3.1 Local Langlands for $\mathrm{GSp}_{4}$

In this section, we will describe the local Langlands correspondence of Gan and Takeda for the group $G := \mathrm{GSp}_{4}/L$ , where $L/\mathbb{Q}_{p}$ is a finite extension. We fix a choice of Whittaker datum $\mathfrak{m} := (B,\psi)$ throughout § 3, where B is the Borel and $\psi$ is a generic additive character of L.

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

\begin{align*} \phi: W_{L} \times \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \hat{G}(\mathbb{C}) = \mathrm{GSpin}_{5}(\mathbb{C}) \simeq \mathrm{GSp}_{4}(\mathbb{C}), \end{align*}

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

Theorem 3.1 [Reference Gan and TakedaGT11a]. There is a surjective finite to one map

\begin{align*} \mathrm{LLC}_{G}: \Pi(\mathrm{GSp}_{4}) \rightarrow \Phi(\mathrm{GSp}_{4}), \end{align*}

\begin{align*} \pi \mapsto \phi_{\pi}, \end{align*}

with the following properties.

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

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

   \[ A_{\phi} := \pi_{0}(S_{\phi}) \simeq \pi_{0}(S_{\phi}/Z(\mathrm{GSp}_{4})), \]
   which is either trivial or equal to $\mathbb{Z}/2\mathbb{Z}$ . When $A_{\phi} = \mathbb{Z}/2\mathbb{Z}$ , exactly one of the two representations in $\Pi_{\phi}(G)$ is generic for the fixed choice of Whittaker datum, and is indexed by the trivial character of $A_{\phi}$ .

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

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

5. (5) If $\pi \in \Pi(\mathrm{GSp}_{4})$ is a generic supercuspidal representation or non-supercuspidal representation then, for any $r \geq 1$ and any smooth irreducible representation $\sigma$ of $\mathrm{GL}_{r}(L)$ , we have that

   \begin{align*} \gamma(s, \pi \times \sigma,\psi) = \gamma(s,\phi_{\pi} \otimes \phi_{\sigma},\psi), \end{align*}
   \begin{align*} L(s, \pi \times \sigma,\psi) = L(s,\phi_{\pi} \otimes \phi_{\sigma},\psi), \end{align*}
   \begin{align*} \epsilon(s, \pi \times \sigma,\psi) = \epsilon(s,\phi_{\pi} \otimes \phi_{\sigma},\psi), \end{align*}
   where the right-hand sides are the Artin local factors associated with the representations of $W_{L} \times \mathrm{SL}_{2}(\mathbb{C})$ , and the left-hand sides are the local factors of Shahidi [Reference ShahidiSha90] with respect to the morphisms of L-groups defined in [Reference Gan and TakedaGT11a, Section 4]. In the case that $r \leq 2$ and $\pi$ is a non-generic supercuspidal representation, we have the same relationship where the left-hand side is given by the local factors defined by Townsend [Reference TownsendTow13].

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

We now make the following definition.

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

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

  \begin{align*} S_{\phi} \simeq \{(a,b) \in \mathbb{C}^{*} \times \mathbb{C}^{*}| a^{2} = b^{2} \} \subset (\mathrm{GL}_{2}(\mathbb{C}) \times \mathrm{GL}_{2}(\mathbb{C}))^{0}, \end{align*}
  where the center $Z(\mathrm{GSp}_{4})(\mathbb{C}) \simeq \mathbb{C}^{*}$ embeds diagonally.

For an L-parameter $\phi$ we see, by Theorem 3.1(2), that the size of the L-packet $\Pi_{\phi}(G)$ is at most 2, this allows us to subdivide into three cases.

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

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

3. (3) The L-packet contains only supercuspidals.

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

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

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

We will say a parameter is mixed supercuspidal if it is of the above form or of Soudry type. More precisely, we say that a parameter is of Soudry type if we have that $\mathrm{std}\circ \phi = \phi_{0} \boxtimes \nu(2)$ , where $\phi_{0}$ is a two-dimensional irreducible representation of $W_{L}$ . In particular, it is stable. In this case, the semisimplification satisfies $\mathrm{std}\circ \phi^{\mathrm{ss}} = \phi_{0} \otimes |\cdot|^{\frac{1}{2}} \oplus \phi_{0} \otimes |\cdot|^{-\frac{1}{2}}$ . This together with the supercuspidal parameters and the almost unipotent parameters (i.e. where $\mathrm{std} \circ \phi = \chi \boxtimes \nu(4)$ for $\chi$ a one-dimensional representation of $W_{L}$ ) gives a complete list of the discrete parameters (i.e. those that do not factor through a proper Levi), and as we will see in the next section the mixed supercuspidal case will be precisely when $\Pi_{\phi}(G)$ and $\Pi_{\phi}(J)$ contain a mix of supercuspidal and non-supercuspidal representations (however, in the Soudry-type case, the L-packet $\Pi_{\phi}(G)$ will be a single non-supercuspidal discrete series representation).

Case (3) in the previous division of parameters (see the discussion preceding Definition 3.3) is the situation where the parameter $\phi$ is supercuspidal as defined in the introduction. In particular, in the supercuspidal case the restriction of the parameter $\phi$ to the $\mathrm{SL}_{2}(\mathbb{C})$ factor is trivial, so the irreducible representations occurring in the decomposition of $\mathrm{std}\circ \phi$ are just representations of $W_{L}$ .

For the purposes of applying the weak form of the Kottwitz conjecture proven in Hansen, Kaletha and Weinstein [Reference Hansen, Kaletha and WeinsteinHKW22], we formulate this correspondence in terms of the refined local Langlands of Kaletha [Reference KalethaKal16] with respect to the fixed choice of Whittaker datum $\mathfrak{m}$ . Now, given a discrete parameter $\phi$ , we have by Theorem 3.1(2) a correspondence between the L-packet $\Pi_{\phi}(G)$ and the set of irreducible characters $A^{\vee}_{\phi}$ . This in turn gives rise to an irreducible character of the group $S_{\phi}$ via the composition

\begin{align*} S_{\phi} \rightarrow \pi_{0}(S_{\phi}) = A_{\phi}. \end{align*}

This allows us to make the following definition.

3.2. Local Langlands for $\mathrm{GU}_{2}(D)$

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

\begin{align*} D^{*} \times \mathrm{GL}_{1}. \end{align*}

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

\begin{align*} \langle \pm 1 \rangle \rightarrow B_{\phi} \rightarrow A_{\phi} \rightarrow 0. \end{align*}

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

Theorem 3.6 [Reference Gan and TantonoGT14]. There is a natural surjective finite-to-one map

\begin{align*} \mathrm{LLC}_{J}: \Pi(\mathrm{GU}_{2}(D)) \rightarrow \Phi(\mathrm{GU}_{2}(D)), \end{align*}

\begin{align*} \rho \mapsto \phi_{\rho}, \end{align*}

with the following properties.

1. (1) A smooth representation $\rho$ is an (essentially) discrete series representation of $\mathrm{GU}_{2}(D)$ if and only if its parameter $\phi_{\rho}$ does not factor through any proper Levi subgroup of $\mathrm{GSp}_{4}(\mathbb{C})$ .

