2. Notation and preliminaries

In this paper, unless otherwise specified, we use $k$ to denote a local field. All the conjectures in this paper are formulated for all local fields, archimedean and non-archimedean, but no proofs are given for archimedean local fields. It is hoped that in due course, not only the few proofs we give in this paper, but also the conjectures that we formulate, will be considered by others for archimedean fields.

For a local field $k$, we let $W(k)$ be the Weil group of $k$ and $\textit {WD}(k)$ the Weil–Deligne group of $k$. Thus,

\[ \textit{WD}(k) = \begin{cases} W(k) & \text{if $k$ is archimedean;} \\ W(k) \times {{\rm SL}}_2(\mathbb{C}), & \text{if $k$ is non-archimedean.} \end{cases} \]

In the non-archimedean case, the ${{\rm SL}}_2(\mathbb {C})$ which arises here is called the Deligne ${{\rm SL}}_2$.

For a reductive algebraic group $G$ over $k$, let ${}^\textrm {L}G = {{\widehat G}}(\mathbb {C}) \rtimes W(k)$ be the L-group of $G$ with ${{\widehat G}}(\mathbb {C})$ the dual group over $\mathbb {C}$. L-parameters for $G$ are admissible homomorphisms

\[ \phi: \textit{WD}(k) \rightarrow {{}^\textrm{L}}G, \]

up to conjugacy by ${{\widehat G}}(\mathbb {C})$. A-parameters for $G$ are admissible homomorphisms

\[ \phi_{\rm A}: \textit{WD}(k) \times {{\rm SL}}_2(\mathbb{C}) \rightarrow {{}^\textrm{L}}G \]

up to conjugacy by ${{\widehat G}}(\mathbb {C})$, where the restriction of $\phi _{\rm A}$ to the Weil–Deligne group $\textit {WD}(k)$ is an admissible homomorphism with bounded image for $W(k)$ and the restriction to ${{\rm SL}}_2(\mathbb {C})$ is algebraic. The extra ${{\rm SL}}_2(\mathbb {C})$ that enters here will be called the Arthur ${{\rm SL}}_2(\mathbb {C})$.

One can obtain an L-parameter from an A-parameter as discussed in the introduction, using the map $\textit {WD}(k) \rightarrow \textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ given by

\[ w \rightarrow (w, {\textrm{diag}}(|w|^{1/2}, |w|^{-1/2})). \]

Composing $\phi _{\rm A}$ with this homomorphism gives an L-parameter $\phi$. The association

\[ \phi_{\rm A} \mapsto \phi \]

is an injective map from the set of A-parameters to the set of L-parameters, and we call its image the set of L-parameters of Arthur type.

In the case of classical groups as discussed in [Reference Gan, Gross and PrasadGGP12a], A-parameters are representations of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ on finite-dimensional complex vector spaces that may come equipped with a symmetric or skew-symmetric bilinear form. For an A-parameter $M_{\rm A}$, with associated L-parameter $M$, we often denote the associated complex vector space (the representation space of $M_{\rm A}$ or $M$) as $M$; sometimes, for sake of clarity, we may use a different symbol $V$ or $W$ for the representation space of $M_{\rm A}$ or $M$.

By the work of Harris and Taylor [Reference Harris and TaylorHT01] for ${{\rm GL}}_n(k)$, Arthur [Reference ArthurArt13] for orthogonal and symplectic groups and Mok [Reference MokMok15] for unitary groups, the local Langlands conjecture for these quasi-split classical groups over local fields is now known. The extended version due to Vogan [Reference VoganVog93], involving pure inner forms of these groups, is also known thanks to the work of Kaletha et al. [Reference Kaletha, Minguez, Shin and WhiteKMSW14] and Mœglin [Reference MœglinMœg07]. We assume this throughout the work, referring to [Reference Gan, Gross and PrasadGGP12a] for precise assertions. For the notion of A-packets, and their relationship to A-parameters, we refer to [Reference ArthurArt89, Reference ArthurArt13], as well as to many works of Mœglin [Reference MœglinMœg06, Reference MœglinMœg09, Reference MœglinMœg11]. Both for L-packet and A-packet, we simultaneously consider all relevant pure inner forms of the pair of groups involved, and use what may be called Vogan L-packet and Vogan A-packet. We do not dwell further on these matters, as they are totally analogous to [Reference Gan, Gross and PrasadGGP12a].

As is well-known, ${{\rm Sym}}^{a}(\mathbb {C}^{2})$ is the unique $(a+1)$-dimensional irreducible representation of ${{\rm SL}}_2(\mathbb {C})$. We often denote this $(a+1)$-dimensional irreducible representation of ${{\rm SL}}_2(\mathbb {C})$ as $[a+1]$. If we regard $[a+1]$ as a representation of the Arthur ${{\rm SL}}_2(\mathbb {C})$, then $[a+1]$ is an A-parameter for ${{\rm GL}}_{a+1}(k)$ whose corresponding A-packet is the singleton set containing only the trivial representation. Hence, by abuse of notation, we also denote by $[a+1]$ the trivial representation of ${{\rm GL}}_{a+1}(k)$. The context will make it clear whether $[a+1]$ denotes an A-parameter or the trivial representation of ${{\rm GL}}_{a+1}(k)$.

If $\pi _1$ is a representation of ${{\rm GL}}_m(k)$ and $\pi _2$ of ${{\rm GL}}_n(k)$, we let $\pi _1 \times \pi _2$ be the representation of ${{\rm GL}}_{m+n}(k)$ parabolically induced from the representation $\pi _1 \boxtimes \pi _2$ of the Levi subgroup ${{\rm GL}}_m(k) \times {{\rm GL}}_n(k)$ of ${{\rm GL}}_{m+n}(k)$. In particular, $[a] \times [b]$ denotes an irreducible (unitary) representation of ${{\rm GL}}_{a+b}(k)$ that has A-parameter $[a] \oplus [b]$. In § 5, we give a more precise description of the A-packets for ${{\rm GL}}_n(k)$.

For any integer $n\geq 1$, we set

\[ \nu := |{\det}|: {{\rm GL}}_n(k) \rightarrow \mathbb{C}^{\times}. \]

When $n=1$ so that ${{\rm GL}}_n(k)=k^{\times }$, we also view $\nu$ as the absolute value map on $\textit {WD}(k)$.

An L-parameter for a classical group $G$ (not ${{\rm GL}}_n(k)$)

\[ \phi: \textit{WD}(k) \rightarrow {\textrm{GL}}(V), \]

is called a discrete L-parameter if it does not factor through a proper Levi subgroup of the dual group $\hat {G}$; equivalently, if it is a multiplicity free sum of selfdual or conjugate-selfdual irreducible representations of $\textit {WD}(k)$ of the correct parity. This terminology originates from the fact that an L-parameter of $G$ is discrete if and only if the associated representations of $G(k)$ are discrete series representations. Similarly, an A-parameter for a classical group (not ${{\rm GL}}_n(k)$)

\[ \phi_{\rm A}: \textit{WD}(k)\times {{\rm SL}}_2(\mathbb{C}) \rightarrow {\textrm{GL}}(V), \]

is called a discrete A-parameter if it is a multiplicity free sum of selfdual or conjugate-selfdual irreducible representations of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$.

For all the number theoretic background on $L$ and $\epsilon$ factors, we refer to the article [Reference TateTat79], usually without explicit mention. All our global $L$-functions will be completed $L$-functions $L(s, \Pi )$ for an automorphic representation $\Pi$ on ${{\rm GL}}_n(\mathbb {A}_F)$ for $F$ a global field, or an automorphic representation on a classical group treated as an automorphic representation $\Pi$ on ${{\rm GL}}_n(\mathbb {A}_F)$ after transfer.

3. Relevant pair of A-parameters

In this section, we formulate the notion of a relevant pair of A-parameters $(M_{\rm A}, N_{\rm A})$ in the context of classical groups (including the case of ${{\rm GL}}$). This notion plays a pivotal role in this paper. We also discuss some basic properties of this notion.

With $k$ a local field, recall that a finite-dimensional complex representation $M_{\rm A}$ of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ which arises from an A-parameter (of a classical group) has a canonical decomposition of the form

\[ M_{\rm A} = \sum_{i = 0}^{d} M_i \otimes {{\rm Sym}}^{i}(\mathbb{C}^{2}), \]

where the $M_i$ are bounded admissible representations of $\textit {WD}(k)$. We say that two representations $M_{\rm A}$ and $N_{\rm A}$ form a relevant pair of A-parameters if we have a decomposition of the respective representations of $\textit {WD}(k)$ as

\[ M_i = M_i^{+} + M_i^{-} \quad \text{and} \quad N_i = N_i^{+} + N_i^{-}, \]

with the property that

\[ M_i^{+} = N_{i+1}^{-}\quad \text{for}\ i \geq 0 \quad \text{and} \quad M_i^{-} = N_{i-1}^{+}\quad \text{for}\ i \geq 1. \]

Therefore, if we write the decomposition of $M_{\rm A}$ as

\[ M_{\rm A} = M_0^{+} + M_0^{-} + \sum_{i = 1}^{d} (M_i^{+} + M_i^{-}) \otimes {{\rm Sym}}^{i} (\mathbb{C}^{2}), \]

we have the following decomposition of $N_{\rm A}$:

\[ N_{\rm A} = N_0^{-} + \sum_{i = 0}^{d} M_i^{+} \otimes {{\rm Sym}}^{i+1} (\mathbb{C}^{2}) + \sum_{i = 1}^{d} M_i^{-} \otimes {{\rm Sym}}^{i-1}(\mathbb{C}^{2}). \]

The notion of a relevant pair $(M_{\rm A},N_{\rm A})$ is symmetric, as we can also write the conditions on the summands as

\[ N_i^{+} = M_{i+1}^{-} \quad \text{for}\ i \geq 0 \quad \text{and} \quad N_i^{-} = M_{i-1}^{+} \quad \text{for}\ i \geq 1. \]

Therefore, if we write the decomposition of $N_{\rm A}$ as

\[ N_{\rm A} = N_0^{+} + N_0^{-} + \sum_{i = 1}^{d} (N_i^{+} + N_i^{-}) \otimes {{\rm Sym}}^{i} (\mathbb{C}^{2}), \]

we have the following decomposition of $M_{\rm A}$:

\[ M_{\rm A} = M_0^{-} + \sum_{i = 0}^{d} N_i^{+} \otimes {{\rm Sym}}^{i+1} (\mathbb{C}^{2}) + \sum_{i = 1}^{d} N_i^{-} \otimes {{\rm Sym}}^{i-1}(\mathbb{C}^{2}). \]

Note that the two summands $M_0^{-}$ and $N_0^{-}$ in a relevant pair are not constrained by the other parameter. In particular, any two bounded representations $(M_0,N_0)$ of $\textit {WD}(k)$ with the trivial action of ${{\rm SL}}_2(\mathbb {C})$ form a relevant pair of A-parameters.

