数理解析研究所講究録 1348 巻 2003 年 118-133

118

Unipotent representations of unitary groups in four
variables
Kazuko Konno$*$

Contents
1 Introduction

1

2 The unitary dual
2.1 Preliminary
2.2 Jacquet-Langlands correspondence for
2.3 Lifting Plancherel measures
2.4 Composition series of induced representations
2.5 The unitary dual

2

3
4
6
7
10

$G_{2}$

3 Construction of an A-packet
3.1 The A-parameter
3.2 Local theta correspondence
3.3 A candidate for the A-packet

1

10
11
11

13

Introduction

Originally, “unipotent representation” stands for those representations of finite reductive

groups whose character is supported on the unipotent set (due to Lusztig). Later thisterm is used in various fields of representation theory, including Vogan’s strategy of describing the unitary dual of real reductive groups and Arthur’s conjecture on the discrete
automorphic spectrum. Accordingly there seems to be some variant of the definition of
unipotent representations. In particular, the characters of unipotent representations of
reductive groups over local fields are not supported on the unipotent orbits.
Since we shall be mainly concerned with the reductive groups over padic fields, we
adopt the definition proposed by Arthur [Art89]. Let
be a connected reductive groupdefined over a padic field . We fix an algebraic closureof so that we have theabsolute Galois group Gal(F/F). Recall that the Weil groupis a variant of
of
$G$

$F$

$\overline{F}$$F$$W_{F}$$F$“JSPS Post Doctoral Fellow at Graduate School of Mathematics, Kyoto University, Oiwake-cho Kitashirakawa Sakyo , Kyoto, 606-8502, Japan-mail: kkonno@math.ky0t0-u.ac.jp$URL$ : http://knmac.math.kyushu-u.a$\mathrm{k}\mathrm{u}$$E$$\mathcal{K}.\mathrm{j}\mathrm{p}/\sim \mathrm{k}\mathrm{k}\mathrm{o}\mathrm{l}\mathrm{m}\mathrm{o}/$IIE)which admits a sufficiently rich complex representations (as opposed to the
Galois group). Writing
for the Langlands dual group (the -group having the root
datum dual to that of ), we have the groupof . In [Art89], Arthur
proposed a conjectural parametrization of irreducible representations of $G(F)$ in terms of
the homomorphisms:,$\mathrm{G}\mathrm{a}1(\overline{F}/F)$$\hat{G}$$\mathbb{C}$$G$$LG=\hat{G}\mathrm{x}$$L$$\psi$$W_{F}$$G$$\mathcal{L}_{F}\cross SL(2, \mathbb{C})arrow LG$

which we call -pararneters. Here $\mathcal{L}_{F}:=W_{F}\cross SU(2)$ is the Langlands group of $F$ , avariant of the Weil-Deligne group. An irreducible representation of $G(F)$ is unipotent
if it is parametrized by an -parameter whose restriction tois trivial. Although
this definition is modulo the conjectural parametrization, the similarity with the Jordan
decomposition is obvious. That is,
is considered as the semisimple part of while
assigns a unipotent element of commuting with the semisimple part (by the
Jacobson-Morozov theorem).
This also resembles Lusztig’s classification of irreducible representations of reductive
group
over finite fields via the orbits in the dual group[Lus84, Ch.13]. In
that case, the semisimple part of a parameter is really a semisimple element (or orbit) in. But in the -adic case, it is already a tempered Langlands parameter which is ofgreat importance in the arithmetic study of automorphic representations. Thus the Aparameters of mixed type (both semisimple and unipotent parts are non-trivial) will also
be worth studying. In this note, we consider the (unique) non-quasisplit unitary groupwith 4 variables associated to a quadratic extension $E$ of $F$ , and construct some examplesof irreducible representations parametrized by such mixed 4-parameters. Although thcy
are not literally unipotent, they are still interesting both representation
theoretically and arithmetically I hope our misuse of the terminology does not confuse
anybody.
The plan of this note is as follows. The -parameters are expected to classify only
those irreducible representations which appear as local components of certain automorphic
representations. This implies that the representations parametrized will be unitarizable.
In \S 2, we prepare some notation on the non-quasisplit unitary group in 4 variables over a-adic field $F$ , and classify the unitary dual of $G(F)$ . In \S 3, we construct representations
parametrized by an -parameter for $G(F)$ . This relies on a detailed study of local 0correspondence for unitary dual pairs, especially between unitary groups in two variables.
Finally the author would like to thank the organizer Prof. Susumu Ariki for giving her
an opportunity to talk at the conference.
$A$

$A$

$\psi$$\mathcal{L}_{F}$$\psi|_{\mathcal{L}_{F}}$$\psi$$\hat{G}$$\psi|_{\mathrm{S}L(2,\mathrm{C})}$$G(\mathrm{F}_{q})$$\hat{G}(\overline{\mathrm{F}}_{q})$$\hat{G}(\overline{\mathrm{F}}_{q})$$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}^{1}\mathrm{o}\mathrm{n}\mathfrak{x}>\backslash$

$A$

$G$

$A$

2

The unitary dual

Here we classify the unitary dual of the non-quasisplit unitary groupin 4 variables.
For
we have to determine the reducible points of parabolically induced from cuspidal
modules. When the inducing representation is generic, the Langlands-Shahidi theory$G$$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{S}_{\}}$

1Of course, the only (elliptic) unipotent representation of unitary groups is the trivial representation.
The representations treated in this note are the analogues of the quadratic unipotent representations inthe sense of Moeglin [Mceg196].

120[Sha90] allows us to reduce the problem to the determination of certain automorphic Lfactors (see [KonOl]). But in the present non-quasisplit case, no such theory is available
and all we can do is to relate the Plancherel measure of the induced representation to theone on the quasisplit inner form of by Langlands’ functoriality.
$G$

2.1

Preliminary

beaquadratic extension of non-archimedean local fields of characteristic zero,):be the non-trivial element in Gal(F/F). We have the norm map.
, and write ) $E/F$ for the non-trivial character of$F$denote the ring of integers of , its unique maximal ideal andand
As usual,of $F$ containing $E$ ,the module of $F$ , respectively [Wei]. Fixing an algebraic closure,for the absolute Galois and Weil groups of
we write
respectively [Tat79]. Similar notation for $E$ will be used with the subscript F.
let us write $\Pi(G(F))$ for the set of
For a connected reductive group defined overisomorphism classes of irreducible admissible representations of $G(F)$ . Also
denote the subsets of unitarizable, tempered,square integrable and (unitarizable) supercuspidal elements in $\Pi(G(F))$ , respectively. Recall that an admissible representation of $G(F)$ is supercuspidal if its matrix coefficientsare compactly supported modulo center.
’
Recall the classification of hermitian spaces of even dimension over $E$ . Fix
and setLet
and$\Gamma\sqrt\lrcorner/F$$\mathrm{N}_{E/F}$$\sigma$$x\mapsto x\sigma(x)\in F^{\mathrm{x}}$$E^{\mathrm{x}}$$F^{\mathrm{x}}/\mathrm{N}_{E/F}(E^{\mathrm{x}})\simeq \mathbb{Z}/2\mathbb{Z}$$||_{F}$$O\supset \mathfrak{p}$$\overline{F}$$\Gamma:=\mathrm{G}\mathrm{a}1(\overline{F}/F)$$F_{i}$$\mathrm{V}V_{F}:=W_{\overline{F}/F}$$G$$F_{\}}$$\Pi_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}}(G(F))\supset$$\Pi_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}(G(F))\supset\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G(F))\supset\Pi_{0}(G(F))$$\wedge[\in F$$\backslash$$\mathrm{N}_{E/F}(E^{\mathrm{x}})$$V_{2}^{+}:=$$(E^{2}, (\begin{array}{ll}0 11 0\end{array}) )$$V_{2}^{-}:=(E^{2}.