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

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

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

5. (5) If $\rho \in \Pi(\mathrm{GU}_{2}(D))$ is a non-supercuspidal representation then, for any smooth irreducible representation $\sigma$ of $\mathrm{GL}_{r}(L)$ , we have that

   \begin{align*} \gamma(s, \rho \times \sigma,\psi) = \gamma(s,\phi_{\rho} \otimes \phi_{\sigma},\psi), \end{align*}
   \begin{align*} L(s, \rho \times \sigma,\psi) = L(s,\phi_{\rho} \otimes \phi_{\sigma},\psi), \end{align*}
   \begin{align*} \epsilon(s, \pi \times \sigma,\psi) = \epsilon(s,\phi_{\rho} \otimes \phi_{\sigma},\psi), \end{align*}
   where the right-hand sides are the Artin local factors associated with the representations of $W_{L} \times \mathrm{SL}_{2}(\mathbb{C})$ and the left-hand sides are the local factors of Shahidi, as defined in [Reference Gan and TantonoGT14, Section 8].

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

   \begin{align*} &\gamma(s,\phi_{\rho}^{\vee} \otimes\phi_{\sigma},\psi) \cdot \gamma(-s,\phi_{\rho} \otimes\phi_{\sigma}^{\vee}, \overline{\psi}) \cdot \gamma(2s,\mathrm{Sym}^{2}\phi_{\sigma} \otimes\mathrm{sim}(\phi_{\rho})^{-1},\psi))\\ &\quad \cdot \gamma(-2s,\mathrm{Sym}^{2}\phi_{\sigma}^{\vee} \otimes \mathrm{sim}(\phi_{\rho}), \overline{\psi}) .\end{align*}
   The map $\mathrm{LLC}_{J}$ is uniquely determined by the properties (1), (3), (5), and (6), with $r \leq 4$ in (5) and (6).

We now further elaborate on the structure of the L-packets $\Pi_{\phi}(J) := \mathrm{LLC}_{J}^{-1}(\phi)$ in the case where the parameter $\phi$ is mixed supercuspidal. If the parameter $\phi$ is of this form, then, it follows from [Reference Gan and TantonoGT14, Proposition 5.4], the description of $\mathrm{LLC}_{J}$ provided in [Reference Gan and TantonoGT14, Section 7], and the description of the non-generic supercuspidal representations of $\mathrm{GSp}_{4}$ provided in [Reference Gan and TakedaGT11b, Theorem 1.1(i)], that the L-packet $\Pi_{\phi}(J)$ has following structure, as alluded to in Remark 3.8(ii).

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

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

3. (3) (Soudry type.) The L-packet $\Pi_{\phi}(J) = \{\rho_{\rm sc}\}$ contains one supercuspidal and the L-packet $\Pi_{\phi}(G) = \{\pi_{\rm disc}\}$ contains one non-supercuspidal (essentially) discrete series representation.

These will be the only cases in which the L-packets contain a mix of supercuspidal and non-supercuspidal representations, justifying the terminology of calling them mixed supercuspidal introduced in the previous section. In particular, the only other examples of discrete non-supercuspidal parameters are those of almost unipotent type, and in the case the packets $\Pi_{\phi}(G)$ (respectively $\Pi_{\phi}(J)$ ) will consist of a single (essentially) discrete series non-supercuspidal representation of G(L) (respectively J(L)) (see [Reference Gan and TantonoGT14, Proposition 7.1(v)]).

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

• – (Stable.) In the case that the L-parameter $\phi$ is stable, we have that $B_{\phi} = \mathbb{Z}/2\mathbb{Z}$ and, as noted in § 3.1, $A_{\phi} = 1$ . This means the set $\hat{B}_{\phi} \setminus \hat{A}_{\phi}$ consists of one element corresponding to the non-trivial character.

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

  \begin{align*} A_{\phi} = \mathbb{Z}/2\mathbb{Z} \xrightarrow{\Delta} B_{\phi} = Z(\mathrm{SL}_{2}) \times Z(\mathrm{SL}_{2}) = \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z,} \end{align*}
  and so the set $\hat{B}_{\phi} \setminus \hat{A}_{\phi}$ has size 2 and is indexed by two characters $\eta_{+-}$ and $\eta_{-+}$ each non-trivial on one of the two $\mathbb{C}^{*}$ -factors under the isomorphism
  \begin{align*} S_{\phi} \simeq \{(a,b) \in \mathbb{C}^{*} \times \mathbb{C}^{*}| a^{2} = b^{2} \} \subset (\mathrm{GL}_{2}(\mathbb{C}) \times \mathrm{GL}_{2}(\mathbb{C}))^{0}, \end{align*}
  from § 3.1.

We now wish to put this local Langlands correspondence for the inner form in the context of the refined local Langlands correspondence of Kaletha [Reference KalethaKal16]. We consider the Kottwitz set B(G) [Reference KottwitzKot85, Reference Rapoport and RichartzRR96] and let $b \in B(G)$ be the basic element whose associated $\sigma$ -centralizer $J_{b} = J$ . We take this to be the basic element whose slope homorphism is the dominant rational cocharacter of G given by $(1/2,1/2,1/2,1/2)$ . Let $Z(\mathrm{GSp}_{4}) \simeq \mathbb{G}_{m}$ be the center. We recall that we have an isomorphism $\pi_{1}(G) \simeq X_*(Z(\hat{G})) \simeq \mathbb{Z}$ and that the $\kappa$ -invariant of b is sent to the element $1 \in \mathbb{Z}$ under this isomorphism. This indexes the identity representation of $\mathbb{G}_{m}$ , denoted $id_{\mathbb{G}_{m}}$ . Thus, given a discrete parameter $\phi: W_{L} \times \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \mathrm{GSp}_{4}(\mathbb{C})$ , the refined local Langlands correspondence asserts bijections

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

$\pi \mapsto \tau_{\pi},$

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

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

where $\mathbf{1}$ is the trivial representation. In § 3.1, we saw how for $\pi \in \Pi_{\phi}(G)$ to construct the desired $\tau_{\pi}$ . Here, it is uniquely pinned down by the property that the trivial representation corresponds to the unique $\mathfrak{m}$ -generic representation. In the case of the inner form, the situation is a bit more tricky. Consider $\rho \in \Pi(J)$ with associated L-parameter $\phi_{\rho}$ . If $\phi_{\rho}$ is stable then $S_{\phi} = \mathbb{G}_{m}$ and $\tau_{\rho}$ is simply $id_{\mathbb{C}^{*}}$ . If $\phi$ is endoscopic, then, as noted in § 3.1, we have an inclusion

\begin{align*} Z(\mathrm{GSp}_{4})(\mathbb{C}) = \mathbb{C}^{*} \xrightarrow{\Delta} S_{\phi} \simeq \{(a,b) \in \mathbb{C}^{*} \times \mathbb{C}^{*}| a^{2} = b^{2} \}. \end{align*}

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

\begin{align*} \rho_{+-} := \theta(\mathrm{JL}^{-1}(\pi_{2}) \boxtimes \pi_{1}) \text{ and } \rho_{-+} := \theta(\mathrm{JL}^{-1}(\pi_{1}) \boxtimes \pi_{2}), \end{align*}

where $\theta$ denotes the non-zero local theta lift from $D^{*} \times \mathrm{GL}_{2}(F)$ to $\mathrm{GU}_{2}(D)$ , as in [Reference Gan and TantonoGT14, Proposition 5.4], and $\mathrm{JL}: \Pi(D^{*}) \rightarrow \Pi(\mathrm{GL}_{2})$ is the Jacquet–Langlands correspondence. Now we would like to match these two representations with $\tau_{1}$ and $\tau_{2}$ under the refined local Langlands of Kaletha. Suppose we fix such a matching. We consider a refined endoscopic datum $\mathfrak{c}$ for the quasi-split reductive group G, which we recall is a tuple $(H,s,\mathcal{H},\eta)$ which consists of

• ’ a quasi-split group H over F;

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

• ’ an element $s \in Z(\widehat{H})^{\Gamma}$ ;

• ’ an L-homomorphism $\eta: \mathcal{H} \to {}^LG$ ;

• ’ satisfying the condition

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

Considering J as the $\sigma$ -centralizer of the unique basic element $b_{1} \in B(G)$ of Kottwitz invariant $1 \in \mathbb{Z}$ , this defines for us an extended pure inner twisting $(\xi,b_{1}): G \rightarrow J_{b_{1}}$ in the sense of [Reference KottwitzKot97, Section 5.2], and we can attach a canonical transfer factor $\Delta[\mathfrak{m},\mathfrak{c},b_{1}]$ to this datum, as defined in [Reference KalethaKal16, Section 4.1]. Given a test function $f \in C^{\infty}_{c}(J(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$ , we can use these transfer factors to say what it means for $f^{\mathfrak{c}} \in C^{\infty}_{c}(H(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$ to be matching in the sense that their stable orbital integrals normalized with respect to these transfer factors match up.

