An approach to the characterization of the local Langlands correspondence

Abstract

Scholze and Shin [J. Amer. Math. Soc. 26 (2013), pp. 261–294] gave a conjectural formula relating the traces on the automorphic and Galois sides of a local Langlands correspondence. Their work generalized an earlier formula of Scholze, which he used to give a new proof of the local Langlands conjecture for $\mathrm{GL}_n$. Unlike the case for $\mathrm{GL}_n$, the existence of non-singleton $L$-packets for more general reductive groups constitutes a serious representation-theoretic obstruction to proving that such a formula uniquely characterizes such a correspondence. We show how to overcome this problem, and demonstrate that the Scholze–Shin equation is enough, together with other standard desiderata, to uniquely characterize the local Langlands correspondence for discrete parameters.

1. Introduction

In Sch13b, Scholze gave a new construction of the local Langlands correspondence for $\mathrm{GL}_n(F)$, for $F$ a $p$-adic field. A key component of Scholze’s analysis is that he was able to characterize his correspondence by an explicit equation relating his construction to certain functions $f_{\tau ,h}$ where $\tau$ is an element of the Weil group $W_F$, and $h$ is a certain cut-off function in the Hecke algebra for $\mathrm{GL}_n$. These functions were defined in terms of the cohomology of certain Berthelot tubes inside of Rapoport–Zink spaces.

A major appeal of Scholze’s characterization of the local Langlands correspondence for $\mathrm{GL}_n(F)$ is that it should be possible to generalize to the setting of a general reductive group $G$. In contrast, the standard characterization for the $\mathrm{GL}_n(F)$ case, as first described fully in Hen00, is specialized to work for $\mathrm{GL}_n(F)$. Similarly, the characterizations for classical groups following from Art13 use twisted endoscopy to reduce to the case of $\mathrm{GL}_n(F)$ where Henniart’s characterization can be applied. Unfortunately, many groups cannot be related to $\mathrm{GL}_n(F)$ via endoscopy and for these cases one needs a more general approach. Moreover, even in cases where this is possible, having a characterization internal to the group $G$ is desirable.

The two major complications of generalizing the results of Sch13b to arbitrary groups $G$ are:

(Q1)

how to generalize the functions $f_{\tau , h}$ of Sch13b and prove they satisfy analogous equations,

(Q2)

decide whether two constructions of the local Langlands correspondence satisfying the generalized equations must coincide.

The question in (Q1) has been considered by several authors. Namely, the functions $f_{\tau , h}$ were generalized in Sch13a to PEL/EL type cases (and in You21 to abelian type cases) and in SS13, Scholze and Shin give a precise conjecture generalizing the main equation in Sch13b. We call the generalized equation in SS13 the Scholze–Shin equation, which roughly is of the following form

$$\begin{equation} {} S\Theta _{\psi }(f_{\tau ,h}^{\mu })=\operatorname {tr}(\tau \mid r_{-{\mu }}\circ \psi ^\mathrm{ss})S\Theta _{\psi }(h). {\tag{1}} \end{equation}$$

Here $r_{-\mu }$ is a certain representation of the $L$-group ${^{L}G}=\widehat{G}\rtimes W_F$, where $S\Theta _\psi$ is the stable character of a Langlands parameter $\psi$, and $\psi ^\mathrm{ss}$ is the ‘semi-simplification’ of $\psi$ (see §3 for precise definitions). In SS13 it is proven that equations of this form hold in EL type cases and in BMY19, the authors prove the Scholze–Shin equations hold for discrete parameters of unramified unitary groups (with the local Langlands conjecture as in Mok15).

That said, the question in (Q2) does not seem to have been studied at all in SS13 or subsequent work. In the context of Sch13b this is not surprising, as (Q2) has a trivial solution there, but difficulties do arise for more general $G$ due to the existence of non-singleton $L$-packets.

The goal of this paper is to resolve (Q2), modulo some extra conditions, for a substantial class of groups in the case of discrete $L$-parameters. These are the $L$-parameters $\psi : W_F \times \mathrm{SL}_2(\mathbb{C}) \to {^{L}G}$ that do not factor through a proper Levi subgroup of ${^{L}G}$. The discrete $L$-parameters are conjecturally those whose $L$-packet consists entirely of discrete series representations. This is a natural class of parameters to consider since their conjectured properties are well understood. We propose a list of axioms for an assignment of packets to discrete $L$-parameters and call such an assignment a discrete local Langlands correspondence (see §3 for a precise definition). Our axioms consist of standard desiderata except that we additionally impose that Equation 1 is satisfied.

Theorem 1.1 (Imprecise version of Theorem 3.4).

Let $G$ be a ‘good’ reductive group over $F$ and suppose $\Pi ^i$ for $i=1,2$ are discrete local Langlands correspondences for $G$ which satisfy the Scholze–Shin equations with respect to the same set of functions $\{f^{\mu }_{\tau ,h}\}$. Then, $\Pi ^1=\Pi ^2$.

The phrase ‘good’ in the above theorem statement means, roughly, that the semi-simplified parameters $\psi ^{\mathrm{ss}}$ of $G$ may be recovered from the compositions $r_{-\mu }\circ \psi ^{\mathrm{ss}}$ (see Definition 3.2). These include classes of groups such as general linear groups, odd special orthogonal groups, and unitary groups.

Remark 1.2.

From the perspective of the work of Kaletha (for instance Kal19) on supercuspidal $L$-packets, it is perhaps natural to consider supercuspidal local Langlands correspondences, instead of discrete ones. These have as their domain the equivalence classes of supercuspidal $L$-parameters (i.e. discrete $L$-parameters with trivial restriction to $\mathrm{SL}_2$) and their target finite subsets of supercuspidal representations. The results of this paper go through, verbatim, for this setup with one key difference. For technical reasons it becomes necessary to axiomatize which supercuspidal representations show up inside the $L$-packets of supercuspidal $L$-parameters. This question is subtle, in distinction to the discrete case, since there are supercuspidal representations which do not appear in the $L$-packet of any supercuspidal $L$-parameter. Kaletha proved in Kal19 that when $G$ splits over a tamely ramified extension of $F$ and $p$ does not divide the order of the Weyl group of $G$, then the answer is given by non-singular supercuspidal representations.

The proof of Theorem 1.1 proceeds by first reducing to the case where the $L$-packet of $\psi$ is a singleton using elliptic endoscopy. For singleton packets, one deduces equality of $\Pi ^1(\psi )$ and $\Pi ^2(\psi )$ from the Scholze–Shin equations, the atomic stability property, and the fact that two discrete parameters with isomorphic semi-simplifications are isomorphic. Atomic stability is roughly the property that the stable distributions $S\Theta _{\psi }$ for $\psi$ a discrete parameter are a basis for the stable distributions arising as linear combinations of discrete characters. We prove that our axiomatization of a discrete Langlands correspondence implies the atomic stability property.

Combining the above theorem with the aforementioned proof that the Scholze–Shin equations hold for the ‘discrete’ local Langlands conjecture as in Mok15, we obtain the following.

Theorem 1.3 (See Theorem 4.2).

The discrete local Langlands correspondence for unramified unitary groups as given by Mok15 is characterized by Theorem 1.1.

2. Notation and terminology

The following notation will be used throughout the rest of the paper unless stated otherwise.