The decomposition of the representations $M_i$ and $N_i$ of $\textit {WD}(k)$ in a relevant pair into components $M_i^{\pm }$ and $N_i^{\pm }$ is unique. Indeed, suppose that the highest (non-zero) summand in the decomposition of $M_{\rm A}$ is $M_d$. Then the highest summand in the decomposition of $N_{\rm A}$ is either $N_{d-1}, N_d$ or $N_{d+1}$. We now consider the three cases separately.

1. (i) Suppose first that it is $N_d$. Then we conclude that $M_d = M_d^{-}$ and $N_d = N_d^{-}$. This implies that $N_{d-1}^{+} = M_d^{-}$, which determines $N_{d-1}^{-}$. Similarly, $M_{d-1}^{+} = N_d^{-}$, which determines $M_{d-1}^{-}$. Continuing to descend in this manner determines all of the decompositions.

2. (ii) Next assume that the highest summand in the decomposition of $N_{\rm A}$ is $N_{d-1}$. Again we conclude that $M_d = M_d^{-}$, but now we also have $M_{d-1} = M_{d-1}^{-}$ as $N_d = 0$. Then $N_{d-1}^{+} = M_d^{-}$, which determines $N_{d-1}^{-}$ and $N_{d-2}^{+} = M_{d-1}^{-}$, which determines $N_{d-2}^{-}$. Continuing to descend in this manner gives the full decomposition.

3. (iii) Finally, if the highest summand in the decomposition of $N_{\rm A}$ is $N_{d+1}$, then one can switch the roles of $M_{\rm A}$ and $N_{\rm A}$ and use the previous argument.

We have occasion to use the following lemma about relevant pair of A-parameters whose simple proof is omitted.

Lemma 3.1 Let

\begin{align*} M_{\rm A} &= M_0^{+} + M_0^{-} + \sum_{i = 1}^{d} M_i \otimes {{\rm Sym}}^{i} (\mathbb{C}^{2}),\\ N_{\rm A} &= N_0^{+} +N_0^{-} + \sum_{i = 1}^{d} N_i \otimes {{\rm Sym}}^{i} (\mathbb{C}^{2}), \end{align*}

be a relevant pair of A-parameters. Then we have the relations

\begin{align*} \sum_{2i-1 \geq 1}M_{2i-1} &= \sum_{2i \geq 0} N_{2i} - N^{-}_0,\\ \sum_{2i-1 \geq 1} N_{2i-1} &= \sum_{2i\geq 0}M_{2i} - M^{-}_{0}. \end{align*}

Suppose now that $(M_{\rm A}, N_{\rm A})$ is a pair of representations with $M_{\rm A}$ symplectic and $N_{\rm A}$ even orthogonal. Then $M_{\rm A}$ is the A-parameter of an odd special orthogonal group whereas $N_{\rm A}$ is that of an even special orthogonal group. If $(M_{\rm A}, N_{\rm A})$ is relevant, the argument we gave for the unicity of the decomposition of $M_i$ and $N_i$ shows that the summands $M_i^{\pm }$ are each symplectic for $i$ even and orthogonal for $i$ odd, and the summands $N_i^{\pm }$ are each orthogonal for $i$ even and symplectic for $i$ odd.

Likewise, suppose that $k/k_0$ is a separable quadratic extension and $(M_{\rm A}, N_{\rm A})$ is a pair of representations of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ with $M_{\rm A}$ conjugate symplectic and $N_{\rm A}$ conjugate orthogonal, then $M_{\rm A}$ is an A-parameter for an even unitary group whereas $N_{\rm A}$ is an A-parameter of an odd unitary group. When $(M_{\rm A}, N_{\rm A})$ is relevant, one sees that the summands $M_i^{\pm }$ and $N_i^{\pm }$ are conjugate symplectic and conjugate orthogonal, respectively.

On the other hand, suppose that $F$ is a global field with conjectural Langlands group $L(F)$, so that global A-parameters of classical groups can be thought of as finite-dimensional complex representations of $L(F) \times {{\rm SL}}_2(\mathbb {C})$. Of course, because the existence of $L(F)$ is not known, one needs to interpret an irreducible $n$-dimensional representation of $L(F)$ as a cuspidal automorphic representation of ${{\rm GL}}_n(\mathbb {A}_F)$. In the context of classical groups, one needs to interpret an irreducible $n$-dimensional symplectic (respectively orthogonal) representation of $L(F)$ as a cuspidal automorphic representation of ${{\rm GL}}_n(\mathbb {A}_F)$ for which the exterior square (respectively symmetric square L-function) has a pole at $s=1$. With this caveat, one can similarly define the notion of a relevant pair of global A-parameters.

Since the work of Ichino and Ikeda, cf. [Reference Ichino and IkedaII10], it is natural to consider in the global setting the ratio of L-functions:

\[ L(M,N, s) = \frac{L(M \otimes N^{\vee}, s + 1/2) \cdot L(M^{\vee} \otimes N, s + 1/2)}{ L(M \otimes M^{\vee}, s + 1)\cdot L(N \otimes N^{\vee}, s + 1)} \]

if $M \times N$ is an L-parameter for ${{\rm GL}}_m \times {{\rm GL}}_n$, or the ratio

\[ L(M,N, s) = \frac{L(M \otimes N, s + ~1/2)}{L({{\rm Sym}}^{2} M \oplus \wedge^{2}N, s + 1)} \]

if $M \times N$ is an L-parameter for ${{\rm SO}}_{2m+1} \times {{\rm SO}}_{2n}$. One may of course consider $L(M,N, s)$ in the local context as well. We conclude this section by highlighting some results about the analytic properties of $L(M,N, s)$ at $s=0$. The proofs of these results, which proceed by explicit computation, will be given in §§ 13 and 14 at the end of the paper.

Theorem 3.2 Let $k$ be a non-archimedean local field and let $(M_{\rm A}, N_{\rm A})$ be a relevant pair of A-parameters for ${{\rm GL}}_m(k) \times {{\rm GL}}_n(k)$ with associated pair of L-parameters $(M,N)$. Then the order of pole at $s=0$ of

\[ L(M,N, s) = \frac{L(M \otimes N^{\vee}, s + 1/2) \cdot L(M^{\vee} \otimes N, s + 1/2)}{ L(M \otimes M^{\vee}, s + 1)\cdot L(N \otimes N^{\vee}, s + 1)} \]

is greater than or equal to zero.

4. Correlator

In this section, we describe an elegant formulation of the notion of a relevant pair of A-parameters which is due to Zhiwei Yun (personal correspondence).

Consider a pair $(M_{\rm A}, N_{\rm A})$ of selfdual finite-dimensional representations of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ with $M_{\rm A}$ symplectic and $N_{\rm A}$ orthogonal, realized on vector spaces $V$ and $W$, respectively. The action of the diagonal torus of ${{\rm SL}}_2(\mathbb {C})$ induces a $\mathbb {Z}$-grading on $V$ and $W$. More precisely, if we identify $\mathbb {G}_m$ with the diagonal torus, taking $t \in \mathbb {G}_m$ to the diagonal matrix $(t, t^{-1})$, then the degree $n$ part of $V$ is the eigenspace for the character $t \mapsto t^{n}$.

Let $e$ (respectively, $e'$) be the nilpotent endomorphism of $V$ (respectively, $W$) given by the image of the usual upper triangular element in the Lie algebra $\mathfrak {sl}_2$ of ${{\rm SL}}_2(\mathbb {C})$. Then the action of $e$ and $e'$ shifts degree by $2$ on $V$ and $W$. In other words, $e$ and $e'$ are degree $2$ elements in ${\textrm {End}\,}(V)$ and ${\textrm {End}\,}(W)$ (equipped with the induced grading).

The key definition of Zhiwei Yun is as follows.

Definition 4.1 For a pair $(M_{\rm A}, N_{\rm A})$ of finite-dimensional representations of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ realized on vector spaces $(V,W)$, together with invariant non-degenerate bilinear forms, a correlator for $(M_{\rm A}, N_{\rm A})$ is a $\textit {WD}(k)$-equivariant linear map

\[ T: V \rightarrow W, \]

such that:

1. (i) $T$ shifts degree by 1, that is, $T$ is an element of degree $1$ in ${\textrm {Hom}}(V, W)$;

2. (ii) $T^{*}T=e$ and $TT^{*}=e'$, where $T^{*}:W \rightarrow V$ is the adjoint of $T$.

Now we have the main observation.

Proof. Given a relevant pair of A-parameters $(M_{\rm A},N_{\rm A})$, we construct a correlator as follows. Writing

\[ V = \bigoplus_n V_n \otimes {{\rm Sym}}^{n}(\mathbb{C}^{2}), \]

the action of the diagonal torus of ${{\rm SL}}_2(\mathbb {C})$ gives a further decomposition of the form

\[ V = \sum_n \sum_a V_n \otimes t_n^{a}, \]

where the inner sum is taken over integers $|a| \leq n$ with $a \equiv n (\bmod 2)$. Moreover, we have written $t_n^{a}$ for a non-zero eigenvector in ${{\rm Sym}}^{n}(\mathbb {C}^{2})$ for the character $t \mapsto t^{a}$. We normalize the choice of $t_n^{a}$ so that the nilpotent element $e$ maps $V_n \otimes t_n^{a}$ to $V_n \otimes t_n^{a+2}$. In particular, the kernel of $e$ on $V$ is $\sum _n V_n \otimes t_n^{n}$.

Now we define a correlator $T$ by giving $\textit {WD}(k)$-equivariant surjective maps

\[ \begin{cases} V_n^{+} \otimes t_n^{a} \longrightarrow W_{n+1}^{-} \otimes t_{n+1}^{a+1}\\ V_n^{-} \otimes t_n^{a} \longrightarrow W_{n-1}^{+} \otimes t_{n-1}^{a+1} \end{cases} \]

for each $n$. In particular,

\[ \textrm{Ker}(T) = \bigoplus_n V_n^{-} \otimes t_n^{n}, \]

and one can easily check that $T$ is a correlator.

Conversely, one argues that a correlator $T$ must send $V_n \otimes t_n^{a}$ to $W_{n+1}^{-} \otimes t_{n+1}^{a+1} \oplus W_{n-1}^{+} \otimes t_{n-1}^{a+1}$ for any $|a| \leq n$. Then $\textrm {Ker}(T)$ determines $V_n^{-}$ for each $n$ and, hence, the representation $V_n^{+}$ (because $V_n$ is semisimple). Hence, the existence of $T$ implies that the pair $(M_{\rm A}, N_{\rm A})$ is relevant.

