in B(G,\mu)$. The $\mu$-ordinary element, defined by the maximal element $b^{max} \in B(G,\mu)$ with respect to the partial ordering on $B(G,\mu)$, and the intermediate strata corresponding to the element lying between the basic element and $b^{max}$ with respect to the partial ordering on $B(G)$. The element $b^{\max}$ admits a reduction to the unique element $b_{T} \in B(T,\mu_{T})$, where $T$ is the maximal torus inside $\GSp_{4}/L$. Similarly, the element $b \in B(G,\mu)$ corresponding to the intermediate strata admits a reduction to the unique basic element $b_{L} \in B(L,\mu_{L})$, where $L$ is the Levi factor of the Klingen parabolic of $\GSp_{4}$.

We now consider some irreducible representation $\xi$ as above and look at the uniformization map

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

furnished by Corollary 4.1 and Theorem 4.2. We let $R\Gamma_{c}(G,b,\mu)_{sc}$ and $R\Gamma_{c}(\mathcal{S}(\mathbf{G},X)_{K^{p}},\mathcal{L}_{\xi})_{sc}$ be the direct summands where $G(\mathbb{Q}_{p})$ acts via a supercuspidal representation. Then we have the following key consequence of the previous lemma, which justifies why we are referring to this as Boyer's trick.

The uniformization map $\Theta$ induces an isomorphism

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

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

We let $\mathcal{F}\ell_{G,\mu^{-1}}^{nbas}$ denote the closed complement of $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ for $b \in B(G,\mu)$ the unique basic element. The space $\mathcal{F}\ell_{G,\mu^{-1}}^{nbas}$ is stratified by $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ for $b \in B(G,\mu)$ non-basic. For varying $K_{p} \subset G(\mathbb{Q}_{p})$, we set $[\mathcal{F}\ell_{G,\mu^{-1}}^{nbas}/\ul{K_{p}}]$ to be the $v$-stack quotient. We let $\mathcal{S}(\mathbf{G},X)_{K^{p}K_{p}}^{nbas}$ denote the preimage of $[\mathcal{F}\ell_{G,\mu^{-1}}^{nbas}/\ul{K_{p}}]$ under the map $\pi_{HT,K_{p}}: \mathcal{S}(\mathbf{G},X)_{K^{p}K_{p}} \ra [\mathcal{F}\ell_{G,\mu^{-1}}/\ul{K_{p}}]$. It follows from the proof of Corollary 4.1 that the cone of $\Theta$ is identified with

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

We want to show that the cohomology is parabolically induced as a $G(\mathbb{Q}_{p})$ representation from proper Levi subgroups of $G$. For this, we consider the Hodge-Tate period morphism:

\[\pi_{HT,K_{p}}:\mathcal{S}(\mathbf{G},X)^{nbas}_{K_{p}K^{p}} \rightarrow [\mathcal{F}\ell_{G,\mu^{-1}}^{nbas}/\ul{K_{p}}] \]

The Cartan-Leray spectral sequence then gives us

\[ E_{2}^{p,q} = H^{p}_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{nbas}/K_{p}],R^{q}\pi_{HT,K_{p}!}(\mathcal{L}_{\xi})) \Longrightarrow H_{c}^{p + q}(\mathcal{S}(\mathbf{G},X)_{K_{p}K^{p}}^{nbas}, \mathcal{L}_{\xi}). \]

Using the $G(\mathbb{Q}_{p})$-equivariance of the Hodge-Tate period map, the colimit over $K_{p} \ra \{1\}$ gives rise to a spectral sequence

\[ \colim_{K_{p} \ra \{1\}} H^{p}_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{nbas}/K_{p}],R^{q}\pi_{HT,K_{p}!}(\mathcal{L}_{\xi})) \Longrightarrow H_{c}^{p + q}(\mathcal{S}(\mathbf{G},X)_{K_{p}}^{nbas}, \mathcal{L}_{\xi}) \]

with $G(\mathbb{Q}_{p})$-equivariant maps. Therefore, we are reduced to showing the following.

For all integers $p,q \geq 0$ the cohomology of $\colim_{K_{p} \ra \{1\}} H^{p}_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{nbas}/\ul{K_{p}}],R^{q}\pi_{HT,K_{p}!}(\mathcal{L}_{\xi}))$ is parabolically induced as a $G(\mathbb{Q}_{p})$-representation from a proper Levi subgroup of $G$.

We apply excision with respect to the locally closed stratification given by $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ for $b \in B(G,\mu)$ which is not basic. We write $j_{b,K_{p}}: [\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\ul{K_{p}}] \hookrightarrow [\mathcal{F}\ell_{G,\mu^{-1}}/\ul{K_{p}}]$ for the associated locally closed immersion at level $K_{p}$. Since the Newton strata $\mathcal{F}\ell_{G,\mu^{-1}}^{b}$ are stable under the $G(\mathbb{Q}_{p})$ action, this reduces us to showing that the cohomology of

\[ \colim_{K_{p} \ra \{1\}} H^{p}_{c}([\mathcal{F}\ell_{G,\mu^{-1}}^{b}/\ul{K_{p}}],j_{b,K_{p}}^{*}R^{q}\pi_{HT,K_{p}!}(\mathcal{L}_{\xi})) \]

is parabolically induced as a $G(\mathbb{Q}_{p})$-representation. However, this now follows from Lemma 4.3, where one uses the second part of the claim to see the unipotent radical acts trivally.

This result will be the key tool in allowing us to describe the $W_{L}$-action on the local Shimura variety by global methods. We start this global analysis by constructing strong transfers between $\GSp_{4}$ and its inner forms over a number field $F$ and proving a strong multiplicity one result.

§ EXISTENCE OF STRONG TRANSFERS AND A STRONG MULTIPLICITY ONE RESULT

In this section, we will show the existence of strong transfers of certain automorphic representations of an inner form of $\GSp_{4}$ over a number field $F$, using the analysis of the trace formula similar to that of <cit.>. We will then combine this with analysis of the simple twisted trace formula of Kottwitz-Shelstad <cit.>, to deduce a kind of strong multiplicity one result for inner forms of $\GSp_{4}$.

§.§ The Simple Trace Formula and Existence of Strong Transfers

In order to describe the Galois action on the global Shimura variety, we will need to construct strong transfers for inner forms of $\GSp_{4}/F$ over a totally real field $F$. This will allow us to compute the traces of Frobenius on the global Shimura variety in terms of the Langlands parameters of the strong transfer. The construction of strong transfers will be accomplished by applying the elliptic part of the stable trace formula with respect to the Lefschetz functions constructed by Kret-Shin <cit.> at the Steinberg/Infinite places and pseudo-coefficients at some finite number places where the representation has supercuspidal $L$-parameter, applying the character identities of Chan-Gan <cit.> to conclude equality of the orbital integrals at these latter places. First, we recall the key results of Kret-Shin on the trace formula with fixed central character. For now, we will work generally. Let $\mathbf{G}$ denote a connected reductive group over a number field $F$ with center $Z$, write $A_{Z}$ for the maximal $\mathbb{Q}$-split torus of $\mathrm{Res}_{F/\mathbb{Q}}Z$, and set $A_{Z,\infty} = A_{Z}(\mathbb{R})^{0}$ to be the connected component of the identity. Let $\mathbb{A}_{F}$ be the adeles of $F$ and write $\mathbf{G}(\mathbb{A}_{F})^{1}$ for a choice of subgroup so that $\mathbf{G}(\mathbb{A}_{F}) = \mathbf{G}(\mathbb{A}_{F})^{1} \times A_{Z,\infty}$, as in <cit.>. We consider a closed subgroup $\mathfrak{X} \subset Z(\mathbb{A}_{F})$ which contains $A_{Z,\infty}$ such that $Z(F)\mathfrak{X}$ is closed inside $Z(\mathbb{A}_{F})$ and a continuous character $\chi: (\mathfrak{X} \cap Z(F))\backslash\mathfrak{X} \rightarrow \mathbb{C}^{*}$.

We write $\Pl_{F}$ for the set of places of $F$, and, for $v \in \Pl_{F}$, let $\mathfrak{X}_{v} \subset Z(F_{v})$ denote a closed subgroup. We let $\chi_{v}: \mathfrak{X}_{v} \rightarrow \mathbb{C}^{*}$ be a smooth character. Write $\mathcal{H}(\mathbf{G}(F_{v}),\chi_{v}^{-1})$ for the space of smooth compactly supported functions modulo center on $\mathbf{G}(F_{v})$ which transform under $\mathfrak{X}_{v}$ via $\chi_{v}^{-1}$. We also require the functions to be $K_{v}$-finite for some maximal compact subgroup $K_{v}$ of $\mathbf{G}(F_{v})$ if $v$ is archimedean. We now take $\mathfrak{X}_{\nu} := \mathfrak{X}(F_{v})$, where $\mathfrak{X}$ is as above. Given a semisimple element $\gamma_{v} \in \mathbf{G}(F_{v})$ and an admissible representation $\pi_{v}$ of $\mathbf{G}(F_{v})$ with central character $\chi_{v}$ on $\mathfrak{X}_{\nu}$, we define the orbital integral and trace character for $f_{v} \in \mathcal{H}(\mathbf{G}(F_{v}),\chi_{v}^{-1})$ as follows. Let $I_{\gamma_{v}}$ denote the connected centralizer of $\gamma_{v}$ in $\mathbf{G}$. We have

\[ O_{\gamma_{v}}(f_{v}) := \int_{I_{\gamma_{v}}(F_{v})\backslash \mathbf{G}(F_{v})} f_{v}(x^{-1}\gamma_{v}x)dx \]

\[ \tr(f_{v}|\pi_{v}) := \tr(\int_{\mathbf{G}(F_{v})/Z(F_{v})} f_{v}(g)\pi_{v}(g)dg) \]

where we have fixed compatible choices of Haar measure on $\mathbf{G}$ and $Z$ throughout. We note that this operator is well defined since the above operator is of finite rank if $v$ is finite and is of trace class if $v$ is infinite, by the $K_{v}$-finiteness assumption.