Let $F$ be a $p$-adic local field. Fix an algebraic closure $\overline{F}$ and let $F^\mathrm{un}$ be the maximal unramified extension of $F$ in $\overline{F}$. Let $L$ be the completion of $F^\mathrm{un}$ and fix an algebraic closure $\overline{L}$.

Let $G$ be a (connected) reductive group over $F$. We denote by $G(F)^\mathrm{reg}$ the set of regular semisimple elements in $G(F)$ and by $G(F)^\mathrm{ell}$ the subset of elliptic regular semisimple elements. We denote by $D$, or $D_G$, the Harish-Chandra discriminant map on $G(F)$. If $\gamma ,\gamma '\in G(F)$ are stably conjugate we denote this by $\gamma \sim _\mathrm{st}\gamma '$.

Let $\widehat{G}$ be the connected Langlands dual group of $G$ and let ${^{L}G}$ be the Weil group version of the $L$-group of $G$ as defined in Kot84b, §1. We denote by $\mathrm{Irr}(G(F))$ the set of equivalence classes of irreducible smooth representations of $G(F)$, by $\mathrm{Irr}^{\mathrm{disc}}(G(F))$ the subset of equivalence classes of essentially square-integrable representations, and by $\mathrm{Irr}^\mathrm{sc}(G(F))$ the subset of equivalence classes of supercuspidal representations. For a finite group $C$ the notation $\mathrm{Irr}(C)$ means all irreducible $\mathbb{C}$-valued representations of $C$. Let $\mathrm{Groth}(G(F))$ denote the Grothendieck group of admissible representations of $G(F)$. Denote by $\mathcal{H}(G(F))$ the Hecke algebra of locally constant and compactly supported functions $G(F)\to \mathbb{C}$.

A discrete Langlands parameter is an $L$-parameter (see Bor79, §8.2) $\psi : W_F \times \mathrm{SL}_2(\mathbb{C}) \to {^{L}G}$ such that the image of $\psi$ is not contained in a proper Levi subgroup of ${^{L}G}$. A supercuspidal Langlands parameter is a discrete $L$-parameter that is also trivial on the $\mathrm{SL}_2(\mathbb{C})$ factor. We say that discrete parameters $\psi$ and $\psi '$ are equivalent if they are conjugate in $\widehat{G}$ and denote this by $\psi \sim \psi '$. Let $C_{\psi }$ be the centralizer of $\psi (W_F \times \mathrm{SL}_2(\mathbb{C}))$ in $\widehat{G}$. Then by Kot84b, §10.3.1 (given the remark at the start of page 648 of loc. cit.), $\psi$ is discrete if and only if the identity component $C^{\circ }_{\psi }$ of $C_{\psi }$ is contained in $Z(\widehat{G})^{\Gamma _F}$. Here by $Z(\widehat{G})$ we mean the center of $\widehat{G}$, and $\Gamma _F$ is the absolute Galois group of $F$. We define the group $\overline{C_{\psi }} \coloneq C_{\psi }/ Z(\widehat{G})^{\Gamma _F}$ which is finite by our assumptions on $\psi$. For the sake of comparison, in Kal16a, Conj. F, Kaletha defines $S^{\natural }_{\psi }\coloneq C_{\psi } / (C_{\psi } \cap [\widehat{G}]_{\mathrm{der}})^{\circ }$. For $\psi$ a discrete parameter, we have $S^{\natural }_{\psi } = C_{\psi }$. Indeed,

$$\begin{equation*} (C_{\psi } \cap [\widehat{G}]_{\mathrm{der}})^{\circ }=(C^{\circ }_{\psi } \cap [\widehat{G}]_{\mathrm{der}})^{\circ } \subset (Z(\widehat{G})^{\Gamma _F} \cap [\widehat{G}]_{\mathrm{der}})^{\circ }=\{1\}, \end{equation*}$$

from where the equality follows.

Define $Z^1(W_F, G(\overline{L}))$ to be the set of continuous cocycles of $W_F$ valued in $G(\overline{L})$ and let $\mathbf{B}(G) \coloneq H^1(W_F, G(\overline{L}))$ be the corresponding cohomology group. Let $\kappa : \mathbf{B}(G) \to X^*(Z(\widehat{G})^{\Gamma _F})$ be the Kottwitz map as in Kot90, Lemma 6.1.

An elliptic endoscopic datum of $G$ (cf. Kot84b, 7.3-7.4) is a triple $(H, s, \eta )$ of a quasisplit reductive group $H$, an element $s \in Z(\widehat{H})^{\Gamma _F}$, and a homomorphism $\eta : \widehat{H} \to \widehat{G}$. We require that $\eta$ gives an isomorphism

$$\begin{equation*} \eta : \widehat{H} \to Z_{\widehat{G}}(\eta (s))^{\circ }, \end{equation*}$$

that the $\widehat{G}$-conjugacy class of $\eta$ is stable under the action of $\Gamma _F$, and that $(Z(\widehat{H})^{\Gamma _F})^{\circ } \subset Z(\widehat{G})$.

An extended elliptic endoscopic datum of $G$ is a triple $(H,s, {{}^{L}\eta })$ such that ${{}^{L}\eta }: {^{L}H}\to {^{L}G}$ and $(H, s, {{}^{L}\eta }|_{\widehat{H}})$ gives an elliptic endoscopic datum of $G$.

An extended elliptic hyperendoscopic datum is a sequence of tuples of data $(H_1, s_1, {{}^{L}\eta }_1),\ldots ,(H_k, s_k, {{}^{L}\eta }_k)$ such that $(H_1, s_1, {{}^{L}\eta }_1)$ is an extended elliptic endoscopic datum of $G$, and for $i>1$, the tuple $(H_i, s_i, {{}^{L}\eta }_i)$ is an extended elliptic endoscopic datum of $H_{i-1}$. An elliptic hyperendoscopic group of $G$ is a quasisplit connected reductive group $H_k$ appearing in an extended elliptic hyperendoscopic datum for $G$ as above.

For simplicity we assume throughout the paper that each elliptic hyperendoscopic group $H$ of $G$ and each elliptic endoscopic datum $(H', s',\eta ')$ of $H$, one can extend $(H', s', \eta ')$ to an extended elliptic endoscopic datum $(H', s', {{}^{L}\eta }')$ such that ${{}^{L}\eta }': {^{L}H}' \to {^{L}H}$.

Remark 2.1.

The authors are not aware of any example for $G$ a group over $F$ where this property does not hold. If $G$ and all its hyperendoscopic groups have simply connected derived subgroup, then this condition is automatic from Lan79, Prop. 1. In particular, unitary groups satisfy this condition. Similarly, all elliptic endoscopic data $(H,s,\eta )$ for $G$ a symplectic or special orthogonal group can also be extended to a datum $(H, s, {{}^{L}\eta })$ (Kal16a, pg.5). Since the elliptic endoscopic groups of symplectic and special orthogonal groups are products of groups of this type (Wal10, §1.8), it follows that symplectic and special orthogonal groups also satisfy this condition. One could remove this assumption altogether at the cost of having to consider $z$-extensions of endoscopic groups (see KS99a).