(-1 \eta[) )$.

is anisotropic hermitian plane, respectively. Thenis hyperbolic andform a complete system of representatives of $2n$-dimensional hermitian space, respectively. To beandandover . We write
for the unitary groups of$F$
explicit, these can be realized as the -algebraic group, which associate to each abelian$F$ -algebra
the groups$V_{2}^{+}$$V_{2n}^{\pm}=V_{2}^{\pm}\oplus$$V_{2}^{-}$$(V_{2}^{+})^{n-1}$$F_{J}$$G_{2r\iota}$$G_{2n}^{*}$$V_{2n}^{-}$$V_{2n}^{+}$

$R$

$l$

$\in$

1(2.1)$n$

Here$I_{n}:=(\begin{array}{llll} 1 1 \cdot 1 \end{array})$.
is the quasisplit inner form of
Now we turn to the case $G=G_{4}$ . We write $P=MU$ for the unique proper parabolic
subgroup, which can be identified with the upper triangular subgroup in the above re. Let $P^{*}=M^{*}U^{*}$ be the corresponding standard
alization. Thus
Note that$G_{2n}$$G_{2n}^{*}$$M\simeq \mathrm{R}_{E[F}\mathrm{G}_{m}\cross G_{2}$121(with respect to the upper triangular Borel subgroup in the realization (2.1)) parabolic
subgroup.
An element in $\Pi_{0}(M(F))$ is of the form$\chi\otimes\tau$with$\chi\in\Pi_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}}(E^{\cross})$$\chi[\lambda]\otimes\tau:=\chi||\mathrm{i}$ $\otimes\tau$: $M(F)\ni$ diag(a, ,$\sigma(a)^{-1}$) $\mapsto\chi(a)\tau(g)\in GL(V)$ ,and
is a realization of $\tau\in\Pi_{0}(G_{2}(F))$ . For, we write. We would like to determine the reducible points of the parabolically$(\tau, V)$

induced representation

2.2

$g$

$\lambda\in \mathbb{C}$.

$I_{P}^{G}(\lambda’[\lambda]\otimes\tau)$,

Jacquet-Langlands correspondence for$G_{-}$

Local correspondence Recall the local Shimizu-Jacquet-Langlands correspondence.
Let $D$ be the unique quaternion division algebra over . We have the reduced normof $D$ , so that we can define the characteristic polynomial of
as $\nu_{D/F}(T-)^{D})$$F[T]$ . We say
is regular semisimple if its characteristic polynomial has two distinct
roots. Also regular semisimple $\gamma\in GL(2, F)$ andcorrespond to each other if
their characteristic polynomials coincide.
$F$$\nu_{D/F}$$\gamma^{D}\in D^{\cross}$$\in$$\gamma^{D}$$\gamma^{D}\in D^{\mathrm{x}}$Fact 2.1 (Jacquet-Langlands). There exists a unique bijection$\tau^{D}\in\Pi(D$ ’ ), which es characterized by the character relation$\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(GL(2, F))\ni\tau\mapsto\sim$$\mathrm{O}-_{\tau}(\gamma)=-\mathrm{O}\tau D$ $(\gamma^{D})$for any regular semisimple$\gamma\in GL(2, F)$,$-/^{*}\in D^{\cross}$correspond to each other.

This can be easily translated into the unitary similitude group case. Letandbe the unitary similitude groups forand, respectively
We have the isomorphisms$\overline{G}_{2}:=$$GU(V_{2}^{-})$$\overline{G}_{\underline{9}}^{*}:=GU(V_{2}^{+})$$V_{2}^{+}$$V_{2}^{-}$$[\mathrm{K}\mathrm{K}, 5.2.1]$$\overline{G}_{\underline{9}}arrow\sim(\mathrm{R}_{E/F}\mathrm{G}_{m}\cross D^{\mathrm{x}})/\triangle F^{\mathrm{x}}$.

$\overline{G}_{2}^{*}-arrow\sim(\mathrm{R}_{E/F}\mathrm{G}_{m}\cross GL(2, F))/\triangle F^{\mathrm{x}}$wherestands for the diagonal embedding
For a characterof thecenterof, we write
andfor the setsof elements inandwith the central character , respectively. We
define the Shimizu-Jacquet-Langlands correspondence$\triangle$$z\mapsto(z, z1_{2})-\cdot$$\overline{Z}_{2}(F)\simeq E^{\mathrm{x}}$$\square (G_{2}(F))_{\omega}$$\overline{G}_{2}^{*}(F),\overline{G}_{2}\underline{(}F)$$\Pi(\overline{G}_{2}(\Gamma\sqrt))$$\omega$$\Pi(\overline{G}_{2}^{*}(F))_{(v}$$\Pi(G_{2}^{*}(F))$$\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(\overline{G}_{2}^{*}(F))_{\omega}\ni\omega$$\omega$$\otimes\tau\mapsto$(2.2)$\otimes\tau^{D}\in\Pi(\overline{G}_{2}(F))_{\omega}$$\omega$, whereis the correspondence in Fact 2.1.
We return to the unitary groups $G_{2}(F)$ ,. $\Pi(G_{2}^{*}(F))$ is partitioned into a disjoint
union of the finite subsets called -packet. By definition [ROg90, 11.1], each -packet of
consists of the irreducible components of. Let usfor some
for the set of -packets of Z2(F), andwrite
for its subset consisting of-packets contained in. For, we writefor$\tilde{G}_{2}(F)$$rightarrow\tau^{D}$$\mathrm{i}$$G_{2}^{*}(F)$

$L$

$L$

$G_{2}^{*}(F)$$\tilde{\pi}|_{G_{2}(F)}*$$\mathrm{i}$$L$$(G_{2}^{*})$$L$$\mathrm{I}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G_{2}^{*})$$\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G_{2}^{*}(F))$$\Pi^{*}\in$$\ominus_{\Pi^{*}}(\gamma):=$$\mathrm{f}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G2*)$$\mathrm{h}$$\tau\in\Pi$$\ominus_{\tau}$$\mathrm{i}\in\Pi(\overline{G}_{2}^{*}(F))$

122
for its stable character, a locally constant function on the regular semisimple set. We also
have a similar description for $G_{2}(F)$ . Now the correspondence (2.2) gives a correspondence$\Phi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G_{\underline{9}}^{*})\ni\Pi^{*}rightarrow$II$\in\Phi(G_{2})$which is characterized by the character identity$\Theta_{\Pi}*(\gamma^{*})=-\mathrm{O}-_{\mathrm{n}}(\gamma)$

which are conjugate in $GL(2, E)$ . We,
for any regular semisimple.
call this the Jacquet-Langlands correspondence for$\gamma^{*}\in G_{2}^{*}(F)$$\gamma\in G_{2}(F)$$G_{2}$

A special global correspondence Next we move to a global situation. One can takeis isomorphic
a quadratic extension $K/k$ of number fields such that$(i=1,2)$,.
theadele ring of
for
Write
placesat
two
non-archimedean
toin two variables with respect to $K/k$. Also we have an anisotropic unitary groupsuch that [C1091, \S 2]$K_{v_{i}}:=(K\otimes_{k}k_{v_{\mathrm{t}}})/k_{v_{i}}$$Vi$$F_{\lrcorner}/F$$\mathrm{A}=\mathrm{A}_{k}$$k$$\underline{G}_{2}$$\circ$with respect to an isomorphism$\underline{G}_{2}\otimes_{k}k_{v_{i}}\simeq G_{2}$$\circ\underline{G}_{2}\otimes_{k}k_{U}$is quasisplit at any place outside of$v_{1}$$k_{v_{i}}\simeq F,$,$v_{2}$$(i=1,2)$ ;.