We define the adelic Hecke algebra $\mathcal{H}(\mathbf{G}(\mathbb{A}_{F}),\chi^{-1})$, as well as the global orbital integrals by taking a restricted tensor product over the local Hecke algebras defined above and products of the local integrals. Write $\Gamma_{ell}(\mathbf{G})$ to be the set of $F$-elliptic conjugacy classes in $\mathbf{G}(F)$. Let $\mathbb{A}_{f,F}$ denote the finite adeles of $F$. For our purposes, it will suffice to consider a central character datum $(\mathfrak{X},\chi)$, where $\mathfrak{X} = Z(\mathbb{A}_{F})$, and we write $\chi = \bigotimes_{v \in \Pl_{F}} \chi_{v}$ for smooth characters $\chi_{v}: Z(F_{v}) \ra \mathbb{C}^{*}$ and all $v \in \Pl_{F}$. We let $L^{2}_{disc,\chi}(\mathbf{G}(F)\backslash \mathbf{G}(\mathbb{A}_{F}))$ denote the space of functions on $\mathbf{G}(F) \backslash \mathbf{G}(\mathbb{A}_{F})$ transforming under $\mathfrak{X}$ by $\chi$ and square-integrable on $\mathbf{G}(F)\backslash \mathbf{G}(\mathbb{A}_{F})^{1}/\mathfrak{X}(\mathbb{A}_{F}) \cap G(\mathbb{A}_{F})^{1}$. Write $\mathcal{A}_{cusp,\chi}(\mathbf{G})$ for the set of isomorphism classes of cuspidal automorphic representations of $\mathbf{G}(\mathbb{A}_{F})$ whose central characters restricted to $\mathfrak{X}$ are $\chi$. For $f \in \mathcal{H}(\mathbf{G}(\mathbb{A}_{F}),\chi^{-1})$, define the invariant distributions $T_{ell,\chi}^{\mathbf{G}}$ and $T_{disc,\chi}^{\mathbf{G}}$ by

\[ T_{ell,\chi}^{\mathbf{G}}(f) := \sum_{\gamma \in \Gamma_{ell}(\mathbf{G})} i(\gamma)^{-1}vol(I_{\gamma}(F)\backslash I_{\gamma}(\mathbb{A}_{F})/\mathfrak{X}(\mathbb{A}_{F}))O_{\gamma}(f) \]

\[ T^{\mathbf{G}}_{disc,\chi}(f) := \tr(f|L^{2}_{disc,\chi}(\mathbf{G}(F)\backslash \mathbf{G}(\mathbb{A}_{F}))) \]

where $i(\gamma)$ is the number of connected components in the centralizer of $\gamma$.
Analogously, we define $T^{\mathbf{G}}_{cusp,\chi}(f)$ by taking the trace on the space of square-integrable cusp forms whose central character restricted to $\mathfrak{X}$ is $\chi$. Let $\mathbf{G}^{*}$ denote the quasi-split inner form of $\mathbf{G}$ over $F$, with a fixed inner twist $\mathbf{G}^{*} \simeq \mathbf{G}$ over $\overline{F}$.  Since $Z$ is canonically identified with the center of $\mathbf{G}^{*}$, we may view the character $\chi$ as a central character datum for $\mathbf{G}^{*}$. We then let $f^{*}$ denote a Langlands-Shelstad transfer of $f$ to $\mathbf{G}^{*}$. One can construct such a transfer by lifting $f$ along the surjection $\mathcal{H}(\mathbf{G}(\mathbb{A}_{F})) \rightarrow  \mathcal{H}(\mathbf{G}(\mathbb{A}_{F}),\chi^{-1})$ applying the transfer with trivial central character due to Waldspurger and then taking the image along the analogous surjection for $\mathbf{G}^{*}(\mathbb{A}_{F})$. We let $\Sigma_{ell,\chi}(\mathbf{G}^{*})$ denote the set of $Z(\mathbb{A}_{F})$ orbits of stable $F$-elliptic conjugacy classes in $\mathbf{G}^{*}(F)$. We define the stable elliptic distribution

\[ ST^{\mathbf{G}^{*}}_{ell,\chi}(f^{*}) := \tau(\mathbf{G}^{*}) \sum_{\gamma \in \Sigma_{ell,\chi}(\mathbf{G}^{*})} \tilde{i}(\gamma)^{-1}SO_{\gamma,\chi}^{\mathbf{G}^{*}}(f) \]

where $SO^{\mathbf{G}^{*}}_{\gamma,\chi}(f^{*})$ denotes the stable orbital integral of $f^{*}$ at $\gamma$, $\tau(\mathbf{G}^{*})$ is the Tamagawa number of $\mathbf{G}^{*}$, and $\tilde{i}(\gamma)$ is the number of Galois fixed connected components of the centralizer of $\gamma$ in $\mathbf{G}^{*}$. Let $\xi$ be an irreducible representation of $\mathbf{G}_{F_{\infty}}$. Denote by $\chi_{\xi}: Z(F_{\infty}) \rightarrow \mathbb{C}^{*}$ the restriction of $\xi$ to $Z(F_{\infty})$. Write $f_{\xi}^{\mathbf{G}} \in \mathcal{H}(\mathbf{G}(F_{\infty}),\chi_{\xi}^{-1})$ for a Lefschetz function associated with $\xi$. In other words, a function such that $\tr(f^{\mathbf{G}}_{\xi}|\pi_{\infty})$ computes the Euler-Poincaré characteristic for the relative Lie algebra cohomology of $\pi_{\infty} \otimes \xi$ for every irreducible admissible representation $\pi_{\infty}$ of $\mathbf{G}(F_{\infty})$ with central character $\chi_{\xi}$. It follows by the Vogan-Zuckerman classification <cit.> that, if $\xi$ is regular, this will be non-zero if and only if $\pi_{\infty}$ is an (essentially) discrete series representation cohomological of regular weight $\xi$. For a finite place $v_{st}$ of $F$, we let $f_{v_{st}} := f_{Lef,v_{st}}^{\mathbf{G}} \in \mathcal{H}(\mathbf{G}(F_{v_{st}}))$ denote a Lefschetz function at $v_{st}$. Morally, this should be characterized by the property that $\tr(f_{v_{st}}|\pi_{v_{st}})$ computes the Euler-Poincaré characteristic of the continuous group cohomology of $\mathbf{G}(F_{v_{st}})$ valued in $\pi_{v_{st}}$. In the case that the center is anisotropic, it follows from the computations in <cit.> that this means that the trace of $f_{Lef,v_{st}}^{\mathbf{G}}$ will only be non-zero if $\pi_{v}$ is $1$-dimensional or an unramified twist of the Steinberg representation, for all irreducible admissible unitary $\pi_{v}$. For finite places $v = v_{st}$, they were originally constructed by Kottwitz; however, these results do not apply for the desired application, since the center of $\GSp_{4}$ is not compact. For the construction of these functions in this case and the proof of the property that their traces detect when a representation is $1$-dimensional or an unramified twist of Steinberg, see <cit.>.

We now assume for simplicity that the center $Z$ of $\mathbf{G}$ is split, which will be the case in all our applications. We consider a character $\eta: \mathbf{G}(F) \ra \mathbb{C}^{*}$ such that $\eta|_{Z(F_{v_{st}})} = \chi_{v_{st}}$. We can define a function $f_{Lef,\eta}^{\mathbf{G}} \in \mathcal{H}(\mathbf{G}(F),\chi_{v_{st}}^{-1})$ as $f_{Lef,\eta}^{\mathbf{G}}(g) := \eta^{-1}(g)f_{Lef}^{G/Z}(\ol{g})$, where $\ol{g} \in G(F_{v_{st}})/Z(F_{v_{st}})$ denotes the image of $g$ under the quotient map. This (up to scaling by a non-zero constant) forms a pseudo-coefficient for the Steinberg representation twisted by $\eta$, as constructed in <cit.>. In particular, by <cit.>, we have, for $\pi_{v_{st}}$ an irreducible (essentially) unitary representation of $\mathbf{G}(F_{v_{st}})$ with central character $\chi_{v_{st}}$, that $\tr(f_{Lef,\eta}^{\mathbf{G}}|\pi_{v_{st}}) \neq 0$ if and only if $\pi_{v_{st}}$ is isomorphic to $\eta$ or Steinberg twisted by $\eta$.

Now we have the key lemma, which tells us that with respect to these choices of test functions we get the following "simple trace formula".
Fix a central character datum $(Z(\mathbb{A}_{F}),\chi)$ with $\chi_{\xi} = \chi|_{Z(F_{\infty})}$ as above for $\chi_{\xi}$ attached to some regular weight $\xi$. For an element $f \in \mathcal{H}(\mathbf{G}(\mathbb{A}_{F}),\chi^{-1})$ in the global Hecke algebra as above, assume that $f_{\infty} = f_{\xi}^{\mathbf{G}} \in \mathcal{H}(\mathbf{G}(F_{\infty}),\chi^{-1})$ is a Lefschetz function at $\infty$ and assume that $f_{v_{st}} = f^{\mathbf{G}}_{Lef,\eta}$ is the Lefschetz function described above at $v_{st}$ for a character $\eta$ of $\mathbf{G}(F_{v_{st}})$ such that $\eta|_{Z(F_{v_{st}})} = \chi_{v_{st}}$. Then we have an equality:

\[ ST_{ell,\chi}^{\mathbf{G}^{*}}(f^{*}) = T^{\mathbf{G}}_{ell,\chi}(f) = T^{\mathbf{G}}_{disc,\chi}(f) = T^{\mathbf{G}}_{cusp,\chi}(f) \]

Strictly speaking, the cited Lemmas only prove this in the case where the central character datum is $(Z(F_{\infty}),\chi_{\xi})$; however, the result in this case easily follows using <cit.> (cf. the proof of <cit.>).

Let $S_{st}$ and $S_{sc}$ be disjoint finite sets of finite places of $F$, and let $S_{0}$ be a finite set of places contained in  $S_{st} \cup S_{sc}$. Let $S_{\infty}$ denote the infinite places of $F$. Set $S$ to be a finite set of places containing $S_{st} \cup S_{sc} \cup S_{\infty}$. We assume that the inner twist $\mathbf{G}^{*}$ of $\mathbf{G}$ is trivialized away from $S_{0}$ and $S_{\infty}$ (i.e for all $v \notin S_{0} \cup S_{\infty}$, the inner twisting gives rise to an isomorphism $\mathbf{G}_{F_{v}} \simeq \mathbf{G}_{F_{v}}^{*}$). In particular, $\mathbf{G}$ is unramified outside $S$ and we can fix a reductive model for $\mathbf{G}$ and $\mathbf{G}^{*}$ over $\mathcal{O}_{F}[1/S]$. By abuse of notation, we write $\mathbf{G}$ and $\mathbf{G}^{*}$ for the integral models of these groups. The inner twist gives an isomorphism $\mathbf{G}^{*}_{F_{v}} \simeq \mathbf{G}_{F_{v}}$ and isomorphisms $\mathbf{G}^{*}_{\mathcal{O}_{F_{v}}}
\simeq \mathbf{G}_{\mathcal{O}_{F_{v}}}$ of the hyperspecial subgroups determined by this model for the finite places $v \notin S$. The notion of unramified local representation on either side will be defined with respect to this fixed choice of hyperspecial level.

For the rest of the section, we will assume that $\mathbf{G}^{*} = \GSp_{4}$. We assume throughout that $\pi$ is a global cuspidal automorphic representation of the group $\mathbf{G}(\mathbb{A}_{F})$ satisfying the following properties:

* $\pi$ is cohomological of some regular weight $\xi$ at infinity.

* $\pi_{v}$ is unramified at all finite places $v \notin S$.

* $\pi_{v}$ has supercuspidal $L$-parameter for $v \in S_{sc}$.

* $\pi_{v}$ is an unramified twist of Steinberg at all finite places $v \in S_{st}$.