3. Statement of main result

We now state our main result. Let us fix $G^\ast$ to be a quasi-split reductive group over $F$. We define a discrete local Langlands correspondence for a group $G^\ast$ to be an assignment

$$\begin{equation*} \Pi _H:\left\{\begin{matrix} \text{Discrete }L\text{-parameters} \\ \text{for } H\end{matrix}\right\} \to \left\{\begin{matrix} \text{Subsets of}\\ \mathrm{Irr}^{\mathrm{disc}}(H(F))\end{matrix}\right\}, \end{equation*}$$

for every elliptic hyperendoscopic group $H$ of $G^\ast$ satisfying the following properties.

(Dis)

If $\Pi _H(\psi ) \cap \Pi _H(\psi ') \neq \emptyset$ then $\psi \sim \psi '$.

(Bij)

For each Whittaker datum $\mathfrak{w}_H$ of $H$, there exists a bijection$$\begin{equation*} \iota _{\mathfrak{w}_H}:\Pi _H(\psi )\to \mathrm{Irr}(\overline{C_\psi }). \end{equation*}$$

This bijection $\iota _{\mathfrak{w}_H}$ gives rise to a pairing$$\begin{equation*} \langle -,-\rangle _{\mathfrak{w}_H}:\Pi _H(\psi )\times \overline{C_\psi }\to \mathbb{C}, \end{equation*}$$

defined as follows:$$\begin{equation*} \langle \pi ,s\rangle _{\mathfrak{w}_H}\coloneq \operatorname {tr}(s \mid \iota _{\mathfrak{w}_H}(\pi )). \end{equation*}$$

(St)

For all discrete $L$-parameters $\psi$ of $H$, the distribution$$\begin{equation*} S\Theta _\psi \coloneq \sum _{\pi \in \Pi _H(\psi )}\langle \pi ,1\rangle _{\mathfrak{w}_H}\Theta _\pi \end{equation*}$$

is stable and does not depend on the choice of $\mathfrak{w}_H$.

(ECI)

For all extended elliptic endoscopic data $(H', s', {{}^{L}\eta }')$ for $H$ and all $h\in \mathcal{H}(H(F))$, suppose $\psi$ is a discrete $L$-parameter of $H$ that factors through ${{}^{L}\eta }'$ by some $L$-parameter $\psi ^{H'}$. Then we assume that $\psi ^{H'}$ satisfies the endoscopic character identity: for any Whittaker datum $\mathfrak{w}_H$$$\begin{equation*} S\Theta _{\psi ^{H'}}(h^{H'})=\Theta ^{s}_{\psi }(h), \end{equation*}$$

where we define $h^{H'}$ to be a transfer of $h$ to $H'$ (e.g. see Kal16a, §1.3) and we define$$\begin{equation*} \Theta ^{s}_{\psi }\coloneq \sum _{\pi \in \Pi _H(\psi )}\langle \pi ,s\rangle _{\mathfrak{w}_H}\Theta _{\pi }, \end{equation*}$$

(the $s$-twisted character of $\psi$).

(Surj)

The union of $\Pi _H(\psi )$ over all discrete $\psi$ equals $\mathrm{Irr}^{\mathrm{disc}}(H(F))$.

Suppose now that $z_\mathrm{iso}\in Z^1(W_F,G(\overline{L}))$ projects to an element of the set $\mathbf{B}(G)_{\mathrm{bas}}$ of basic elements of $\mathbf{B}(G)$ (see Kot97, §5.1). We denote by $\overline{z_\mathrm{iso}}$ the image of $z_\mathrm{iso}$ in $H^1(W_F,G(\overline{L}))$. Let $G$ be the inner form of $G^\ast$ corresponding to the projection of $z_\mathrm{iso}$ to $Z^1(W_F,\mathrm{Aut}(G)(\overline{F}))$. We then define a discrete local Langlands correspondence for the extended pure inner twist $(G,z_\mathrm{iso})$ (cf. Kal16a, §2.5) to be a discrete local Langlands correspondence for $G^\ast$ as well as a correspondence

$$\begin{equation*} \Pi _{(G,z_\mathrm{iso})}:\left\{\begin{matrix} \text{Discrete }L\text{-parameters}\\ \text{for }G\end{matrix}\right\}\to \left\{\begin{matrix} \text{Subsets of}\\ \mathrm{Irr}^\mathrm{disc}(G(F))\end{matrix}\right\}, \end{equation*}$$

satisfying

(Bij’)

For each Whittaker datum $\mathfrak{w}_G$ of $G$, a bijection$$\begin{equation*} \iota _{\mathfrak{w}_G}:\Pi _G(\psi )\to \mathrm{Irr}(C_{\psi },\chi _{z_\mathrm{iso}}), \end{equation*}$$

where $\mathrm{Irr}(C_{\psi }, \chi _{z_{\mathrm{iso}}})$ denotes the set of equivalence classes of irreducible algebraic representations of $C_{\psi }$ with central character on $Z(\widehat{G})^{\Gamma _F}$ equal to $\chi _{z_{\mathrm{iso}}} \coloneq \kappa (\overline{z_{\mathrm{iso}}})$. This gives rise to a pairing$$\begin{equation*} \langle -,-\rangle _{\mathfrak{w}_G}:\Pi _G(\psi ) \times C_\psi \to \mathbb{C}, \end{equation*}$$

defined as$$\begin{equation*} \langle \pi ,s\rangle _{\mathfrak{w}_G}\coloneq \operatorname {tr}(s\mid \iota _{\mathfrak{w}_G}(\pi )). \end{equation*}$$

(ECI’)

For all discrete parameters $\psi$ of $G$ and all extended elliptic endoscopic data $(H,s,{{}^{L}\eta })$ of $G$ such that $\psi$ factors as $\psi = {{}^{L}\eta }\circ \psi ^H$, there is an equality$$\begin{equation*} \Theta ^s_{\psi }(h)=S\Theta _{\psi ^H}(h^H), \end{equation*}$$

where $h\in \mathcal{H}(G(F))$ and $S\Theta _{\psi }$ is independent of choice of Whittaker datum in (Bij’).

(Surj’)

The union of $\Pi _G(\psi )$ over all discrete $\psi$ equals $\mathrm{Irr}^{\mathrm{disc}}(G(F))$.

The above axioms are not enough to uniquely specify a discrete local Langlands correspondence $\Pi$ for $G^*$. For instance, if $G^\ast =T$ is a torus, then any bijection between $\mathrm{Irr}(T(F))$ and the set of $L$-parameters for $T$ satisfies these conditions because there are no non-trivial elliptic endoscopic groups for $T$ and the $L$-packets are singletons. The goal of our main theorem is to explain a sufficient extra condition which does uniquely specify a discrete local Langlands correspondence.

In the statement of this condition we need to assume an extra property of $G$. To state it, we first define a map

$$\begin{equation*} \iota : W_F \to W_F \times \mathrm{SL}_2(\mathbb{C}), \end{equation*}$$

given by

$$\begin{equation*} w \mapsto \left(w, \begin{pmatrix} \|w\|^{\frac{1}{2}} & 0 \\ 0 & \|w\|^{-\frac{1}{2}} \end{pmatrix}\right). \end{equation*}$$

We refer to $\psi \circ \iota$ as the semi-simplification⁠¹ of $\psi$.