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

\begin{align*} \Theta^{1}_{\phi^{\mathfrak{c}}}(f^{\mathfrak{c}}) = \sum_{\pi \in \Pi_{\phi^{\mathfrak{c}}}(H)} \mathrm{tr}(1|\tau_{\pi})\theta_{\pi}(f^{\mathfrak{c}}) = e(J)\sum_{\rho \in \Pi_{\phi}(J)} \mathrm{tr}(s|\tau_{\rho}) \theta_{\pi}(f) = \Theta_{\phi}^{s}(f), \end{align*}

where e(J) is the Kottwitz sign of J, as defined in [Reference KottwitzKot83] and $\theta_{\pi}$ denotes the Harish-Chandra character of $\pi$ . Using the linear independence of the distributions $\theta_{\pi}$ and the fact that the packets $\Pi_{\phi}(J)$ are disjoint, we can see that the matching between $\rho \mapsto \tau_{\rho}$ is uniquely characterized by these relations. To show that there exists a matching between the representations $\tau_{1}$ and $\tau_{2}$ and $\rho_{+-}$ and $\rho_{-+}$ , we need to show that these identities are satisfied under the parametrization of Gan, Takeda, and Tantono. This will follow from the endoscopic character identities verified by Chan and Gan [Reference Chan and GanCG15]. Namely, in the case that the parameter $\phi$ is stable, one only needs to consider the trivial endoscopic datum $\mathfrak{c}_{\mathrm{triv}} = (G,1,\phantom{}^{L}G,\mathrm{id})$ , and these identities follow from [Reference Chan and GanCG15, Proposition 11.1(1)]. In the case that the parameter $\phi$ is endoscopic, the case $\mathfrak{c}_{\mathrm{triv}} = (G,1,\phantom{}^{L}G,\mathrm{id})$ follows from [Reference Chan and GanCG15, Proposition 11.1(1)], but one also needs to consider the refined endoscopic datum given by $\mathfrak{c} = (\mathrm{GSO}_{2,2},(1,-1),\mathrm{GSpin}_{4},i: \mathrm{GSpin}_{4} \rightarrow \mathrm{GSpin}_{5} \simeq \mathrm{GSp}_{4})$ , where i is the map from before and $(1,-1) \in S_{\phi} \subset (\mathrm{GL}_{2}(\mathbb{C}) \times \mathrm{GL}_{2}(\mathbb{C}))^{0}$ . Here, we have committed an abuse of notation in the endoscopic datum, writing $\mathrm{GSpin}_{4}$ instead of its split extension $\mathrm{GSpin}_{4} \times W_{F}$ by the Weil group and similarly for $\mathrm{GSp}_{4}$ .

In this case, the identities (up to a sign) follow from combining [Reference Chan and GanCG15, Proposition 11.1(1)] as before and [Reference Chan and GanCG15, Proposition 11.1(2)], where we note that since $\tau_{1}$ and $\tau_{2}$ are by definition given by the two projection maps their traces against $(1,-1)$ have the opposite sign, which is consistent with [Reference Chan and GanCG15, Proposition 11.1(2)].

In order for the above argument for deducing the refined local Langlands from the results of Chan and Gan to work, one needs to compare the transfer factors used in [Reference Chan and GanCG15, Section 4] with those of Kaletha, so that the notion of matching functions between f and $f^{\mathfrak{c}}$ coincide. We comment briefly on this now. When discussing the transfer factors $\Delta[\mathfrak{m},\mathfrak{c}]$ used for transferring between two quasi-split groups both Kaletha and Chan and Gan work with the Whittaker normalized transfer factors considered in [Reference Kottwitz and ShelstadKS12]. Kaletha on the one hand works with the renormalized factors $\Delta'_{\lambda}$ in the notation of [Reference Kottwitz and ShelstadKS12, Section 5.5] (see [Reference KalethaKal16, p. 221]) while Chan and Gan work with the original factors denoted $\Delta_{\lambda}$ considered in [Reference Kottwitz and ShelstadKS99, Section 5.3] (see [Reference Chan and GanCG15, Section 4.1]); however, as noted at the end of [Reference Kottwitz and ShelstadKS12, Section 5.5], when the element s in the refined endoscopic datum satisfies $s = s^{-1}$ , the two transfer factors agree, and this will always be true for the endoscopic datum mentioned above. We also note that Chan and Gan unlike Kaletha do not include the Weyl discriminant term $\Delta_{IV}$ in their definition of the Whittaker normalized transfer factors, as noted in [Reference Chan and GanCG15, Section 4.1]. Instead they include it in their definition of the orbital integrals used when defining the notion of matching functions. Moreover, both authors use the arithmetic normalization of local class field theory. These observations allow one to see that that the transfer factors $\Delta[\mathfrak{m},\mathfrak{c}]$ considered in [Reference KalethaKal16, p. 221] and the transfer factors considered [Reference Chan and GanCG15] give rise to the same definition of matching functions when transferring between two quasi-split groups.

Therefore, the main difficulty when comparing the two conventions of transfer factors lies when considering transfer between a quasi-split group and a non-quasi-split inner form. In particular, one needs to compare the transfer factors considered in [Reference Chan and GanCG15, Section 4.3] with the transfer factor $\Delta[\mathfrak{m},\mathfrak{c},b_{1}]$ mentioned above. This is determined up to a norm one element in $\mathbb{C}$ , as mentioned in [Reference Chan and GanCG15, Section 4]. For Chan and Gan, fixing this normalization is accomplished by matching the Levi subgroups $D^{*}\times \mathrm{GL}_{1} \subset \mathrm{GU}_{2}(D)$ and $\mathrm{GL}_{2} \times \mathrm{GL}_{1}\subset \mathrm{GSp}_{4}$ under the inner twisting, and then fixing the normalization by setting the transfer factor to be equal to the Kottwitz sign $e(\mathrm{D}^{*} \times \mathrm{GL}_{1}) = -1$ on the pairs $(\gamma,\delta)$ of strongly regular elements, where $\delta \in(D^{*} \times \mathrm{GL}_{1})(L)$ is the norm of an element $\gamma \in(\mathrm{GL}_{2} \times \mathrm{GL}_{1})(L)$ , where we reiterate that Chan and Gan incorporate the Weyl determinant into their orbital integrals. Then one chooses normalization of the transfer factors between $\mathrm{GSp}_{4}$ and $\mathrm{GU}_{2}(D)$ to be compatible with the one on the Levi, as defined in [Reference Chan and GanCG15, Section 4.3]. Similarly, when contemplating the transfer factors between $\mathrm{GU}_{2}(D)$ and $\mathrm{GSO}_{2,2}$ for the endoscopic case, Chan and Gan perform the same procedure where one matches up the Levi $D^{*} \times \mathrm{GL}_{1} \subset \mathrm{GU}_{2}(D)$ with either $\mathrm{GL}_{2} \times \mathrm{GL}_{1} \subset \mathrm{GSO}_{2,2}$ or $\mathrm{GL}_{1} \times \mathrm{GL}_{2} \subset \mathrm{GSO}_{2,2}$ , since both of these match up with $D^{*} \times \mathrm{GL}_{1}$ after passing to dual groups and applying the map $i: \mathrm{GSpin}_{4} \rightarrow \mathrm{GSp}_{4}$ . After fixing the choice, we can normalize the transfer factors such that their value is equal to $-1$ when evaluated on the strongly regular elements in norm correspondence between $D^{*} \times \mathrm{GL}_{1}$ and the other choice of Levi. However, the two choices determine transfer factors that differ up to a sign, since they are interchanged under the unique outer automorphism of $\mathrm{GSO}_{2,2}$ which switches the two factors, as remarked in [Reference Chan and GanCG15, Section 4.4]. Then for the endoscopic character identities in [Reference Chan and GanCG15, Proposition 11.1(2)] an ad hoc choice is made for which Levi to match it with at the beginning of [Reference Chan and GanCG15, Section 11]. On the other hand, as mentioned above Kaletha normalizes his choice by realizing $\mathrm{GU}_{2}(D)$ as an extended pure inner twist of $\mathrm{GSp}_{4}$ , where one obtains an identity