Note, as in Definition 2.1, we can further partition $S_{sc}$ into $S_{ssc}$, where $\mathrm{std}\circ \phi_{\pi_{v}}$ is irreducible (ssc = stable supercuspidal) and $S_{esc}$, where $\mathrm{std}\circ \phi_{\pi_{v}}$ is reducible (esc = endoscopic supercuspidal).

The main result of this section shows the existence of strong transfers from $\mathbf{G}$ to $\mathbf{G}^{*}$ at the places in $S_{sc} \cup S_{st} \cup S_{\infty}$ for a certain class of automorphic representations of $\mathbf{G}$. It is essentially a more refined version of <cit.> in the particular case that $\mathbf{G}^{*} = \GSp_{4}$.

Suppose that $S_{st}$ is non-empty. Given a $\pi$ as above, there exists a cuspidal automorphic representation $\tau$ of $\mathbf{G}^{*}(\mathbb{A}_{F})$ satisfying the following:

* $\tau^{S} \simeq \pi^{S}$.

* At all $v \in S_{sc} \cup S_{st} \cup S_{\infty}$, $\tau_{v}$ has the same Langlands parameter as $\pi_{v}$.

Moreover, we can choose $\tau$ to be globally generic. For the first part, the same is true with the roles of $\tau$ and $\pi$ reversed.

First off note that, since $\pi$ is an unramified twist of Steinberg at some finite place, $\tau$, if it exists, is automatically (essentially) tempered at all places (cf. Remark 5.1). It follows, by <cit.>, that the global $L$-packet of $\tau$ therefore contains a globally generic representation. So, if we can show the existence of some $\tau$ with the desired properties, that means we can find $\tau$ globally generic with the same properties. For the former, we now apply the trace formula.

Let $\mathfrak{X} = Z(\mathbb{A}_{F})$ and $\chi$ be the central character of $\pi$. We set $f = \bigotimes_{v \in F} f_{v}$ to be a test function on $\mathbf{G}(\mathbb{A}_{F})$ satisfying the following:

* $f_{\infty} = f_{\xi}^{\mathbf{G}}$ is a Lefschetz/Euler-Poincaré function of weight $\xi$ of $\mathbf{G}(F_{\infty})$.

* At $v \in S_{st}$, $f_{v} = f_{Lef,\eta_{v}}^{\mathbf{G}}$ is a Lefschetz function for $\mathbf{G}(F_{v})$, where $\eta_{v}$ is the unique character such that $\pi_{v}$ is the Steinberg twisted by $\eta_{v}$.

* At $v \in S_{ssc}$, $f_{v} = f_{\pi_{v}}$ is the pseudo-coefficient of $\pi_{v}$, as constructed in <cit.>.

* At $v \in S_{esc}$, $f_{v} = f_{\pi_{v}^{+}} + f_{\pi_{v}^{-}}$, where $\{\pi_{v}^{+},\pi_{v}^{-}\}$ is the $L$-packet over $\phi_{\pi_{v}}$ and $f_{\pi_{v}^{\pm}}$ is the pseudo-coefficient of $\pi_{v}^{\pm}$.

* At the finite places $v \notin S$, $f_{v}$ is an arbitrary element of the unramified Hecke algebra.

* For $v \in S \setminus S_{st} \cup S_{\infty} \cup S_{sc}$, choose $f_{v}$ to be an arbitrary function such that $\tr(f_{v}|\pi_{v}) > 0$.

Given such a $f$, we choose a test function $f^{*} = \bigotimes_{v \in F} f_{v}^{*}$ on $\mathbf{G}^{*}(\mathbb{A}_{F})$ satisfying the following:

* $f_{\infty}^{*} = f_{\xi}^{\mathbf{G}^{*}}$ is a Lefschetz/Euler-Poincaré for the representation $\xi$ of $\mathbf{G}^{*}(F_{\infty})$.

* At $v \in S_{st}$, $f_{v}^{*} = f^{\mathbf{G}^{*}}_{Lef,\eta_{v}}$ is the Lefschetz function for $\mathbf{G}^{*}(F_{v})$.

* At $v \in S_{ssc}$, $f_{v}^{*} = f_{\tau_{v}}$ is a pseudo-coefficient of $\tau_{v}$, where $\tau_{v}$ is the unique supercuspidal representation of $\mathbf{G}^{*}(F_{v})$ with Langlands parameter $\phi_{\pi_{v}}$.

* At $v \in S_{esc}$, $f_{v}^{*} = f_{\tau_{v}^{+}} + f_{\tau_{v}^{-}}$, where $\{\tau_{v}^{+},\tau_{v}^{-}\}$ is the $L$-packet over $\phi_{\pi_{v}}$ of $\mathbf{G}^{*}(F_{v})$ and $f_{\tau_{v}^{\pm}}$ is the pseudo-coefficient of $\tau_{v}^{\pm}$.

* At the finite places $v \notin S$, $f_{v}^{*} = f_{v}$ is the same element of the unramified Hecke algebra.

* For $v \in S \setminus S_{st} \cup S_{\infty} \cup S_{sc}$, choose $f_{v}^{*} = f_{v}$.

Now we wish to check that $f$ and $f^{*}$ are matching up to a non-zero constant $c$, in the sense of <cit.>. We can check this place by place. For the finite places $v \notin S_{st} \cup S_{sc} \cup S_{\infty}$, this is tautological. For all $v \in S_{\infty} \cup S_{st}$, this follows from <cit.>. For $v \in S_{sc}$, this follows from the character identities of Chan-Gan <cit.>. Namely, recall for a regular semisimple elliptic element $\gamma$ the orbital integrals of the pseudo-coefficients of a discrete series representation $\pi$ is given by the Harish-Chandra character of $\pi$ evaluated at $\gamma$, and it vanishes if $\gamma$ is non-elliptic <cit.> <cit.>. It follows that the orbital integrals for a sum of pseudo-coefficients over an $L$-packet of $\GSp_{4}$ or its inner form is already a stable distribution, by <cit.> combined with <cit.> for the inner form. Therefore, the desired equality of stable orbital integrals reduces to showing an equality of the sum of Harish-Chandra characters over the $L$-packets of supercuspidal parameters, and this is precisely <cit.>.

Since the test functions are matching and $S_{st} \neq \emptyset$, we can apply Lemma 5.1 to conclude:

\[ T^{\mathbf{G}^{*}}_{cusp,\chi}(f^{*}) = ST^{\mathbf{G}^{*}}_{ell,\chi}(f^{*}) =  cT^{\mathbf{G}}_{cusp,\chi}(f) \]

Then, by linear independence of characters, we have a relationship

\[ \sum_{\substack{\Pi' \in \mathcal{A}_{cusp,\chi}(\mathbf{G}^{*}) \\\\ \Pi'^{S} \simeq \pi^{S}}} m(\Pi')\tr(f_{S}^{*}|\Pi'_{S})  = c \cdot \sum_{\substack{\Pi \in \mathcal{A}_{cusp,\chi}(\mathbf{G}) \\\\ \Pi^{S} \simeq \pi^{S}}} m(\Pi)\tr(f_{S}|\Pi_{S})\]

where $m(\Pi)$ (resp. $m(\Pi')$) denotes the multiplicity of $\Pi$ (resp. $\Pi'$) in the cuspidal spectrum of $\mathbf{G}$ (resp. $\mathbf{G}^*$). Now at the infinite places, as soon as $\tr(f_{v}|\Pi_{v}) \neq 0$ at $v \mid \infty$ the regularity condition on $\xi$ implies that $\Pi_{v}$ is an (essentially) discrete series representation cohomological of regular weight $\xi$ and that $\tr(f_{v}|\Pi_{v}) = (-1)^{q(\mathbf{G}_{v})}$ by the Vogan-Zuckerman classification of unitary cohomological representations, where $q(\mathbf{G})$ is the $F$-rank of the derived group of $\mathbf{G}$. At $v_{st} \in S_{st}$ it follows by <cit.> that $\Pi_{v_{st}}$ is either the $\eta_{v_{st}}$-twist of the Steinberg or trivial representation. If $\Pi_{v_{st}}$ were one-dimensional then the global representation would also be one-dimensional by a strong-approximation argument <cit.>, implying that $\Pi_{\infty}$ cannot be tempered, which would contradict the fact $\Pi_{\infty}$ is an (essentially) discrete series representation. Therefore, $\Pi_{v_{st}}$ is always the $\eta_{v_{st}}$ twist of the Steinberg representation. At the remaining $v \in S_{ssc}$ (resp. $v \in S_{esc}$), it follows from the definition of pseudo-coefficients that, if $\tr(f_{v}|\Pi_{v}) > 0$, we have $\Pi_{v} \simeq \pi_{v}$ (resp. $\Pi_{v} \in \{ \pi_{v}^{+},\pi_{v}^{-} \}$). Similar considerations apply for $\Pi' \in \mathcal{A}_{cusp,\chi}(\mathbf{G}^{*})$ occurring non-trivially in the LHS. In summary, by the above analysis, we can deduce that the RHS of the previous equation is non-zero for the term corresponding to $\pi$ and that all the non-trivial terms on the RHS have the same sign. Therefore, the LHS is also non-zero, and we see, by choosing any non-zero term, that we obtain the desired $\tau$. The converse direction works similarly, where the role of $\mathbf{G}$ and $\mathbf{G}^{*}$, are swapped.

We note that we crucially used at the places $v \in S_{sc}$ that $\phi_{\pi_{v}}$ was supercuspidal. Otherwise, $\tr(f_{v}|\Pi_{v}) \neq 0 $ wouldn't necessarily imply that $\Pi_{v}$ lies in the $L$-packet over $\phi_{\pi_{v}}$ without assuming that $\Pi_{v}$ is tempered. However, by <cit.> any representation of $\GSp_{4}$ that is Steinberg at some non-empty finite set of places is tempered at all places. Therefore, we can relax this assumption at least for the forward direction of Theorem 5.2 to just assuming that $\tau_{v}$ is a discrete series representation at all $v \in S_{sc}$.