¹

Note that for any representation $r$ of ${^{L}G}$ the representation $r\circ \psi \circ \iota$ is semi-simple. Indeed, by BH06, Proposition 28.7 it suffices to show that

$$\begin{equation*} (r\circ \psi )\left(\mathrm{Frob}, \left(\begin{smallmatrix} q^{\frac{1}{2}} & 0 \\ 0 & q^{-\frac{1}{2}} \end{smallmatrix}\right)\right)=(r\circ \psi )\left(\mathrm{Frob}\right)(r\circ \psi )\left( \left(\begin{smallmatrix} q^{\frac{1}{2}} & 0 \\ 0 & q^{-\frac{1}{2}} \end{smallmatrix}\right)\right) \end{equation*}$$

is semi-simple. But, this is evidently a commuting product of semi-simple elements, so is semi-simple.

✖

Definition 3.1.

Then we say that $G^*$ is good if for every elliptic hyperendoscopic group $H$ of $G^\ast$ we have that there exists a set $S^H$ of dominant cocharacters of $H_{\overline{F}}$ with the following property: for any pair $\psi ^H_1$ and $\psi ^H_2$ of discrete parameters of $H$ such that for all dominant cocharacters $\mu \in S^H$, we have an equivalence $r_{-\mu } \circ (\psi ^{H}_1 \circ \iota )|_{W_{E_\mu }} \sim r_{- \mu } \circ (\psi ^H_2 \circ \iota )|_{W_{E_\mu }}$, then $\psi ^H_1 \circ \iota \sim \psi ^H_2 \circ \iota$. Here $r_{-\mu }$ is the representation of ${^{L}H}$ as defined in Kot84a, (2.1.1) and $E_\mu$ is the reflex field of $\mu$. We say that $G$ is good if $G^\ast$ is.

It is worth noting that many classes of groups satisfy this assumption and $S^H$ can often be taken to be a single minuscule cocharacter (see §4).

Definition 3.2.

Suppose $G$ is a good group and fix $S^H$ as above. A Scholze–Shin datum $\{f_{\tau ,h}^{\mu }\}$ for $G$ (and $S^H$) consists of the following data for each elliptic hyperendoscopic group $H$ of $G$:

•

A compact open subgroup $K^H \subset H(F)$,

•

For each $\mu \in S^H$ with reflex field $E_{\mu }$, each $\tau \in W_{E_{\mu }}$, and each $h \in \mathcal{H}(K^H)$, a function $f^{\mu }_{\tau ,h} \in \mathcal{H}(H(F))$.

Let us say that a discrete local Langlands correspondence for $G$ satisfies the Scholze–Shin equations relative to the Scholze–Shin datum $\{f_{\tau ,h}^{\mu }\}$ if the following holds:

(SS)

For all elliptic hyperendoscopic groups $H$, all $h\in \mathcal{H}(K^H)$, all ${\mu }\in S^H$, and all parameters $\psi$ of $H$ one has the equality$$\begin{equation*} S\Theta _{\psi }(f_{\tau ,h}^{\mu })=\operatorname {tr}(\tau \mid (r_{-{\mu }}\circ (\psi \circ \iota )|_{W_{E_\mu }})(\chi _\mu ))S\Theta _{\psi }(h), \end{equation*}$$

where $\chi _\mu \coloneq |\cdot |^{-\langle \rho ,\mu \rangle }$ and $\rho$ is the half-sum of the positive roots of $H$ (for a representation $V$ and character $\chi$ we denote by $V(\chi )$ the character twist of $V$ by $\chi$).

Remark 3.3.

As mentioned in the introduction, equations of this form were originally conjectured to hold in SS13 (with the obvious extension to the abelian type case in the forthcoming work You21). These equations use Scholze–Shin data constructed from the cohomology of certain tubular neighborhoods inside of Rapoport–Zink spaces (cf. Sch13a). It seems conceivable to the authors that such Scholze–Shin data could also be constructed in a much greater generality using Scholze’s theory of moduli spaces of mixed characteristic shtuka. For this reason, and not knowing the precise form of these generalizations, we have chosen to work in the maximal generality for which our arguments work.

We then have the following result:

Theorem 3.4.

Let $G$ be a good group and suppose $\Pi ^i$ for $i=1,2$ are discrete local Langlands correspondences for $(G,z_\mathrm{iso})$ which satisfy the Scholze–Shin equations relative to the same Scholze–Shin data $\{f^{\mu }_{\tau ,h}\}$. Then,

•

we have $\Pi ^1=\Pi ^2$,

•

and for every choice of Whittaker datum $\mathfrak{w}_H$, the bijections $\iota _{\mathfrak{w}_H}^i$ for $i=1,2$ agree.

Remark 3.5.

In this paper we have considered only $G$ that arise as extended pure inner twists of $G^*$ (e.g. see Kal16a). In general, the map $\mathbf{B}(G^*)_{\mathrm{bas}} \to \mathrm{Inn}(G^*)$, where $\mathrm{Inn}(G^*)$ denotes the set of inner twists of $G^*$, need not be surjective. However, when $G^*$ has connected center, this map is surjective (see Kal16a, pg.20). In general, one can consider all inner twists by adapting the arguments of this paper to the language of rigid inner twists as in Kal16b (cf. Kal16a).

4. Examples

In this section we discuss some examples of where the conditions necessary to apply Theorem 3.4 are satisfied.

4.1. The characterization in the unitary case

We start by discussing the case of unitary groups which was mentioned in the introduction. Namely, let $F$ be an extension of $\mathbb{Q}_p$ and let $E$ be a quadratic extension of $F$. Let us set $U_{E/F,n}^\ast$ to be the quasi-split unitary group associated to the extension $E/F$. We call a group $G$ over $F$ a unitary group if $G^\ast =U^\ast _{E/F,n}$ for some $n$, $F$, and $E$. Note though that every inner form of $U_{E/F,n}^\ast$ can be upgraded to an extended pure inner twist, and we leave such choice implicit. We say that $G$ is unramified if the extension $E/\mathbb{Q}_p$ is unramified.

Proposition 4.1.

Let $G$ be a unitary group. Then, $G$ is good.

Proof.

Without loss of generality we may assume that $G=U^\ast _{E/F,n}$. Note that every elliptic endoscopic group of $G$ is of the form $U_{E/F,a}^\ast \times U_{E/F,b}^\ast$ for some $a,b\in \mathbb{N}$ such that $a+b=n$ (e.g. see Rog90, Proposition 4.6.1). From this we deduce that the elliptic hyperendoscopic groups of $G$ are given by $U_{E/F,P}^\ast$ where $P=(n_1,\ldots ,n_k)$ is a partition of $n$ and

$$\begin{equation*} U_{E/F,P}^\ast \coloneq U_{E/F,n_1}^\ast \times \cdots \times U_{E/F,n_m}^\ast . \end{equation*}$$

Then $G$ is good because can take $S^{U_{E/F,P}^\ast }$ to be $\{\mu _{P,\mathrm{std}}\}$ where

$$\begin{equation*} \mu _{P,\text{std}}=\mu _{n_1,\text{std}}\times \cdots \times \mu _{n_m,\text{std}} \end{equation*}$$

where $\mu _{n_i,\text{std}}$ is the cocharacter corresponding to the standard representation of $U_{E/F,n_i}^\ast$. Indeed, the field of definition of this cocharacter is $E$ and we can recover $\psi \circ \iota$ from its restriction to $W_E$ by GGP12, Theorem 8.1.

