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%  Author Package
%% Translation via Omnimark script a2l, April 6, 2004 (all in one day!)
\controldates{21-APR-2004,21-APR-2004,21-APR-2004,21-APR-2004}
 
\RequirePackage[warning,log]{snapshot}
\documentclass{era-l}
\issueinfo{10}{05}{}{2004}
\dateposted{April 23, 2004}
\pagespan{39}{50}
\PII{S 1079-6762(04)00128-3}
\usepackage{amssymb}

\newcommand{\ulambda}{\pmb {\lambda }}
\newcommand{\into}{\hookrightarrow }
\newcommand{\LG}{{}^LG}
\newcommand{\LH}{{}^LH}
\newcommand{\wh}{\widehat }
\newcommand{\A}{\mathbb{A}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\G}{\mathbb{G}}
\newcommand{\1}{\boldsymbol{1}}
\newcommand{\uB}{\mathbf{B}}
\newcommand{\uC}{\mathbf{C}}
\newcommand{\uG}{\mathbf{G}}
\newcommand{\uH}{\mathbf{H}}
\newcommand{\uN}{\mathbf{N}}
\newcommand{\uT}{\mathbf{T}}
\newcommand{\GSp}{\operatorname{GSp}}
\newcommand{\Int}{\operatorname{Int}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\PGL}{\operatorname{PGL}}
\newcommand{\PGSp}{\operatorname{PGSp}}
\newcommand{\PSp}{\operatorname{PSp}}
\newcommand{\PSL}{\operatorname{PSL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\SO}{\operatorname{SO}}
\newcommand{\mySp}{\operatorname{Sp}}
\newcommand{\mysp}{\operatorname{sp}}
\newcommand{\St}{\operatorname{St}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\diag}{\operatorname{diag}}

\theoremstyle{plain}
\newtheorem*{theorem1}{Local Theorem (PGSp(2) to PGL(4))}
\newtheorem*{theorem2}{Global Theorem${}^{\ast }$ (PGSp(2) to PGL(4))}

\begin{document}
\title{Automorphic forms on $\operatorname{PGSp}(2)$}
\author{Yuval Z. Flicker}
\address{Department of Mathematics, The Ohio State University,
231 W. 18th Ave., Columbus, OH 43210-1174}
\email{flicker@math.ohio-state.edu}
\thanks{Partially supported by a Lady Davis Visiting Professorship at the
Hebrew University, 2004, and Max-Planck-Institut f\"{u}r Mathematik,
Bonn, 2003.}
\subjclass[2000]{Primary 11F70; Secondary 22E50, 22E55, 22E45}
\copyrightinfo{2004}{American Mathematical Society}
\date{March 4, 2004}
\commby{David Kazhdan}
\keywords{Automorphic representations, symplectic group, liftings,
twisted endoscopy, packets, quasi-packets, multiplicity one, rigidity,
functoriality, twisted trace formula, character relations}
\begin{abstract}The theory of lifting of automorphic and admissible
representations is developed in a new case of great classical interest:
Siegel automorphic forms. The self-contragredient representations of PGL(4)
are determined as lifts of representations of either symplectic PGSp(2) or
orthogonal SO(4) rank two split groups. Our approach to the lifting uses
the global tool of the trace formula together with local results such as the
fundamental lemma. The lifting is stated in terms of character relations.
This permits us to introduce a definition of packets and quasi-packets of
representations of the projective symplectic group of similitudes PGSp(2),
and analyse the structure of all packets. All representations, not only
generic or tempered ones, are studied. Globally we obtain a multiplicity one
theorem for the discrete spectrum of the projective symplectic group PGSp(2),
a rigidity theorem for packets and quasi-packets, determine all counterexamples
to the naive Ramanujan conjecture, and compute the multiplicity of each member
in a packet or quasi-packet in the discrete spectrum. The lifting from SO(4)
to PGL(4) amounts to establishing a product of two representations of GL(2)
with central characters whose product is 1. The rigidity theorem for SO(4)
amounts to a strong rigidity statement for a pair of representations of
$\operatorname{GL}(2,\mathbb{A})$.
\end{abstract}
\maketitle

According to the ``principle of functoriality'', ``Galois'' representations
$\rho :L_{F}\to \LG $ of the hypothetical Langlands group $L_{F}$ of a global field
$F$ into the complex dual group $\LG $ of a reductive group $\uG $ over $F$
should parametrize ``packets'' of automorphic representations of the ad\`{e}le
group $\uG (\A )$. Thus a homomorphism $\lambda :\LH \to \LG $ of complex dual
groups should give rise to lifting of automorphic representations $\pi _{H}$ of
$\uH (\A )$ to those $\pi $ of $\uG (\A )$.

We report here on the work of \cite{F1}. It
proves the existence of the expected lifting of automorphic representations
of the projective symplectic group of similitudes $\uH =\PGSp (2)$ to those
on $\uG =\PGL (4)$. The image is the set of the self-contragredient
representations of PGL(4) which are not lifts of representations of the
rank two split orthogonal group SO(4).

