On multiplicity in restriction of tempered representations of p-adic groups

Abstract

We establish an equality between two multiplicities: one in the restriction of tempered representations of a p-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in Langlands dual sides, so-called $\mathcal {S}$-groups, under working hypotheses about the tempered local Langlands conjecture and the internal structure of tempered L-packets. This provides a formula of the multiplicity for p-adic groups by means of dimensions of irreducible representations of their $\mathcal {S}$-groups.

Appendix A. Examples

We present some examples related to the results established in Section 4. We continue with the notation in the previous sections.

Example A.1

This example is based on [10, Sects. 7.6 and 7.7]. Let $\widetilde{\mathbf{G}}=\text {G}Sp_{1,1}$ be the non-quasi-split inner form of $\text {G}Sp_4,$ and let $\mathbf G=\text {S}p_{1,1}$ be the non-quasi-split inner form of $\text {S}p_4.$ Let $\widetilde{\varphi }= \widetilde{\varphi }_0 \oplus (\widetilde{\varphi }_0 \otimes \chi ) \in \Phi (\text {G}Sp_{1,1})$ be given, where $\chi$ is a quadratic character, $\widetilde{\varphi }_0 \in \Phi (\text {G}L_2)$ is primitive (by definition, $\widetilde{\varphi }_0$ is not of the form $\text {I}nd_{W_E}^{W_F} \tau$ for any non-trivial finite extension E / F and irreducible representation $\tau$), and $\widetilde{\varphi }_0 \not \simeq \widetilde{\varphi }_0 \otimes \chi .$

We have

$$\begin{aligned} {\text {R}es}^{\text {G}Sp_{1,1}}_{\text {S}p_{1,1}}(\widetilde{\sigma }'_1) = {\text {R}es}^{\text {G}Sp_{1,1}}_{\text {S}p_{1,1}}(\widetilde{\sigma }'_2)= \{ \sigma ' \}, \end{aligned}$$

and $\widetilde{\sigma }'_2 \simeq \widetilde{\sigma }'_1 \chi .$ Moreover, from the fact that $X(\widetilde{\varphi }) \simeq \{ \mathbb {1}, \chi \}$ (see [12, Proposition 6.3(iii)(b)]), it follows that

$$\begin{aligned} I(\widetilde{\sigma }'_1) = I(\widetilde{\sigma }'_2) = \{ \mathbb {1} \}. \end{aligned}$$

(A.1)

Thus, the L-packet $\Pi _{\varphi }({\text {S}p}_{1,1})$ of $\text {S}p_{1,1}$ attached to the L-parameter $\varphi$ is $\{ \sigma ' \}.$ We recall from [10], Sections 7.6 and 7.7] that

$$\begin{aligned} 1 \longrightarrow \mu _2(\mathbb {C}) \longrightarrow S_{\widetilde{\varphi }, \text {s}c}(\widehat{{\text {G}Sp}_{1,1}}) \simeq (\mathbb {Z}/2\mathbb {Z})^2 \longrightarrow S_{\widetilde{\varphi }}(\widehat{{\text {G}Sp}_{1,1}}) \simeq \mathbb {Z}/2\mathbb {Z}\longrightarrow 1, \end{aligned}$$

$$\begin{aligned} 1 \longrightarrow \mu _2(\mathbb {C}) \longrightarrow S_{\varphi , \text {s}c}(\widehat{{\text {S}p}_{1,1}}) \simeq \mathcal {D}_8 \longrightarrow S_{\varphi }(\widehat{{\text {S}p}_{1,1}}) \simeq (\mathbb {Z}/2\mathbb {Z})^2 \longrightarrow 1, \end{aligned}$$

where $\mathcal {D}_8$ denotes the dihedral group of order 8. Further, we note that $\mathrm{Irr}(\mathcal {D}_8)$ consists of four 1-dimensional characters and one 2-dimensional irreducible representation. We denote by $\rho '$ the 2-dimensional irreducible representation. Setting $\mathrm{Irr}(\mu _2(\mathbb {C}))=\{ \mathbb {1}, \textsf {sgn}\},$ the map $\sigma ' \mapsto \rho '$ from $\Pi _{\varphi }({\text {S}p}_{1,1})$ to $\mathrm{Irr}(S_{\varphi , \text {s}c}(\widehat{{\text {S}p}_{1,1}}), \textsf {sgn})$ provides an equality