We writefor the quasi-split unitary group in two variables with respect to $K/k$ . For
contains an unramified
each familyof local -packets $\Pi_{v}\in\Phi(Gv)$ such thatrcprescntation2 at all but finite number of non-archimedean places, we can form the:associated global -packet for$\underline{G}_{2}^{*}$$L$$(\Pi_{\cup})_{v}$$L$$\Pi_{\{1}$$\underline{G}_{2}(\mathrm{A})$$\Pi_{\mathrm{A}}:=\{\otimes_{v}’\tau_{U}|(\mathrm{i}\mathrm{i})(\mathrm{i})$As usual,$\tau_{?\mathrm{J}}\mathrm{i}\mathrm{s}\mathrm{u}\mathrm{n}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d}\tau_{v}\in\Pi_{v},\mathrm{a}\mathrm{t}\forall v$

at$l’v\}$

stands for the restricted tensor product
Using a standard argument on Deligne-Kazhdan’s simple version
can deduce the following from [ROg90, Ch.13] (or rather [LL79]).
$\otimes_{v}’$Proposition 2.2. For each $\tau\in\Pi_{0}(G_{2}(F))_{f}$ there exists a global
which have the following properties.
of$L$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$-packetformula, one.

$\Pi_{\mathrm{A}}=\otimes_{v}’\Pi_{v}$$\mathrm{V}_{2}(\mathrm{A})$

(i)
(ii)

$\Pi_{v_{i}}$contains$\mathrm{r}$for $i=1,2$ .

is automorphic; that is, there exists an irreducible subrepresentation
is automaticallyat all . Notice thatsuch thatofcan be chosen
cuspidal since it has supercuspidal local component at . Moreover$\tau_{\mathrm{A}}=\otimes_{v}’\tau_{v}$$\Pi_{\mathrm{A}}$$\tau_{v}\in\Pi_{v}$$L^{2}(\underline{G}_{2}(k)\backslash \underline{G}_{2}(\mathrm{A}))$$v$$\tau_{\mathrm{A}}$$v_{\mathrm{j}}$$\tau_{\mathrm{A}}$in such a way thatis unramified if it is quasisplit and split
reductive group over a non-archimedean local fieldover a non-archimedean extension of . Then $G(F)$ admits a hyperspecial maximal compact subgroup ,$\sim \mathrm{A}9$

$F$

$G$

$F$

$\mathrm{K}$wllic.ll is unique up to $G(F)$ -conjugation. An irreducible admissible representation of $G(F)$ is unramified
if it contains a non-zero -fixed vector.
$\mathrm{K}$

123
(a)

$\tau_{v_{i}}\simeq\tau$

for $i=1,2$ ;(b) There eists an irreducible cuspidal automorphic representation
contained in the Jacquet-Langlands correspondentof$\underline{G}_{2}^{*}(\mathrm{A})$

$i$

$ii$

.

$\tau_{\mathrm{A}}^{*}=\otimes_{v}’r_{v}^{*}$$\Pi_{\mathrm{A}}$$\Pi_{\mathrm{A}}^{*}$

at any other than , 1,2;
. There exists a non-trivial character$v$$\tau_{v}\simeq\tau_{v}^{*}$

7;

is

of

such that$v_{1}$$\psi_{\mathrm{A}}- gene\dot{n}c^{3}$$\psi_{\mathrm{A}}=\otimes_{v}\psi_{v}$:$\mathrm{A}/karrow \mathbb{C}^{\mathrm{x}}$such that.

Lifting Plancherel measures

2.3

Going back to the local situation, we put, where$F$-rational character group of . Since has a discrete valuation,$M(F)^{1}:= \bigcap_{\chi}\mathrm{k}\mathrm{e}\mathrm{r}|\chi|_{F}$$F$$\chi$

runs over tbe$\lambda \mathrm{f}$$\hat{A}_{\Lambda \mathit{4}}:=$$\mathrm{H}\mathrm{o}\mathrm{m}(M(F)/\Lambda f(F)^{1}, \mathbb{C}^{\cross})$-torus. Let us write(A#(F))}.
{A yr $\in 4_{M}$ , rr$(F)/A_{\Lambda
I}(F)$ ) is a
Thenacts on this by(\Lambda f-orbit in
A@r. Eachhomogeneous space underand hence a C-variety.
Fix such-orbit ] . Writing for the parabolic subgroup of opposite to $P$ withrespect to A#, consider the intertwining operator
ls a$\mathbb{C}$$\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(M(F)/A_{\lambda I},(F)):=$$\chi\in\hat{A}_{M}$$\otimes$$|\mathrm{X}$$\hat{A}_{M}$$\tau\mapsto$$\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}$$\mathrm{p}$$\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}$$\hat{A}_{\Lambda I}$$\hat{A}_{M}$$G$$\overline{P}$$3$$J_{P|P}(\pi)$:$I_{P}^{G}(\pi)arrow I_{\overline{P}}^{G}(\pi)$,$\pi\in$$\mathrm{J}3$.

This is defined by an absolutely convergent integral on an open subset of , and defines
a rational function on [Wa103, ]. It is known that the induced representation 1
is irreducible for in some Zariski dense subset of . Hence, by Schur’s lemma, there
exists a rational function on J3 such that$\mathrm{I}^{\mathrm{J}}$$P(G\pi)$$\mathrm{I}\mathrm{V}$$l\mathit{7}\mathit{3}$$\pi$$\mathrm{p}$$j$$J_{P|\overline{P}}(\pi)\circ J_{\overline{P}|P}(\pi)=j(\pi)\cdot \mathrm{I}\mathrm{d}$The Plancherel measure of$\pi\in$) is defined by [Wa103, V.2.]$3$$\mu^{G}(\pi):=j(\pi)^{-1}\gamma(G/M)^{2}$

Here$\gamma(G/\lambda l)$.

is a certain volume factor defined in [Wa103, 1.1.5].

Proposition 2.3. For$\chi\otimes\tau\in\Pi_{0}(\lambda f(F))$

and s$\in \mathbb{C}_{f}$we have$\gamma(G_{4}/M)^{-2}\mu c_{4}(\chi[s]\otimes\tau)=\gamma(G_{4}^{*}/M^{*})^{-2}\mu^{G_{4}^{*}}(\chi[s]\otimes\tau^{*})$

Hereis the unique -packet of $G_{2}(F)$ containing , andJacquet-Langlands correspondentof . Notice thatthe choice of
.
$L$

$\Pi$

$\tau$$\Pi^{*}$

$\Pi$

is,

$\tau_{v}^{*}$

admits a$\psi_{v}$is any element in theis independent of$\mu^{G_{4}^{*}}(\chi[s]\otimes\tau^{*})$$\tau^{*}\in\Pi^{*}$$3\mathrm{T}\mathrm{h}\mathrm{a}\mathrm{t}$$\tau^{*}$-Whittaker model at each .
$v$

.

124
Proof. (Sketch) The key point is that we have a complete functional equation for global

intertwining operators$(l)=J_{P|\overline{P}}(\chi_{\mathrm{A}}[s]$&$($$\tau_{\mathrm{A}})\circ J_{\overline{P}|P}(\chi \mathrm{A}[s]\otimes\tau_{\mathrm{A}})\phi$