For $(M_{\rm A},N_{\rm A})$ a pair of local A-parameters for ${{\rm GL}}_n \times {{\rm GL}}_m$, we can apply the same reformulation above to the selfdual A-parameters $M_{\rm A}+M_{\rm A}^{\vee }$ and $N_{\rm A}+N_{\rm A}^{\vee }$ (on vector spaces $V,W$ of dimensions $2n,2m$). A correlator in this situation becomes a pair of maps of $\textit {WD}(k)$-representations $T:V +V^{\vee } \rightarrow W+W^{\vee }$ and $T^{*}: W+W^{\vee } \rightarrow V+V^{\vee }$, each of degree 1, such that $T^{*}T=e$ and $TT^{*}=e'$. One can then check by a similar argument as given previously that the existence of such a correlator is equivalent to the relevance of $(M_{\rm A},N_{\rm A})$.

The notion of a correlator $T$ defined above should remind the reader familiar with the theory of reductive dual pairs (in the sense of Howe) of the moment map that arises there. Let us explicate this connection here. Given a pair $(M_{\rm A}, N_{\rm A})$ of A-parameters where $M_{\rm A}$ (respectively, $N_{\rm A})$ is a symplectic (respectively, orthogonal) representation of $\textit {WD}(k)$ on a vector space $V$ (respectively, $W$), one has a symplectic representation $V \otimes W$ of $\textit {WD}(k)$:

\[ \textit{WD}(k) \longrightarrow {\textrm{Sp}}(V) \times O(W) \longrightarrow {\textrm{Sp}}(V \otimes W). \]

The ${{\rm Sp}}(V) \times O(W)$-symplectic variety $V \otimes W$ is Hamiltonian and possesses a $\textit {WD}(k)$-equivariant moment map

\[ \mu = \mu_V \times \mu_W: V \otimes W \longrightarrow \mathfrak{sp}(V) \times \mathfrak{so}(W), \]

where $\mathfrak {sp}(V)$ (respectively, $\mathfrak {so}(W)$) is the Lie algebra of ${{\rm Sp}}(V)$ (respectively, ${{\rm SO}}(W)$), which we prefer to write as a double fibration:

As $V$ and $W$ are equipped with non-degenerate bilinear forms, we have isomorphisms

\[ V \otimes W \cong {{\rm Hom}}(V,W) \cong {{\rm Hom}}(W,V). \]

The maps $\mu _V$ and $\mu _W$ are then given by

\[ \mu_V(T) = T^{*} T \quad \text{and} \quad \mu_W(T) = T T^{*} \quad \text{for $T \in {{\rm Hom}}(V,W)$.} \]

The double fibration diagram gives one a correspondence between the set of ${{\rm Sp}}(V)$-adjoint orbits and $O(W)$-adjoint orbits: say that a ${{\rm Sp}}(V)$-adjoint orbit $\mathcal {O}_V$ and an $O(W)$-adjoint orbit $\mathcal {O}_W$ correspond if there exists $T \in V \otimes W$ such that

\[ \mu_V(T) \in \mathcal{O}_W \quad \text{and} \quad \mu_W(T) \in \mathcal{O}_W. \]

Now we may reformulate the notion of a correlator in terms of this moment map. Given a pair of A-parameters $(M_{\rm A}, N_{\rm A})$, recall that one has a pair of nilpotent elements

\[ e \in \mathfrak{sp}(V) \quad \text{and} \quad e' \in \mathfrak{so}(W) \]

which are fixed by $\textit {WD}(k)$. Recalling that $V \otimes W \cong {\textrm {Hom}}(V,W)$ has a $\mathbb {Z}$-grading, we see that a correlator for $(M_{\rm A}, N_{\rm A})$ is a degree $1$ element $T \in {\textrm {Hom}}(V,W)$ that is fixed by $\textit {WD}(k)$ and such that

\[ \mu_V(T) = e \quad \text{and} \quad \mu_W(T) = e'. \]

Hence, we have a nice geometric formulation of the notion of relevance: $(M_{\rm A}, N_{\rm A})$ is relevant if and only if the nilpotent orbits of $e$ and $e'$ are in the moment map correspondence via a $\textit {WD}(k)$-fixed degree $1$ element $T$.

We note that such correlator maps $T$ have also appeared in the work of Gomez and Zhu [Reference ZhuZhu19] where they studied the transfer of generalized Gelfand–Graev models (attached to nilpotent orbits) under the local theta correspondence. In their work, the correlator maps were considered on the side of representation theory of the real or $p$-adic groups, whereas in our work, they appeared on the side of the Langlands dual groups.

5. Local conjecture for ${{\rm GL}}_{n}$

Having introduced the key notion of relevant pairs of A-parameters, we can begin the consideration of restriction problems. In this section, we discuss the restriction problem for ${{\rm GL}}_n(k) \subset {{\rm GL}}_{n+1}(k)$ in the context of L-parameters of Arthur type. We first explicate the representations of ${{\rm GL}}_n(k)$ which belong to such L-packets.

An A-parameter of ${{\rm GL}}_n(k)$ is an $n$-dimensional representation of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ of the form

\[ M_{\rm A} = \bigoplus_{i=1}^{r} M_i \boxtimes {{\rm Sym}}^{d_i}(\mathbb{C}^{2}), \]

where $M_i$ is an irreducible $m_i$-dimensional representation of $\textit {WD}(k)$. As discussed in the introduction, $M_{\rm A}$ gives rise to an L-parameter $M$. Moreover, the local A-packet associated to $M_{\rm A}$ is equal to the local L-packet associated to $M$ that is a singleton set. We can describe this unique representation $\pi _M$ as follows. By the local Langlands correspondence, each $M_i$ corresponds to a discrete series representation $\pi _{M_i}$ of ${{\rm GL}}_{m_i}(k)$. The representation in the A-packet associated to the A-parameter $M_i \otimes {{\rm Sym}}^{d_i}(\mathbb {C}^{2})$ is a Speh representation $\textrm {Speh}(M_i, d_i)$, which is the unique irreducible quotient of the standard module

\[ \pi_{M_i} |{\det}|^{d_i/2} \times \pi_{M_i} |{\det}|^{d_i/2 -1} \times\cdots\times \pi_{M_i} |{\det}|^{-d_i/2}, \]

where we have used the standard notation for parabolic induction. Then the representation $\pi _M$ is given by the irreducible parabolic induction

\[ \pi_M = \textrm{Speh}(M_1, d_1) \times \textrm{Speh}(M_2, d_2) \times\cdots\times \textrm{Speh}(M_r, d_r). \]

As an example, when $M = \chi \boxtimes {{\rm Sym}}^{n-1}(\mathbb {C}^{2})$ with $\chi$ a one-dimensional character of $\textit {WD}(k)$, the associated representation $\pi _M$ is the character $\chi \circ \det$ of ${{\rm GL}}_n(k)$.

We can now consider the restriction problem. Suppose that $M_{\rm A}$ is an A-parameter of ${{\rm GL}}_{n+1}(k)$ and $N_{\rm A}$ one for ${{\rm GL}}_n(k)$. Let $\pi _M$ and $\pi _N$ be the irreducible representations in the respective L-packets (of Arthur type), which we have described previously. By [Reference Aizenbud, Gourevitch, Rallis and SchiffmannAGRS10], it is known that

\[ \dim {{\rm Hom}}_{{{\rm GL}}_n(k)}( \pi_M, \pi_N) \leq 1. \]

When $M_{\rm A}$ and $N_{\rm A}$ are tempered A-parameters, so that $\pi _M$ and $\pi _N$ are both tempered (and generic), it is well-known that the Hom space given in this equation is one-dimensional. In general, we make the following conjecture.

Conjecture 5.1 In the context given here ($k$ any local field, archimedean or non-archimedean),

\[ {{\rm Hom}}_{{{\rm GL}}_n(k)}(\pi_M, \pi_N) \ne 0 \Longleftrightarrow \text{the pair $(M_{\rm A}, N_{\rm A})$ of A-parameters is relevant.} \]

Of course, one could have formulated a restriction problem for an arbitrary ${{\rm GL}}_n \times {{\rm GL}}_m$, that is, including the case of Bessel models and Fourier–Jacobi models. We highlight this particular case here because of its intrinsic simplicity, and defer the general case to the next section where we deal with classical groups.

Before proceeding further, it is good to do a reality check. Suppose that

\[ M_{\rm A} = {{\rm Sym}}^{n}(\mathbb{C}^{2}) \]

is the A-parameter of the trivial representation of ${{\rm GL}}_{n+1}(k)$, so that $\pi _M$ is the trivial representation ${{\rm GL}}_{n+1}(k)$. Then its restriction to ${{\rm GL}}_n(k)$ is certainly the trivial representation, that is, the only A-parameter $N_{\rm A}$ of ${{\rm GL}}_n(k)$ for which ${\textrm {Hom}}_{{{\rm GL}}_n(k)}(\pi _M, \pi _N) \ne 0$ is

\[ N_{\rm A} = {{\rm Sym}}^{n-1}(\mathbb{C}^{2}). \]

The reader can easily verify that the only $N_{\rm A}$ (of dimension $n$) such that $({{\rm Sym}}^{n}(\mathbb {C}^{2}), N_{\rm A})$ is relevant is $N_{\rm A} = {{\rm Sym}}^{n-1}(\mathbb {C}^{2})$.

On the other hand, if we start with $N_{\rm A} = {{\rm Sym}}^{n-1}(\mathbb {C}^{2})$, then the A-parameters $M_{\rm A}$ (of dimension $(n+1)$) of ${{\rm GL}}_{n+1}(k)$ such that $(M_{\rm A}, N_{\rm A})$ is relevant are:

1. (i) $M_{\rm A} = {{\rm Sym}}^{n}(\mathbb {C}^{2})$; or

2. (ii) $M_{\rm A} = {{\rm Sym}}^{n-2}(\mathbb {C}^{2}) \oplus M_0$ where $M_0$ is a bounded two-dimensional representation of $\textit {WD}(k)$.

This reflects the known fact that the irreducible admissible representations of ${{\rm GL}}_{n+1}(k)$ which are of Arthur type and which are ${{\rm GL}}_{n}(k)$-distinguished are either trivial or theta lifts of tempered representations of ${{\rm GL}}_2(k)$ (see [Reference VenketasubramanianVen13] for the precise results for all irreducible representations of ${{\rm GL}}_{n+1}(k)$ for $k$ a non-archimedean local field).