■

In particular, let us define a Scholze–Shin datum $\{f_{\tau ,h}^{\mu _{P,\text{std}}}\}$ as in You21. Let us then set $\Pi ^\mathrm{Mok}$ to be the discrete local Langlands correspondence associated to an unramified unitary group as in Mok15, Theorem 2.5.1. By BMY19, Theorem 1, $\Pi ^\mathrm{Mok}$ satisfies the Scholze–Shin equations relative to the Scholze–Shin datum $\{f_{\tau ,h}^{\mu _{P,\text{std}}}\}$. This together with Theorem 3.4 show the following:

Theorem 4.2.

Let $G$ be an unramified unitary group. Then, $\Pi ^\mathrm{Mok}$ is characterized by the Scholze–Shin datum $\{f_{\tau ,h}^{\mu _{P,\mathrm{std}}}\}$ and the set of singly $\Pi ^\mathrm{Mok}$-accessible representations.

4.2. The odd orthogonal case

We now discuss the case of odd special orthogonal groups. Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $n\geqslant 1$ be an integer. By an odd special orthogonal group we mean a group $G$ over $F$ of the form $\mathrm{SO}(V,q)$ where $(V,q)$ is a quadratic space over $F$ of odd dimension. We denote by $\mathrm{SO}_{2n+1,F}$ the special odd orthogonal group of the split quadratic space of dimension $2n+1$ which is a split group. If $G$ is any odd special orthogonal group then $G^\ast =\mathrm{SO}_{2n+1,F}$ for $2n+1=\dim (V)$. We call an odd special orthogonal group unramified if $F$ is unramified.

Proposition 4.3.

Let $G$ be an odd special orthogonal group. Then, $G$ is good.

Proof.

Since $G$ being good depends only on $G^\ast$, and $G^\ast =\mathrm{SO}_{2n+1,F}$ for some $n$ we may assume that $G=\mathrm{SO}_{2n+1,F}$. Let us denote by $\mu _n$ the unique non-trivial minuscule cocharacter of $\mathrm{SO}_{2n+1,F}$. By GGP12, Theorem 8.1, we can recover $\psi$ from $r \circ \psi$ where $\psi$ any admissible homomorphism $\psi :W_F\to \mathrm{Sp}(2n)(\mathbb{C})$ and $r$ is the standard representation. Let us also note that $\widehat{G}=\mathrm{Sp}_{2n}(\mathbb{C})$ and $r_{-\mu _n}=r$. Now, since every such elliptic hyperendoscopic group of $G$ is a product of odd special orthogonal groups (e.g. by Wal10, §1.8), we are done as in the definition of good we may take $S^H=\{\mu _H\}$ where if

$$\begin{equation*} H=\mathrm{SO}_{2n_1+1,F}\times \cdots \times \mathrm{SO}_{2n_k+1,F}, \end{equation*}$$

we denote by $\mu _H$ the cocharacter $\mu _{n_1}\times \cdots \times \mu _{n_k}$.

■

Remark 4.4.

We end by remarking to what extent one might hope that the result Theorem 4.2 extends to the case of odd special orthogonal groups. A construction of the local Langlands correspondence for odd special orthogonal groups is complete by Art13. Moreover, the works of Arthur and Taï19 prove the global multiplicity formula results. Such multiplicity results play a pivotal role in the proof in BMY19 that Mok’s Langlands correspondence for unramified unitary groups satisfies the Scholze–Shin equations for the data $\{f_{\tau ,h}^{\mu _{P,\text{std}}}\}$ from the last section.

Moreover, there are well-studied Shimura data associated to the odd special orthogonal groups over number fields (see Zhu18). The cocharacter associated to this Shimura datum is $\mu _n$. In You21 there are constructed functions $f^{\mu _H}_{\tau , h}$ which serve as candidate Scholze–Shin data. Combining this geometric input with the aforementioned results of Arthur and Taïbi it then seems conceivable to prove that Arthur’s local Langlands correspondence for unramified odd special orthogonal group satisfies the Scholze–Shin equations relative to the functions in You21. This would then allow one to prove the analogue of Theorem 4.2 for unramified odd special orthogonal groups.

5. Atomic stability of $L$-packets

Before we begin the proof of Theorem 3.4 in earnest, we first discuss the following extra assumption one might make on a discrete local Langlands correspondence $\Pi$ for the group $G$ which, for this section, we assume is quasi-split.

Definition 5.1.

Suppose that for any hyperelliptic endoscopic group $H$ of $G$ and for all finite subsets $S\subseteq \mathrm{Irr}^{\mathrm{disc}}(H(F))$ and complex numbers $\{a_\pi \}_{\pi \in S}\subseteq \mathbb{C}$ such that $\Theta \coloneq \sum a_\pi \pi$ is a stable distribution, there is a partition $S=\bigsqcup _{\psi }\Pi _{H}(\psi )$ such that $\Theta =\sum b_\psi S\Theta _\psi$ for some $\{b_\psi \}\subseteq \mathbb{C}$. We then say that $\Pi$ satisfies atomic stability.

Somewhat surprisingly, this condition on $\Pi$ is automatically implied by the axioms we have already imposed.

Proposition 5.2.

Let $\Pi$ be a discrete local Langlands correspondence for a group $G$. Then, $\Pi$ satisfies atomic stability.

For notational convenience, we may assume that $H=G$ in the proof of the above. For discrete $L$-parameters $\psi _1, \ldots , \psi _n$ we denote by $D(\psi _1, \ldots , \psi _n)$ the $\mathbb{C}$-span of the distributions $\Theta _\pi$ for $\pi \in \Pi _G(\psi _1)\cup \cdots \cup \Pi _G(\psi _n)$ and let $S(\psi _1, \ldots , \psi _n)$ be the subspace of stable distributions in $D(\psi _1, \ldots , \psi _n)$. To prove Proposition 5.2, it suffices to show that $\{S\Theta _{\psi _1}, \ldots , S\Theta _{\psi _n}\}$ is a basis for $S(\psi _1, \ldots ,\psi _n)$. Indeed, we can enlarge $S$ to be a union $\Pi _{\psi _1}(G)\sqcup \cdots \sqcup \Pi _{\psi _n}(G)$ of $L$-packets. Proposition 5.2 is then clear since every stable distribution in the span of $S$ is contained in $S(\psi _1,\ldots ,\psi _n)$.

Before we proceed with the proof of Proposition 5.2 we establish some further notation and basic observations. For an element $\pi$ of $\mathrm{Irr}^{\mathrm{disc}}(G(F))$ we denote by $f_\pi$ the locally constant $\mathbb{C}$-valued function on $G(F)^\mathrm{reg}$ given by the Harish-Chandra regularity theorem (see HC65, Theorem 2). We then obtain a linear map

$$\begin{equation*} R:D(\mathrm{Irr}^{\mathrm{disc}}(G(F)))\to C^\infty (G(F)^\mathrm{ell},\mathbb{C}) \end{equation*}$$

given by linearly extending the association $\Theta _\pi \mapsto f_\pi \mid _{G(F)^\mathrm{ell}}$. Here we denote by $D(\mathrm{Irr}^{\mathrm{disc}}(G(F)))$ the $\mathbb{C}$-span of the distributions on $\mathcal{H}(G(F))$ of the form $\Theta _\pi$ for $\pi$ in $\mathrm{Irr}^{\mathrm{disc}}(G(F))$.