§.§ The Stable and $\sigma$-twisted Simple Trace Formula

For the proof of strong multiplicity one, we will need some more refined analysis of trace formulae. Namely, we will be interested in the discrete part of the stable trace formula in the particular case of $\mathbf{G}^{*} = \GSp_{4}/F$, as discussed in <cit.>. To this end, fix a central character datum $(\mathfrak{X},\chi) = (Z(\mathbb{A}_{F}),\chi)$ as before. We recall that the unique elliptic proper endoscopic group of $\GSp_{4}/F$ is $C = \GSO_{2,2} \simeq (\GL_{2} \times \GL_{2})/\{(t,t^{-1})| t \in \GL_{1}\}$. Then, for a test function $f^{*}$ on $\mathbf{G}^{*}(\mathbb{A}_{F})$ as above, the discrete part of the stable trace formula is an equality:

\[ I^{\mathbf{G}^{*}}_{disc,\chi}(f^{\mathbf{*}}) = ST^{\mathbf{G}^{*}}_{disc,\chi}(f^{*}) + \frac{1}{4}ST^{C}_{disc,\chi}(f^{C}) \]

for $f^{C}$ a matching test function on $C$. Here

\[ I_{disc,\chi}^{\mathbf{G}^{*}}(f^{*}) = \sum_{M} |W(G,M)|^{-1}\cdot \sum_{s \in W(M,G)_{reg}}|\det(s - 1)_{\mathfrak{a}_{M}/\mathfrak{a}_{G}}|^{-1} \cdot \tr(M_{P}(s,0)\cdot I^{P}_{disc,\chi}(0,f^{*})) \]

is a sum indexed over classes of standard Levi subgroups of $\mathbf{G}^{*}$. The precise definition of the terms will not be important for our purposes, but the interested reader can look at <cit.>. We simply note that the term corresponding to $M = \mathbf{G}^{*}$ is precisely equal to
$T_{disc,\chi}^{\mathbf{G}^{*}}(f^{*})$, as defined in section 5.1. $ST^{\mathbf{G}^{*}}_{disc,\chi}$ is a stable distribution on $\mathbf{G}^{*}$, similar to $ST_{ell,\chi}^{\mathbf{G}^{*}}$, and $ST^{C}_{disc,\chi}$ is the analogous stable distribution on $C(\mathbb{A}_{F})$. However, since $C$ has no proper elliptic endoscopic group, we have

\[ ST^{C}_{disc,\chi}(f^{C}) = I_{disc,\chi}^{C}(f^{C}) = T_{disc,\chi}^{C}(f^{C}) + (\text{other terms}) \]

with the other terms indexed by proper standard Levi subgroups of $C$ as above.

We will be interested in combining this with the elliptic part of the twisted trace formula as described by Kottwitz-Shelstad <cit.> for the particular group $\tilde{\mathbf{G}} := \GL_{4} \times \GL_{1}/F$ with respect to involution

\[ \sigma: (g,e) \mapsto (J^{t}g^{-1}J^{-1},e\det(g))\]

\[
J :=
\left[ {\begin{array}{cccc}
&  &  & 1 \\
&  & 1 &  \\
& -1 &  &   \\
-1 &  &  &   \\
\end{array} } \right]
\]

We can enumerate the elliptic $\sigma$-twisted endoscopic groups as follows.

* $\mathbf{G}^{*} = \GSp_{4}$

* $C_{E} = \mathrm{Res}_{E/F}\GL_{2}' := \{(g_{1},g_{2}) \in \mathrm{Res}_{E/F}\GL_{2} | \det(g_{1}) = \det(g_{2})\}$

* $C_{+}^{E} = (\GL_{2} \times \mathrm{Res}_{E/F}\GL_{1})/\GL_{1}$

where $E$ is an étale quadratic $F$-algebra and $E$ is not split in case (3). The simple stable twisted trace formula says that if $\tilde{f}$ is a test function on $\tilde{\mathbf{G}}$ whose twisted orbital integral is supported on the regular elliptic set at at least $3$ finite places, then we have an identity

\[ I^{\tilde{\mathbf{G}},\sigma}_{disc,\chi}(\tilde{f}) = \frac{1}{2}ST_{disc,\chi}^{\mathbf{G}^{*}}(f^{*}) + \frac{1}{4}\sum_{E}ST^{C_{E}}_{disc,\chi}(f^{C_{E}}) + \frac{1}{8}\sum_{E \neq F^{\oplus 2}} ST^{C_{+}^{E}}_{disc,\chi}(f^{C_{+}^{E}}) \]

* $\sum_{E}$ is a sum over étale quadratic $F$-algebras $E$,

* $\tilde{f}$, $f^{*}$, $f^{C_{E}}$, and $f^{C_{+}^{E}}$ are matching test functions,

* $ST^{\mathbf{G}^{*}}_{disc,\chi}$, $ST^{C_{E}}_{disc,\chi}$, and $ST^{C_{+}^{E}}_{disc,\chi}$ are the stable distributions appearing in the discrete part of the stable trace formula, as described above,

* $I^{\tilde{\mathbf{G}},\sigma}_{disc,\chi}$ is the invariant distribution which is the twisted analogue of $I^{\mathbf{G}^{*}}_{disc,\chi}$. It is given by <cit.> and has the form

\[ I_{disc,\chi}^{\tilde{\mathbf{G}},\sigma}(\tilde{f}) = \sum_{M} |W(G,M)|^{-1}\cdot \sum_{s \in W(M,G)_{reg}}|\det(s - 1)_{\mathfrak{a}_{M}/\mathfrak{a}_{G}}^{s\cdot\sigma}|^{-1} \cdot \tr(M_{P}(s,0)\cdot I^{P}_{disc,\chi}(0,\tilde{f})I_{P,disc}(\sigma)) \]

where the sum runs over standard Levi subgroups $M$ of $G$.

Now we want to apply these trace formulae with respect to appropriately chosen test functions. We will assume that $S_{st}$ is a finite set of places such that $|S_{st}| \geq 3$. Then, for all $v \in S_{st}$, we let $f^{*}_{v}$ be a pseudo-coefficient for the unramified twist of Steinberg by $\eta_{v}$, where $\eta_{v}$ is a character of $\mathbf{G}(F_{v})$ such that the restriction to $Z(F_{v})$ is $\chi_{v}$. In particular, we will take $f^{*}_{v}$ to be the Lefschetz function $f_{Lef,\eta_{v}}^{\mathbf{G}^{*}} \in \mathcal{H}(\mathbf{G}^{*}(F_{v}),\chi_{v}^{-1})$ considered in the previous section which is a pseudo-coefficient for Steinberg up to scaling. It follows, by <cit.>, that we can choose the local constituent of the matching function $\tilde{f}$ at $v$ to be the $\sigma$-twisted pseudo-coefficient of the Steinberg twisted by $\eta_{v}$, as defined in <cit.>. These functions orbital integrals are supported on the regular elliptic set (See the construction in <cit.>) and therefore we can apply the simple twisted trace formula. Moreover, the twisted orbital integral of $\tilde{f}_{v}$ is a stable function, and hence the $\kappa$-orbital integral of $\tilde{f}_{v}$ is zero for all $\kappa \neq 1$. Therefore, it follows that the transfers of $f_{v}$ to all elliptic twisted endoscopic groups $(\tilde{\mathbf{G}}_{F_{v}},\sigma)$ vanish, except possibly for $\mathbf{G}^{*}_{F_{v}}$. Thus, the simple twisted trace formula simplifies giving an equality:

\[ I^{\tilde{\mathbf{G}},\sigma}_{disc,\chi}(\tilde{f}) = \frac{1}{2}ST_{disc,\chi}^{\mathbf{G}^{*}}(f^{*}) \]

Now we apply the discrete part of the stable trace formula for $\mathbf{G}^{*}$ to the RHS this gives us an equality:

\[ I^{\mathbf{G}^{*}}_{disc,\chi}(f^{\mathbf{*}}) - \frac{1}{4}ST^{C}_{disc,\chi}(f^{C}) = ST^{\mathbf{G}^{*}}_{disc,\chi}(f^{*})  \]

However, for any $v \in S_{st}$, the orbital integral of $f_{v}^{*}$ is again stable, so we see that its transfer to the endoscopic group $C$ is zero. Hence, the second term vanishes. All in all, we obtain the following lemma.

For $S_{st}$ a finite set of finite places with $|S_{st}| \geq 3$, $f^{*}$ and $\tilde{f}$ matching test functions on $\mathbf{G}^{*}$ and $\tilde{\mathbf{G}}$, respectively, such that $f^{*}_{v}$ is a pseudo-coefficient for the (essentially) discrete series Steinberg representation twisted by an unramified character $\eta_{v}$ such that $\eta_{v}|_{Z(F_{v})} = \chi_{v}$ and $\tilde{f}_{v}$ is the $\sigma$-twisted pseudo-coefficient for the Steinberg representation of $\tilde{\mathbf{G}}_{F_{v}}$ twisted by $\eta_{v}$, we have an equality:

\[ \frac{1}{2}I^{\mathbf{G}^{*}}_{disc,\chi}(f^{\mathbf{*}}) =  I_{disc,\chi}^{\tilde{\mathbf{G}},\sigma}(\tilde{f}) \]

relating spectral information on $\mathbf{G}^{*}$ to $\tilde{\mathbf{G}}$.

§.§ Strong Multiplicity One

We now would like to combine the analysis of sections 5.1 and 5.2 to deduce a strong multiplicity one result for $\mathbf{G}^{*} = \GSp_{4}/F$ and certain inner forms. Our analysis is very similar to <cit.> and benefited from reading the proofs of <cit.> in a paper of Rosner and Weissauer, where they prove a similar multiplicity one result using Weselmann's topological twisted trace formula <cit.> instead of the simple twisted trace formula of Kottwitz-Shelstad. Let $S_{st}$ and $S_{sc}$ be disjoint finite sets of finite places. Let $S_{\infty}$ denote the set of infinite places. Set $S_{0} \subset S_{st} \cup S_{sc}$ and $S_{sc} \cup S_{st} \cup S_{\infty} \subset S$ to be finite sets of places as before. We let $\mathbf{G}$ be an inner form over $F$, as in Theorem 5.2, trivialized outside of $S_{0} \cup S_{\infty}$. We have the following.

Assume that $|S_{st}| \geq 3$. Let $\pi$ be a cuspidal automorphic representation of $\mathbf{G}^{*} = \GSp_{4}/F$ or the above inner form $\mathbf{G}$ satisfying the following:

* $\pi$ is cohomological of regular weight $\xi$ at infinity,

* $\pi$ is unramified outside of $S$,

* $\pi$ is an unramified twist of Steinberg at all places in $S_{st}$,

* $\pi$ has supercuspidal $L$-parameter at all places in $S_{sc}$.

If $\pi'$ is a cuspidal automorphic representation of $\mathbf{G}^{*}$ satisfying (1), (3), and $\pi'^{S} \simeq \pi^{S}$ then its Langlands parameter at all places in $S$ agrees with $\pi$. If $\pi'$ is a cuspidal automorphic representation of $\mathbf{G}$ satisfying conditions (1), (3), and $\pi'^{S} \simeq \pi^{S}$ then its Langlands parameter at all places in $S_{st} \cup S_{sc} \cup S_{\infty}$ agrees with $\pi$.

Set $\chi$ to be the central character of $\pi$. We apply the above trace formulae with central character datum $(Z(\mathbb{A}_{F}),\chi)$. If $\pi$ is a representation of $\mathbf{G}^{*} = \GSp_{4}/F$, we take $\tau$ to be a globally generic member of the global $L$-packet of $\pi$, as in the proof of Theorem 5.2. If $\pi$ is a representation of the inner form $\mathbf{G}$ then, using Theorem 5.2, we take $\tau$ to be a globally generic strong transfer $\tau$ of $\pi$ to a cuspidal automorphic representation of $\mathbf{G}^{*}$, with Langlands parameter equal to $\phi_{\pi_{v}}$ at all places in $v \in S_{sc} \cup S_{st} \cup S_{\infty}$. Now we apply <cit.> to $\tau$ to deduce the existence of a strong transfer to a globally generic automorphic representation of $\GL_{4}(\mathbb{A}_{F})$, denoted $\tilde{\tau}$. It satisfies the following:

* $\tilde{\tau}$ is a global theta lift of $\tau$.