The rest of this section is devoted to proving a special case of Conjecture 5.1.

Theorem 5.2 Let $k$ be a non-archimedean local field. Consider a pair of local A-parameters $(M_{\rm A}, N_{\rm A})$ for ${{\rm GL}}_{n+1}(k) \times {{\rm GL}}_n(k)$ satisfying the following extra properties:

1. (a) on every irreducible summand of $M_{\rm A}$ or $N_{\rm A}$, at least one of the two ${{\rm SL}}_2(\mathbb {C})$ in

   \[ \textit{WD}(k) \times {{\rm SL}}_2(\mathbb{C}) = W(k)\times {{\rm SL}}_2(\mathbb{C}) \times {{\rm SL}}_2(\mathbb{C}) \]
   acts trivially;

2. (b) for any irreducible representation $\rho$ of $W(k)$ and $a \geq 1$ an integer,

   \[ \rho \otimes \mathbb{C} \otimes {{\rm Sym}}^{a}(\mathbb{C}^{2}) \oplus \rho \otimes {{\rm Sym}}^{a}(\mathbb{C}^{2}) \otimes \mathbb{C} \nsubseteq M_{\rm A}, \]
   and likewise for $N_{\rm A}$.

Then

\[ {\rm Hom}_{{{\rm GL}}_n(k)}(\pi_M,\pi_N) \not = 0 \Longleftrightarrow \text{$(M_{\rm A}, N_{\rm A})$ is relevant.} \]

In particular, the two properties are satisfied and, hence, the result holds in the case when the Deligne ${{\rm SL}}_2(\mathbb {C})$ (i.e. the ${{\rm SL}}_2(\mathbb {C})$ in $\textit {WD}(k)$) acts trivially on both $M_{\rm A}$ and $N_{\rm A}$.

Proof. The proof proceeds by analyzing the restriction of the irreducible representation $\pi _M$ of ${{\rm GL}}_{n+1}(k)$ to its mirabolic subgroup. This restriction was described in a classic theorem of Bernstein and Zelevinsky in [Reference Bernstein and ZelevinskyBZ77], in terms of certain induced representations arising from the Bernstein–Zelevinsky derivatives $\pi _M^{i}$ of the representation $\pi _M$. As the mirabolic subgroup contains ${{\rm GL}}_n(k)$, this gives one a way to understand the restriction of $\pi _M$ to ${{\rm GL}}_n(k)$.

By this theorem of Bernstein and Zelevinsky, if $\textrm {Hom}_{{{\rm GL}}_n(k)}(\pi _M,\pi _N) \not = 0$, then for some $i \geq 0$, there exists:

1. (i) an irreducible composition factor $A$ of $\nu ^{1/2} (\pi _M)^{i+1}$;

2. (ii) an irreducible composition factor $B$ of $((\pi _N^{\vee })^{i})^{\vee }$;

such that

\[ {\rm Hom}_{{{\rm GL}}_{n-i}(k)} (A, B) \not = 0. \]

For the converse, given some $(i, A, B)$ as described previously, suppose that we are lucky enough to have the further property:

1. (i) for any $j\leq i$ and any $(C,D) \not = (A,B)$, where $C$ is an irreducible composition factor of $\nu ^{1/2} (\pi _M)^{j+1}$ and $D$ is an irreducible composition factor of $((\pi _N^{\vee })^{j})^{\vee }$,

   \[ \textrm{Ext}^{1}_{{\textrm{GL}}_{n-j}(k)}[ C, D] = 0. \]

Then the non-zero homomorphism from a certain submodule of $\pi _M$ to $\pi _N$ represented by a homomorphism in $\textrm {Hom}_{{{\rm GL}}_{n-i}(k)}[ A, B] \not = 0,$ extends to all of $\pi _M$.

To have this vanishing of ${\textrm {Ext}}^{1}[C,D]$, the simplest reason would be when the cuspidal support of any irreducible composition factor of $\nu ^{1/2} (\pi _N^{\vee })^{j}$ and of $(\pi _M^{j+1})^{\vee }$ are different (except for a unique choice $(A,B)$ among all pairs $(C,D)$). The conditions (a) and (b) in the theorem are imposed to ensure this.

Denote by $[d]$ the trivial representation of ${{\rm GL}}_d(k)$, and by $[d]^{\times m}$ the irreducible admissible representation of ${{\rm GL}}_{d\cdot m}(k)$, which is $[d] \times \cdots \times [d]$. If $\sigma$ is a cuspidal representation of ${{\rm GL}}_r(k)$, let $\sigma [d]$ be the Speh representation of ${{\rm GL}}_{r\cdot d}(k)$ constructed from $\sigma$, that is, in the Zelevinsky notation $\sigma [d]= Z[\sigma \nu ^{-(d-1)/2},\ldots , \sigma \nu ^{(d-1)/2} ]$. It is known that the only non-zero derivative of $\sigma [d]$ is $\sigma [d]^{i}$ for $i=0, r$, and $\sigma [d]^{r} = \nu ^{-1/2}\sigma [d-1]= Z[\sigma \nu ^{-(d-1)/2},\ldots , \sigma \nu ^{(d-3)/2}]$.

Denote by ${\textrm {St}}_d[\rho ]$, the generalized Steinberg representation of ${{\rm GL}}_{dd'}(k)$ where $\rho$ is a cuspidal unitary representation of the general linear group ${{\rm GL}}_{d'}(k)$. The derivatives of ${\textrm {St}}_d[\rho ]^{i}$ are known to be non-zero only for $i = jd'$ for some $0\leq j \leq d$ with ${\textrm {St}}_d[\rho ]^{jd'} = \nu ^{j/2} {\textrm {St}}_{d-j}[\rho ]$. A tempered representation of ${{\rm GL}}_m(k)$ is built as a product of the generalized Steinberg representations ${\textrm {St}}_d[\rho ]$.

For a cuspidal representation $\rho$ of ${{\rm GL}}_m(k)$ (or the corresponding irreducible representation of $W(k)$), we call the set of representations $\{\rho \nu ^{i} \mid i \in \mathbb {Z}\}$, the cuspidal line passing through $\rho$. By Leibnitz rule, if a representation has cuspidal support contained in the cuspidal line passing through $\rho$, all its derivatives have the same property. Therefore, the following analysis which mostly involves dealing with the non-tempered parts of $\pi _M$ and $\pi _N$, to find an irreducible composition factor $A$ of $\nu ^{1/2} (\pi _M)^{i+1}$, and an irreducible composition factor $B$ of $((\pi _N^{\vee })^{i})^{\vee }$, such that $\textrm {Hom}_{{{\rm GL}}_{n-i}(k)}[ A, B] \not = 0,$ we can focus attention on a cuspidal line passing through a fixed cuspidal representation $\rho$, that is, we can assume that the non-tempered parts of $M_{\rm A}$ and $N_{\rm A}$ when restricted to $W(k)$ are multiples of $\rho$, without any constraint on the tempered parts of $M_{\rm A}$ and $N_{\rm A}$. Replacing $\rho$ (an irreducible bounded representation of $W(k)$) by the trivial representation of $W(k)$ has no effect on the following analysis, and this is what we assume in the rest of the proof, that is, we assume in the rest of the proof that the non-tempered parts of $M_{\rm A}$ and $N_{\rm A}$ when restricted to $W(k)$ are multiples of the trivial representation.

Let

\begin{align*} \pi_M &= [a_1] \times \cdots \times [a_r] \times [b_1] \times \cdots \times [b_s] \times {\textrm{St}}[c_1] \times \cdots \times {\textrm{St}}[c_t]\times {\textrm{St}}[d_1] \times \cdots \times {\textrm{St}}[d_u], \\ \pi_N &= [a'_1] \times \cdots \times [a'_{r'}] \times [b'_1] \times \cdots \times[b'_{s'}] \times {\textrm{St}}[c'_1] \times \cdots \times {\textrm{St}}[c'_{t'}] \times {\textrm{St}}[d'_1] \times \cdots \times {\textrm{St}}[d'_{u'}], \end{align*}

where ${\textrm {St}}[a]$ denotes the Steinberg representation of ${{\rm GL}}_a(k)$. There could be a further non-Steinberg tempered part in $\pi _M$, and in $\pi _N$, which we do not write here out for brevity of notation and which will not play any role in the arguments given in the following. Note also that ${\textrm {St}}[1]=[1]$, the trivial representation of ${{\rm GL}}_1(k)$; in fact, we use the notation ${\textrm {St}}[a]$ in the expressions given previously for $\pi _M,\pi _N$ only for $a>1$.

This notation is so made that for calculating $(j+1)$th derivative of $\pi _M$, and $j$th derivative of $\pi _N$, we differentiate all the terms $[a_i], [a'_i], {\textrm {St}}[d_i], {\textrm {St}}[d'_i]$ at least once, and no more for $[a_i], [a'_i]$; whereas none of the terms involving $[b_i], [b'_i], {\textrm {St}}[c_i], {\textrm {St}}[c'_i]$ are to be differentiated. If the cuspidal support of an irreducible composition factor $C$ of $\nu ^{1/2}\pi _M^{j+1}$ and $D$ of $(\pi _N^{\vee }){{}^{j}}^{\vee }$ are to be the same, we must have equality of cuspidal supports of $\tilde {C}$, and $\tilde {D}$ defined as follows:

\begin{align*} \tilde{C} &= \nu^{1/2} \{ \nu^{-1/2}[a_1-1] \times \cdots \times \nu^{-1/2}[a_r-1] \times [b_1] \times \cdots \times [b_s]\\ &\quad \times {\textrm{St}}[c_1] \times \cdots \times {\textrm{St}}[c_t] \times \nu^{e_1/2}{\textrm{St}}[d_1-e_1] \times \cdots \times \nu^{e_u/2}{\textrm{St}}[d_u-e_u]\}, \\ \tilde{D} &= \nu^{1/2}[a'_1-1] \times \cdots \times \nu^{1/2}[a'_{r'}-1] \times [b'_1] \times \cdots \times[b'_{s'}] \times \\ &\quad \times {\textrm{St}}[c'_1] \times \cdots \times {\textrm{St}}[c'_{t'}] \times \nu^{-f_1/2}{\textrm{St}}[d'_1-f_1] \times \cdots \times \nu^{-f_{u'}/2}{\textrm{St}}[d'_{u'}-f_{u'}]. \end{align*}

If the cuspidal supports of the representations $\tilde {C}$ and $\tilde {D}$ are to be the same, we draw the following conclusions from Lemma 5.3. Let us recall that the cuspidal support of ${\textrm {St}}[a]$ and $[a]$ is the same, so when using Lemma 5.3, we replace all ${\textrm {St}}[a]$ in the previous expressions for $\tilde {C}$ and $\tilde {D}$ by the corresponding $[a]$. Further, the hypothesis in Lemma 5.3 is easily seen to be the same as the hypothesis in the Theorem 5.2 that for any integer $a \geq 1$, the representation $\mathbb {C} \otimes {{\rm Sym}}^{a}(\mathbb {C}^{2}) \oplus {{\rm Sym}}^{a}(\mathbb {C}^{2}) \otimes \mathbb {C}$ does not appear in $M_{\rm A}$ or in $N_{\rm A}$.