We then have the following likely well-known lemma concerning $R$.

Lemma 5.3.

The linear map $R$ is injective.

Proof.

Suppose that $R\left(\sum _i a_i \Theta _{\pi _i}\right)$ is zero. By KHW21, Theorem C.1.1 this implies that we may write

$$\begin{equation*} \sum _i a_i \pi _i=\sum _j b_j \mathrm{Ind}_{P_i}^G(\rho _j) \end{equation*}$$

for proper parabolic subgroups $P_j$ of $G$, and $\rho _i$ representations on $L_j(F)$ where $L_j$ is a Levi factor of $P_j$. Up to replacing $P_j$ by smaller parabolic subgroups, we may assume by the $p$-adic Langlands classification that each $\rho _i$ is an essentially tempered representation (see BW00, §IX.2). By Wal03, Proposition III.4.1 we may find a unique (up to conjugacy) parabolic subgroup $Q_j\subseteq L_j$ with Levi factor $M_j\subseteq Q_j$ and an essentially square integrable representation $\sigma _j$ of $M_j(F)$ such that $\rho _j$ is a direct summand of $\mathrm{Ind}_{Q_j}^{L_j}(\sigma _j)$. Let us write $\delta _j$ as the complement of $\rho _j$ in $\mathrm{Ind}_{Q_j}^{L_j}(\sigma _j)$. So then, we see that

$$\begin{equation*} \sum _j b_j \mathrm{Ind}_{Q_j}^{L_j}(\sigma _j)=\sum _j b_j \delta _j+\sum _i a_i\pi _i. \end{equation*}$$

From the independence theory of characters, and the fact that each $\mathrm{Ind}_{Q_j}^{L_j}(\sigma _j)$ is semi-simple (cf. loc. cit.) we deduce that each $\pi _i$ must be a direct summand of some $\mathrm{Ind}_{Q_j}^{L_j}(\sigma _j)$. But, since $Q_j$ is a proper parabolic subgroup of $G$, this contradicts the unicity part of loc. cit.

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We also have an averaging map

$$\begin{equation*} \mathrm{Avg}:C^\infty (G(F)^\mathrm{ell},\mathbb{C})\to C^\infty (G(F)^\mathrm{ell},\mathbb{C}) \end{equation*}$$

given by

$$\begin{equation*} \mathrm{Avg}(f)(\gamma )\coloneq \frac{1}{n_\gamma }\sum _{\gamma '}f(\gamma ') \end{equation*}$$

where $\gamma '$ runs over representatives of conjugacy classes of $G(F)$ whose stable conjugacy class is equal to $\gamma$, and $n_\gamma$ is the number of such classes.

Lemma 5.4 describes the relationship between this averaging map and the map $R$. The proof of this follows from the well-known fact that since $\Theta$ is a stable distribution, the function $R(\Theta )$ is constant on stable conjugacy classes in $G(F)^{\mathrm{ell}}$.

Lemma 5.4.

Let $\Theta \in D(\mathrm{Irr}^{\mathrm{disc}}(G(F)))$ be stable as a distribution. Then,

$$\begin{equation*} \mathrm{Avg}(R(\Theta ))=R(\Theta ). \end{equation*}$$

We may now proceed to the proof of Proposition 5.2.

Proof of Proposition 5.2.

In the following, we fix a Whittaker datum $\mathfrak{w}_G$.

Step 1.

Recall the notation fixed immediately after the statement of Proposition 5.2. Note that by assumption (Bij), the set of virtual characters $\Theta ^s_{\psi _i}$, as $s$ runs through representatives for the conjugacy classes in $\overline{C_\psi }$ and $i$ runs through $\{1,\ldots ,n\}$, is a basis of $D(\psi _1,\ldots ,\psi _n)$. To see this, it suffices to show this in the case when $n=1$. Write $\psi$ instead of $\psi _1$. Since $\{\Theta ^s_{\psi }\}$ is the result of applying the matrix $(a_{\pi ,s})$ where $a_{\pi ,s}=\langle \pi ,s\rangle _{\mathfrak{w}_G}$ to the basis $\{\Theta _\pi \}$, we are done if we can show that this matrix is invertible. But, this matrix is in fact unitary by the orthogonality relations of characters for the finite group $\overline{C_\psi }$.

Step 2.

We next show that for any discrete $L$-parameter $\psi$ and any non-trivial $s$ in $\overline{C_\psi }$, the function $\mathrm{Avg}(R(\Theta ^s_\psi ))$ is zero. Let $(H,s,{{}^{L}\eta },\psi ^H)$ be the quadruple corresponding to the pair $(\psi ,s)$ as in BMY19, Proposition I.2.15. To begin, a simple application of the Weyl integration formula to the equations in (ECI) yields the following (cf. HS12, Lemma 6.20)

$$\begin{equation*} \mathrm{Avg}(R(\Theta ^s_\psi ))(\gamma ) = \frac{1}{n_\gamma }\sum _{\gamma '}\sum _{\gamma _H\in X(\gamma ')/\sim _\mathrm{st}}\Delta (\gamma _H,\gamma ')\left|\frac{D_H(\gamma _H)}{D_G(\gamma ')}\right|S\Theta _{\psi ^H}(\gamma _H), \end{equation*}$$

where here $\gamma '$ travels over the set of conjugacy classes of $G(F)$ stably equal to the conjugacy class of $\gamma$ and, as in loc. cit., $X(\gamma ')$ is the set of conjugacy classes in $H(F)$ that transfer to $\gamma$, and $\Delta (\gamma _H,\gamma ')$ is the $\mathfrak{w}_G$-normalized transfer factor of LS87, and $D_G$ (resp. $D_H$) denotes the Weyl discriminant function for $G$ (resp. $H$). Let us note that we can rewrite this sum as

$$\begin{equation*} \frac{1}{n_\gamma }\sum _{\gamma _H\in X(\gamma )/\sim _\mathrm{st}}\left(\sum _{\gamma '}\Delta (\gamma _H,\gamma ')\left|\frac{D_H(\gamma _H)}{D_G(\gamma ')}\right|\right)S\Theta _{\psi ^H}(\gamma _H), \end{equation*}$$

because $X(\gamma ')/\sim _\mathrm{st}$ is independent of the choice of $\gamma '$.

As $D_G(\gamma ')$ is defined in terms of the characteristic polynomial for $\mathrm{Ad}(\gamma ')$, we have that $D_G(\gamma ')=D_G(\gamma )$ for all $\gamma '$ stably conjugate to $\gamma$. Thus, we can further rewrite this as

$$\begin{equation*} \frac{1}{n_\gamma }\sum _{\gamma _H\in X(\gamma )/\sim _\mathrm{st}}\left|\frac{D_H(\gamma _H)}{D_G(\gamma )}\right|\left(\sum _{\gamma '}\Delta (\gamma _H,\gamma ')\right)S\Theta _{\psi ^H}(\gamma _H), \end{equation*}$$

and so it suffices to show that this inner sum is zero.