,

’
is as
, $(i=1,2)$ andis an idele class character of $K$ such that
where
$=iv$
in Prop. 2.2. It is understood that the vector 6 is chosen to have an Euler decomposition. Using the Gindikin-Karpelevich formula (as for the padic case, see [Lan71])
are unramified, we can write the right hand side asat the places where& and$\chi_{v_{i}}=\chi$$\chi_{v}$$\chi_{\mathrm{A}}$$\tau_{\mathrm{A}}$$\phi=\otimes_{v}’\phi_{v}$$\chi_{v}$$\phi_{v}$$\tau_{v}$$\frac{L^{\mathrm{b}^{1}}(s,\tau_{\mathrm{A}}^{\vee}\cross\chi_{\mathrm{A}})L^{S}(2s,\chi_{\mathrm{A}}|_{\mathrm{A}_{k}^{\cross}})L^{S}(-s,\tau_{\mathrm{A}}\cross\chi_{\mathrm{A}}^{-1})L^{S}(-2s,\chi_{\mathrm{A}}^{-1}|_{\mathrm{A}_{k}^{\cross}})}{L^{S}(1-s,\tau_{\mathrm{A}}\cross\chi_{\mathrm{A}}^{-1})L^{S}(1-2s,\chi_{\mathrm{A}}^{-1}|_{\mathrm{A}_{k}^{\mathrm{X}}})L^{S}(s+1,\tau_{\mathrm{A}}^{\vee}\cross\chi_{\mathrm{A}})L^{S}(2s+1,\chi_{\mathrm{A}}|_{\mathrm{A}_{k}^{\mathrm{X}}})}\phi^{S}$$\otimes\otimes j^{G_{v}}(\chi_{v}[s]\otimes\tau_{v})\phi_{v}v\in S^{\cdot}$

andis ramified. $L^{S}(s,$is the finite set of places where at least one of ,
( ,
, and
is the partial standard -function of twisted by
A ) is Just theof. We have a similar description for(partial) Hecke -function of
Prop. 2.2:

Here

$S$

$6_{v}$$\tau_{v}$$\chi_{v}$$L$$\chi_{\mathrm{A}}^{-1})$$\cross$$7_{\mathrm{A}}$$L^{S}$$X_{\mathrm{A}}^{-1}$$\tau_{\mathrm{A}}$$s$$\chi_{\mathrm{A}}|$$k\cross$$L$$I_{\mathrm{r}}^{\frac{G}{P}}*(\chi_{\mathrm{A}}[s]\otimes\tau_{\mathrm{A}}^{*})$$\chi_{\mathrm{A}}$$\phi=’\frac{L^{S}(s,\tau_{\mathrm{A}}^{\vee}\cross\chi_{\mathrm{A}})I_{J}^{S}(2s,\chi_{\mathrm{A}}|_{\mathrm{A}_{k}^{\cross}})L^{S}(-s,\tau_{\mathrm{A}}\cross\chi_{\mathrm{A}}^{-1})L^{S}(-2s,\chi_{\mathrm{A}}^{-1}|_{\mathrm{A}_{k}^{\cross}})}{L^{S}(1-s,\tau_{\mathrm{A}}\cross\chi_{\mathrm{A}}^{-1})L^{S}(1-2_{\mathrm{S}},\lambda_{\mathrm{A}}^{-1}|_{\mathrm{A}_{k}^{\cross}})L^{S}(s+1,\tau_{\mathrm{A}}^{\vee}\cross\chi_{\mathrm{A}})L^{S}(2s+1,\lambda \mathrm{A}|_{\mathrm{A}_{k}^{\cross}})}\phi^{S}$$\otimes\otimes j^{G_{\mathrm{t})}}v\in S$$(\chi_{v}[s]\otimes r:)_{C}ft_{v}$

.

since
Notice, for example,
Comparing these two formulae, we obtain the equality between the$L^{S}(s, \tau_{\mathrm{A}}^{*\vee}’\cross\chi_{\mathrm{A}})=L^{S}(s, \tau_{\mathrm{A}}^{\vee}\cross\chi_{\mathrm{A}})$$\gamma(G_{4}/\Lambda I)_{7^{\mathrm{Z}}}^{-4}G_{4}(\chi[s]\otimes \mathrm{y})2=\gamma(G_{4}^{*}/\mathrm{A}\prime I^{*})_{7^{\mathrm{t}^{G}}}^{-4}$$\tau_{v}^{*}=\tau_{v}$;$v_{1\}}v2$$(\chi[s]\otimes\tau^{*})^{2}$at any $v\not\in S.$-components:.

This combined with the positivity of the Plancherel measure on the unitary axis yields
the assertion on the unitary axis. Then this equality of rational functions on such ZariskiEldense subset must extends to the whole connected component.

2.4

Composition series of induced representations

Here we determine the reducible points of the induced representations (& r) :$\tau\in\Pi_{0}(G_{2}(F))$). To describe the result, we need a classification of, A
in terms of the Jacquet-Langlands correspondence (\S 2.2) and Rogawski’s description of$I_{P}^{G}$$\tau\in\Pi_{0}(\mathit{1}1I(F))$$\Phi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G_{2}^{*})$$\chi[\lambda]$$(\chi\otimes$$\in \mathbb{R}$.

for the unitary group in one
Supercuspidal representations of $G_{2}(F)$ Writeandsuch thatvariable. We conventionally write and for characters of
etc. For such 7/, let. Thus another such characters are denoted by ,be the character $G_{1}(F)$ $x/\sigma(x)\mapsto\eta(x)\in$ Cx. We abbreviate the one-dimensionalare). The elements of(resp.
) asrepresentation(resp.
ofdescribed as follows [ROg90, \S 12.1].
$G_{1}$$\eta|_{F}\cross=1$$E^{\mathrm{x}}$$\eta$

$\mu$

$fI’$

$\mu|_{F^{\cross}}=\omega_{E/F}$$\mu’$)$\ni \mathrm{t}$$\eta_{u}$$\eta_{u}\circ\det$$G_{n}^{*}$$G_{n}$$\mathrm{I}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G_{2}^{*})$$\eta_{G_{n}^{*}}$$\eta_{G_{n}}$125(i) The stable cuspidal -packet
consists of one irreducible cuspidal representation
whose standard base change lift is a supercuspidal representationof$GL(2, E)$ .
$L$$\Pi^{*}(\pi)$$\mathrm{y}^{*}$$\pi$(ii) The endoscopic -packet
withconsists of two distinct irreducible
cuspidal representations. This is the endoscopic lift of a characterof. [ROg90, 4.8]:
with respect to theto$L$$\eta\neq 1$$\lambda_{\overline{\mu}}^{G_{2}^{*}}(1, \eta)$$1\otimes$$G_{2}^{*}(F)$$z_{1}\mu(w)$$z_{2}\mu(w))\mathrm{x}w$:$( LG_{1}^{2})\ni(z_{1}, z_{2})\mathrm{n}$$\lambda_{\overline{\mu}}^{G_{2}^{*}}$$G_{1}(F)^{2}$$\eta_{u}$$L- \mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}^{4}$

if $w\in W_{E)}$$w\mapsto\{\{$$\in LG^{*}\underline,$$-z_{1})\mathrm{n}w$

,

if

$w$

.

$=w,$

$z_{2}$(iii)$\{\eta_{G_{2}^{*}}\delta^{G_{2}^{*}}\}$, where$\delta^{G_{2}^{*}}$

is the Steinberg representation of$G_{2}^{*}(F)$.

Recall thatconsists of the Jacquet-Langlands correspondent of these L-packets,
we denote each of which as follows.
$\Phi(G_{2})$(1)$\Pi(\pi)$(ii), containing only one irreducible supercuspidal representation.

$\lambda_{\mu}^{G_{2}}(1,7)$(iii)$\{\eta_{G_{2}}\}$

with$\eta\neq 1.$. Note that$\eta_{G_{2}}$This consists of two distinct supercuspidal representations.
is supercuspidal.