\begin{align*} \Delta[\mathfrak{m},\mathfrak{c},b_{1}](\gamma,\delta) = \Delta[\mathfrak{m},\mathfrak{c}](\gamma,\delta^{*})\langle \mathrm{inv}[b_{1}](\delta^{*},\delta), s \rangle. \end{align*}

Here, $\gamma$ and $\delta$ are strongly regular elements of H(L) and $\mathrm{GU}_{2}(L)$ , respectively, and the terms $\delta^{*} \in \mathrm{GSp}_{4}(F)$ and $\langle \mathrm{inv}[b_{1}](\delta^{*},\delta), s \rangle$ are as constructed in the discussion around [Reference KalethaKal16, Equation 6] (here, Kaletha is discussing pure inner twists as opposed to extended pure inner twists, but the constructions are almost identical). The element $\langle \mathrm{inv}[b_{1}](\delta^{*},\delta), s \rangle$ is a certain cocycle evaluated on the element s, and, by virtue of s being an element of order 2 in the cases of interest, this is given by $\pm 1$ . This sign can in general depend on $(\delta^{*},\delta)$ , but if $H = G$ in the endoscopic datum $\mathfrak{c}$ then its value is determined by evaluating $\kappa(b_{1}) \in X^{*}(Z(\hat{G})^{\Gamma})$ on the element $s \in Z(\hat{G})^{\Gamma}$ (see [Reference Hansen, Kaletha and WeinsteinHKW22, Fact 3.2.3]). By studying the restriction of the construction of $\langle \mathrm{inv}[b_{1}](\delta^{*},\delta), s \rangle$ to Levi subgroups (where one can check that if $(\gamma,\delta)$ lie in Levis which transfer to each other then $\delta^{*}$ can also be chosen to lie in the associated Levi of $\mathrm{GSp}_{4}$ ) and using the previous observations, one can conclude that the transfer factors used in [Reference KalethaKal16] and [Reference Chan and GanCG15] differ by a sign. More specifically, we can for example look at the inclusion of the Levi $M = \mathrm{GL}_{2} \times \mathrm{GL}_{1} \subset \mathrm{GSp}_{4}$ corresponding to the Siegel parabolic. This induces a map of Kottwitz sets $B(\mathrm{GL}_{2} \times \mathrm{GL}_{1}) \rightarrow B(\mathrm{GSp}_{4})$ . The element $b_{1}$ admits a (necessarily unique) basic lift to an element $b_{1,M} \in B(M)_{\mathrm{basic}}$ with the property that $J_{b_{1,M}} \simeq D^{*} \times \mathrm{GL}_{1}$ (see [Reference Hamann and ImaiHI24, Lemma 4.24]). This defines an inner twisting $(\xi_{M},b_{1,M}): \mathrm{GL}_{2} \times \mathrm{GL}_{1} \rightarrow D^{*} \times \mathrm{GL}_{1}$ , which will be compatible with the inner twisting by $(\xi,b_{1})$ in the obvious sense. If $\delta^{*} \in \mathrm{GSp}_{4}(F)$ is an element which lies in the Siegel Levi $\mathrm{GL}_{2}(F) \times \mathrm{GL}_{1}(F)$ then the element $\delta$ can also be chosen to lie in $D^{*} \times \mathrm{GL}_{1}$ , and the value of $\langle \mathrm{inv}[z](\delta^{*},\delta), s \rangle$ in this case is equal to $\kappa(b_{1,M}) \in X^{*}(Z(\hat{M}))$ on the element $s \in Z(\hat{G}) \rightarrow Z(\hat{M})$ as in the case where $H = G$ , and again this will just be some sign since s has order 2. We then can match the sign of the transfer functors up with the ad hoc choice fixed by Chan and Gan in the endoscopic case after restricting to Levis by evaluating both transfer factors on strongly regular elements $(\gamma,\delta)$ which are in norm correspondence between $\mathrm{GL}_{2} \times \mathrm{GL}_{1} \subset \mathrm{GSO}_{2,2}$ or $\mathrm{GL}_{2} \times \mathrm{GL}_{1} \subset \mathrm{GSO}_{2,2}$ , depending on the rather inexplicit choice Chan and Gan make for the statement of [Reference Chan and GanCG15, Proposition 11.1(2)] and then dividing out the Weyl discriminant term in Kaletha’s normalization. Determining the precise sign, however, would require carefully tracing through all the constructions.

The above argument shows the refined local Langlands correspondence of Kaletha holds for the group G; however, in order to describe the exact matching between $\rho_{+-}$ and $\rho_{-+}$ with $\tau_{1}$ and $\tau_{2}$ one needs to exactly compare the transfer factors. The above argument only shows that there is some matching; nonetheless, for our purposes the choice ends up being irrelevant, so we denote the representations in the L-packet $\Pi_{\phi}(J)$ corresponding to the projections $\tau_{1}$ and $\tau_{2}$ by $\rho_{1}$ and $\rho_{2}$ , respectively. Similarly, for the representations obtained by pre-composing a character with the composition

\begin{align*} S_{\phi} \rightarrow A_{\phi}, \end{align*}

we denote the elements of the L-packet $\Pi_{\phi}(G)$ corresponding to the trivial (non-trivial) character of $A_{\phi}$ by $\pi^{+}$ (respectively $\pi^{-}$ ). We note that, by Theorem 3.1(2), $\pi^{+}$ can be characterized by the unique $\mathfrak{m}$ -generic representation of this L-packet.