$$\begin{aligned} \dim \rho ' = 2, \end{aligned}$$

(A.2)

while the multiplicity $\langle \sigma ', \widetilde{\sigma }'_i \rangle _{\text {S}p_{1,1}}$ in ${\text {R}es}_{\text {S}p_{1,1}}^{\text {G}Sp_{1,1}}(\widetilde{\sigma }'_i)$ for $i=1,2$ satisfies

$$\begin{aligned} \langle \sigma ', \widetilde{\sigma }'_i \rangle _{\text {S}p_{1,1}} = 1. \end{aligned}$$

(A.3)

We now consider $(\text {G}Sp_{1,1}(F))_{\sigma '}$ which turns out to be equal to $\text {G}Sp_{1,1}(F)$ and

$$\begin{aligned} \widetilde{G}/ \widetilde{G}_{\sigma '} = \{1\}. \end{aligned}$$

We also note that

$$\begin{aligned} I(\rho ') = \{ \eta \in \big (\mathcal {D}_8\big /(\mathbb {Z}/2\mathbb {Z})^2\big )^\vee : \rho ' \eta \simeq \rho '\} = \big (\mathcal {D}_8\big /(\mathbb {Z}/2\mathbb {Z})^2\big )^\vee , \end{aligned}$$

which implies that

$$\begin{aligned} (S_{\varphi , \text {s}c}/S_{\widetilde{\varphi }, \text {s}c})^\vee \big / I(\rho '). \end{aligned}$$

Hence, this coincides with Theorem 4.13.

For Theorem 4.17, we first have

$$\begin{aligned} X(\widetilde{\varphi }) = \{1, \chi \} ~~ \text { and } ~~ I(\widetilde{\sigma }') = {1}. \end{aligned}$$

We note that, for any Klein-four subgroup H of $\mathcal {D}_8$ (in fact, there are two), we have

$$\begin{aligned} {\text {R}es}^{\mathcal {D}_8}_{H}(\rho ') = \{\chi _1, \chi _2\}, \end{aligned}$$

where $\chi _1$ and $\chi _2$ are distinct 1-dimensional characters of H,  since the trace of $\rho '$ vanishes outside such subgroup H (see [17, (20.13)(2)]). Fix $\chi _1$ corresponding to $\widetilde{\sigma }'_1$ via the bijection between $\Pi _{\widetilde{\varphi }}({\text {G}Sp}_{1,1})$ and $\mathrm{Irr}(\mathcal S_{\widetilde{\varphi }, \text {s}c}(\widehat{{\text {G}Sp}_{1,1}}), \textsf {sgn}).$ Computing the stabilizer $(\mathcal {D}_8)_{\chi _1}=\{s \in \mathcal {D}_8 : {^s}\chi _1 = \chi _1 \} ,$ we have $\mathcal {D}_8/(\mathcal {D}_8)_{\chi _1} \simeq \mathbb {Z}/2\mathbb {Z}.$ Hence, this coincides with Theorem 4.17.

Lastly, for the multiplicity in the restriction, given a lifting $\widetilde{\sigma }' \in \Pi _{\widetilde{\varphi }}(\text {G}Sp_{1,1})$ of $\sigma ' \in \Pi _{\varphi }(\text {S}p_{1,1}),$ we have

$$\begin{aligned} \langle \sigma ', \widetilde{\sigma }' \rangle _{G} = \frac{\dim \rho '}{\dim \chi _1} |\Pi _{\rho '}(S_{\widetilde{\varphi }, \text {s}c})|^{-1} = \frac{2}{1} 2^{-1} = 1, \end{aligned}$$

which coincides with (A.3). Hence, this coincides with Proposition 4.22. This applies to all the others in $\Pi _{\varphi }(\text {S}p_{1,1}).$