Reducible points and Jordan-Holder series The reducibility of, an
induced representation from supercuspidal module, is controlled by the behavior of its
Plancherel measure as a function in :$I_{P}^{G}(\chi[\lambda]\otimes\tau)$$\lambda$

Fact 2.4 (Harish-Chandra, Silberger). (i)is reducible atonly if (and(ii) [Si180, Lem. 1.2]
is reducible at $\lambda=s>0$ if and only if
has aat A $=s.$$I_{P}^{G}(\chi[\lambda] \ \tau)$

if. and$\chi=\sigma(\chi)^{-1}$$I_{P}^{G}(\chi[\lambda]\otimes\tau)$$\mu^{G}(\chi[\lambda]\otimes\tau)$$\lambda=0$$\mu^{G}(\chi\otimes\tau)\neq 0.$$=\sigma(\chi)^{-1}$and$p\mathrm{o}/e$

Combining Proposition 2.3 with [Sha90, Cor. 3.6], we can calculate the Plancherel
measure. The result is”$\mu^{G}(\chi[s]\otimes\tau)=\frac{\gamma(G/M)^{2}}{\gamma(G^{*}/M^{*})^{2}}\mu^{G}$$=\gamma(G/M)^{2}\epsilon(s, \tau^{*\vee}\cross\chi,\overline{\psi}_{F})\epsilon$( $2s$ ,$\chi|F^{*}$, F)$\overline{\psi}$$(\chi[s]\otimes\tau^{*})$$\epsilon(-s, \tau^{*}\cross\chi^{-1}, \psi_{F})\epsilon(-2s, \chi|_{F}^{-}\mathrm{p}, \psi_{F})$$\cross\frac{L(1-s,\tau^{*}\cross\chi^{\vee})L(1-\underline{9}s,\chi|_{F^{\mathrm{X}}}^{-1})L(s+1,\tau^{*\vee}\mathrm{x}\chi)L(2s+1,\chi|_{F^{\mathrm{X}}})}{L(s,\tau^{*\mathrm{v}}\mathrm{x}\chi)L(2s,\chi|_{F^{\cross}})L(-s,\tau^{*}\cross\chi^{-1})L(-2s,\chi|_{F^{\mathrm{X}}}^{-1})}$Here
is a fixed non-trivial character of F.
,
and -factors of. We know from [KonOl, Prop. 3.2] that
$L(s, \chi|_{F}\cross)$$(7)_{F}$$\epsilon$

are the Hecke$\chi|_{F^{\mathrm{X}}}$$L(s, \tau^{*}\cross\chi)=L_{E}(s, \pi \mathrm{x} \chi)$$4\mathrm{F}\mathrm{o}\mathrm{r}$$\epsilon(s, \chi|_{F}\cross, I_{F})$.

,the general definition of the functorial lifting of representations, please consult [BOr79].

$L$

126
where $\pi\in\Pi(GL(2, E))$ is the standard base change lift of the -packet containing .
is the standard -factor for $GL(2, E)$ defined in [JL70]. Finally note thatthc -factors are all exponential functions so that they do not affect the poles or zeros of
the Plancherel measure. These calculations give the following result.
$L$$\tau^{*}$$L$$L_{F_{\lrcorner}}(s, \pi\cross\chi)$$\epsilon$

Theorem 2.5 (1)(2)(& r) with A$I_{P}^{G}$

(i)
(ii)

(3)

$\tau$

belongs to either II
belongs to$I\mathrm{j}(\eta[\lambda] \ \tau)$

(i)

(ii)

$\geq 0$$\mu[\lambda]$$\tau$

is irreducible unlesswith A
is irreducible except for the following cases.

with Aor$\{\eta_{G_{2}}\}$

ancl A $=0.$

with$\eta\neq 1$

and$(\pi)$$\lambda^{G}l^{l^{-1}}(1, \eta)$$\geq 0$$\chi=\sigma(\chi)^{-1}$$\geq 0$

$I_{P}^{G}(\chi[\lambda]\otimes\tau)$$\cdot$.

$\lambda=1.$

is irreducible excePt for the following cases.

and ) $=1/2$ .

$\tau\not\simeq\eta_{G_{2}}$and A $=3/2$ .

$\tau\simeq\eta_{G_{2}}$with $\tau\in\Pi_{0}(G_{2}(F))$ is at most two [BZ77], its comp0Since the length ofsition series at the reducible points are easy to describe. In fact, Langlands’ classification
and Casselman’s criterion of square integrability [Wa103, Prop.III.1.1] yield the following
complete description of composition series at the reducible points in the previous theorem.
$I_{P}^{G}(\chi[\lambda]\otimes\tau)$

Corollary 2.6. Let , be as above. For $s>0_{f}$ we write
quotient of.
(i) Ifwe have$\chi$$\tau$$J_{P}^{G_{4}}(\chi[s]\otimes\tau)$

for the Langlands$I_{P}^{G}(\chi[s] \ \tau)$$\tau\in\Pi(\pi \mathrm{I}f$$I_{P}^{G}(\mu\otimes\tau)=$$\mathrm{r}G(\mu, \mathrm{r})_{+}\oplus\tau^{G}(\mu, \mathrm{y} )_{-}$.

$0arrow\delta^{G}(\eta, \tau)arrow I_{P}^{G}(\eta[1/2]\otimes\tau)arrow J_{P}^{G}(\eta[1/2]\otimes\tau)arrow 0,$

where(ii) If$\tau^{G}(\mu, \tau)_{\pm}\in\Pi_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}(G(F))$$\tau\in\lambda_{\mu^{-1}}(1, \eta)$

with$\eta\neq 1,$$\backslash \Pi_{\mathrm{d}i\mathrm{s}\mathrm{c}}(G(F))$

and$\delta^{G}(\eta, \tau)\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G(F))$.

we have$0arrow\delta^{G}(\mu, \tau)arrow I_{P}^{G}(\mu[1]\otimes\tau)arrow J_{P}^{G}(\mu[1]\otimes\tau)arrow 0,$$0arrow\delta^{G}(\eta’, \tau)-I_{P}^{G}(\eta’[1/2]\otimes\tau)-J_{P}^{G}(\eta’[1/2]\otimes\tau)arrow 0.$Here(ii) If$\delta^{\mathrm{C}_{x}^{l}}(\mu, \tau)$,$\delta^{G}(\eta’, \tau)\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G(F))$$\tau=\eta_{G_{2}}$.

, we have$I_{P}^{G}(\mu\otimes \mathit{7}G2)$$=\tau^{G}(\mu, \eta_{G_{2}})_{+}$$0arrow\delta^{G}(’/’, \eta_{G_{2}})arrow I_{P}^{G}(\eta’[1/2]$

&
$($

I

$\tau^{G}(/"\eta_{G_{2}})_{-}$,$\eta_{G_{2}})arrow J_{P}^{G}(\eta’[1/2]\otimes\eta_{G_{2}})" \mathrm{p}$$0$,$0arrow\delta^{G}(\eta,\eta_{G_{2}})arrow I_{P}^{G}(\eta[3/2]\otimes\eta G_{2})arrow J_{P}^{G}(\eta[3/2]\otimes\eta c_{2})arrow 0,$

where$\eta’\neq\eta$$\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G(F))$.

. Here$\tau^{G}(\mu, \eta_{G_{2}’})_{\pm}\in\Pi_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}(G(F))\backslash \Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G(F))$

and$\delta^{G}(\eta’, \eta_{G_{2}})$,$\delta^{G}(\eta, \eta_{G_{2}})$

$\in$

127
The unitary dual

2.5

Now the classification of the unitary dual of $G(F)$ is an easy task. We have only to remark
that, ifsatisfies
but is irreducible, there appears the complementary
series representations
for A between 0 and the reducible point. Otherwise,
no complementary series appears. We summarize the result as a corollary.
$\sigma(\chi)^{-1}=\chi$$I_{P}^{G}(\chi\otimes\tau)$$I_{P}^{G}(\chi[\lambda]\otimes\tau)$

Corollary 2.7. The reducible unitarizable representations of $G(F)$ are the followings.
(i) Elements of.
(ii) The $non- s^{r}upercuspidal$ square integrable modules
,
,
,
),
, ,).
with
(iii) The tempered non square integrable representations(the -packet of is
not of the for,,,$\tau\in\Pi_{0}(G_{2}(F)))$ .
(iv) The following non-tempered unitt wezable representations.
$i$$\square _{0}(G(F))$$\delta^{G}(7, \tau)$$\eta\neq 1$$\lambda_{\overline{\mu}}^{G_{2}}(1, \eta)$$\delta^{G}(/"\tau)$$(\tau\not\simeq\eta_{G_{2}})$$(\tau\in$$\delta^{G}\mathrm{O}\mathrm{y}$$\eta_{G_{-}}$$L$$\tau^{G}(\mu, \tau)_{\pm_{f}}$$rm\lambda_{\overline{\mu}}^{G_{2}}(1, \eta))_{f}I_{P}^{G}(\mu\otimes\tau)$