Definition 3.7. Given a discrete L-parameter $\phi$ as above and $\rho \in \Pi_{\phi}(J)$ , we let $\tau_{\rho}$ be the irreducible representation of $S_{\phi}$ associated with it via the matching described above. Given $\pi \in \Pi_{\phi}(G)$ and $\rho \in \Pi_{\phi}(J)$ , we set

\begin{align*} \delta_{\pi,\rho} := \tau_{\pi}^{\vee} \otimes \tau_{\rho}, \end{align*}

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

4.1 Overview of the Fargues–Scholze local Langlands correspondence

For now, let G be any connected reductive group over $\mathbb{Q}_{p}$ . Since we are going to be using geometric Satake, we fix a choice of the square root of p in $\overline{\mathbb{Q}}_{\ell}$ , so that half Tate twists are well defined. For us, we will always take $i^{-1}(\sqrt{p})$ , where $i: \overline{\mathbb{Q}}_{\ell} \xrightarrow{\simeq} \mathbb{C}$ is the fixed isomorphism. Fargues and Scholze [Reference Fargues and ScholzeFS21] consider the moduli space of G-bundles on the Fargues–Fontaine curve X, denoted $\mathrm{Bun}_{G}$ . This moduli space is an Artin v-stack (in the sense of [Reference Fargues and ScholzeFS21, Section IV.I]) and has the structure that the underlying points of its topological space $|\mathrm{Bun}_{G}|$ are in natural bijection with elements of the Kottwitz set B(G), where the slopes of the G-isocrystal associated with $b \in B(G)$ are the negatives of the slopes of the associated vector bundle $\mathcal{E}_{b}$ . Moreover, the specializations between points of $|\mathrm{Bun}_{G}|$ ire dictated by the partial ordering on B(G) induced by the kappa invariant and the slope homomorphism [Reference ViehmannVie21]. In particular, the connected components of $\mathrm{Bun}_{G}$ are in bijection with $B(G)_{\rm basic} \xrightarrow{\simeq} \pi_{1}(G)_{\Gamma}$ . Specifically, for any $b \in B(G)_{\rm basic}$ , there is a unique open HN-strata $\mathrm{Bun}_{G}^{b} \subset \mathrm{Bun}_{G}$ dense inside the associated connected component. We recall that the elements of $B(G)_{\rm basic}$ parametrize extended pure inner forms of G, via sending an element $b \in B(G)_{\rm basic}$ to its $\sigma$ -centralizer $J_{b}/\mathbb{Q}_{p}$ . For such a basic b, we have an identification $\mathrm{Bun}_{G}^{b} \simeq [\ast/\underline{J_{b}(\mathbb{Q}_{p})}] =: B\underline{J_{b}(\mathbb{Q}_{p})}$ of the HN-strata defined by b and the classifying stack of $J_{b}(\mathbb{Q}_{p})$ . For any Artin v-stack Z, Fargues and Scholze define a triangulated category $\mathrm{D}_{\blacksquare}(Z,\overline{\mathbb{Q}}_{\ell})$ of solid $\overline{\mathbb{Q}}_{\ell}$ -sheaves [Reference Fargues and ScholzeFS21, Section VII.1] and isolate a nice full subcategory $\mathrm{D}_{\mathrm{lis}}(Z,\overline{\mathbb{Q}}_{\ell}) \subset \mathrm{D}_{\blacksquare}(Z,\overline{\mathbb{Q}}_{\ell})$ of lisse-étale $\overline{\mathbb{Q}}_{\ell}$ -sheaves [Reference Fargues and ScholzeFS21, Section VII.6.], which may be roughly (up to the difference between the $\ell$ -adic and discrete topology) thought of as the unbounded derived category of étale $\overline{\mathbb{Q}}_{\ell}$ sheaves on Z, where one has made an enlargement to capture information about the topology of p-adic groups. In any case, the key point for us is that we have the following basic result.

Lemma 4.1 [Reference Fargues and ScholzeFS21, Proposition VII.7.1]. There is an equivalence of categories

\begin{align*} \mathrm{D}_{\mathrm{lis}}(B\underline{J_{b}(\mathbb{Q}_{p})}),\overline{\mathbb{Q}}_{\ell})\simeq \mathrm{D}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell}), \end{align*}

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

Lemma 4.1 tells us that, given an irreducible smooth representation $\pi$ of $G(\mathbb{Q}_{p})$ , we can consider the associated sheaf, denoted $\mathcal{F}_{\pi}$ , on $\mathrm{Bun}_{G}^{\mathbf{1}}$ the open HN-strata corresponding to the trivial element $\mathbf{1} \in B(G)$ , and take the extension by zero along the open inclusion $j_{!}(\mathcal{F}_{\pi})$ Footnote ² . This realizes the representation $\pi$ in terms of a sheaf on the moduli space $\mathrm{Bun}_{G}$ in an analogous way to how the function-sheaf dictionary realizes cuspidal automorphic forms as functions associated with sheaves in the context of curves over finite fields. Following Lafforgue [Reference LafforgueLaf18], Fargues and Scholze construct a semisimple L-parameter associated with this sheaf by looking at the action of the excursion algebra on this category $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ . This relies on a form of the geometric Satake correspondence for the $B_{dR}^{+}$ -affine Grassmannians. For any finite set I, let $X^{I}$ be the product of I-copies of the diamond $X = \mathop{\textrm{Spd}}(\breve{\mathbb{Q}}_{p})/\mathrm{Frob}^{\mathbb{Z}}$ . We then have the global Hecke stack

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

\begin{align*} \beta:\mathcal{E}_{1}|_{X_{S} \setminus \bigcup_{i \in I} S_{i}^{\sharp}} \xrightarrow{\simeq} \mathcal{E}_{2}|_{X_{S} \setminus \bigcup_{i \in I} S_{i}^{\sharp}},\end{align*}

meromorphic along the $S_{i}^{\sharp}$ , where $h^{\leftarrow}((\mathcal{E}_{1},\mathcal{E}_{2},i,(S_{i}^{\sharp})_{i\in I})) = \mathcal{E}_{1}$ and $h^{\rightarrow} \times supp((\mathcal{E}_{1},\mathcal{E}_{2},\beta, (S_{i}^{\sharp})_{i \in I})) = (\mathcal{E}_{2},(S_{i}^{\sharp})_{i \in I})$ , as in [Reference Fargues and ScholzeFS21, p.320]. We set $\phantom{}^{L}G^{I}$ to be I-copies of the Langlands dual group of G, i.e. $^{L}G = Q \ltimes \hat{G}(\overline{\mathbb{Q}}_{\ell})$ , where $\hat{G}$ is the reductive group having dual root datum to G and is viewed as a reductive group over $\overline{\mathbb{Q}}_{\ell}$ . The Weil group acts on $\hat{G}$ via the induced action on root datum through some finite quotient Q, which we now fix. Let $\mathrm{Rep}_{\overline{\mathbb{Q}}_{\ell}}(^{L}G^{I})$ denote the category of algebraic $\overline{\mathbb{Q}}_{\ell}$ -representations of I-copies of $^{L}G$ . For each element $W \in \mathrm{Rep}_{\overline{\mathbb{Q}}_{\ell}}(^{L}G^{I})$ , the geometric Satake correspondence of Fargues and Scholze [Reference Fargues and ScholzeFS21, Chapter VI] furnishes a solid $\overline{\mathbb{Q}}_{\ell}$ -sheaf $\mathcal{S}_{W}$ on the global Hecke Hck [Reference Fargues and ScholzeFS21, p.319]. This is given by pulling back along the natural map $\mathrm{Hck} \rightarrow \mathcal{H}\mathrm{ck}$ from the global Hecke stack to the local one, defined by restricting a modification to a formal neighborhood of the Cartier divisor (as in [Reference Fargues and ScholzeFS21, p. 16]), where the Satake category is by definition an abelian subcategory of all sheaves on $\mathcal{H}\mathrm{ck}_{G}$ . This sheaf on the global Hecke stack allows us to define Hecke operators.