1. (i) There is no differentiation for ${\textrm {St}}[d_i]$ and ${\textrm {St}}[d'_i]$.

2. (ii) For each $[a_i]$ in $\pi _M$ (which get differentiated once), there is exactly one of $[a_i-1]$ or ${\textrm {St}}[a_i-1]$ amongst $[b'_i]$ or ${\textrm {St}}[c'_i]$ for $\pi _N$ (these are not differentiated in achieving $\pi _N^{j}$).

3. (iii) For each $[a'_i]$ in $\pi _N$ (which get differentiated once), there is exactly one of $[a'_i-1]$ or ${\textrm {St}}[a'_i-1]$ amongst $[b_i]$ or ${\textrm {St}}[c_i]$ for $\pi _M$ (these are not differentiated in achieving $\pi _M^{j+1}$).

4. (iv) Each of the terms in $\tilde {C}$ and $\tilde {D}$ (in expressing these representations as a sum, in the Grothendieck group of representations, of products of representations of the form $[a]$ and ${\textrm {St}}[b]$) is matched through conclusions (i), (ii), and (iii).

It follows that if the cuspidal supports of $\tilde {C}$ and $\tilde {D}$ are to be the same, we must have the following structure for $\pi _M$ and $\pi _N$ (up to multiplication by a tempered representation $\pi _M^{t}$ in $\pi _M$ and up to multiplication by a tempered representation $\pi _N^{t}$ in $\pi _N$; in using the Leibnitz rule to calculate $\pi _M^{j+1}$ and $\pi _N^{j}$, one takes the highest possible derivatives for $\pi _M^{t}$ and $\pi _N^{t}$, which has the effect that the terms $\pi _M^{t}$ and $\pi _N^{t}$ do not show up in $\pi _M^{j+1}$ and $\pi _N^{j}$):

\begin{align*} \pi_M &\stackrel{({\rm i})}= [b'_1+1] \times \cdots \times[b'_{s'}+1] \times [c'_1+1] \times \cdots \times [c'_{t'}+1] \times [b_1] \times \cdots \times [b_s] \\ &\quad \times {\textrm{St}}[c_1] \times \cdots \times {\textrm{St}}[c_t], \\ \pi_N &\stackrel{({\rm ii})} = [b_1+1] \times \cdots \times [b_{s}+1] \times [c_1+1] \times \cdots \times[c_{t}+1] \times [b'_1] \times \cdots \times[b'_{s'}] \\ &\quad \times {\textrm{St}}[c'_1] \times \cdots \times {\textrm{St}}[c'_{t'}]. \end{align*}

For these representations $\pi _M,\pi _N$, if there is a composition factor $C$ of $\nu ^{1/2} \pi _M^{j+1}$, and $D$ of ${(\pi _N^{\vee })^{j}}^{\vee },$ for which $C,D$ have the same cuspidal support as representations of ${{\rm GL}}_{n-j}(k)$, the representations $\tilde {C}$ and $\tilde {D}$ must be

\begin{align*} \tilde{C} &\stackrel{({\rm iii})}= [b'_1] \times \cdots \times[b'_{s'}] \times [c'_1] \times \cdots \times [c'_{t'}] \times \nu^{1/2} \{ [b_1] \times \cdots \times [b_s] \times {\textrm{St}}[c_1] \times \cdots \times {\textrm{St}}[c_t]\}, \\ \tilde{D} &\stackrel{({\rm iv})}= \nu^{1/2}\{ [b_1] \times \cdots \times [b_{s}] \times [c_1] \times \cdots \times[c_{t}] \} \times [b'_1] \times \cdots \times[b'_{s'}] \times {\textrm{St}}[c'_1] \times \cdots \times {\textrm{St}}[c'_{t'}]. \end{align*}

By Lemma 5.5, such representations $\tilde {C}, \tilde {D}$ are irreducible, and for $\textrm {Hom}_{{{\rm GL}}_{n-j}(k)}[ \tilde {C}, \tilde {D}]$ to be non-zero, we must have

\begin{align*} c_i &= 1 \quad \text{for all}\ i,\\ c'_i &= 1 \quad \text{for all}\ i. \end{align*}

Thus, as a consequence, we find that if any of the spaces $\textrm {Hom}_{{{\rm GL}}_{n-j}(k)}[ C, D]$ is non-zero, then $M_{\rm A},N_{\rm A}$ are a relevant pair of A-parameters.

Next, we prove that ${\textrm {Hom}}_{{{\rm GL}}_n(k)}[\pi _M,\pi _N] \not = 0$ when the A-parameter $M_{\rm A}$ for an irreducible representation $\pi _M$ of ${{\rm GL}}_{n+1}(k)$ and $N_{\rm A}$ for an irreducible representation $\pi _N$ of ${{\rm GL}}_n(k)$ satisfies all conditions in the statement of this theorem with $M_{\rm A},N_{\rm A}$, a relevant pair of A-parameters. With the structure of $\pi _M,\pi _N$ given in conclusions (i) and (ii), the only way to obtain an irreducible composition factor $\tilde {C}$ of $\nu ^{1/2}\pi _M^{j+1}$ and $\tilde {D}$ of $(\pi _N^{\vee }){{}^{j}}^{\vee }$ to have same cuspidal supports is that none of the terms ${\textrm {St}}(c_i), {\textrm {St}}(c'_i)$ should be present in the expressions for $\pi _M$ and $\pi _N$ given in conclusions (i) and (ii), and also in $\tilde {C},\tilde {D}$ given in conclusions (iii) and (iv), that is, we must take full derivative of all tempered factors of $\pi _M$ that do not match-up with presence of a $[2]$ in $\pi _N$ and we must take full derivative on all tempered factors of $\pi _N$ which do not match-up with presence of a $[2]$ in $\pi _M$.

Thus, the strategy outlined in the beginning of the proof of the theorem that there is a unique $j$, and a unique irreducible composition factor $C$ of $\nu ^{1/2}\pi _M^{j+1}$ and $D$ of $(\pi _N^{\vee }){{}^{j}}^{\vee }$ with the same cuspidal supports holds; we elaborate more on this now.

Write

\begin{align*} \pi_M &= [b'_1+1] \times \cdots \times[b'_{s'}+1] \times [b_1] \times \cdots \times [b_s] \times \tau_1, \\ \pi_N &= [b_1+1] \times \cdots \times [b_{s}+1] \times [b'_1] \times \cdots \times[b'_{s'}] \times \tau_2, \end{align*}

where $\tau _1,\tau _2$ are tempered representations, not generalized Steinberg, of ${{\rm GL}}_{d_1}(k)$ and ${{\rm GL}}_{d_2}(k)$, for certain integers $d_1,d_2$ with

\begin{align*} s'+d_1 + \sum_{i=1}^{s} b_i + \sum_{i=1}^{s'} b'_i &= n+1, \\ s+d_2 + \sum_{i=1}^{s} b_i + \sum_{i=1}^{s'} b'_i &= n. \end{align*}

We see that there is exactly one integer $j$, which, by the previous equations, has the property that

\begin{align*} j+1 &= s'+ d_1,\\ j &= s+d_2, \end{align*}

and for this integer $j$, there is only one composition factor, say $A$, of $\nu ^{1/2} \pi _M^{j+1}$ and one of ${(\pi _N^{\vee })^{j}}^{\vee }$, say $B$, for which $\textrm {Hom}_{{{\rm GL}}_{n-j}(k)}[A, B]$ is non-zero, and all the other composition factors of $\nu ^{1/2} \pi _M^{j+1}$ and of ${(\pi _N^{\vee })^{j}}^{\vee }$ have different cuspidal supports, and this is also the case for any composition factors of $\nu ^{1/2} \pi _M^{i+1}$ and of ${(\pi _N^{\vee })^{i}}^{\vee }$ for $i \not = j$. (A subtlety in this argument may be pointed out, which is that a particular composition factor may appear with higher (Jordan–Hölder) multiplicity, for example, the first derivative of the representation $1 \times 1$ of ${{\rm GL}}_2(k)$ is the trivial representation of ${{\rm GL}}_1(k)$ with multiplicity two, one which appears as a submodule, and the other which appears as a quotient. This subtlety will show up as soon as $b_i=b_j'+1$ for some $i,j$. As we are not trying to prove multiplicity one theorem for ${\textrm {Hom}}(\pi _M,\pi _N)$, but rather only the existence of a non-zero homomorphism, this subtlety does not create a problem for us.)

Therefore,

\[ \textrm{Ext}^{\ell}_{{\textrm{GL}}_{n-i}(k)}[ C, D] = 0,\quad \text{for}\ (A,B) \not = (C,D),\quad \text{and for all}\ \ell \geq 0, \]

completing the proof of the ‘existence’ part of the theorem, that is, when the parameters $M_{\rm A},N_{\rm A}$ are a relevant pair of A-parameters, and are representations of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ as in the statement of the theorem, ${\textrm {Hom}}_{{{\rm GL}}_{n}(k)}[ \pi _M, \pi _N] \not = 0$. The other part of the theorem was already proved earlier, thus the proof of the theorem is complete.

Lemma 5.3 Suppose that the cuspidal supports of

\begin{align*} V &= [a_1] \times \cdots \times [a_r] \times \nu^{e_1/2}[b_1] \times \cdots \times \nu^{e_s/2}[b_s] , \\ W &= [c_1] \times \cdots \times[c_{t}] \times \nu^{1/2}\{ [d_1] \times \cdots \times [d_{u}] \} \times \nu^{-f_1/2}[g_1] \times \cdots \times \nu^{-f_{v}/2}[g_{v}], \end{align*}

are the same (where $e_i > 0$, $f_i> 0$ are integers). Assume that the following two conditions $(V)$, $(W)$ are satisfied:

1. (V) $a_i +1 \not = b_j+e_j-1$ for any pair $(i,j)$ with $e_j>1$;

2. (W) $d_i+1 \not = g_j+f_j$ for any pair $(i,j)$.