Representation
cg r)($J_{P}^{G}$$\mu[s]$$(\tau\in\lambda_{\overline{\mu}}^{G_{2}}(1, /),$

$I_{P}^{G}(\chi \otimes\tau)$$\tau$$(\chi|_{F^{\cross}}\neq)E/Fz$

Conditions$\tau\in\lambda_{\overline{\mu}}^{\mathrm{G}_{2}^{1}}(1, \eta)f$$(\eta[s] \ \tau)$

V7$\neq 1,$

$1\geq s>0$$3/2\geq s>0$

$J_{P}(\eta[s]\otimes\eta_{G_{2}})$$J_{P}$$\eta f\neq 1)$$\mathrm{r}$$\not\in\{\eta_{G_{2}}\}_{f}1/2\geq s>0$

Remark 2.8. For the purpose of compar ison, we recall the corresponding classificationand A
wefor $G^{*}(F)$ . For\S , \S 5.3]
$Ij$(1)
with A
is irreducible, ifa.
utith Ais irreducible, unless:$\chi\otimes$$(\chi[\lambda]S)$$r\in\Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(M^{*}(F))$$(\mathit{2})I_{P^{*}}^{G^{*}}(\mu[\lambda]S)$ $\tau)$

(i)

(ii)
(3)

$\geq 0,$$\geq 0$$\tau)$$\chi\not\simeq$$have/Ko^{r}n\mathit{0}\mathit{1},$$\mathit{4}\cdot \mathit{3}$$(\chi)^{-1}$$\geq 0$$\tau\in\Pi^{*}(\pi)$$or+rtc_{2}*\delta^{G_{2}^{*}}\}_{2}$$\tau\in\lambda_{\overline{\mu}}^{G_{2}^{*}}(1, \eta)$$I_{P^{*}}^{G^{*}}(\eta[\lambda]\otimes\tau)$

withwith Aand A $=0.$$\eta\neq 1,$$\geq 0$and A $=1.$

is irreducible, unless(i) A $=1/2$ .
(ii)$\tau=\eta_{G_{2}}*\delta^{G_{2}^{*}}$In this c\^a ,$e$$I_{P^{*}}^{G^{*}}$and A $=3/2$ .
$(\eta[1/2] \mathrm{C}\otimes\eta_{G_{2}^{*}}\delta^{G_{2}^{*}})$

is reducible [KonOl, Prop. 5.8]:$0arrow\tau^{G^{*}}(\eta_{G_{2}^{*}}\delta^{G_{2}^{*}})arrow I_{P^{*}}^{G^{*}}(\eta[1/2]\otimes\eta_{G_{2}^{*}}\delta^{G_{2}^{*}})arrow J_{P^{\mathrm{s}}}^{G^{*}}(\eta[1/2]\otimes\eta_{G_{2}^{*}}\delta^{G_{2}^{*}})arrow 0.$

Here$\tau^{G^{*}}(\eta c_{2}*\delta^{G_{2}^{*}})\in\Pi_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}(G^{*}(F))$

3

Construction of an A-packet$\backslash \Pi_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(G(F))$.

For the moment, letbe a general connected reductive group defined overparameter is a continuous homomorphism$G$$\psi$:$\mathcal{L}_{F}\cross SL(2, \mathbb{C})arrow L$

10

Cv,

$F$

. An A-

128
is defined in \S 1. Two A-parameters are equivalent if they
where the Langlands groupare -conjugate. Let us write $\Psi(G)$ for the set of -conjugacy classes of A-parametersfor . According to Arthur’s conjecture, there associates to each $\psi\in\Psi(G)$ a finite
called an -packet. In the cases where these A-packets were
subsetdetermined, namely, $G=GL(n)$ , $SL(n)$ and $U_{E/F}(3)$ , these played a fundamental roleboth in local harmonic analysis and in the description of discrete automorphic spectrum.
In [KK], we determined a system of candidates for the A-packets of the quasisplit unitarygroup $G^{*}(F)$ in 4 variables, by means of the global condition imposed on them. Here we
examine a similar construction for its inner form $G(F)$ .
$\mathcal{L}_{F}$$\hat{G}$$\hat{G}$

$G$

$A$

$\Pi_{\psi}(G)\subset\Pi_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}}(G(F))$

The A-parameter

3.1

which we described in
is an inner form of . the set $\Psi(G)$ coincides with[
, Prop. 3.2]. (Precisely speaking, there should be also certain “relevance conditions”. But the parameteron -parameter , so that $\Psi(G)$ might be a proper subset ofwhich we consider below certainly occurs in I $((;).)$ Here let us consider the parameterof type (G.2.b.ii) in [ $loc.cit.,$ \S j 3.3]:

Since

$G$

$\Psi(G^{*})$$G^{*}$$\mathrm{K}\mathrm{K}$$A$$\Psi(G^{*})$$’\psi|_{\mathcal{L}_{E}’ SL(2)}=$$((\eta \ \rho_{2,SL(2)})\mathrm{e}(\eta’\otimes\rho_{2.SU(2)}))\cross p_{W_{E}}$,$\psi(w_{\sigma})=(1_{2}+^{1_{2}})n$$w_{\sigma}$) is the two dimensional standard representation. Here) (resp.
with$SL(2)$$SU(2)$. Also we have fixed) and(resp.
denotes the projectionof. In the following} we shall construct a candidate of the A-packet
of associated to this parameter.
Associated to this, we have a non-tempered Langlands parameter [Art89, fi 4]$\eta’\neq\eta$$\rho_{2,SU}(2)$$\mathrm{p}2,\mathrm{s}\mathrm{u}2$$\mathcal{L}_{E}arrow W_{E}$$p_{W_{E}}$$\Pi_{\psi}(G)$$w_{\sigma}\in W_{F}\backslash W_{E}$$G$$\phi_{\psi}|_{C_{E}}=(\eta[1/2]\mathit{3})$and the corresponding, we know that$\Pi_{\phi_{\psi}}(G)$$L$$\psi$$\eta[-1/2]$

EC

Since
occurs in $\Psi(G)$ . On the other hand, the 5-gr0up-packet$\Pi_{\phi_{\psi}}(G)=\{J_{P}^{G}(\eta[1/2] \ \eta_{G_{2}}’)\}$$S_{\psi}(G):=S_{\psi}(G)/S_{\psi}(G)^{0}Z(\hat{G})^{\Gamma}$

is isomorphic to
for thisshould be a pairing,$\rangle=\langle$.

$5_{7}(G):=$.

Cent(Q,$\Pi_{\psi}(G)$$\hat{G}$contains)

Prop. 3.7]. According to the conjecture, there$\mathbb{Z}/2\mathbb{Z}[\mathrm{K}\mathrm{K}]$$\psi$$\langle$.

$(\eta’\mathrm{C}\otimes\rho_{2,SU(2)}))\cross p_{W_{E}}$$|J_{P}^{G}(\mathrm{Q}[1/2]_{E}\ \mathrm{r}7_{G}’\sim\tau))$$)$:$S_{\psi}(G)\cross\Pi_{\psi}(G)arrow \mathbb{C}^{1}$.

has another member.
We do not know if this is perfect. But if such is the case,
Actually, we shall construct a candidate for that representation in what follows.
$\Pi_{\psi}(G)$3.2

Local theta correspondence

Let $(V$, (: )be the 2-dimensional hermitian space over $E$ \S 2.1. Fix a$F$$E$. Then any 4-dimensional skew-hermitian
generator of over such that
, wherespace over $E$ is of the form it , , )$)$$=(V_{2}^{\pm}, (, )_{2}^{\pm})$$\triangle:=\delta^{2}\in F^{\cross}$$\delta$

$\pm$

$($

$\mathbb{H}=$$\langle$$:=W_{2}^{\pm}\oplus \mathbb{H}$$)^{\pm}$$(W_{2}^{+}, \langle\cdot, \cdot\rangle_{2}^{+}):=(E^{2}\backslash (_{-1}^{0} 01))$,$(W_{2}^{-}, \langle\cdot, \cdot\rangle_{2}^{-}):=$

11$(E^{2}, \langle\cdot, \cdot\rangle \mathrm{j} =\delta\cdot\sigma((\cdot, \cdot)_{2}^{-}))$.