Definition 4.3. For each $W \in \mathrm{Rep}_{\overline{\mathbb{Q}}_{\ell}}(^{L}G^{I})$ , we define the Hecke operator

\begin{align*} T_{W}: \mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathrm{D}_{\blacksquare}(\mathrm{Bun}_{G} \times X^{I}), \end{align*}

\begin{align*} A \mapsto R(h^{\rightarrow} \times supp)_{\natural}(h^{\leftarrow *}(A) \otimes^{\mathbb{L}} \mathcal{S}_{W}),\end{align*}

where $\mathcal{S}_{W}$ is a solid $\overline{\mathbb{Q}}_{\ell}$ -sheaf and the functor $R(h^{\rightarrow} \times supp)_{\natural}$ is the natural push forward. That is, the left adjoint to the restriction functor in the category of solid $\overline{\mathbb{Q}}_{\ell}$ -sheaves [Reference Fargues and ScholzeFS21, Proposition VII.3.1].

Remark 4.4. These satisfy various compatibilities with respect to composition and restriction to the diagonal. In particular, given two representations $V,W \in \mathrm{Rep}_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G)$ , we have that

\begin{align*} (T_{V} \times {\rm id})(T_{W})(\cdot)|_{\Delta} \simeq T_{V \otimes W}(\cdot), \end{align*}

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

We then consider $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})^{BW_{\mathbb{Q}_{p}}^{I}}$ , the category of objects in $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ with continuous action by $W_{\mathbb{Q}_{p}}^{I}$ . Examples of objects in this category are objects of $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ tensored by a continuous representation of $W_{\mathbb{Q}_{p}}^{I}$ , for a more precise description see [Reference Fargues and ScholzeFS21, Section IX.1]. With this in hand, we then have the following theorem of Fargues and Scholze.

Theorem 4.5 [Reference Fargues and ScholzeFS21, Theorem I.7.2, Proposition IX.2.1, Corollary IX.2.3]. The Hecke operator $T_{W}$ for $W \in \mathrm{Rep}_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G^{I})$ ,

\begin{align*} T_{W}: \mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathrm{D}_{\blacksquare}(\mathrm{Bun}_{G} \times X^{I}), \end{align*}

induces a functor

\begin{align*} \mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})^{BW_{\mathbb{Q}_{p}}^{I}} \end{align*}

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

From now on, when talking about Hecke operators we shall always refer to this induced functor, which we will also abusively denote by $T_{W}$ . Theorem 4.5 has direct implications for the cohomology of local Shimura varieties. To study this, consider a minuscule cocharacter $\mu$ with field of definition E, and let $b \in B(G,\mu)$ be the unique basic element in the $\mu$ -admissible locus (see [Reference Rapoport and ViehmannRV14, Definition 2.3]). We say that the triple $(G,b,\mu)$ defines a local Shimura datum in the sense of Rapoport and Viehmann [Reference Rapoport and ViehmannRV14]. Attached to such data, Scholze and Weinstein [Reference Scholze and WeinsteinSW20] construct a tower of diamonds

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

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

Definition 4.7. Let $\mathrm{Sht}(G,b,\mu)_{K,\mathbb{C}_{p}}$ be the base change of the above tower to $\mathbb{C}_{p}$ . We define the complex

\begin{align*} R\Gamma_{c}(G,b,\mu) := \mathop{\textrm{colim}}\nolimits_{K \rightarrow \{1\}} R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{K,\mathbb{C}_{p}},\mathbb{Z}_{\ell}), \end{align*}

a colimit of $W_{E} \times J_{b}(\mathbb{Q}_{p})$ -modules with a $G(\mathbb{Q}_{p})$ -action, where $W_{E}$ is the Weil group of E. The complex $R\Gamma_{c}(\mathrm{Sht}(G,b,\mu)_{K,\mathbb{C}_{p}},\mathbb{Z}_{\ell})$ is naturally a smooth $J_{b}(\mathbb{Q}_{p})$ -representation with a continuous action of $W_{E}$ , by [Reference Fargues and ScholzeFS21, Theorem IX.3.1] (cf. [Reference HansenHan20, Proposition 2.3]). A priori it only has an action by the inertia group, but this space admits a non-effective Frobenius descent datum. We then define, for $\rho$ a smooth admissible $J_{b}(\mathbb{Q}_{p})$ -representation, the complex

\begin{align*} R\Gamma_{c}(G,b,\mu)[\rho] := \mathop{\textrm{colim}}\nolimits_{K \rightarrow \{1\}} R\Gamma_{c}(G,b,\mu)_{K} \otimes^{\mathbb{L}}_{\mathcal{H}(J_{b})} \rho, \end{align*}

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

\begin{align*} R\Gamma_{c}^{\flat}(G,b,\mu)[\rho] := \mathop{\textrm{colim}}\nolimits_{K \rightarrow \{1\}} R\mathcal{H}om_{J_{b}(\mathbb{Q}_{p})}(R\Gamma_{c}(G,b,\mu)_{K},\rho)[-2d](-d). \end{align*}

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

To study these complexes, we specialize the above discussion of Hecke operators to the case where $W = V_{\mu^{-1}}$ is specified by the highest weight representation of highest weight $\mu^{-1}$ a dominant inverse of $\mu$ and $I = \{\ast\}$ is a singleton. The sheaf $\mathcal{S}_{W}$ will then be supported on the closed subspace $\mathrm{Hck}_{\leq \mu^{-1}} = \mathrm{Hck}_{\mu^{-1}}$ of Hck, parametrizing modifications with meromorphy bounded by or equal to $\mu^{-1}$ , where the equality follows by the minuscule assumption. The space $\mathrm{Hck}_{\mu^{-1}}$ is cohomologically smooth of dimension $d := \langle 2\rho_{G},\mu \rangle$ , and the sheaf $\mathcal{S}_{W}$ , as in the geometric Satake correspondence of [Reference Mirković and VilonenMV07], behaves like the intersection cohomology of this space, so we have $\mathcal{S}_{W} \simeq \overline{\mathbb{Q}}_{\ell}[d]({d}/{2})$ . This implies that, to study the action of the Hecke operator $T_{W}$ on $\mathrm{Bun}_{G}$ , we can look at the restriction of the diagram defining the Hecke correspondence to this subspace.

In particular, we have an isomorphism:

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

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

$j_{!}(\mathcal{F}_{\pi}).$

Then the fiber of $\mathrm{Hck}_{\mu^{-1}}$ of $h_{\mu^{-1}}^{\leftarrow}$ over $\mathrm{Bun}_{G}^{\mathbf{1}}$ is identified with

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

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

\begin{align*} T_{\mu^{-1}}j_{!}(\mathcal{F}_{\pi}) \end{align*}

is then supported on the HN-strata given by the Kottwitz elements in $B(G,\mu)$ since, by [Reference RapoportRap18, Proposition A.9], any G-bundle occurring as a modification of type $\mu^{-1}$ of the trivial bundle has associated Kottwitz element lying in this set. We then consider the restriction

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

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

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

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

\begin{align*} \mathrm{Sht}(G,b,\mu)_{\infty} \rightarrow Gr_{G,\mu^{-1}}^{b} \end{align*}

over this Newton strata. Using this description of the infinite level Shimura variety, it then follows from base change (see [Reference Fargues and ScholzeFS21, Chapter IX.3] for details) that we have an isomorphism,

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

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

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

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

Corollary 4.9. Given a local Shimura datum $(G,b,\mu)$ as above and $\pi$ (respectively $\rho$ ) a smooth irreducible representation of $G(\mathbb{Q}_{p})$ (respectively $J_{b}(\mathbb{Q}_{p})$ ). There exists an isomorphism

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

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

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

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

We have the following basic structural result which, in more generality, follows from the analysis in Fargues and Scholze, but, in the case of a local Shimura datum, also partially follows from standard finiteness results for rigid spaces (see [Reference Rapoport and ViehmannRV14, Section 6]). In particular, one can show the following.