Then:

1. (i) $e_i =1$ for all $i$;

2. (ii) $f_j=0$ for all $j$;

3. (iii) the set of irreducible representations appearing in writing $V$ as a product and $W$ as a product is unique up to permutation of the representations involved.

Proof. We prove the lemma by choosing a component $[a_i]$ or $\nu ^{e_j/2}[b_j]$ from $V$ with the property that $V$ contains $\nu ^{x}$, with $x$ the maximum half integer such that $\nu ^{x}$ is contained in the cuspidal support of $V$, and proving that this component ($[a_i]$ or $\nu ^{e_j/2}[b_j]$) lies in $W$ too. Removing this common component from $V$ and $W$ proves the lemma by an inductive process. (Recall that the cuspidal support of $V$ is the union with multiplicities of the support of individual factors in it, and that the cuspidal support of $[a]$ is $\{\nu ^{-(a-1)/2}, \ldots , \nu ^{(a-1)/2} \}$.

The proof of the lemma will be carried out in the following two cases.

Case 1. Suppose $V$ contains $\nu ^{x}$, with $x$ the maximum half integer such that $\nu ^{x}$ is contained in the cuspidal support of $V$, and that $\nu ^{x} \in [a_i] \subset V$.

In this case, $\nu ^{-x}$ also belongs to the support of $V$, $\{-x,x \}$ are the extreme points of the support of $[a_i]$ as well as $V$ (because $e_i>0$). We would like to prove that $[a_i] = [c_j]$ for some $[c_j] \subset W$. Assume the contrary, that is, $x \not = c_j$ for any $[c_j] \subset W$.

As $f_j>0$, if $\nu ^{x} \in \nu ^{-f_j/2}[g_j] \subset W$, then $\nu ^{-y} \in W$ for some $y>x$, a contradiction to the fact that $\nu ^{-y},\ y>x$ does not belong to the support of $V$.

If $\nu ^{x} \in \nu ^{1/2} [d_j] \subset W$, then $x=d_j/2$. As $\nu ^{-x} \in V$, $\nu ^{-x}$ also belongs to $W$, but $\nu ^{-x}$ does not belong to terms of the form $[c_j], \nu ^{1/2} [d_j] \subset W$, so the only option is that $\nu ^{-x}$ belongs to $\nu ^{-f_j/2}[g_j] \subset W$, that is, $2x = f_j+g_j-1$. On the other hand, $x=d_j/2$.

Thus, $W$ contains both $\nu ^{1/2} [d_j]$ and $\nu ^{-f_j/2}[g_j]$ with

\[ 2x = d_j = f_j+g_j-1, \]

which is not allowed by the condition ($W$) in the statement of the lemma. Thus, the only option left is that $\nu ^{x} \in [c_j] \subset W$.

Case 2. Suppose $V$ contains $\nu ^{x}$, with $x$ the maximum half integer such that $\nu ^{x}$ is contained in the cuspidal support of $V$, and that $\nu ^{x} \in \nu ^{e_i/2} [b_i] \subset V$ with $e_i$ minimal positive integer, and with $\nu ^{x} \not \in [a_i]$ for any $[a_i] \subset V$.

In this case, $\nu ^{-x}$ does not belong to the support of $V$, hence does not belong to the support of $W$ either. It follows that $\nu ^{x}$, which is supposed to belong to $W$, must belong to either $\nu ^{1/2}[d_j]$ or $\nu ^{-f_j/2}[g_j]$. Clearly, $\nu ^{x}$ cannot belong to $\nu ^{-f_j/2}[g_j]$ because otherwise $\nu ^{-x}$ will belong to the support of $W$.

If $\nu ^{x}$ belongs to $\nu ^{1/2}[d_j] \subset W$, $x$ being the largest exponent in $W$, $x= d_j/2$ and $\nu ^{-x+1} \in \nu ^{1/2}[d_j]$, hence $\nu ^{-x+1} \in V$. If $e_i>1$, the only option is that $\nu ^{-x+1} \in [2x-1] \subset V$. However, we already have $\nu ^{x} \in \nu ^{e_i/2} [b_i] \subset V$, so $V$ contains $[2x-1]$ as well as $\nu ^{e_i/2} [b_i]$ with

\[ 2x=e_i+b_i-1, \]

once again a contradiction to our condition ($V$) in the statement of the lemma. Thus, $e_i=1$, $\nu ^{x} \in \nu ^{1/2}[b_i] \subset V$ and $\nu ^{x}\in \nu ^{1/2}[d_j] \subset W$, completing the proof of the lemma.

We thank A. Minguez for the proof of the following lemma.

Lemma 5.5 The representations

\begin{align*} V &= [a_1] \times \cdots \times [a_r] \times \nu^{1/2}\{[b_1] \times \cdots \times [b_s] \times {{\rm St}}[d_1] \times \cdots \times {{\rm St}}[d_u] \}, \\ W &= \nu^{1/2}\{ [a'_1] \times \cdots \times [a'_{r'}] \} \times [b'_1] \times \cdots \times[b'_{s'}] \times {{\rm St}}[d'_1] \times \cdots \times {{\rm St}}[d'_{u'}], \end{align*}

are irreducible, and are isomorphic if and only if the expressions for $V, W$ as products are the same up to a permutation, in particular, if $V\cong W$, then $d_i,d'_i \leq 1$.

Proof. Irreducibility is part of Theorem 3.9 of the paper [Reference Badulescu, Lapid and MínguezBLM13] due to Badulescu et al., which gives a sufficient condition for a product $Z({\frak m}) \times L( {\frak m} ')$ to be irreducible: when no segment of ${\frak m}$ is juxtaposed (in the obvious sense) to a segment in ${\frak m} '$.

To prove that $V\cong W$ if and only if the expressions for $V, W$ as products are the same up to a permutation, define

\[ R= \sum_{n \geq 0} R_n, \]

where $R_0=\mathbb {Z}$, and $R_d$ is the Grothendieck group of finite length representations of ${{\rm GL}}_d(k)$. Then $R$ is a ring under the product $R_n \times R_m \rightarrow R_{n+m}$ given by parabolic induction.

The ring $R$ is known to be a polynomial ring $\mathbb {Z}[S(C)]$ on indeterminates $S(C)$, where $S(C)$ is the set of all segments $\{\rho , \rho \nu ,\ldots , \rho \nu ^{n-1} \}$ where $\rho$ is a cuspidal representation of some ${{\rm GL}}_d(k)$. There is a natural map from the polynomial ring $\mathbb {Z}[S(C)]$ to $R$ taking $S(C)$ to the unique irreducible submodule of the full parabolic induction: $\rho \times \rho \nu \times \cdots \times \rho \nu ^{n-1}$, which is an isomorphism of rings. In particular, $R$ is a unique factorization domain (UFD). Further, it is known (by Tadic [Reference TadićTad86, Section 3]) that the representations $[a]$ and ${\textrm {St}}[a]$ are prime elements of the ring $R$, proving the second part of the lemma.

Remark 5.6 An explicit example of a representation $\pi _M$ of ${{\rm GL}}_5(k)$ and $\pi _N$ of ${{\rm GL}}_4(k)$ was provided by Chan [Reference ChanCha19] where the previous proof cannot be carried out to prove that ${\textrm {Hom}}_{{{\rm GL}}_4(k)}(\pi _M,\pi _N) \not = 0$. For this, take the representation $\pi _M$ of ${{\rm GL}}_5(k)$ and $\pi _N$ of ${{\rm GL}}_4(k)$ which in the previous notation are

\begin{align*} \pi_M &= [3] \times [1] \times [1], \\ \pi_N &= {\textrm{St}} [2] \times [2]. \end{align*}

In this case, it is easy to see $\nu ^{1/2} (\pi _M)^{i+1}$, and $((\pi _N^{\vee })^{i})^{\vee }$ have the same cuspidal support for $i = 1, 2$, and the strategy of the proof of the theorem to conclude ${\textrm {Hom}}_{{{\rm GL}}_4(k)}(\pi _M,\pi _N) \not = 0$ fails. In fact, we do not know at this point whether ${\textrm {Hom}}_{{{\rm GL}}_4(k)}(\pi _M,\pi _N) = 0$ or is non-zero although the representations have a pair of relevant A-parameters.

The proof of the following proposition follows from the proof of Theorem 5.2.

Proposition 5.7 Let $\pi _M$ be an irreducible admissible representation of ${{\rm GL}}_{n+1}(k)$ with A-parameter $M_{\rm A}$ of dimension $(n+1)$, and $\pi _N$ be an irreducible admissible representation of ${{\rm GL}}_{n}(k)$ with A-parameter $N_{\rm A}$ of dimension $n$. Assume that the representations $M_{\rm A}, N_{\rm A}$ of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ are, in fact, representations of $W(k) \times {{\rm SL}}_2(\mathbb {C})$. Then, if

\[ {\rm Ext}^{i}_{{{\rm GL}}_n(k)}(\pi_M,\pi_N)\not = 0,\quad \text{for some}\ i \geq 0, \]

$M_{\rm A},N_{\rm A}$ are a relevant pair of A-parameters.

6. Local conjecture for classical groups

In this section, we consider the restriction problem for classical groups. Hence, we shall work in the context of [Reference Gan, Gross and PrasadGGP12a]. More precisely, we fix local fields $k_0 \subset k$ with $[k:k_0] \leq 2$. Then, with $\epsilon = \pm$, we consider a pair of non-degenerate $\epsilon$-Hermitian spaces $W \subset V$ over $k$ such that:

1. (i) $\epsilon \cdot (-1)^{\dim W^{\perp }} = -1$;

2. (ii) $W^{\perp }$ is a split space.

Setting $G(V)$ to be the identity component of the automorphism group of the space $V$, we assume further that $G(V)$ is quasi-split. We thus have a diagonal embedding

\[ G(W)^{\Delta} \hookrightarrow G(V) \times G(W). \]

Indeed, as was explained in [Reference Gan, Gross and PrasadGGP12a], one has a subgroup

\[ G(W)^{\Delta} \subset H = G(W)^{\Delta} \cdot N \subset G(V) \times G(W) \]

where $N$ is a certain unipotent subgroup of $G(V)$ normalized by $G(W)^{\Delta }$. Moreover, a certain representation $\nu$ of $H(k)$ was defined in [Reference Gan, Gross and PrasadGGP12a, § 14]: it is a one-dimensional character if $\dim W^{\perp }$ is odd and is essentially a Weil representation when $\dim W^{\perp }$ is even. When $\dim W^{\perp } =1$, for example, $\nu$ is simply the trivial representation.