* $\tilde{\tau}^{\vee} \otimes \chi \simeq \tilde{\tau}$.

* For all places $v$, we have that $\phi_{\tilde{\tau}_{v}} = \mathrm{std}\circ \phi_{\tau_{v}} $ as conjugacy classes of parameters.

* Its form falls into the two cases:

* $\tilde{\tau}$ is cuspidal

* $\tilde{\tau} = \sigma \boxplus \sigma'$ for $\sigma \neq \sigma'$ a cuspidal automorphic representation of $\GL_{2}$.

In the latter case, $\tau$ is the theta lift of a cusp form $\sigma \otimes \sigma'$ on $C= \GSO_{2,2}$.

To distinguish these two cases, we say that $\tilde{\tau}$ is a stable or endoscopic lift. We choose matching test functions $\tilde{f}$ and $f^{*}$ on $\tilde{\mathbf{G}}$ and $\mathbf{G}^{*}$, respectively, such that, for $v \in S_{st}$, they are pseudo-coefficients for Steinberg twisted by $\eta_{v}$, as in Lemma 5.3, where $\eta_{v}$ the unramified character that $\pi_{v_{st}}$ is a twist of Steinberg of, where we can, up to scaling, take this to be the Lefschetz function $f_{Lef,\eta_{v_{st}}}^{\mathbf{G}^{*}}$ considered in section 5.1. We let $f^{*}_{\infty}$ be a Lefschetz function for the discrete series $L$-packet given by $\xi$ as before. Lemma 5.3 then gives us an equality:

\[ \frac{1}{2}I^{\mathbf{G}^{*}}_{disc,\chi}(f^{\mathbf{*}}) =  I_{disc,\chi}^{\tilde{\mathbf{G}},\sigma}(\tilde{f}) \]

We first treat the case where $\tilde{\tau}$ is a stable lift, and consider the part of the RHS corresponding to the cuspidal representation $\tilde{\tau}$ constructed above. By using linear independence of the unramified characters and the strong multiplicity one property for $\tilde{\mathbf{G}}$, the above identity implies an equality

\begin{equation}
c_{1}\cdot\sum_{\Pi' \simeq \pi^{S}} m(\Pi')\tr(f^{*}_{S}|\Pi'_{S}) =  \tr_{\sigma}(\tilde{\tau}_{S}|\tilde{f}_{S})
\end{equation}

where $c_{1}$ is a non-zero constant. Here $\tr_{\sigma}(\tau_{S}|\tilde{f}_{S})$ is the $\sigma$-twisted trace, as defined in <cit.>. The LHS runs over automorphic representations satisfying the following:

* $\Pi'$ has non-zero contribution to the discrete part of the trace formula $I_{disc,\chi}^{\mathbf{G}^{*}}$.

* The coefficient $m(\Pi')$ is the coefficient associated with the trace of $\Pi'$ in $I_{disc,\chi}^{\mathbf{G}^{*}}$.

We can further simplify the LHS of (3) by noting that non-discrete spectrum representations which intervene in $I_{disc,\chi}^{\mathbf{G}^{*}}$ are parabolically induced from the discrete spectrum of proper Levi subgroups of $\mathbf{G}^{*}$. By <cit.>, we know that parabolically induced representations of $\mathbf{G}^{*}$ lift to parabolically induced representations of $\tilde{\mathbf{G}}$. Therefore, since $\tilde{\tau}$ is a stable lift and therefore cuspidal, all terms occurring in in the LHS must all come from the discrete spectrum $T_{disc,\chi}^{\mathbf{G}^{*}}(f^{*})$, by strong multiplicity one for $\tilde{\mathbf{G}}$. Moreover, the coefficients $m(\Pi')$ must then be the multiplicities of $\Pi'$ in the discrete spectrum. However, as in Lemma 5.1, we have an equality:

\[ T_{disc,\chi}^{\mathbf{G}^{*}}(f^{*}) = T_{cusp,\chi}^{\mathbf{G}^{*}}(f^{*}) \]

In other words, we may assume that the sum on the LHS of (3) ranges over  $\Pi' \in \mathcal{A}_{cusp,\chi}(\mathbf{G}^{*})$, and that $m(\Pi')$ denotes the multiplicity in the cuspidal automorphic spectrum. In other words, we can rewrite the LHS as

\[ \sum_{\substack{\Pi' \in \mathcal{A}_{cusp,\chi}(\mathbf{G}^{*}) \\\\
\Pi'^{S} \simeq \pi^{S}}} m(\Pi')\tr(f_{S}^{*}|\Pi'_{S}) \]

Now, for the RHS, we apply the local character identities of Chan-Gan <cit.>, this tells us that we have an equality:

\[ \tr_{\sigma}(\tilde{\tau}_{S}|\tilde{f}_{S}) = c_{2}\prod_{v \in S} \sum_{\pi'_{v} \in \Pi_{\phi_{\tau_{v}}}(\mathbf{G}^{*}_{F_{v}})} \tr(f_{v}^{*}|\pi'_{v}) \]

for some non-zero constant $c_{2}$, where we have used property (3) of the representation $\tilde{\tau}$. In summary, we have concluded

\[ \sum_{\substack{\Pi' \in \mathcal{A}_{cusp,\chi}(\mathbf{G}^{*}) \\\\
\Pi'^{S} \simeq \pi^{S}}} m(\Pi')\tr(f_{S}^{*}|\Pi'_{S}) = c \cdot \prod_{v \in S} \sum_{\pi'_{v} \in \Pi_{\phi_{\tau_{v}}}(\mathbf{G}^{*}_{F_{v}})} \tr(f_{v}^{*}|\pi'_{v})  \]

for some non-zero constant $c$. If $\pi$ was a representation of $\mathbf{G}^{*}$, we know by our choice of $\tau$ that $\phi_{\tau_{v}} = \phi_{\pi_{v}}$ for all $v \in S$, so, by linear independence of characters at the places $v \in S \setminus S_{\infty} \cup S_{st}$, this tells us that the local constituents of some $\Pi'$ occurring in the LHS with non-zero trace at $S_{\infty} \cup S_{st}$ are described by members of the $L$-packet over $\phi_{\pi_{v}}$ occurring with some multiplicity. Since the representation $\pi'$ is by assumption cohomological of regular weight $\xi$ and an unramified twist of Steinberg at all places in $S_{st}$, by arguing as in proof of Theorem 5.2, we have that $\tr(f_{S_{st} \cup S_{\infty}}|\pi'_{S_{st} \cup S_{\infty}}) \neq 0$, and this gives us the desired claim for $\mathbf{G}^{*} = \GSp_{4}$. Now, if $\pi'$ is a representation of the inner form, we apply the character identities of Chan-Gan <cit.>. This tells us that the RHS of the previous equation is equal to

\[ c_{3}\prod_{v \in S} \sum_{\rho_{v} \in \Pi_{\phi_{\tau_{v}}}(\mathbf{G}_{F_{v}})} \tr(f_{v}|\rho_{v}) \]

for some non-zero constant $c_{3}$. Now, to rewrite the LHS, we apply the trace formula as in the proof of Theorem 5.2. By linear independence of characters, we obtain a relationship

\begin{equation*}
\sum_{\substack{\Pi' \in \mathcal{A}_{cusp,\chi}(\mathbf{G}^{*}) \\\\
\Pi'^{S} \simeq \pi^{S}}} m(\Pi')\tr(f_{S}^{*}|\Pi'_{S})  = c_{4} \cdot \sum_{\substack{\Pi \in \mathcal{A}_{cusp,\chi}(\mathbf{G}) \\\\ \Pi^{S} \simeq \pi^{S}}} m(\Pi)\tr(f_{S}|\Pi_{S})
\end{equation*}

for some non-zero constant $c_{4}$. All in all, we obtain that

\[ \cdot \sum_{\substack{\Pi \in \mathcal{A}_{cusp,\chi}(\mathbf{G}) \\\\ \Pi^{S} \simeq \pi^{S}}} m(\Pi)\tr(f_{S}|\Pi_{S}) = c'\prod_{v \in S} \sum_{\rho_{v} \in \Pi_{\phi_{\tau_{v}}}(\mathbf{G}_{F_{v}})} \tr(f_{v}|\rho_{v}) \]

for some non-zero constant $c'$. We know by our choice of $\tau$ that $\phi_{\tau_{v}} = \phi_{\pi_{v}}$ for all $v \in S_{sc} \cup S_{st} \cup S_{\infty}$. From here, the claim follows.

In the case that $\tilde{\tau}$ is an endoscopic lift. We apply the stable trace form for $\mathbf{G}^{*}$ and a test function $f^{*}$ of $\mathbf{G}^{*}(\mathbb{A}_{F})$. We recall that this is an identity:

\[ ST^{\mathbf{G}^{*}}_{disc,\chi}(f^{*}) = I^{\mathbf{G}^{*}}_{disc,\chi}(f^{\mathbf{*}}) - \frac{1}{4}ST^{C}_{disc,\chi}(f^{C})  \]

for a $f^{C}$ a matching test function on $C(\mathbb{A}_{F})$. Moreover, since $C$ has no proper elliptic endoscopic groups, we have

\[ ST^{C}_{disc,\chi}(f^{C}) = I_{disc,\chi}^{C}(f^{C}). \]

We look at the $\sigma_{1} \otimes \sigma_{2}$-isotypic part, where $\sigma_{1} \otimes \sigma_{2}$ is the representation of $C$ whose theta lift is $\tau$. and use linear independence of characters at the unramified places to obtain a semi-local identity

\[ ST_{\sigma_{1} \otimes \sigma_{2}}^{\mathbf{G}^{*}}(f^{*}_{S}) = \sum_{\Pi'^{S} \simeq \pi^{S}} m(\Pi')\tr(f_{S}^{*}|\Pi'_{S}) - \frac{1}{4}\sum_{(\sigma'_{1} \otimes \sigma_{2}')^{S} \simeq (\sigma_{1} \otimes \sigma_{2})^{S}} m(\sigma \otimes \sigma')\tr(f^{C}_{S}|(\sigma_{1}' \otimes \sigma_{2}')_{S}) \]

where the LHS is a stable distribution on $\mathbf{G}^{*}(\mathbb{A}_{S})$, as in <cit.>. The first term in the RHS is a sum over discrete automorphic representations $\Pi'$ of $\mathbf{G}^{*}(\mathbb{A}_{F})$ with the coefficient $m(\Pi')$ being the multiplicity in the discrete spectrum, and the second term is a sum over all automorphic representations $\sigma_{1}' \otimes \sigma_{2}'$ of $C(\mathbb{A}_{F})$. It follows by <cit.> that the LHS is equal to

\[ \frac{1}{2} \prod_{v \in S} \sum_{\pi'_{v} \in \Pi_{\phi_{\tau_{v}}}(\mathbf{G}^{*}_{F_{v}})} \tr(f^{*}_{v}|\pi'_{v})   \]

On the other hand, if, for $v \in S_{st}$, we take $f_{v}^{*} = f_{Lef,\eta_{v}}^{\mathbf{G}^{*}}$ to be the Lefschetz function as above, we see that

\[ -\frac{1}{4}\sum_{(\sigma_{1} \otimes \sigma_{2}')^{S} \simeq (\sigma_{1} \otimes \sigma_{2})^{S}} m(\sigma \otimes \sigma')\tr(f^{C}_{S}|(\sigma_{1}' \otimes \sigma_{2}')_{S}) \]

vanishes, since the orbital integral of $f_{v}^{*}$ is stable and $S_{st} \neq \emptyset$. In summary, we have concluded an identity

\[ c''\prod_{v \in S} \sum_{\pi'_{v} \in \Pi_{\phi_{\tau_{v}}}(\mathbf{G}^{*}_{F_{v}})} \tr(f^{*}_{S}|\pi'_{v}) =  \sum_{\Pi'^{S} \simeq \pi^{S}} m(\Pi')\tr(f_{S}^{*}|\Pi'_{S})   \]

for some non-zero constant $c''$. Now taking $f_{\infty}^{*}$ to be our Lefschetz function at $\infty$, we can argue just as in the stable case.