Fargues and Scholze use the endofunctors defined by the Hecke algebra on $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ to define the excursion algebra.

Definition 4.12. For a finite set I, a representation $W \in \mathrm{Rep}_{\overline{\mathbb{Q}}_{\ell}}(^LG^{I})$ , maps $\alpha: \overline{\mathbb{Q}}_{\ell} \rightarrow \Delta^{*}W$ and $\beta: \Delta^{*}W \rightarrow \overline{\mathbb{Q}}_{\ell}$ , and elements $\gamma_{i} \in W_{\mathbb{Q}_{p}}$ for $i \in I$ , one defines the excursion operator on $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ to be the composition

\begin{align*} {\rm id} = T_{\overline{\mathbb{Q}}_{\ell}}\xrightarrow{\alpha} T_{\Delta^{*}W} = T_{W}\xrightarrow{(\gamma_{i})_{i \in I}} T_{W} = T_{\Delta^{*}W}\xrightarrow{\beta} T_{\overline{\mathbb{Q}}_{\ell}} = {\rm id}, \end{align*}

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

This defines a natural endomorphism of the identity functor on $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ . If one looks at the induced endofunctor given by the inclusion $\mathrm{D}(G(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell}) \subset \mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ induced by the open immersion $j: \mathrm{Bun}_{G}^{\mathbf{1}} \hookrightarrow \mathrm{Bun}_{G}$ then one obtains a natural endomorphism of the identity functor on $\mathrm{D}(G(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$ . In other words, we get a family of compatible endomorphisms for all complexes of smooth representations of $G(\mathbb{Q}_{p})$ ; namely, an element of the Bernstein center. One can verify that this excursion algebra satisfies similar properties to that considered by Lafforgue, so, using Lafforgue’s reconstruction theorem [Reference LafforgueLaf18, Proposition 11.7], one can show the following.

Theorem 4.13. To an irreducible smooth $\overline{\mathbb{Q}}_{\ell}$ -representation $\pi$ of $G(\mathbb{Q}_{p})$ (or more generally $A \in \mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ any Schur-irreducible object (i.e. $\mathrm{End}(A) = \overline{\mathbb{Q}}_{\ell}$ )), there is a unique continuous semisimple map

\begin{align*} \phi^{\mathrm{FS}}_{\pi}: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \end{align*}

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

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

By further studying the geometry of $\mathrm{Bun}_{G}$ and the Hecke stacks, one can deduce various good properties of this correspondence.

Theorem 4.14 [Reference Fargues and ScholzeFS21, Theorem I.9.6]. The mapping defined above,

\begin{align*} \pi \mapsto \phi^{\mathrm{FS}}_{\pi}, \end{align*}

enjoys the following properties:

1. (1) (Compatibility with local class field theory.) If $G = T$ is a torus, then $\pi \mapsto \phi_{\pi}$ is the usual local Langlands correspondence

2. (2) The correspondence is compatible with character twists, passage to contragredients, and central characters.

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

   \begin{align*} \pi_{1} \boxtimes \pi_{2} \mapsto \phi^{\mathrm{FS}}_{\pi_{1}} \times \phi^{\mathrm{FS}}_{\pi_{2}}\end{align*}
   under the Fargues–Scholze local Langlands correspondence for $G_{1} \times G_{2}$ .

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

   \begin{align*} W_{\mathbb{Q}_{p}}\xrightarrow{\phi^{\mathrm{FS}}_{\pi_{M}}} {} ^{L}M(\overline{\mathbb{Q}}_{\ell}) \rightarrow ^{L}G(\overline{\mathbb{Q}}_{\ell}), \end{align*}
   where the map $\phantom{}^{L}M(\overline{\mathbb{Q}}_{\ell}) \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$ is the natural embedding.

5. (5) (Compatibility with Harris–Taylor/Henniart LLC.) For $G = \mathrm{GL}_{n}$ or an inner form of G the semisimple L-parameter associated with $\pi$ is the (semisimplified) parameter $\phi_{\pi}^{\mathrm{ss}}$ associated with $\pi$ by Harris and Taylor/Henniart.

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

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

4.2 The spectral action

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

\begin{align*} \phi: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \end{align*}

commuting with the natural projection to Q. Assuming the Frobenius semisimplicity of $\phi$ , one can use the classical construction of Deligne and Grothendieck to see that this coincides with the definition given in § 3 for $\mathrm{GSp}_{4}$ after applying the isomorphism i, where the monodromy operation is recovered through the exponential of the action of $W_{\mathbb{Q}_{p}}$ on the $\ell$ -power roots of unity. Such a continuous map can be thought of as a continuous 1-cocycle $W_{\mathbb{Q}_{p}} \rightarrow \hat{G}(\overline{\mathbb{Q}}_{\ell})$ , with respect to the action of $W_{\mathbb{Q}_{p}}$ on $\hat{G}(\overline{\mathbb{Q}}_{\ell})$ . If we let $A/\mathbb{Z}_{\ell}$ be any $\mathbb{Z}_{\ell}$ -algebra endowed with a topology given by writing $A = \mathop{\textrm{colim}}\nolimits_{A' \subset A} A'$ , where A’ is a finitely generated $\mathbb{Z}_{\ell}$ -module with its $\ell$ -adic topology, then we can defined a moduli space, denoted $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})$ , over $\mathbb{Z}_{\ell}$ , whose A-points are the continuous 1-cocycles $W_{\mathbb{Q}_{p}} \rightarrow \hat{G}(A)$ with respect to the natural action of $W_{\mathbb{Q}_{p}}$ on $\hat{G}(A)$ . This defines a scheme considered in [Reference Dat, Helm, Kurinczuk and MossDHKM20] and [Reference ZhuZhu20] which, by [Reference Fargues and ScholzeFS21, Theorem I.8.1], can be written as a union of open and closed affine subschemes $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}}/P,\hat{G})$ as P runs through subgroups of wild inertia of $W_{E}$ , where each $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}}/P,\hat{G})$ is a flat local complete intersection over $\mathbb{Z}_{\ell}$ of dimension $\mathrm{dim}(G)$ . This allows us to consider the Artin stack quotient $[\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})/\hat{G}]$ , where $\hat{G}$ acts via conjugation. We then consider the base change to $\overline{\mathbb{Q}}_{\ell}$ , denoted $[\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]$ and referred to as the stack of Langlands parameters, as well as the category $\mathrm{Perf}([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])$ of perfect complexes on this space. We let $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ denote the triangulated sub-category of compact objects in $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ (which are precisely the objects with quasi-compact support on $\mathrm{Bun}_{G}$ and which restrict to compact objects in $\mathrm{D}(J_{b}(\mathbb{Q}_{p}),\overline{\mathbb{Q}}_{\ell})$ for all $b \in B(G)$ by [Reference Fargues and ScholzeFS21, Theorem V.4.1, Proposition VII.7.4]). We then have the key theorem of Fargues and Scholze.

Theorem 4.16 [Reference Fargues and ScholzeFS21, Corollary X.I.3]. There exists a natural compactly supported $\overline{\mathbb{Q}}_{\ell}$ -linear action of $\mathrm{Perf}(\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G})$ on $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ satisfying the property that the restriction along the map

\begin{align*} \mathrm{Rep}_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G) \rightarrow \mathrm{Perf}([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}])^{BW_{\mathbb{Q}_{p}}} \end{align*}

induces the action of Hecke operators

\begin{align*} \mathrm{Rep}_{\overline{\mathbb{Q}}_{\ell}}(\phantom{}^{L}G^{I}) \rightarrow \mathrm{End}(\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})^{\omega})^{BW_{\mathbb{Q}_{p}}^{I}} \end{align*}

for a varying finite index set I.