Given an irreducible representation $\pi _V \otimes \pi _W$ of $G(V) \times G(W)$, we defined in [Reference Gan, Gross and PrasadGGP12a, § 14] a multiplicity $d(\pi _V, \pi _W)$, given by

\[ d(\pi_V, \pi_W) = \dim {{\rm Hom}}_{G(W)} (\pi_V \otimes \pi_W, \nu). \]

It is known that

\[ d(\pi_V, \pi_W) \leq 1. \]

We have also introduced in [Reference Gan, Gross and PrasadGGP12a] a notion of relevant pure inner forms $G(V') \times G(W')$ of $G(V) \times G(W)$, and one can likewise consider the multiplicity $d(\pi _{V'}, \pi _{W'})$ for representations of the pure inner form. Our goal is to determine the multiplicity $d(\pi _V, \pi _W)$ when $\pi _V \otimes \pi _W$ belongs to local L-packets of Arthur type. Before formulating the conjecture, we recall some basic facts about A-packets of classical groups.

Suppose that $M_{\rm A}$ is a local A-parameter for $G(V)$, with associated L-parameter $M$. By the local Langlands correspondence, $M$ gives rise to a Vogan L-packet $\Pi _M$ consisting of irreducible representations of pure inner forms $G(V')$ of $G(V)$. Moreover, fixing generic data as in [Reference Gan, Gross and PrasadGGP12a], there is a bijection

\[ \Pi_M \longleftrightarrow \textrm{Irr}(A_M), \]

where $A_M$ is the component group of the L-parameter $M$ and is an elementary abelian 2-group with a canonical basis. Hence,

\[ \Pi_M = \{\pi_{\eta} \in \textrm{Irr}(G(V')): \eta \in \textrm{Irr}(A_M)\}. \]

On the other hand, the A-parameter $M_{\rm A}$ gives rise to a local A-packet $\Pi _{M_{\rm A}}$, which is a priori a multi-set of unitary representations of the pure inner forms of $G(V)$ in correspondence with the irreducible characters of the component group $A_{M_{\rm A}}$ (which is also an elementary abelian 2-group with a canonical basis). In other words, one has

\[ \Pi_{M_{\rm A}} = \{ \pi_{\eta} : \eta \in \textrm{Irr}(A_{M_{\rm A}}) \} \]

where now $\pi _{\eta }$ is a (possibly zero and possibly reducible) finite length unitary representation of some pure inner form $G(V')$ of $G(V)$. The work of Mœglin [Reference MœglinMœg06, Reference MœglinMœg09, Reference MœglinMœg11] and Mœglin and Renard [Reference Mœglin and RenardMR18] has shown that (except possibly over $\mathbb {R}$) the local A-packets $\Pi _{M_{\rm A}}$ are sets rather than multi-sets. This means that the representations $\pi _{\eta }$ are multiplicity-free and $\pi _{\eta }$ and $\pi _{\eta '}$ have no common constituents if $\eta \ne \eta '$. One expects similar results over $\mathbb {R}$ for which special cases have been shown by Mœglin and Renard [Reference Mœglin and RenardMR17, Reference Mœglin and RenardMR19].

Regarding $\Pi _{M_{\rm A}}$ as a set of irreducible representations (by considering the irreducible summands of $\pi _{\eta }$), one thus has a map

\[ \Pi_{M_{\rm A}} \longrightarrow \textrm{Irr}(A_{M_{\rm A}}), \]

which is neither surjective nor injective in general. There is a natural surjective map

\[ A_{M_{\rm A}} \longrightarrow A_M, \]

giving rise to a natural injective map

\[ \textrm{Irr}(A_M) \hookrightarrow \textrm{Irr}(A_{M_{\rm A}}), \]

and a commutative diagram

In other words, $\Pi _M$ is a subset of $\Pi _{M_{\rm A}}$ (this is actually part of the defining property of an A-packet) and the labelling of its elements by characters of the component groups $A_M$ or $A_{M_{\rm A}}$ is consistent, cf. Conjecture 6.1(iv) of [Reference ArthurArt89].

After this short preparation, we can now formulate the conjecture.

The recipe for $\chi$ is sufficiently intricate that we have no desire to repeat it here; we refer the reader to [Reference Gan, Gross and PrasadGGP12a] for its precise definition. Let us make a few remarks concerning this conjecture.

7. A conjecture for A-packets

In the previous section, Conjecture 6.1 addressed the restriction problem for the local L-packet $\Pi _M \times \Pi _N$ contained in a local A-packet $\Pi _{M_{\rm A}} \times \Pi _{N_{\rm A}}$. In this section, we attempt to address the restriction problem for all members of A-packets, but our conjecture in this section is not as precise. Ideally, one would like to know the multiplicities for all relevant pairs of representations in the A-packet. Namely, one would like to have a prediction for the sum

\[ d(M_{\rm A}, N_{\rm A}):= \sum_{ (\pi, \pi') \in (\Pi_{M_{\rm A}} \times \Pi_{N_{\rm A}})^\textrm{rel}} d(\pi, \pi') \]

and a prediction of which relevant pairs $(\pi , \pi ')$ give non-zero contribution. Examples of restriction problems for low-rank groups (some of which we discuss later) indicate that the sum can be greater than one; see Remark 7.8.

Recall that unlike the case of ${{\rm GL}}_n$ where $\Pi _{M_{\rm A}} = \Pi _M$ (i.e. an A-packet is equal to its associated L-packet) are singletons and where A-packets corresponding to distinct A-parameters are disjoint, the A-packets of classical groups are more complicated and more interesting.

1. (i) First, it is rarely the case that the containment $\Pi _M \subset \Pi _{M_{\rm A}}$ is an equality.

2. (ii) Second, the sets $\Pi _M$ and $\Pi _{M_{\rm A}}$ are typically not singletons.

3. (iii) Third, A-packets are not necessarily disjoint.

These complications make the problem of predicting the multiplicities in an A-packet rather tricky. At this moment, we can only offer the following (optimistic) conjecture, which may be treated more as a question.

We make a few remarks about this conjecture.

Remark 7.3 It appears to us that higher multiplicities in a given A-packet is the result of other A-packets (with relevant A-parameters) intersecting this A-packet. To be more precise, we feel that for $(M_{\rm A}, N_{\rm A})$, a pair of A-parameter for $G(V) \times G(W)$,

\[ d(M_{\rm A}, N_{\rm A}) \leq |X|, \]

where $X$ is a set of maximal cardinality, consisting of pairs $(\pi _1' \times \pi _2', M_{\rm A}' \times N_{\rm A}')$ such that:

1. (i) $M_{\rm A}' \times N_{\rm A}'$ is any relevant pair of A-parameter for $G(V) \times G(W)$;

2. (ii) $\pi _1' \times \pi _2'$ is a representation belonging to both the A-packets $\Pi _{M_{\rm A}} \times \Pi _{N_{\rm A}}$ and $\Pi _{M'_{\rm A}} \times \Pi _{N'_{\rm A}}$

3. (iii) the projections from $X$ to both the first and the second factor are injective.

We now offer some evidence for Conjecture 7.1, but before doing so, we need to recall some results of Mœglin. Given a discrete L-parameter $M$ of a classical group, Mœglin determined precisely which elements of its L-packet $\Pi _M$ can be a supercuspidal representation. To formulate the result, we introduce the following two notions.

1. (i) Say that a discrete L-parameter $M$ is without gaps (or holes) if the following holds: for any selfdual irreducible representation $\rho$ of $W(k)$ and $a \geq 1$,

   \[ \rho \boxtimes [a+2] \subset M \Longrightarrow \rho \boxtimes [a] \subset M. \]
   Equivalently, let
   \[ \rho \boxtimes [1] + \rho \boxtimes [3] +\cdots \quad \text{or} \quad \rho \boxtimes [2] + \rho \boxtimes [4] +\cdots \]
   be a maximal chain appearing as a direct summand in $M$. Then $M$ is without gaps if these chains (as $\rho$ varies) span $M$.

2. (ii) Let $M$ be a discrete L-parameter without gaps and let $A_M$ be its component group, so that $\Pi _M$ is in bijection with $\textrm {Irr}(A_M)$. Recall that $A_M$ is a vector space over $\mathbb {Z}/2\mathbb {Z}$ with a canonical basis indexed by the irreducible summands of $M$. Say that $\alpha \in \textrm {Irr}(A_M)$ is alternating if the following holds: for any $\rho \boxtimes [a] + \rho \boxtimes [a+2] \subset M$,

   \[ \alpha(\rho \boxtimes [a]) = -\alpha (\rho \boxtimes [a+2]), \]
   with the convention that if $a=0$, then $\rho \boxtimes [a]=0$, and $\alpha (\rho \boxtimes [ 0])=1$, so that we must have $\alpha (\rho \boxtimes [2])=-1$.

Here then is Mœglin's first result [Reference MœglinMœg09] that we use.

The second result of Mœglin [Reference MœglinMœg06, Reference MœglinMœg09] we need concerns the possible overlaps of different local A-packets. If $M_{\rm A}$ is the local A-parameter of a (classical) group $G$ over a non-archimedean local field $k$, let $M_{\rm A}^{\Delta }$ be the tempered L-parameter defined by

where $\Delta$ is the diagonal embedding of ${{\rm SL}}_2(\mathbb {C})$.

Theorem 7.5 Let $M_{\rm A}$ and $M'_{\rm A}$ be two local A-parameters for a classical group over a $p$-adic field with associated A-packets $\Pi _{M_{\rm A}}$ and $\Pi _{M'_{\rm A}}$. Then

\[ \Pi_{M_{\rm A}} \cap \Pi_{M'_{\rm A}} \ne \emptyset \Longrightarrow M_{\rm A} ^{\Delta} \cong M_{\rm A}'^{\Delta}. \]

Further, if $\,\Pi _{M_{\rm A}^{\Delta }}^{\rm sc}$ denotes the set of supercuspidal representations in $\Pi _{M_{\rm A}^{\Delta }}$, then

\[ \Pi_{M_{\rm A} ^{\Delta}}^{\rm sc} \subset \Pi_{M_{\rm A}}. \]

The first part of this theorem of Mœglin is eventually related to the observation that two A-parameters $M_{\rm A}$ and $M'_{\rm A}$ of ${{\rm GL}}_n(k)$ have the same cuspidal support if and only if $M_{\rm A}^{\Delta }$ and ${M'_{\rm A}}^{\Delta }$ are equivalent.

At this point, let us take note of the following consequence of tempered GGP, which was pointed out to us by Waldspurger.