128
Let $W$ be any 2 or 4-dimensional skew-hermitian space over $E$ . We have the correand W. respectively. We introduce thesponding unitary groups $G(V)$ and $G(W)$ forspace:
symplectic
16-dirnensional$V$$(\mathrm{W}, \langle\langle, \rangle\rangle)$$\mathrm{W}:=V\otimes_{E}W$

and writein:$Sp(\mathrm{W})$,$\langle\langle$

f7 $\otimes w$ ,$v’\otimes w’\rangle\rangle$$:= \frac{1}{2}\mathrm{T}\mathrm{r}_{E/F}[(v, v’)\sigma(\langle w, \mathrm{f}\mathrm{f}’))]$,

for its symplectic group. Then $(G(V), G(W))$ is a dual reductive pair

$Sp(\mathrm{W})$$\iota_{V,W}=\iota_{W}\cross\iota_{V}$: $G(V)\cross G(W)\ni(_{\backslash }g, g’)\mapsto g\otimes g’\in$ Sp(W).

Fixing a non-trivial charactera central extension$\psi_{F}$of $F$ , we have the metaplectic group of$1arrow \mathbb{C}^{1}arrow Mp_{\psi_{F}}(\mathrm{W})arrow$

This admits a unique Weil representation
multiplication [RR93] . For a character of:oflifting$E^{\mathrm{x}}$$\eta$$G(V)\cross G(W)arrow\lambda Ip_{\psi_{F}}(\mathrm{W})$which is.

on which the subgroup
acts by the, there corresponds awith$\mathbb{C}^{1}$$\omega\psi_{F}$$\mathrm{i}_{V,W,\eta}$$arrow 1$$\mathrm{S}\mathrm{G}(\mathrm{W})$$Sp(\mathrm{W})$$\eta|_{F}\cross=1_{F^{\cross}}$$\iota_{V,W}$:$G(V)\cross G(W)arrow\overline{\iota}_{V,W,\eta}Mp_{\psi_{F}}(\mathrm{W})$$||$$\downarrow$$G(V)\cross G(\uparrow V)arrow\iota_{V,W}$$Sp(\mathrm{W})$is the Weil representation of
associated
The composite,$G(V)$,, $G(W)$ ,for its restriction toto 7, a smooth representation. We write
respectively.
for the set of isomorphism classes of irreducible admissible
We write. We denote
representations of $G(V)$ which appear as quotients ofa realization,
quotients.
forthemaximalof
we writeofForon which $G(V)$ acts by some copies of . Thus we have smooth representationsof $G(W)$ such that
$G(V)\cross C_{\gamma}(F)$$\ )_{V.w_{\eta}=\omega\psi_{F}\circ\tilde{\iota}_{V.W,\eta}}$$\omega_{W,1}$$\omega_{V,\eta}$$\mathscr{B}(G(V), \omega_{W,1})$$S_{V,W,\eta}$$\omega_{W,1}$$\omega_{V,\nu V,\eta}$$S_{V,W,\eta}(\tau_{V})$$\tau_{V}\in \mathscr{B}(G(V), \omega_{\mathrm{T}\Psi,1}.)$$S_{V,W,\eta}$$\ominus_{\eta}(\tau_{V}, \mathrm{Y} )$$\mathrm{I}_{V}$$S_{V,W,\eta}(\tau_{V})\simeq\tau_{V}\otimes\ominus_{\eta}(\tau_{V}, W)$.

. Now the local
andfor
Similarly, we have
Howe duality conjecture, proved by Waldspurger if the residual characteristic of is odd,
asserts the following.
$\ominus_{\eta}(\tau_{W}, V)$$\mathrm{J}(G(W)_{:}\mathrm{u}_{V,\eta})$$\tau_{W}\in$$\mathrm{V}(G(W), \omega_{V,\eta})$$F$$\circ\ominus_{\eta}(\tau_{V}, W)$(resp.

$\ominus_{\eta}$($\tau_{W}$, $V$ )) is an admissible representation of finite length.

( , )) admits a unique maximal submodule and hence a(resp. ( , $V$ )) .
unique irreducible quotient$\circ\ominus_{\eta}(\tau_{V}, W)$(resp.

$\mathrm{O}-_{\eta}$$\tau_{W}$$V$$\theta_{\eta}(\tau_{V}, \mathrm{I}4/)$$\circ\tau_{V}\mapsto\theta_{\eta}(\tau_{V}, W)$$\mathrm{f}(G(V),\omega_{W,1})$

and
and

$\tau_{W}\mapsto\theta,,(\tau_{W}, V)$$\mathscr{B}(G(W), \mathrm{w}’))$$\theta_{\eta}$$\tau_{W}$are bijections converse to each other between.

Moreover, when $\dim_{E}W=2$ we have the following. Notice that $G(V_{2}^{+})\simeq G(W_{2}^{+})\simeq$.
and$G_{2}^{*}(F)$$G(V_{2}^{-})$$\simeq G(W_{-}^{-},)$$\simeq G_{2}(F)$

12

130
.
Theorem 3.1 ( -dichotomy [KK], Th. 5.4). We write. For a 2-dimensional hermitian space(i) Letandbe an -packet ofover $E$ ,if and only if$\epsilon(V_{\pm}):=$$\epsilon$$L$$\Pi^{*}$$\mathrm{f}1$$\tau^{*}\in\Pi^{*}$$G_{2}^{*}(F)$$V$$\mathrm{O}-_{\eta}(\tau^{*}, V)\neq 0$$\epsilon(1/2, \Pi^{*}\cross\eta^{-1} :.

\psi_{F})\omega_{\mathrm{n}}*(-1)\lambda(E/F, \psi_{F})^{-2}=\epsilon(V)$Heredenotes the standard -factor fortrnisted by
the Langlands-Shahidi theory [Sha90],
is the central character ofis Langlands ’ -factor $/Lan$ 70J.
(i) If this is the case, we have a bisection$\epsilon(s, \Pi^{*}\cross\omega, \psi_{F})$$\Pi^{*}$

$\in$

(3.1)

$\omega\in\Pi(E^{\mathrm{x}})$$\tau^{*}$$\omega_{\mathrm{n}*}$anddefined by$\lambda(E/F, \psi_{F})$$\lambda$$\eta_{G}\Pi^{*,\mathrm{V}}$if$\eta_{G}\Pi^{\vee}$otherwise,$\Pi^{*}\ni\tau^{*}\mapsto\ominus_{\eta}(\tau^{*}.V)\in\{$

where

$\Pi$

3.3

is the Jacquet-Langlands correspondent of$\Pi^{*}$$\epsilon(V)=1,$.

A candidate for the A-packet

Now we are in position to determine the partner ofTh. 3.1 to. We know from

$J_{P}^{G}(\eta[1/2]\otimes\eta_{G_{2}}’)$$[\mathrm{K}\mathrm{K}, \mathit{3}i6.1.2(2.\mathrm{b}.\mathrm{i}\mathrm{i})]$$\tau=\eta_{G_{2}^{*}}’\delta^{G_{2}^{*}}$$\acute{\circ}(1/2, ?]_{G_{2}^{*\delta^{G_{2}^{*}}}}’\cross\eta^{-1},$$p_{F})$. First we apply

that

A $(E/F, \psi_{F})^{-2}=1.$and hence,$\ominus_{\eta}(\eta_{G_{2}^{*}}’\delta^{G_{2}^{*}}, V_{2}^{+})=(\eta\overline{\eta}’)_{G_{2}^{\mathrm{s}}}\delta^{G_{2}^{*}}$$\mathrm{O}-_{\eta}$(’$7_{G_{2}^{*\delta^{G_{2}^{*}},V_{2}^{-})}}’=\{0\}$

.