Strong multiplicity one for globally generic automorphic representations of $\GSp_{4}$ has been proven by Jiang-Soudry <cit.>. So, in the particular case that $\pi$ and $\pi'$ are representations of $\GSp_{4}$, we could have just assumed that $S_{st}$ is non-empty and then applied their results to a globally generic member in the global $L$-packets of $\pi$ and $\pi'$ to deduce the desired claim.

§ GALOIS REPRESENTATIONS IN THE COHOMOLOGY OF SHIMURA VARIETIES

We now would like to combine the results of the previous section with results of Sorensen <cit.> on the Galois representations associated to automorphic representations of $\mathbf{G}^{*} = \GSp_{4}/F$ to say something about the Galois action of the global Shimura varieties occurring in basic uniformization. Let $F/\mathbb{Q}$ be a totally real field and $\mathbb{A}_{f,F}$ the finite adeles of $F$. Throughout, we will assume that $\tau$ is a cuspidal automorphic representation of $\mathbf{G}^{*}$ satisfying the same properties as in the previous section.

* $\tau_{\infty}$ is cohomological of some regular weight $\xi$ of $\mathbf{G}^{*}(F_{\infty})$.

* $\tau_{v}$ is unramified at all finite places outside of $S$.

* $\tau_{v}$ is an unramified twist of Steinberg at some finite set of finite places $S_{st}$.

We have the following key result of Sorensen.
Fix a globally generic $\tau$ as above such that $S_{st}$ is non-empty. Then there exists, a unique (after fixing the isomorphism $i: \overline{\mathbb{Q}}_{\ell} \xrightarrow{\simeq} \mathbb{C}$) irreducible continuous representation $\rho_{\tau}: \Gal(\overline{F}/F) \rightarrow \GSp_{4}(\overline{\mathbb{Q}}_{\ell})$ characterized by the property that, for each finite place $v \nmid \ell$ of $F$, we have

\[ i\WD(\rho_{\tau}|_{W_{F_{v}}})^{F-s.s} \simeq \phi_{\tau_{v}} \otimes |\cdot|^{-3/2}\]

where $(-)^{F-s.s}$ denotes the Frobenius semisimplification and $\phi_{\tau_{v}}$ is the Gan-Takeda parameter of $\tau_{v}$.

Now let us fix $\tau$ with associated $\rho_{\tau}$ as above and assume that $\tau$ is a strong transfer of some cuspidal automorphic representation $\pi$ of $\mathbf{G}$, as in Theorem 5.2. We assume that $S_{st}$ contains $q$ an odd inert prime in the number field $F$ and choose the inner form $\mathbf{G}$ to be of the following form, as in Kret-Shin <cit.>,

* $\mathbf{G}(\mathbb{R}) \simeq \GSp_{4}(\mathbb{R}) \times \GU_{2}(\mathbb{H})^{[F:\mathbb{Q}] - 1}$,

* $\mathbf{G}_{F_{v}} \simeq \GSp_{4}/F_{v}$ at all finite places $v$ if $[F:\mathbb{Q}]$ is odd,

* $\mathbf{G}_{F_{v}} \simeq \GSp_{4}/F_{v}$ at all but the finite place $q$ if $[F:\mathbb{Q}]$ is even,

where $\mathbb{H}$ is the Hamilton quaternions.
Let $A(\pi)$ be the set of isomorphism classes of cuspidal automorphic representations $\Pi$ of $\mathbf{G}$ such that, for all $v \in S_{st}$, $\Pi_{v}$ is an unramified twist of Steinberg, $\Pi_{\infty}$ is $\xi$ cohomological, and, for all $v \notin S_{\infty} \cup S_{st}$, $\Pi_{v} \simeq \pi_{v}$. Our main task now is to show that $\rho_{\tau}$ is realized in the $\pi^{\infty}$ isotypic component of the Shimura variety associated to a Shimura datum $(\mathbf{G},X)$, where $X$ is as in <cit.>. Let $\Sh(\mathbf{G},X)_{K,\overline{F}}$ be the associated Shimura variety over $\overline{F}$ which we recall is $3$-dimensional. We set $\mathcal{L}_{\xi}$ to be the $\overline{\mathbb{Q}}_{\ell}$ local system associated to a irreducible representation of $\mathbf{G}$ over $F$ of highest weight $\xi$ on it as before, and let $H^{i}_{c}(\Sh(\mathbf{G},X)_{K},\mathcal{L}_{\xi})_{ss}$ denote the semisimplification as a Hecke module of the compactly supported etale cohomology valued in $\mathcal{L}_{\xi}$. Choose $K \subset \mathbf{G}(\mathbb{A}_{f,F})$ a sufficiently small compact open subgroup such that $\pi^{\infty}$ has a non-zero $K$-invariant vector. Let $S_{bad}$ denote the set of prime numbers $p$ for which either $p = 2$, the group $\mathbf{G}$ is ramified, or $K_{p} = \prod_{v|p} K_{v}$ is not hyperspecial. Then we define the virtual Galois representation

\begin{equation}
\rho^{\pi}_{shim} := (-1)^{3}\sum_{\Pi \in A(\pi)} \sum_{i = 0}^{6} (-1)^{i}[Hom_{\mathbf{G}(\mathbb{A}_{f,F})}(\Pi^{\infty},H^{i}_{c}(\Sh(\mathbf{G},X)_{K,\overline{F}},\mathcal{L}_{\xi})_{ss})] \in K_{0}(\overline{\mathbb{Q}}_{\ell}(\Gamma_{F}))
\end{equation}

where $K_{0}(\overline{\mathbb{Q}}_{\ell}(\Gamma_{F}))$ denotes the Grothendieck group of continuous $\Gamma_{F} := \Gal(\overline{F}/F)$-representations with coefficients in $\overline{\mathbb{Q}}_{\ell}$. We now define the rational number

\[ a(\pi) := (-1)^{3}N_{\infty}^{-1}\sum_{\Pi \in A(\pi)} m(\Pi)\cdot ep(\Pi_{\infty} \otimes \xi) \]

* $m(\Pi)$ is the multiplicity of $\Pi$ in the automorphic spectrum of $\mathbf{G}$,

* $N_{\infty} = |\Pi_{\xi}^{G(F_{\infty})}|\cdot|\pi_{0}(\mathbf{G}(F_{\infty})/Z(F_{\infty})| = 4$, where $\Pi_{\xi}^{G(F_{\infty})}$ denotes the discrete series $L$-packet of representations of $G(F_{\infty})$ cohomological of weight $\xi$,

* $ep(\Pi_{\infty} \otimes \xi) := \sum_{i = 0}^{\infty} (-1)^{i}\dim(H^{i}(Lie(\mathbf{G}(F_{\infty})),K_{\infty};\Pi_{\infty} \otimes \xi)$.

Then we have the following proposition of Kret-Shin.
With notation as above, for almost all finite $F$-places $v$ not dividing a prime number in $S_{bad}$ and all sufficiently large integers $j$, we have

\[ \mathrm{tr}(\rho^{\pi}_{shim}(\Frob_{v}^{j})) = a(\pi)q_{v}^{j\frac{3}{2}}\mathrm{tr}(\mathrm{std}\circ \phi_{\pi_{v}})(\Frob^{j}_{v}) \]

Moreover, the virtual representation $\rho^{\pi}_{shim}$ is a true representation. In particular, the only non-zero term appearing in the above alternating sum occurs in middle degree ($= 3$).

The claim about it occurring in middle degree is part of the proof of the Proposition not the statement. (See the discussion after equation (8.13) in <cit.>)

We use this to deduce the following corollary.

The $\pi^{\infty}$-isotypic component of $R\Gamma_{c}(\Sh(\mathbf{G},X)_{K^{p},\overline{F}},\mathcal{L}_{\xi})$ is concentrated in degree $3$ and has $\Gamma_{F}$-action given (up to multiplicity) by the semi-simplification of $\mathrm{std}\circ \rho_{\tau}$.

The first part follows immediately from the previous Proposition, and the second part follows from the identification of the traces. In particular, by the Brauer-Nesbitt Theorem, Cheboratev density theorem, and the condition characterizing $\rho_{\tau}$, we can identify (up to multiplicity) the Galois representation $\rho^{\pi}_{shim}$ with the semi-simplifaction of the Galois representation $\mathrm{std} \circ \rho_{\tau}$.

§ PROOF OF THE KEY PROPOSITION

We will now combine the results of the previous three sections to deduce some key consequences that will be used to derive Proposition 1.4. For this, using Krasner's lemma, we now fix a totally real number field $F$ with two odd totally inert primes $p$ and $q$ such that $F_{p} \simeq L$ the fixed unramified extension of $\mathbb{Q}_{p}$. We fix the $\mathbb{Q}$-inner form $\mathbf{G}$ of $\mathbf{G}^{*} = \mathrm{Res}_{F/\mathbb{Q}}\GSp_{4}$ defined in section $6$, and let $\mathbf{G}'$ be the inner form of $\mathbf{G}$ seen in Definition 4.1. We take $(\mathbf{G},X)$ to be the Shimura datum considered in <cit.> as in section 6. We note that this forces the associated geometric dominant cocharacter $\mu$ of $G := \mathbf{G}_{\mathbb{Q}_{p}}$ to be the Siegel cocharacter; in particular, we can apply the results of section 4.2. Set $\xi$ to be a regular weight of an algebraic representation of $\mathbf{G}$ over $\mathbb{Q}$. Let $K^{p} \subset \mathbf{G}(\mathbb{A}^{p\infty})$ be an open compact subgroup. We set $S_{sc} = \{p\}$ and $S_{st}$ to be a disjoint finite set of finite places of $\mathbb{Q}$ containing $q$. We consider the uniformization map

\begin{equation}
\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})
\end{equation}

supplied by Theorem 4.2 and Corollary 4.1. Now fix a smooth irreducible supercuspidal representation $\rho$ of $J = \mathrm{Res}_{L/\mathbb{Q}_{p}}(\GU_{2}(D))(\mathbb{Q}_{p}) = \GU_{2}(D)(L)$. We have the following lemma.