Proposition 7.6 Let $M_0 \times N_0$ be a tempered L-parameter for ${{\rm SO}}_{2n+1} \times {{\rm SO}}_{2m}$ over a non-archimedean local field $k$ with associated L-packets $\Pi _{M_0} \times \Pi _{N_0}$. Let $(\pi , \sigma )$ be the unique member of $\Pi _{M_0} \times \Pi _{N_0}$ such that $d(\pi , \sigma ) \ne 0$, so that $(\pi , \sigma )$ corresponds to the distinguished character $\chi := \chi _{M_0, N_0}$ of the component group by [Reference Gan, Gross and PrasadGGP12a].

(i) Suppose that $\rho$ is an irreducible selfdual parameter for ${{\rm GL}}_d(k)$ such that for some $a \geq 3$, both the representations $\rho \otimes [a]$ and $\rho \otimes [a-2]$ of $\textit {WD}(k)=W(k) \times {{\rm SL}}_2(\mathbb {C})$ appear with multiplicity 1 in $M_0$. In this case, these two summands give rise to two basis elements of the component group associated to $M_0$. Let

\[ \chi(\rho,a), \chi(\rho,a-2) \in \{\pm 1\} \]

be the value of the distinguished character $\chi$ on these two elements respectively. Then

\[ \chi(\rho,a) = - \chi(\rho,a-2) \]

if and only if the parameter $N_0$ contains $\rho \otimes [a-1]$ with odd multiplicity.

(ii) Further, if $\rho \otimes [2]$ appears with multiplicity 1 in $M_0$, then

\[ \chi(\rho,2) = - 1 \]

if and only if the parameter $N_0$ contains $\rho$ with odd multiplicity.

Furthermore, there are similar assertions when both the representations $\rho \otimes [a]$ and $\rho \otimes [a-2]$ of $\textit {WD}(k)=W(k) \times {{\rm SL}}_2(\mathbb {C})$ appear with multiplicity 1 in $N_0$.

Proof. Let us begin by recalling Conjecture 20.1 of [Reference Gan, Gross and PrasadGGP12a], giving a recipe for $\chi (\rho ,a)$:

\[ \chi(\rho,a) = \epsilon(\rho \otimes [a] \otimes N_0)\cdot (\det \rho)(-1)^{(a\dim N_0)/2}\cdot (\det N_0)(-1)^{(a\dim \rho)/2}. \]

Define more generally for any selfdual representation $\mu$ of $W(k) \times {{\rm SL}}_2(\mathbb {C})$ with even parity (i.e. the same as that of $N_0$),

\[ \chi_{\mu}(\rho, a)= \epsilon(\rho \otimes [a] \otimes \mu) \cdot (\det \rho)(-1)^{(a\dim N_0)/2} \cdot (\det \mu)(-1)^{(a\dim \rho)/2}, \]

so that if $N_0 = \oplus _i \mu _i$, a sum of irreducible selfdual representations $\mu _i$ of $W(k) \times {{\rm SL}}_2(\mathbb {C})$ with even parity (i.e. the same as that of $N_0$), then

\[ \chi(\rho, a)= \prod_i \chi_{\mu_i}(\rho, a). \]

Thus, we need to compare $\chi _{\mu _i}(\rho , a)$ and $\chi _{\mu _i}(\rho , a-2)$ for any $\mu _i$ as before.

Recall from [Reference TateTat79] that for an irreducible representation $\lambda \otimes [n]$ of $\textit {WD}(k)=W(k) \times {{\rm SL}}_2(\mathbb {C})$, one has

\[ \epsilon( \lambda \otimes [n]) = \epsilon(\lambda)^{n} \cdot \det (-F, \lambda^{I})^{n-1}, \]

where $\lambda ^{I}$ denotes the subspace of $\lambda$ fixed by the inertia group $I$ and $F$ denotes the Frobenius element of $W(k)/I$. In particular, if $\lambda$ is an irreducible selfdual representation of $W(k)$, then

\[ \epsilon( \lambda \otimes [n]) = \begin{cases} \epsilon(\lambda)^{n} & \textrm{if } \lambda \not =1; \\ (-1)^{n-1} & \textrm{if } \lambda =1 . \end{cases} \]

Using the Clebsch–Gordon theorem,

\[ [a] \otimes [b] = [a+b-1] \oplus [a+b-3] \oplus \cdots \oplus [|a-b|+1], \]

it follows that if $\rho , \tau$ are irreducible selfdual representations of $W(k)$, then

\[ \epsilon( \rho \otimes [a] \otimes \tau \otimes [b]) = \begin{cases} \epsilon(\rho \otimes \tau )^{ab} & \text{ if $\rho \not \cong \tau$,} \\ \epsilon(\rho \otimes \tau )^{ab} (-1)^{n(a,b)} & \text{ if $\rho \cong \tau$,} \end{cases} \]

where $n(a,b) = \textrm {min}\{a,b\} \cdot [ \textrm {max}\{a, b\} -1]$. In particular, observe that for $a,b$ positive integers of different parity,

\[ \epsilon( [a] \otimes [b]) \cdot \epsilon( [a-2] \otimes [b])= \begin{cases} 1 & \text{ if $b \not = a-1$,} \\ -1 & \text{ if $b = a-1$.} \end{cases} \]

More generally, it is easily seen that for any irreducible representation $\mu$ of $W(k) \times {{\rm SL}}_2(\mathbb {C})$ as previously and $a \geq 3$,

\[ \chi_{\mu}(\rho, a) = \begin{cases} \chi_{\mu}(\rho, a-2), & \text{ if $\mu \ne \rho \otimes [a-1]$,} \\ - \chi_{\mu}(\rho, a-2), & \text{ if $\mu = \rho \otimes [a-1]$.} \end{cases} \]

It follows that $\chi (\rho , a) = - \chi (\rho , a-2)$ if and only if $\rho \otimes [a-1]$ occurs in $N_0$ with odd multiplicity, thus completing the proof of part (i). The proof of part (ii), where $a=2$, follows along the same lines; we omit the details.

After this preparation, we can prove the following theorem. Part (b) of the theorem is a contribution to Conjecture 7.1(i) and (ii).

Proof. (a) This is a consequence of the tempered GGP [Reference WaldspurgerWal12], Theorems 7.4 and 7.5, and Proposition 7.6.

(b) This follows from part (a) on noting that the Aubert–Zelevinsky involution

\[ \pi \rightarrow D(\pi) \]

on the category of admissible representations of a group $G(k)$ takes the tempered L-packet $\Pi _{M_0} \times \Pi _{N_0}$ to $D(\Pi _{M_0}) \times D(\Pi _{N_0})$, which is the A-packet associated to the A-parameter $D(M_0) \times D(N_0)$. Now $\pi = D(\pi )$ (up to a sign) for supercuspidal representations $\pi$. Hence, if $(\pi , \sigma )$ is a supercuspidal member of $\Pi _{M_0} \times \Pi _{N_0}$ such that $d(\pi , \sigma ) \ne 0$, then $(\pi , \sigma )$ is also a member of the A-packet $D(\Pi _{M_0}) \times D(\Pi _{N_0})$ with $d(\pi , \sigma ) \ne 0$.

In the rest of this section, we construct examples of non-relevant A-parameters $(M_{\rm A}, N_{\rm A})$ for which $d(M_{\rm A},N_{\rm A}) > 0$, a subtlety that Conjecture 7.1(ii) takes into account.

For any subset $J \subset \{1,2,\ldots ,n\}$ (for $n$ fixed), consider the representation of ${{\rm SL}}_2(\mathbb {C}) \times {{\rm SL}}_2(\mathbb {C})$ given by

\[ M_{{\rm A},J} = \bigoplus_{i \notin J} [2i] \otimes [1] \oplus \bigoplus_{j \in J} [1] \otimes [2j]. \]

Then $M_{{\rm A},J}$ is a multiplicity-free sum of symplectic representation of $\textit {WD}(k) \times {{\rm SL}}_2(\mathbb {C})$ with $W(k)$ acting trivially. Hence, $M_{{\rm A},J}$ is a discrete A-parameter for an odd special orthogonal group of the appropriate size.

Now observe that $M_{{\rm A},J}^{\Delta }$ is independent of $J$ and is a discrete L-parameter without gaps:

\[ M_{{\rm A},J}^{\Delta} = M_{A, \emptyset} = \bigoplus_{j=1}^{n} 1_{W(k)} \otimes [2j]. \]

By Theorem 7.4, the L-packet of $M_{{\rm A},J}^{\Delta }$ contains a unique supercuspidal representation $\pi _{\rm sc}$ for an odd special orthogonal group of the appropriate size, attached to the unique alternating character $\alpha _{\rm sc}$ of the component group:

\[ \alpha_{\rm sc}([2j]) = (-1)^{j}. \]

By Theorem 7.5, we see that

\[ \pi_{\rm sc} \in \Pi_{M_{{\rm A},J}} \quad \text{for any $J$.} \]

We now consider the restriction problem for the pair $(M_{{\rm A},J}, N)$ where $M_{{\rm A},J}$ is as previously and $N$ is the discrete L-parameter

\[ N := [1] \oplus [3] \oplus \cdots \oplus [2n-1] \]

on which $W(k)$ acts trivially. One may apply Proposition 7.6 (together with Theorem 7.4) to deduce that

\[ d(\pi_{\rm sc}, N) =1, \]

where we recall that $\pi _{\rm sc}$ is the unique supercuspidal representation with L-parameter $M_{A,\emptyset }$. Since $\pi _{\rm sc} \in \Pi _{M_{{\rm A},J}}$ for any $J$, we thus conclude that

\[ d(M_{{\rm A},J}, N) \ne 0 \quad \text{for any $J$.} \]

It is easy to see that

\[ (M_{{\rm A},J}, N)\ \text{is relevant} \Longleftrightarrow J = \emptyset,\ \text{or}\ J = \{1 \}. \]

Hence, if $n > 1$, we have produced non-relevant pairs $(M_{{\rm A},J}, N)$ such that $d(M_{{\rm A},J}, N) > 0$. In other words, ‘relevance’ is not a necessary condition for branching in A-packets.

The simplest example one can take is $n=2$. In this case, we have the A-parameters

\[ M_{\rm A} = [1] \boxtimes [4] + [2] \boxtimes [1] \quad \text{or} \quad [1] \boxtimes [4] + [1] \boxtimes [2], \]

and

\[ N = [3] \boxtimes [1] + [1] \boxtimes [1]. \]

The pair $(M_{\rm A},N)$ is not relevant, yet $d(M_{\rm A}, N) > 0$. (We note that the A-packets of ${{\rm SO}}_7$ corresponding to the two choices of $M_{\rm A}$ given previously can be constructed by theta lifting from an appropriate L-packet and an A-packet on ${\rm Mp}_2$, respectively, and the unique supercuspidal representation in them is simply the theta lift (with respect to $\psi$) of the odd Weil representation $\omega _{\psi }$).