To study this spectral action, we consider, as in [Reference Fargues and ScholzeFS21, Section VIII.3.], the coarse quotient in the category of schemes

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

of $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}$ by the action of $\hat{G}$ via conjugation. Given an L-parameter $\phi:W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$ , it follows by [Reference Dat, Helm, Kurinczuk and MossDHKM20, Proposition 4.13] or [Reference Fargues and ScholzeFS21, Proposition VIII.3.2], that the $\hat{G}$ -orbit of $\phi$ defines a closed $\overline{\mathbb{Q}}_{\ell}$ -point of $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}} / / \hat{G}$ if and only if $\phi$ is a semisimple parameter. Moreover, the natural map

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

evaluated on a $\overline{\mathbb{Q}}_{\ell}$ -point in the stack quotient defined by an L-parameter $\phi$ defines a closed $\overline{\mathbb{Q}}_{\ell}$ -point in the coarse moduli space given by its semisimplification $\phi^{\mathrm{ss}}$ . We can fit excursion operators into this picture as follows. We let $\mathcal{Z}^{\rm spec}(G,\overline{\mathbb{Q}}_{\ell}) := \mathcal{O}(\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}})^{\hat{G}}$ be the ring of functions on the stack of L-parameters/the coarse moduli space, which we refer to as the spectral Bernstein center. As noted above, the excursion operators define a family of commuting endomorphisms of the identity functor on $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ . We let $\mathcal{Z}^{\rm geom}(G,\overline{\mathbb{Q}}_{\ell})$ be the ring of such endomorphisms as in [Reference Fargues and ScholzeFS21, Definition IX.0.2], which we refer to as the geometric Bernstein center. In [Reference Fargues and ScholzeFS21, Corollary IX.0.3], Fargues and Scholze construct a canonical map of rings

\begin{align*} \mathcal{Z}^{\rm spec}(G,\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathcal{Z}^{\rm geom}(G,\overline{\mathbb{Q}}_{\ell}), \end{align*}

which is given by excursion operators in the following sense. By [Reference Fargues and ScholzeFS21, Theorem VIII.3.6], there is an identification between $\mathcal{Z}^{\rm spec}(G,\overline{\mathbb{Q}}_{\ell})$ and the algebra of excursion operators. In particular, an excursion operator, as in Definition 4.12, associated with the datum I, W, $\alpha$ , $\beta$ , and $\gamma_{i} \in W_{\mathbb{Q}_{p}}$ for $i \in I$ defines a function $f_{I,W,\alpha,\beta,(\gamma_{i})_{i \in I}} \in \mathcal{O}_{[Z^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G}]} = \mathcal{Z}^{\rm spec}(G,\overline{\mathbb{Q}}_{\ell})$ on $\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}} / / \hat{G}$ , whose evaluation on the closed point of the coarse moduli space associated with a semisimple parameter $\phi: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell})$ is precisely the scalar that results from the endomorphism

\begin{align*} \overline{\mathbb{Q}}_{\ell} \xrightarrow{\alpha} \Delta^{*}W = W \xrightarrow{(\phi(\gamma_{i}))_{i \in I}} W = \Delta^{*}W \xrightarrow{\beta} \overline{\mathbb{Q}}_{\ell}. \end{align*}

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

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

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

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

which will be precisely the scalar given by evaluating $\phi_{A}^{\mathrm{FS}}$ on the excursion datum (see [Reference ZouZou22, Theorem 5.2]). In this way, we see that the action of excursion operators can be obtained through the spectral action. We will leverage this interpretation of excursion operators to prove the following key lemma.

Proof. If we look at the action on $A \in \mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$ via the map

\begin{align*} \mathcal{Z}^{\rm spec}(G,\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathcal{Z}^{\rm geom}(G,\overline{\mathbb{Q}}_{\ell}), \end{align*}

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

\begin{align*} C \star A \in \mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell}), \end{align*}

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

\begin{align*} C \star A \rightarrow C \star A, \end{align*}

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

\begin{align*} \mathcal{Z}^{\rm spec}(G,\overline{\mathbb{Q}}_{\ell}) \rightarrow \mathcal{Z}^{\rm geom}(G,\overline{\mathbb{Q}}_{\ell}), \end{align*}

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

To take advantage of this lemma, we now introduce the following endofunctors of $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ , which are analogues of the averaging operators considered by [Reference Anschütz and Le-BrasALB21] in the Fargues–Scholze geometric Langlands correspondence for $\mathrm{GL}_{n}$ and by [Reference Frenkel, Gaitsgory and VilonenFGV02, Reference GaitsgoryGai04] in the classical geometric Langlands correspondence over function fields.

Definition 4.19. Let $\phi$ be a representation of $W_{\mathbb{Q}_{p}}$ and V a representation of $\phantom{}^{L}G$ with $T_{V}$ the associated Hecke operator. We consider the endofunctor of $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})$

\begin{align*} A \mapsto R\Gamma(W_{\mathbb{Q}_{p}},T_{V}(A) \otimes \phi^{\vee}),\end{align*}

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

We now would like to realize the functor $Av_{V,\phi}$ as the spectral action of an object in $\mathrm{Perf}([\mathcal{Z}^{1}(W_{\mathbb{Q}_{p}},\hat{G})_{\overline{\mathbb{Q}}_{\ell}}/\hat{G})])$ similar to [Reference Anschütz and Le-BrasALB21, Section 5.5]. An obvious guess would be that one should take the vector bundle $C_{V}$ corresponding to the Hecke operator $T_{V}$ , as in Remark 4.17(1), and then twist this by the constant sheaf defined by $\phi^{\vee}$ , which we denote by

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

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

\begin{align*} \tilde{\phi}: W_{\mathbb{Q}_{p}} \rightarrow \phantom{}^{L}G(\overline{\mathbb{Q}}_{\ell}) \xrightarrow{r_{V}} GL(V),\end{align*}

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

\begin{align*} (C_{V} \otimes \phi^{\vee})^{P}, \end{align*}

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

(4.1)

This gives a presentation of $\mathcal{A}v_{\phi,V}$ as a perfect complex, as in [Reference Anschütz and Le-BrasALB21, p. 21]. To see this, note that $\sigma$ and $\tau$ generate a dense discrete subgroup of $W_{\mathbb{Q}_{p}}/P$ subject to the relationship that $\sigma^{-1}\tau\sigma = \tau^{p}$ , and it follows that the limit of the diagram is just $((C_{V} \otimes \phi^{\vee})^{P})^{W_{\mathbb{Q}_{p}}/P}$ and that the homotopy limit is in turn the derived functor $R\Gamma(W_{\mathbb{Q}_{p}},C_{V} \otimes \phi^{\vee}) \simeq R\Gamma(W_{\mathbb{Q}_{p}}/P,(C_{V} \otimes \phi^{\vee})^{P})$ , where we have used the vanishing of the higher Galois cohomology with respect to P for the last isomorphism.

Then we have the following lemma which is a verbatim generalization of [Reference Anschütz and Le-BrasALB21, Lemma 5.7].

Lemma 4.20. There exists a canonical identification

\begin{align*} Av_{V,\phi}(-) \simeq \mathcal{A}v_{V,\phi} \star (-)\end{align*}

of endofunctors of $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\overline{\mathbb{Q}}_{\ell})^{\omega}$ .

However, if we take the homotopy limit of the diagram in (4.1), the claim follows from the fact that the spectral action commutes with homotopy limits, as noted in Remark 4.17(3).

With this identification in hand, we can apply Lemma 4.18 to prove the following key consequence.