Suppose $\rho$ is a supercuspidal representation of $J(\mathbb{Q}_{p})$, then, for sufficiently regular $\xi$ and sufficiently small $K^{p}$, we can find a lift $\Pi'$ to a cuspidal automorphic representation of $\mathbf{G}'$, such that $\Pi'^{\infty}$ occurs as a $J(\mathbb{Q}_{p})$-stable direct summand of $\mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f}))/K^{p},\mathcal{L}_{\xi})$. Moreover, for all places in $S_{st}$, we can assume that the local constituents at $v \in S_{st}$ are unramified twists of the Steinberg representation.

This follows from an argument using the simple trace formula. See for example <cit.> or <cit.>. We note in particular that cuspidality is vacuous, since $\mathbf{G}'(\mathbb{R})$ is compact modulo center by construction.

So let $\Pi'$ be a globalization of a fixed supercuspidal $\rho$ to a cuspidal automorphic representation of $\mathbf{G}'$ for some sufficiently regular $\xi$ and sufficiently small $K^{p}$. We can and do regard $\Pi'^{p\infty}$ as a representation of $\mathbf{G}(\mathbb{A}^{p}_{f}) \simeq \mathbf{G}'(\mathbb{A}^{p}_{f})$. We set $K^{p} = K^{p}_{S}K^{S}$, where $K^{S} \subset \mathbf{G}(\mathbb{A}_{f}^{S})$ is an open compact in the finite adeles away from $S$, for $S \subset \Pl_{F}$ some finite set of places of $\mathbb{Q}$ containing $S_{st} \cup \{p\} \cup \{\infty\}$, as in section 5. We assume that $S$ is sufficiently large such that outside of $S$ the automorphic representation $\Pi'$ is unramified, so, in particular, the subgroup $K^{S} \subset \mathbf{G}(\mathbb{A}_{f}^{S})$ is a product of hyperspecial subgroups away from $S$. We consider the abstract commutative Hecke algebra

\[ \mathbb{T}^{S} := Z[\mathbf{G}(\mathbb{A}_{f}^{S})//K^{S})]  \]

of bi-invariant compactly supported smooth functions on $\mathbf{G}(\mathbb{A}_{f}^{S})$. We regard both sides of (5) as $\mathbb{T}^{S}$-modules and consider the maximal ideal $\mathfrak{m}$ defined by the Hecke eigenvalues of $\Pi'^{S}$. We then localize both sides of $(5)$ at $\mathfrak{m}$ to obtain 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 would like to apply Propositions 4.4 and 5.4 to the representations occurring on both sides of this map. However, to apply these results we need to make some more modifications. In particular, the automorphic representations of $\mathbf{G}'$ occurring in the LHS (resp. RHS) of $\Theta_{\mathfrak{m}}$ are not necessarily unramified twists of Steinberg at all places in $S_{st}$.

To remedy this, we set $K^{p} = K^{p}_{S_{st}}K^{\{p\} \cup S_{st}}$, where $K^{\{p\} \cup S_{st}} \subset \mathbf{G}(\mathbb{A}_{f}^{\{p\} \cup S_{st}}) \simeq \mathbf{G}'(\mathbb{A}_{f}^{\{p\} \cup S_{st}})$ is an open compact subgroup. Then we consider the colimits

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

\[ \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{\{p\} \cup S_{st}},\mathcal{L}_{\xi}) := \colim_{K^{p}_{S_{st}} \rightarrow \{1\}} \mathcal{A}(\mathbf{G}'(\mathbb{Q})\backslash \mathbf{G}'(\mathbb{A}_{f})/K^{p}_{S_{st}}K^{\{p\} \cup S_{st}},\mathcal{L}_{\xi}) \]

Since $S_{st} \subset S$, the map $\theta_{\mathfrak{m}}$ gives rise to a map

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

Now, for all $v \in S_{st}$, we can project to the Bernstein component of $\mathbf{G}(F_{v}) \simeq \mathbf{G}'(F_{v})$ corresponding to the unramified twists of the Steinberg representation. This gives a map:

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

Moreover, passing to the Bernstein component where $G(\mathbb{Q}_{p})$ acts via a supercuspidal representation, by Proposition 4.4, we obtain an isomorphism:

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

We now apply Proposition 5.4 to obtain the following.

Let $\rho \in \Pi(J)$ be a representation with supercuspidal Gan-Tantono parameter $\phi$. Assume that $|S_{st}| \geq 3$. Then, for $\Pi'$ a choice of globalization of $\rho$ as in Lemma 7.1, unramified outside $S$ with associated maximal ideal $\mathfrak{m} \subset \mathbb{T}^{S}$ in the Hecke algebra defined by the Hecke eigenvalues of $\Pi'^{S}$, the representations of $\mathbf{G}'(\mathbb{A}_{f})$ occurring in the LHS of the map

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

are isomorphic to $\ol{\Pi}'^{\infty}$ for $\ol{\Pi}'$ a cuspidal automorphic representation of $\mathbf{G}'(\mathbb{A})$ satisfying the following:

* $\ol{\Pi}'^{S} \simeq \Pi'^{S}$,

* $\ol{\Pi}'$ is cohomological of regular weight $\xi$ at $\infty$,

* $\ol{\Pi}'$ is unramified twist of Steinberg at all $v \in S_{st}$,

* $\ol{\Pi}'$ has local constituent at $p$ with associated $L$-parameter $\phi$.

We localized at $\mathfrak{m}$ corresponding to $\Pi'^{S}$ and are considering algebraic automorphic representations of $\mathbf{G}'$ valued in the algebraic representation defined by $\xi$. Therefore, it is clear that any representation occurring in the LHS is of the form $\ol{\Pi}'^{\infty}$, where $\ol{\Pi}'$ is a cuspidal automorphic form of $\mathbf{G}'$ satisfying (1) and (2). Here cuspidality is automatic since $\mathbf{G}'(\mathbb{R})$ is compact modulo center. Moreover, by construction, it follows that $\mathbf{G}'(\mathbb{Q}_{v})$ acts on the LHS via representations in the Bernstein component of the unramified twists of Steinberg for $v \in S_{st}$. Therefore, $\ol{\Pi}'$ has local constituent at $v \in S_{st}$ given by either an unramified twist of the trivial or Steinberg representation at these places. However, since $\xi$ is regular by assumption, we can argue as in the proof of Theorem 5.2 using strong approximation as in <cit.> to show that the former case cannot occur, since it would imply that the representation $\ol{\Pi}'$ is $1$-dimensional contradicting the regularity of $\xi$. This allows us to apply proposition 5.4, since $|S_{st}| \geq 3$ by assumption. Proposition 5.4 applied to the inner form $\mathbf{G}'$ of $\mathbf{G}^{*}$ and the cuspidal automorphic representation $\Pi'$ of $\mathbf{G}'$ tells us that $\ol{\Pi}'$ must have Langlands parameter at $\{p\} = S_{sc}$ given by $\phi$, which was the desired claim.

We now combine this with Corollary 6.3 to deduce the following.

With notation as above, the map

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

is an isomorphism of complexes of $G(\mathbb{Q}_{p}) \times W_{L}$-modules concentrated in degree $3$ with $W_{L}$-action given, up to multiplicity and semi-simplification as a $W_{L}$-module, by $\mathrm{std}\circ \phi \otimes |\cdot|^{-3/2}$.

Proposition 7.2 tells us that the LHS of $\Theta_{\mathfrak{m},sc}^{st}$ breaks up as a direct sum of $G(\mathbb{Q}_{p}) \times W_{L}$-modules of the form

\[ R\Gamma_{c}(G,b,\mu)_{sc} \otimes_{\mathcal{H}(J_{b})}^{\mathbb{L}} \ol{\Pi}'^{\{p,\infty\} \cup S_{st}} \]

for $\ol{\Pi}'$ a cuspidal automorphic representation of $\mathbf{G}'$ that has $L$-parameter $\phi$ at $p$, and is also cohomological of regular weight $\xi$ at infinity and an unramified twist of Steinberg at all places in $S_{st}$. It suffices to prove the claim for each one of these summands. This summand will map to the $\ol{\Pi}'^{\{p,\infty\} \cup S_{st}}$-isotypic part of the RHS by construction. Let $\tau$ denote a strong transfer of $\Pi'$ to a cuspidal automorphic representation of $\mathbf{G}^{*}$ given by Theorem 5.2, with associated Galois representation $\rho_{\tau}$ given by Theorem 6.1. Applying Theorem 5.2 again, we consider a strong transfer of $\tau$ to $\mathbf{G}$ given by $\Pi$. We note, by Corollary 6.3, that the $\ol{\Pi}'^{\{p,\infty\} \cup S_{st}} \simeq \Pi^{\{p,\infty\} \cup S_{st}}$-isotypic part will be concentrated in degree $3$ and have $W_{L}$-action given (up to multiplicity and semi-simplification) by $\phi \otimes |\cdot|^{-3/2}$, by the property characterizing $\rho_{\tau}$ and the fact that $\tau$ was a strong transfer.

With this in hand, we are finally ready conclude our key Proposition.

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 $\mathrm{std}\circ \phi \otimes |\cdot|^{-3/2}$.

This is an immediate consequence of Proposition 7.2 and Corollary 7.3.

In particular, using Corollary 3.21, we can deduce the following.

If $p > 2$ and $L/\mathbb{Q}_{p}$ is an unramified extension, then, for all $\rho \in \Pi(J)$ with supercuspidal Gan-Tantono parameter $\phi_{\rho}$, the Fargues-Scholze and Gan-Tantono correspondences are compatible.

§ APPLICATIONS

We will now apply Corollary 7.5 to deduce some applications to the strong form of the Kottwitz conjecture and conclude the proof of Theorem 1.1. We begin with the latter.

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.

Part (1) follows by Corollary 3.12 and Corollary 3.16. Part (2) for the Gan-Tantono local Langlands is precisely Corollary 7.5. It remains to show that for $L/\mathbb{Q}_{p}$ unramified and $p > 2$, $\pi$ a smooth irreducible representation of $\GSp_{4}/L$ with supercuspidal Gan-Takeda $\phi_{\pi}$ parameter that the two correspondences are compatible. To show this, we consider the complex

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

of $J(\mathbb{Q}_{p}) \times W_{L}$-representations. We know, by Theorem 3.13, that this admits sub-quotients as a $J(\mathbb{Q}_{p})$-module given by $\rho$, for all $\rho$ whose Gan-Tantono parameter $\phi_{\rho}$ is equal to the Gan-Takeda parameter $\phi_{\pi}$ of $\pi$. However, by Corollary 3.15, we know that these representations must have Fargues-Scholze parameter equal to $\phi_{\pi}^{\mathrm{FS}}$. Therefore, we get a chain of equalities

\[ \phi_{\pi}^{\mathrm{FS}} = \phi_{\rho}^{\mathrm{FS}} = \phi_{\rho} = \phi_{\pi} \]

where we have used compatibility of the Gan-Tantono and the Fargues-Scholze correspondence for the middle equality.