Example A.2

Let $\widetilde{\mathbf{G}}=\text {G}Sp_4,$$\mathbf G=\text {S}p_{4},$ and $\widetilde{\varphi }$ be given as above in Example A.1. From [10, Sects. 7.6 and 7.7] we have

$$\begin{aligned} S_{\widetilde{\varphi }}(\widehat{{\text {G}Sp}_{4}}) \simeq \mathbb {Z}/2\mathbb {Z}, \quad S_{\varphi }(\widehat{{\text {S}p}_{4}}) \simeq \mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z}, \end{aligned}$$

$$\begin{aligned} {\text {R}es}^{{\text {G}Sp}_4}_{{\text {S}p}_4}(\widetilde{\sigma }_1) = \{ \sigma _1^+, \sigma _1^- \}, \quad {\text {R}es}^{{\text {G}Sp}_4}_{{\text {S}p}_4}(\widetilde{\sigma }_2) = \{ \sigma _2^+, \sigma _2^- \}, \end{aligned}$$

$$\begin{aligned} \Pi _{\widetilde{\varphi }}({\text {G}Sp}_4) = \{ \widetilde{\sigma }_1, \widetilde{\sigma }_2 \}, \quad \Pi _{\varphi }({\text {S}p}_4) = \{\sigma _1^+, \sigma _1^-, \sigma _2^+, \sigma _2^- \}. \end{aligned}$$

From the bijection $\Pi _{\varphi } \overset{1-1}{\longleftrightarrow } \mathrm{Irr}(S_{\varphi }(\widehat{{\text {S}p}_4})),$ we correspond $\sigma =\sigma _1^+$ to $\rho =\mathbb {1}.$ Recall that there is a bijection ${\text {G}Sp}_4(F)/({\text {G}Sp}_4(F))_{\sigma } \overset{1-1}{\longleftrightarrow } \Pi _{\widetilde{\sigma }_1}(\text {S}p_4).$ Then we note that

$$\begin{aligned} \{ \eta \in \big (S_{\varphi }(\widehat{{\text {S}p}_{4}})\big /S_{\widetilde{\varphi }}(\widehat{{\text {G}Sp}_{4}})\big )^\vee : \mathbb {1} \eta = \mathbb {1}\} = \{ \mathbb {1} \} \end{aligned}$$

and we have

$$\begin{aligned} {\text {G}Sp}_4(F)\big /({\text {G}Sp}_4(F))_{\sigma } \simeq \mathbb {Z}/2\mathbb {Z}, \end{aligned}$$

$$\begin{aligned} \big (S_{\varphi }\simeq \mathbb {Z}/2\mathbb {Z},(\widehat{{\text {S}p}_{4}})/S_{\widetilde{\varphi }}(\widehat{{\text {S}p}_{4}})\big )^\vee \big / \{ \eta \in \big (S_{\varphi }(\widehat{{\text {S}p}_{4}})/S_{\widetilde{\varphi }}(\widehat{{\text {S}p}_{4}}) \big )^\vee : \mathbb {1} \eta = \mathbb {1}\} \simeq \mathbb {Z}/2\mathbb {Z}. \end{aligned}$$

Hence, this coincides with Theorem 4.13. Moreover, from the bijection $\Pi _{\widetilde{\varphi }} \overset{1-1}{\longleftrightarrow } \mathrm{Irr}(S_{\widetilde{\varphi }}(\widehat{{\text {G}Sp}_4})),$ we correspond $\widetilde{\sigma }=\widetilde{\sigma }_1$ to $\widetilde{\rho }=\mathbb {1}.$ Then we have

$$\begin{aligned} X(\widetilde{\varphi })/\bar{I}(\widetilde{\sigma }) = \{ \mathbb {1} \},~~ S_{\widetilde{\varphi }}(\widehat{{\text {G}Sp}_4})\big /S_{\widetilde{\varphi }}(\widehat{{\text {G}Sp}_4})_{\mathbb {1}} =\{ 1 \}. \end{aligned}$$

Hence, this coincides with Theorem 4.17.

Lastly, for the multiplicity in the restriction, given a lifting $\widetilde{\sigma }$ of $\sigma ,$ we have

$$\begin{aligned} \langle \sigma , \widetilde{\sigma }\rangle _{G} = \frac{\dim \mathbb {1}}{\dim \mathbb {1}} |\Pi _{\mathbb {1}}(S_{\widetilde{\varphi }, \text {s}c})|^{-1} = \frac{1}{1} 1^{-1} = 1. \end{aligned}$$

Hence, this coincides with Proposition 4.22. This applies to all the others in $\Pi _{\varphi }(\text {S}p_4).$