Let $\mathfrak{K}(I_\gamma /F)^D$ be the Pontryagin dual of the Kottwitz group associated to $\gamma$ (see Kot86, §4.6). As in Kot84b, §5.6 the element $s$ defines an element $\mathfrak{K}(I_\gamma /F)$ in a natural way, and associated to any $\gamma '\sim _\mathrm{st}\gamma$ is an element $\mathrm{inv}(\gamma ,\gamma ')$ of $\mathfrak{K}(I_\gamma /F)^D$. By KS99b, Theorem 5.1.D, the equality

$$\begin{equation*} \Delta (\gamma _H, \gamma ')=\langle \mathrm{inv}(\gamma , \gamma '), s \rangle \Delta (\gamma _H, \gamma ) \end{equation*}$$

holds, where $\langle -,-\rangle$ denotes the tautological pairing between a finite group and its Pontryagin dual. Since $\gamma$ is elliptic, $\gamma ' \mapsto \mathrm{inv}(\gamma , \gamma ')$ gives a bijection between $F$-conjugacy classes in the stable conjugacy class of $\gamma$ and $\mathfrak{K}(I_{\gamma }/F)^D$ (see Kot86, §4). Hence,

$$\begin{equation*} \sum _{\gamma '}\Delta (\gamma _H,\gamma ')= \Delta (\gamma _H, \gamma ) \sum _{\chi \in \mathfrak{K}(I_{\gamma })^D} \chi (s). \end{equation*}$$

In particular, it suffices to show that $s$ gives a nontrivial element of $\mathfrak{K}(I_{\gamma }/F)$. Since $(H,s,\eta )$ is a nontrivial elliptic endoscopic datum and $\gamma$ is elliptic, this follows from Shi10, Lemma 2.8 (again cf. BMY19, Proposition I.2.15).

Step 3.

Note that to show $\{S\Theta _{\psi _1},\ldots ,S\Theta _{\psi _n}\}$ is a basis of $S(\psi _1,\ldots ,\psi _n)$, we need only show it spans since $\{S\Theta _{\psi _1},\ldots ,S\Theta _{\psi _n}\}$ is independent by the independence of characters and assumption (Dis). But, this is now clear since if $\Theta \in S(\psi _1,\ldots ,\psi _n)$ then we know by Lemma 5.4 that $\mathrm{Avg}(R(\Theta ))=R(\Theta )$. On the other hand, since we have already observed that $\{\Theta _{\psi _i}^s\}$ is a basis of $D(\psi _1,\ldots ,\psi _n)$ we may write

$$\begin{equation*} \Theta =\sum _{i=1}^n \sum _s a_{is}\Theta ^s_{\psi _i}. \end{equation*}$$

We see from the above discussion, as well as combining assumption (St) with Lemma 5.4, that

$$\begin{equation*} \mathrm{Avg}(R(\Theta ))=\sum _{i=1}^n R(a_{ie}S\Theta _{\psi _i})=R\left(\sum _{i=1}^n a_{ie}S\Theta _{\psi _i}\right) \end{equation*}$$

where we have identified $S\Theta _{\psi _i}$ with $\Theta ^e_{\psi _i}$ (where $e$ is the identity conjugacy class in $\overline{C_\psi }$). Thus, putting everything together we see that

$$\begin{equation*} R(\Theta )=R\left(\sum _{i=1}^n a_{ie}S\Theta _{\psi _i}\right), \end{equation*}$$

and so the claim then follows from Lemma 5.3.

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6. Proof of main result

We now begin the proof of our main result. Our notation throughout this section will be as in §3. For discrete local Langlands correspondences $\Pi ^i$ for $i=1,2$ for a group $G$, we shall use the superscript $i$ to indicate which local Langlands correspondence the object refers to (e.g. $S\Theta _\psi ^i$ denotes the stable character associated to $\psi$ by $\Pi ^i$).

We begin by establishing that it suffices to assume that $G$ is quasi-split.

Lemma 6.1.

Suppose that the Theorem 3.4 holds for $(G^\ast ,1)$, then it holds for $(G,z_\mathrm{iso})$.

Proof.

Let $\psi$ be any discrete $L$-parameter for $G$. By assumption (ECI’) we have that

$$\begin{equation*} S\Theta ^1_\psi (h)=S\Theta ^1_{\psi ^{G^{\ast }}}(h^{G^\ast })=S\Theta ^2_{\psi ^{G^\ast }}(h^{G^\ast })=S\Theta ^2_\psi (h) \end{equation*}$$

for all $h \in \mathcal{H}(G(F))$, where $h^{G^\ast }$ denotes the transfer of $h$ to $G^\ast$. By the independence of characters, this implies that $\Pi ^1_{(G,z_\mathrm{iso})}(\psi )$ is equal to $\Pi ^2_{(G,z_\mathrm{iso})}(\psi )$. It remains to show that $\iota ^1_{\mathfrak{w}_G}$ is equal to $\iota ^2_{\mathfrak{w}_G}$, and hence that for all $\pi \in \Pi ^1_{(G,z_\mathrm{iso})}(\psi )=\Pi ^2_{(G,z_\mathrm{iso})}(\psi )$ and $s \in C_{\psi }$, one has that $\langle \pi ,s\rangle _{\mathfrak{w}_G}^1=\langle \pi ,s\rangle _{\mathfrak{w}_G}^2$. By independence of characters, it suffices to show that $\Theta ^{1,t}_\psi =\Theta ^{2,t}_\psi$ for all $t\in C_\psi$. But, there exists an elliptic endoscopic triple $(H,s,{{}^{L}\eta })$ and a parameter $\psi ^H$ for $H$ such that ${{}^{L}\eta }\circ \psi ^H$ is equivalent to $\psi$, and such that $s$ has the same image in $\overline{C_\psi }$ as $t$ (cf. BM21, Prop. 2.10). The claim then easily follows from (ECI’) since $\Theta ^{i,t}_{\psi }(h)$ equals $S\Theta ^i_{\psi }(h^H)$ and by assumption, we have that $S\Theta ^1_{\psi }(h^H)$ is equal to $S\Theta ^2_{\psi }(h^H)$.

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Due to Lemma 6.1, it is sufficient to prove Theorem 3.4 in the quasi-split case. Thus, throughout the entire rest of this subsection we assume that $G$ is quasi-split and that $z_\mathrm{iso}$ is trivial.

Before we begin the proof in the quasi-split case, we note that since the Scholze–Shin equations only involve the semi-simplified parameter $\psi \circ \iota$, we will certainly need the following fact which says that a discrete parameter $\psi$ may be recovered from its semi-simplified parameter.

Proposition 6.2 (BMIY22, Proposition 3.18).

Let $G$ be a reductive group over $F$ and suppose that $\psi _1$ and $\psi _2$ are discrete parameters for $G$. If $\psi _1 \sim \psi _2$, then $\psi _1 \circ \iota \sim \psi _2 \circ \iota$.

We now begin the proof of the quasi-split case of Theorem 3.4 in earnest. We wish to, in particular, show that $\Pi ^1_H(\psi )$ is equal to $\Pi ^2_H(\psi )$ for every hyperelliptic endoscopic group $H$, and every parameter $\psi$ of $H$. Our proof will involve an inductive-type argument to reduce to the case when $\Pi ^1_H(\psi )$ is a singleton. This case is then handled in Lemma 6.3.

Lemma 6.3.