Now, with this out of the way, we turn our attention to proving some strong forms of the Kottwitz conjecture for these representations, verifying Theorem 1.3.

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 $\phi: W_{L} \rightarrow \GSp_{4}(\overline{\mathbb{Q}}_{\ell})$. 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 $\Pi_{\phi}(J) = \{\rho_{1},\rho_{2}\}$, 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 $\mathrm{std}\circ \phi_{\rho} \simeq \phi_{1} \oplus \phi_{2}$, 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.

We show the proof in the endoscopic case, with the stable case being strictly easier. We first note that, since $\phi = \phi_{\rho}^{\mathrm{FS}}$ by Theorem 8.1, it follows by assumption that the Fargues-Scholze parameter $\phi_{\rho}^{\mathrm{FS}}$ of $\rho$ is supercuspidal. Therefore, by Remark 3.12, we have an isomorphism

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

of $G(\mathbb{Q}_{p}) \times W_{L}$-modules. Moreover, by Theorem 3.17, we see that both are concentrated in middle degree $3$. Applying Theorem 3.13, we get the following chain of equalities in the Grothendieck group of admissible $G(\mathbb{Q}_{p})$-representations of finite length

\[ -[H^{3}(R\Gamma_{c}(G,b,\mu)[\rho])] = [R\Gamma_{c}(G,b,\mu)[\rho]] = [R\Gamma^{\flat}_{c}(G,b,\mu)[\rho]] = -\sum_{\pi \in \Pi_{\phi}(G)} Hom_{S_{\phi}}(\delta_{\pi,\rho},std\circ\phi_{\rho})\pi \]

Now we saw in the discussion proceeding Theorem 3.13 that the RHS takes the form:

\[ -2\pi^{+} - 2\pi^{-} \]

We set $p_{1}$ and $p_{2}$ to be the two representations of $S_{\phi} \simeq \{(a,b) \in \overline{\mathbb{Q}}_{\ell}^{*} \times \overline{\mathbb{Q}}_{\ell}^{*}| a^{2} = b^{2} \} \subset (\GL_{2}(\overline{\mathbb{Q}}_{\ell}) \times \GL_{2}(\overline{\mathbb{Q}}_{\ell}))^{0}$ given by projecting to the first and second $\overline{\mathbb{Q}}_{\ell}^{*}$-factor, respectively. By Corollary 3.3, Corollary 3.11, and Theorem 8.1, we have an isomorphism of $G(\mathbb{Q}_{p}) \times W_{L}$-modules

\[ R\Gamma_{c}(G,b,\mu)[\rho]  \simeq   \Act_{p_{1}}(\rho)[-3] \boxtimes \phi_{1} \otimes |\cdot|^{-3/2} \oplus \Act_{p_{2}}(\rho)[-3] \boxtimes \phi_{2} \otimes |\cdot|^{-3/2} \]

where $\Act_{p_{1}}(\rho)$ and $\Act_{p_{2}}(\rho)$ are a priori direct sum of shifts of supercuspidal representations of $G(\mathbb{Q}_{p})$ with Fargues-Scholze (= Gan-Takeda) parameter equal to $\phi$. However, since we know that the LHS is a complex concentrated in middle degree $3$, this implies, by the above description in the Grothendieck group, that one of the $\Act_{p_{1}}(\rho)$ and $\Act_{p_{2}}(\rho)$ is isomorphic to $\pi^{+}$ and the other is isomorphic to $\pi^{-}$. Without loss of generality, assume that

\[ \Act_{p_{1}}(\rho_{1}) \simeq \pi^{+} \]

\[ \Act_{p_{2}}(\rho_{1}) \simeq \pi^{-} \]

We let $p_{+}$ and $p_{-}$ be the representation of $S_{\phi}$ determined by the trivial and non-trivial characters of the component group, respectively. Now, given two representations of $S_{\phi}$, denoted $W$ and $W'$, it follows from Remark 3.7 (3) that we have an isomorphism:

\[ \Act_{W} \circ \Act_{W'}(\cdot) \simeq \Act_{W \otimes W'}(\cdot) \]

In turn, we get

\[ \Act_{p_{1}^{\vee}}(\pi^{+}) \simeq \Act_{p_{1}^{\vee}} \circ \Act_{p_{1}}(\rho_{1}) \simeq \Act_{p_{+}}(\rho_{1}) \simeq \rho_{1} \]

where the last isomorphism follows since $p_{+}$ is the trivial representation. Similarly, depending on the values of $\Act_{p_{1}}(\rho_{2})$ and $\Act_{p_{2}}(\rho_{2})$ we can deduce that $\Act_{p_{2}^{\vee}}(\pi^{+})$ is isomorphic to $\rho_{1}$ or $\rho_{2}$. Now by Corollary 3.3, Corollary 3.11, and Theorem 8.1, we have an isomorphism

\[ R\Gamma_{c}(G,b,\mu)[\pi^{+}] \simeq \Act_{p_{1}^{\vee}}(\pi^{+})[-3] \boxtimes \phi_{1}^{\vee} \otimes |\cdot|^{-3/2} \oplus \Act_{p_{2}^{\vee}}(\pi^{+})[-3] \boxtimes \phi_{2}^{\vee} \otimes |\cdot|^{-3/2} \]

Since $\Act_{p_{1}^{\vee}}(\pi^{+}) \simeq \rho_{1}$ it therefore follows, by Theorem 3.13 and Remark 3.11, that $\Act_{p_{2}^{\vee}}(\pi^{+})$ must be isomorphic to $\rho_{2}$. Moreover, we know that $\Act_{p_{-}} \circ \Act_{p_{1}}(\rho_{1}) \simeq \Act_{p_{2}}(\rho_{1})$ and  $\Act_{p_{-}} \circ \Act_{p_{2}}(\rho_{1}) \simeq \Act_{p_{1}}(\rho_{1})$. Therefore, we obtain that $\Act_{p_{-}}(\pi^{+}) \simeq \pi^{-}$ and $\Act_{p_{-}}(\pi^{-}) \simeq \pi^{+}$. This allows us to determine that

\[ \Act_{p_{1}^{\vee}}(\pi^{-}) \simeq \Act_{p_{2}^{\vee}}\circ \Act_{p_{-}}(\pi^{-}) \simeq  \Act_{p_{2}^{\vee}}(\pi^{+}) \simeq \rho_{2} \]

\[ \Act_{p_{2}^{\vee}}(\pi^{-}) \simeq \Act_{p_{1}^{\vee}}\circ \Act_{p_{-}}(\pi^{-}) \simeq  \Act_{p_{1}^{\vee}}(\pi^{+}) \simeq \rho_{1} \]

which will determine the cohomology of $R\Gamma_{c}(G,b,\mu)[\pi^{-}]$. It only remains to show that the value of $R\Gamma_{c}(G,b,\mu)[\rho_{2}]$ is determined. However, this follows since

\[ \Act_{p_{2}}(\rho_{2}) \simeq \Act_{p_{2}}\circ \Act_{p_{2}^{\vee}}(\pi^{+}) \simeq \pi^{+} \]

\[ \Act_{p_{1}}(\rho_{2}) \simeq \Act_{p_{1}} \circ \Act_{p_{1}^{\vee}}(\pi^{-}) \simeq \pi^{-} \]

To conclude this section, we use Theorem 8.1 to deduce compatibility with the local Langlands correspondence for $\Sp_{4}$ and its unique non quasi-split inner form $SU_{2}(D)$. These correspondences are described in the papers <cit.> and <cit.> by Gan-Takeda and Choiy, respectively. For $\Sp_{4}$, this is described as the unique correspondence which sits in the commutative diagram:

\[ \begin{tikzcd}
& \Pi(\GSp_{4}) \arrow[r,"\LLC_{\GSp_{4}}"] \arrow[d] & \Phi(\GSp_{4}) \arrow[d,"\alpha"] \\
& \Pi(\Sp_{4}) \arrow[r,"\LLC_{\Sp_{4}}"] & \Phi(\Sp_{4})
\end{tikzcd} \]

Here, the left vertical arrow is not a map at all, it is a correspondence defined by the subset of $\Pi(\GSp_{4}) \times \Pi(\Sp_{4})$ consisting of pairs $(\pi,\omega)$ such that $\omega$ is a constituent of the restriction of $\pi$ to $\Sp_{4}$, and the right vertical arrow is the map on $L$-parameters induced by the natural map $\GSpin_{5}(\mathbb{C}) \rightarrow \SO_{5}(\mathbb{C})$. One has a similar characterization of the local Langlands correspondence for $\SU_{2}(D)$. With this description of the correspondence, compatibility for $\Sp_{4}$ and $\SU_{2}(D)$ follows from Theorem 8.1 and Theorem 3.6 (7).

For $\pi$ (resp. $\rho$) a smoooth irreducible representation of $\Sp_{4}/L$ (resp. $\SU_{2}(D)$), with associated Gan-Takeda (resp. Choiy) parameter $\phi_{\pi}: W_{L} \times \SL_{2}(\mathbb{C}) \rightarrow \SO_{5}(\mathbb{C})$ (resp. $\phi_{\rho}$) we have that:

* The Fargues-Scholze and Gan-Takeda (resp. Choiy) local Langlands correspondences are compatible for any representation  $\pi$ (resp. $\rho$) such that $\phi_{\pi}$ (resp. $\phi_{\rho}$) is not supercuspidal.

* If $L/\mathbb{Q}_{p}$ is unramified and $p > 2$ then the Fargues-Scholze and Gan-Takeda (resp. Choiy) local Langlands correspondences are compatible for any representation $\pi$ (resp. $\rho$) such that $\phi_{\pi}$ (resp. $\phi_{\rho}$) is supercuspidal.

We note that Corollary 3.11, Theorem 3.18, Remark 3.11, and <cit.> apply to a triple $(G,b,\mu)$, where $\mu$ is any cocharacter and $b \in B(G,\mu)$ is the unique basic element. Therefore, by applying the same kind of analysis as in the proof of Theorem 8.2, we can prove the analogue of Theorem 8.2 in the Grothendieck group of finite length admissible representations with a smooth action of $W_{E}$ (cf. <cit.>) for the cohomology of the local Shtuka spaces defined by the triple $(G,b,\mu)$. By Corollary 8.3, this works even in the case when $G = \Sp_{4}$ and there are no Shimura varieties that these spaces uniformize. Moreover, in the case that $G = \GSp_{4}$, one can also deduce from Theorem 8.2 that it is concentrated in middle degree $\langle 2\rho_{G}, \mu \rangle$, using the monoidal property of the $\Act$-functors and Corollary 3.11, for $\mu$ any cocharacter. This in particular will imply some form of Fargues' Conjecture for these groups (See e.g. <cit.>, for this worked out in the more complicated case of $G = \U_{n}$).

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