(3.2)

Then this determines the local &-correspondent of. In fact, thedichotomy property of local Howe duality for supercuspidal representations of generalunitary groups over padic fields is established in [HKS96]. This means that exactly one$\neq\{0\}$ , it must beofis non zero. If
since theHowe duality is a correspondence. This contradicts (3.2) so that we have$(\eta\overline{\eta}’)_{G_{2}}\in\Pi_{0}(G_{2}(F))$$\ominus_{\eta}((\eta\overline{\eta}’)_{G_{2}}, W_{2}^{\pm})$$\ominus_{\eta}((’/\eta-’)_{G_{2)}}\mathrm{L}V_{2}^{+})$$\Theta_{\eta}((\eta\overline{\eta}’)_{G_{2’}}W_{2}^{+})=\{0\}$,$\eta_{G_{2}^{*}}’\delta^{G_{2}^{*}}$$\Theta_{\eta}((\eta\overline{\eta}’)_{G_{2}}, W_{2}^{-})=\eta_{G_{2}}’$.

Let us calculate. For this, we use the induction principle for local-correspondence [MVW87, Ch.3 IV.4]. This tells us:$\theta_{\eta}((\eta\overline{\eta}’)_{G_{2}}, W^{-})$$\theta$(i) Since$\ominus_{\eta}((\eta\overline{\eta}’)_{G_{2}}, W_{2}^{-})$

is non-zero, so is$\ominus_{\eta}((’/\overline{\eta}’)G_{2}, W^{-})$(ii) Its Jacquet module along $P$ is $\eta[-1/2]$ (&Sincemodule$J_{P}^{G}(\eta[1/2]\otimes\eta_{G_{2}}’)$$\eta[-1/2]\otimes\eta_{G_{2}}’$$\eta_{G_{A}}’$;.

is the unique irreducible representation of $G(F)$ with the Jacquet
along , we see$P$$\ominus((’/\overline{\eta}’)G_{2} , \mathrm{Y}V^{-})$$\simeq J_{P}^{G}(\eta[1/2]\otimes\eta_{Gr2}’)$.

To construct another element in, we consider the Jacquet-Langlands correspon$=\{0\}(3.2)$ , one can show thatdent
. Althoughof(( )c;JG) is non-trivial by a global argument. This must be the desired representation. On the other hand, the induction principle cited above asserts the following.
$\Pi_{\psi}(G)$$(\eta\overline{\eta}’)_{G_{2}}$$(\eta\overline{\eta}’)_{G_{2}^{*}}\delta^{G_{2}^{*}}$$’$$\ominus_{\eta}$$2*,$$7/\mathrm{j}$$\ominus_{r/}((\eta\overline{\eta}’)G2*\delta^{G_{\dot{2}}}, W_{2}^{-})$$W^{-}$

13

131
is an admissible representation of finite length, so that thisadmits an irreducible quotient (but not necessarily unique).

(a)$\ominus_{\eta}((’/\eta-’)G2*\mathrm{e}5^{G_{2}^{*}}, W^{-})$

in the realization (2.1). The(b-ii) Let
be the lower triangular Borel subgroup of. Then any irreducible quotient
Jacquet module ofalong 2 isis a quotient of ( $\eta’[-1/2]\otimes$ ’7c.2).
of$\overline{B}_{2}$$G_{2}^{*}$$\overline{B}$$\eta\overline{\eta}’[-1/2]$$(\eta\overline{\eta}’)_{G_{2}^{*}}\delta^{G_{2}^{*}}$$I_{P}^{G}$$\ominus_{\eta}((\eta\overline{\eta}’)_{G_{2}}*\delta^{G_{2}^{*}}, W^{-})$

We know from the classification Cor. 2.6 that
. We concludeof$\delta^{G}(\eta’, \eta_{G_{2}})$is the unique irreducible quotient$I_{P}^{G}(\eta’[-1/2]\otimes\eta c_{2})$$\theta_{\eta}((\eta\overline{\eta}’)_{G_{2}^{*}}\delta^{G_{2}^{*}}, W^{-})\simeq\delta^{G}(’ 7’, \eta c_{2})$and set$\Pi_{\psi}(G):=\{J_{P}^{G}(\eta[1/2]\otimes\eta_{C\mathrm{v}}’2), \delta^{G}(\eta’, \eta_{G_{2}})\}$.

Our construction can be summarized as the following figure.

7

71

2

$\mathrm{T}\mathrm{I}^{r}$$\mathrm{l}V$

Figure 1: Construction of the A-packet

Remark 3.2. By transposing$7c_{2})$,$\delta^{G}(\eta, \eta_{G_{2}}’)\}$$\psi’|$$(\eta, \eta’)$, we have another$A$-packetfor

A $E=$ $(\eta’\otimes\rho_{2,SL(2)})\oplus(\eta\otimes\rho_{2,SU()}\underline’)$Also $t/zere$ is a discrete$L$-packet$?^{A|_{L_{E}}=(\eta \mathrm{B}}$$\psi’(w_{\sigma})=(_{1_{2}}+^{1_{2}})\mathrm{x}$$\Pi_{\varphi}(G)=\{\delta^{G}(\eta, \eta_{G_{2}}’), \delta^{G}(\eta’, \eta_{G_{2}})\}$$\eta’)\otimes\rho_{2,SU(2)}$,$\mathit{6}.\mathit{1}.\mathit{2}]$$(\mathit{2}.b.ii)$$G^{*}$:,$\Pi_{\psi}(G^{*})=\{J_{P^{*}}^{G^{*}}(\eta[1/2]\otimes\eta_{G_{2}^{*}}’\delta^{G_{2}^{*}}), \pi_{c}\}$$\Pi_{\varphi}(G^{*})=\{\delta^{G}(\eta, \eta’), \pi_{\mathrm{c}}\}$

14

.

.

They were given in.

$\Pi_{\psi’}(G^{*})=\{J_{P^{*}}^{G^{*}}(\eta’[1/2]\otimes\eta_{G_{2}^{*}}\delta^{G_{2}^{*}}), \pi_{\mathrm{c}}\}$$w_{\sigma}$for$\varphi(w_{\sigma})=(_{1_{2}}+^{1_{\underline{9}}})\mathrm{x}lw_{\sigma}$

Let us compare these with the corresponding packets for$\mathit{3}i$$\Pi_{\psi’}(G)=\{J_{P}^{G}(\eta’[1/2]\mathrm{S}$,$/KK_{f}$132
is an iniucible supercuspidal representation. Notice thatin the-case. Thusin the -case but$G^{*}(F)$investigate.
theinteresting
be
to
It
will
appears
supercuspidal representationforcharacter relations between these packets. We would like to retur$m$ this question in someHere$\pi_{c}:=\Theta_{\eta}((\eta\overline{\eta}’)_{G(V_{-})}, \mathrm{Y}^{+}.)$$\delta^{G}(\eta, \eta_{G_{2}}’)$

’

’

’

$\not\simeq\delta^{G}$$G$$(\eta’, \eta c_{2})$$\delta^{G}$$(\eta, \eta’)\simeq\delta^{G}$$G^{*}$$(\eta’, \eta)$$a$$\pi_{c}$near future.

References[Art89]James Arthur, Unipotent automorphic representations: conjectures, Ast\’erisque(1989), no. 171-172, 13-71, Orbites unipotentes et repr\’esentations, II. MR
91f:22030[BOr79]

A. Borel, Automorphic -functions, Automorphic forms, representations and-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore.,
1977), Part 2, Amer. Math. Soc, Providence, R.I., 1979, pp. 27-61. MR
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