Example A.3

Let $\widetilde{\mathbf{G}}=\text {G}L_2,$$\mathbf G=\text {S}L_2,$ and $\widetilde{\varphi }\in \Phi (\widetilde{G})$ be dihedral with respect to three quadratic extensions. Then we have

$$\begin{aligned} S_{\widetilde{\varphi }}(\widehat{{\text {G}L}_2}) = \{ 1 \}, \quad S_{\varphi }(\widehat{{\text {S}L}_2}) \simeq \mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z}, \end{aligned}$$

$$\begin{aligned} \Pi _{\widetilde{\varphi }}({\text {G}L}_2) = \{ \widetilde{\sigma }\}, \quad {\text {R}es}^{{\text {G}L}_2}_{{\text {S}L}_2}(\widetilde{\sigma }) = \Pi _{\varphi }({\text {S}L}_4) = \{ \sigma _1, \sigma _2, \sigma _3, \sigma _4 \}. \end{aligned}$$

From the bijection $\Pi _{\varphi } \overset{1-1}{\longleftrightarrow } \mathrm{Irr}(S_{\varphi }(\widehat{{\text {S}L}_2})),$ we correspond $\sigma =\sigma _1$ to $\rho =\mathbb {1}.$ Note that there is a bijection ${\text {G}L}_2(F)/({\text {G}L}_2(F))_{\sigma } \overset{1-1}{\longleftrightarrow } \Pi _{\widetilde{\sigma }}(\text {S}L_2).$ We then have

$$\begin{aligned} {\text {G}L}_2(F)\big /({\text {G}L}_2(F))_{\sigma } \simeq \mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z}, \end{aligned}$$

$$\begin{aligned} \big (S_{\varphi }(\widehat{{\text {S}L}_2})/S_{\widetilde{\varphi }}(\widehat{{\text {S}L}_2})\big )^\vee \big / \{ \eta \in \big (S_{\varphi }(\widehat{{\text {S}L}_2})/S_{\widetilde{\varphi }}(\widehat{{\text {S}L}_2})\big )^\vee : \mathbb {1} \eta = \mathbb {1}\} \simeq \mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z}. \end{aligned}$$

Hence, this coincides with Theorem 4.13. Moreover, from the bijection $\Pi _{\widetilde{\varphi }} \overset{1-1}{\longleftrightarrow } \mathrm{Irr}(S_{\widetilde{\varphi }}(\widehat{{\text {G}L}_2})),$ we correspond $\widetilde{\sigma }=\widetilde{\sigma }_1$ to $\widetilde{\rho }=\mathbb {1}.$ Then we have

$$\begin{aligned} X(\widetilde{\varphi })/\bar{I}(\widetilde{\sigma }) = \{ \mathbb {1} \},~~ S_{\widetilde{\varphi }}(\widehat{{\text {G}L}_2})\big /S_{\widetilde{\varphi }}(\widehat{{\text {G}L}_2})_{\mathbb {1}} =\{ 1 \}. \end{aligned}$$

Hence, this coincides with Theorem 4.17.

Lastly, for the multiplicity in the restriction, given a lifting $\widetilde{\sigma }$ of $\sigma ,$ we have

$$\begin{aligned} \langle \sigma , \widetilde{\sigma }\rangle _{G} = \frac{\dim \mathbb {1}}{\dim \mathbb {1}} |\Pi _{\mathbb {1}}(S_{\widetilde{\varphi }, \text {s}c})|^{-1} = \frac{1}{1} 1^{-1} = 1. \end{aligned}$$

Hence, this coincides with Proposition 4.22. This applies to all the others in $\Pi _{\varphi }(\text {S}L_2).$

Remark A.4

From Example A.3, we note that two sizes $|I(\rho )|=1$ and $|I(\widetilde{\sigma })|=4$ does not necessarily equal each other. This implies that $|\Pi _{\widetilde{\sigma }}(G)|=4$ does not need to be identical with $|\Pi _{\rho }(\mathcal S_{\widetilde{\varphi }, \text {s}c})|=1.$