Suppose that $H$ is an elliptic hyperendoscopic group of $G$ and suppose that $\Pi ^1_H(\psi )$ is a singleton set $\{\pi \}$. Then, in fact, $\{\pi \}=\Pi ^2_H(\psi )$.

Proof.

Since $\{\pi \}$ is a discrete packet for $\Pi ^1_H$, we have by assumption (St) that $\Theta _\pi$ is stable. By Proposition 5.2 applied to $\Pi ^2_H$, we have $\{\pi \}=\Pi ^2_H(\psi ')$ for some discrete $L$-parameter $\psi '$ of $H$. Then, by the assumption of the theorem we have that

$$\begin{equation*} \operatorname {tr}(\tau \mid (r_{-\mu }\circ \psi \circ \iota )(\chi _\mu ))\operatorname {tr}(h\mid \pi )\!=\!\operatorname {tr}(f_{\tau ,h}^\mu \mid \pi )\!=\!\operatorname {tr}(\tau \mid (r_{-\mu }\circ \psi ' \circ \iota )(\chi _\mu ))\operatorname {tr}(h\mid \pi ). \end{equation*}$$

In particular, choosing $h\in \mathcal{H}(K^H)$ such that $\operatorname {tr}(h\mid \pi )\ne 0$ and letting $\tau$ vary we deduce that

$$\begin{equation} \operatorname {tr}(\tau \mid (r_{-\mu }\circ \psi \circ \iota )(\chi _\mu ))=\operatorname {tr}(\tau \mid (r_{-\mu }\circ \psi ' \circ \iota )(\chi _\mu )) \tag{2} \end{equation}$$

for all $\tau \in W_E$. Since the representations $r_{-\mu }\circ \psi$ and $r_{-\mu }\circ \psi '$ are semi-simple, by the Brauer–Nesbitt theorem we deduce that $r_{-\mu }\circ \psi \circ \iota$ is equivalent to $r_{-\mu }\circ \psi ' \circ \iota$ for all $\mu \in S^H$. By our assumption on $S^H$, we deduce that $\psi \circ \iota$ is equivalent to $\psi ' \circ \iota$. Then by Proposition 6.2, we deduce that $\psi$ is equivalent to $\psi '$. In particular, $\{\pi \}=\Pi ^2_\psi (H)$ as desired.

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We now proceed to the proof of Theorem 3.4 which, as mentioned before, is an induction-type argument to reduce to the singleton packet case.

Proof of Theorem 3.4.

We prove this by inducting on the number of absolute roots $k$ for elliptic hyperendoscopic groups $H$ of $G$. If $k=0$ then $H$ is a torus. Since every distribution on $H$ is stable, one deduces from assumption (Dis) and assumption (St) that $\Pi ^1_H(\psi )$ is a singleton and thus we are done by Lemma 6.3. Suppose now that the result is true for elliptic hyperendoscopic groups of $G$ with at most $k$ roots. Let $H$ be an elliptic hyperendoscopic group of $G$ with $k+1$ roots and let $\psi$ be a discrete parameter of $H$. We wish to show that $\Pi ^1_H(\psi )=\Pi ^2_H(\psi )$. If $\Pi ^1_H(\psi )$ is a singleton, then we are done again by Lemma 6.3. Otherwise, we show that $\Pi ^1_H(\psi ) \subset \Pi ^2_H(\psi )$, which by (Bij) will imply that $\Pi ^1_H(\psi )=\Pi ^2_H(\psi )$.

To show this containment, let $\pi$ be an element of $\Pi ^1_H(\psi )$ and fix a Whittaker datum $\mathfrak{w}_G$ of $G$. We may choose a non-trivial $\overline{s}\in \overline{C_\psi }$ and a lift $s \in C_{\psi }$ such that $\langle \pi , \overline{s}\rangle _{\mathfrak{w}_G}\ne 0$.⁠² By definition of $\overline{C_{\psi }}$, we have that $s \notin Z(\widehat{G})$. Now, it suffices to show that $\Theta ^{1,s}_\psi =\Theta ^{2,s}_\psi$ for this particular $s$ since then by independence of characters, we may deduce that $\pi \in \Pi ^2_H(\psi )$ as desired. We in fact claim that $\Theta ^{1,s}_\psi =\Theta ^{2,s}_\psi$ for all non-trivial $s\in \overline{C_\psi }$. To show this, we begin by noting that by BMY19, Proposition I.2.15, there exists a quadruple $(H',s,{{}^{L}\eta },\psi ^{H'})$ with $\psi ^{H'}$ so that $\psi ={{}^{L}\eta }\circ \psi ^{H'}$. The parameter $\psi ^{H'}$ is discrete since if it factored through a Levi subgroup of ${}^LH$, its image would commute with a nontrivial torus of ${}^LH$ and hence also of ${}^LG$. This would contradict the discreteness of $\psi$. One then has from Assumption (ECI) that $\Theta ^{1,s}_\psi =\Theta ^{2,s}_\psi$ if and only if $S\Theta ^1_{\psi ^{H'}}=S\Theta ^2_{\psi ^{H'}}$. Moreover, since $s$ is non-central, we know that $H'$ has a smaller number of roots than $H$ and thus $S\Theta ^1_{\psi ^{H'}}=S\Theta ^2_{\psi ^{H'}}$ by induction. The conclusion that $\Pi ^1=\Pi ^2$ follows.

²

If $M$ is a finite group and $\chi$ the character of the irreducible representation $\rho$ of $M$, then $\chi (m)\ne 0$ for some non-trivial $m$ in $M$. Indeed, evidently $\rho$ is non-trivial, but also by the orthogonality relations we have that $\dim (\rho )^2=|M|$. Since $\displaystyle |M|=1+\sum \dim (\rho ')^2$, where $\rho '$ travels over the non-trivial elements of $\mathrm{Irr}(M)$, this is a contradiction.

✖

Let us now show that for any discrete $L$-parameter $\psi$ one has that $\iota _{\mathfrak{w}_H}^1=\iota _{\mathfrak{w}_H}^2$ for all elliptic hyperendoscopic groups $H$ of $G$ and Whittaker data $\mathfrak{w}_H$ of $H$. By the orthogonality relations, it suffices to show that $\langle \pi ,s\rangle _{\mathfrak{w}_H}^1=\langle \pi ,s\rangle _{\mathfrak{w}_H}^2$ for all $\pi \in \Pi ^1_\psi (H)=\Pi ^2_\psi (H)$. By independence of characters, it suffices to show that $\Theta ^{1,s}_\psi =\Theta ^{2,s}_\psi$ for all $s\in \overline{C_\psi }$. As $s$ is an element of $\overline{C_\psi }$, there again exists associated to the pair $(\psi ,s)$ a quadruple $(H',s,{{}^{L}\eta },\psi ^{H'})$, where $H'$ is an elliptic endoscopic group of $H$ and $\psi ^{H'}$ is a parameter such that $\psi ={{}^{L}\eta }\circ \psi ^{H'}$. By assumption (ECI) it suffices to show that $S\Theta ^1_{\psi ^{H'}}=S\Theta ^2_{\psi ^{H'}}$, but this follows from the previous part of the argument since we know that $\Pi ^1_{H'}({\psi ^{H'}})=\Pi ^2_{H'}(\psi ^{H'})$.

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