Example A.5

Let $\widetilde{\mathbf{G}}=\text {G}L_1(D),$$\mathbf G=\text {S}L_1(D),$ where D is the quaternion division algebra over F,  and $\widetilde{\varphi }\in \Phi (\widetilde{\mathbf{G}})$ be as in Example A.3. Then we have

$$\begin{aligned} 1 \longrightarrow \mu _2(\mathbb {C}) \longrightarrow S_{\widetilde{\varphi }, \text {s}c}(\widehat{{\text {G}L}_1(D)}) \simeq \mathbb {Z}/2\mathbb {Z}\longrightarrow S_{\widetilde{\varphi }}(\widehat{{\text {G}L}_1(D)}) ={1} \longrightarrow 1, \end{aligned}$$

$$\begin{aligned} 1 \longrightarrow \mu _2(\mathbb {C}) \longrightarrow S_{\varphi , \text {s}c}(\widehat{{\text {S}L}_1(D)}) \simeq Q_8 \longrightarrow S_{\varphi }(\widehat{{\text {S}L}_1(D)}) \simeq \mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z}\longrightarrow 1, \end{aligned}$$

where $Q_8$ denotes the quaternion group of order 8. Recall that

$$\begin{aligned} \Pi _{\widetilde{\varphi }}({\text {G}L}_1(D)) = \{ \widetilde{\sigma }'\}, ~~{\text {R}es}^{{\text {G}L}_1(D)}_{{\text {S}L}_1(D)}(\widetilde{\sigma }') = \Pi _{\varphi }({\text {S}L}_1(D)) = \{ \sigma ' \}, \end{aligned}$$

$$\begin{aligned} \mathrm{Irr}(Q_8) = \{ \chi _1, \chi _2, \chi _3, \chi _4, \rho ' \}, \end{aligned}$$

where $\chi _i$’s are distinct 1-dimensional representations, and $\rho '$ is the 2-dimensional representation of $Q_8.$ From the bijection $\Pi _{\varphi }(\text {S}L_1(D)) \overset{1-1}{\longleftrightarrow } \mathrm{Irr}(S_{\varphi , \text {s}c}(\widehat{{\text {S}L}_1(D)}), \textsf {sgn}) = \{ \rho ' \},$ we correspond $\sigma '$ to $\rho '.$ Note that there is a bijection ${\text {G}L}_1(D)/({\text {G}L}_1(D))_{\sigma '} \overset{1-1}{\longleftrightarrow } \Pi _{\widetilde{\sigma }'}(\text {S}L_1(D)).$ We then have

$$\begin{aligned} {\text {G}L}_1(D)\big /({\text {G}L}_1(D))_{\sigma '} = \{1\},~~ \big (Q_8/(\mathbb {Z}/2\mathbb {Z})\big )^\vee \big / \{ \eta \in \big (Q_8/(\mathbb {Z}/2\mathbb {Z})\big )^\vee : \rho '\eta = \rho '\} \simeq \{ \mathbb {1} \}, \end{aligned}$$

since $\{ \eta \in (Q_8/(\mathbb {Z}/2\mathbb {Z}))^\vee : \rho '\eta = \rho '\} = \{ \chi _1, \chi _2, \chi _3, \chi _4 \} .$ Hence, this coincides with Theorem 4.13. Moreover, from the bijection $\Pi _{\widetilde{\varphi }} \overset{1-1}{\longleftrightarrow } \mathrm{Irr}(S_{\widetilde{\varphi }, \text {s}c}(\widehat{{\text {G}L}_1(D)}), \textsf {sgn}),$ we correspond $\widetilde{\sigma }'$ to $\textsf {sgn}.$ Then we have

$$\begin{aligned} X(\widetilde{\varphi })/I(\widetilde{\sigma }') = \{ \mathbb {1} \},~~ Q_8/(Q_8)_{\rho '} =\{ 1 \}. \end{aligned}$$

Hence, this coincides with Theorem 4.17.

Lastly, for the multiplicity in the restriction, given a lifting $\widetilde{\sigma }'$ of $\sigma ',$ we have

$$\begin{aligned} \langle \sigma ', \widetilde{\sigma }' \rangle _{G} = \frac{\dim \rho '}{\dim \mathbb {1}} |\Pi _{\mathbb {1}}(S_{\widetilde{\varphi }, \text {s}c})|^{-1} = \frac{2}{1} 1^{-1} = 2. \end{aligned}$$

Hence, this coincides with Proposition 4.22. This applies to all the others in $\Pi _{\varphi }(\text {S}L_1(D)).$