The global lifting is defined by means of local lifting. We define the local
lifting in terms of character relations. This permits us to introduce a
definition of packets and quasi-packets of representations of PGSp(2) as the
sets of representations that occur in these relations. Our main local result
is that packets exist and partition the set of tempered representations.
We give a detailed description of the structure of packets (in Sect. 4).

Our global results include a detailed description
of the structure of the global packets and quasi-packets (the latter are
almost everywhere nontempered). We obtain a {\em multiplicity one theorem for
the discrete spectrum of} PGSp(2), a {\em rigidity theorem for packets and
quasi-packets}, determine all {\em counterexamples to the naive Ramanujan
conjecture}, and compute the {\em multiplicity of each member in a packet
or quasi-packet in the discrete spectrum}, in Sect. 5.

Interesting phenomena
of instability are described in Sect. 6. Sect. 7 discusses the special case
of generic representations. It serves as another introduction. The first
three sections are preparatory. They describe the relevant parts of abstract
functoriality in our case.

We also prove the lifting from SO(4) to PGL(4). This amounts to
establishing a product of two representations of GL(2) with
central characters whose product is 1. Our rigidity theorem for
SO(4) amounts to a strong rigidity statement for a pair of
representations of $\GL (2,\A )$; see \cite{F6}. Our method uses the
global tool of the trace formula and the local tool of the
fundamental lemma. We deal with all, not only generic or tempered,
representations.

For applications to decomposition of cohomology of Shimura
varieties see \cite{F7}.


\section*{1. Homomorphisms of dual groups }

Let $\uG $ be the projective general linear group $\PGL(4)=\PSL(4)$
over a number field $F$. Our initial purpose is to determine the
automorphic representations $\pi $
of $\uG (\A )$, $\A $ being the ring of ad\`{e}les of $F$, which are
self-contragredient: $\pi \simeq \check \pi $, equivalently
(\cite{BZ1}), $\theta $-invariant:
$\pi \simeq \,^{\theta }\!\pi $. Here $\theta $, $\theta (g)=J^{-1}{}^{t} g^{-1}J$,
is the involution defined by
${J=(\begin{smallmatrix}~0&w\\-w&0\end{smallmatrix})
, w=
(\begin{smallmatrix}0&1\\1&0\end{smallmatrix})
}$,
where $^{t}g$ denotes the transpose of $g\in \uG $, and ${}^{\theta }\pi (g)
=\pi (\theta (g))$. According to the principle of functoriality (\cite{Bo},
\cite{A}) these automorphic representations are essentially described by
representations of the Weil group $W_{F}$ of $F$ into the dual
group $\wh {G}=\SL (4,\C )$ of $\uG $ which are $\hat {\theta }$-invariant,
namely representations of $W_{F}$ into centralizers
$Z_{\wh {G}}(\hat {s}\hat {\theta })$ of $\Int (\hat {s})\hat {\theta }$ in $\wh {G}$.
Here $\hat {\theta }$ is the dual involution
$\hat {\theta }(\hat {g})=J^{-1}{}^{t}\hat {g}^{-1}J$, and $\hat {s}$ is a
semisimple element in $\wh {G}$. These centralizers are the duals of the twisted
(by $\hat {s}\hat {\theta })$ endoscopic groups (\cite{KS}).

A twisted endoscopic group is called elliptic if its dual is not
contained in a proper parabolic subgroup of $\wh {G}$. Representations
of nonelliptic endoscopic groups can be reduced by parabolic
induction to known ones of smaller rank groups. For our $\wh {G}$,
up to conjugacy the elliptic twisted endoscopic groups have as duals the
symplectic group $\wh {H}=Z_{\wh {G}}(\hat {\theta })=\mySp (2,\C )$ and
the special orthogonal group
$\wh {C}=Z_{\wh {G}}(\hat {s}\hat {\theta })=\text{``}\SO (4,\C )\text{"}=
\left \{g\in \SL (4,\C );g\hat {s}J^{t}g=\hat {s}J=
(\begin{smallmatrix}0&\omega \\ \omega ^{-1}&0\end{smallmatrix})
\right \}$,
of all $A\otimes B=(\begin{smallmatrix}aB&bB\\cB &dB\end{smallmatrix})
$; $\!\left (A=(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})
, B\right )\!\in (\GL (2,\C )
\times \GL (2,\C ))/\C ^{\times }$
which satisfy $\det A\cdot \det B=1$.
Here $z\in \C ^{\times }$ embeds as $(z,z^{-1})$, $\hat {s}=\diag (-1,1,-1,1)$
and $\omega ={(\begin{smallmatrix}0&-1\\1&~0\end{smallmatrix})
}$.

The group $\wh {H}$ is the dual group of the simple $F$-group
$\uH =\PSp (2)=\PGSp (2)$, the projective group of symplectic similitudes.
It is the quotient of the group GSp(2) of $(g,\ulambda )\in \GL (4)\times \G _{m}$
with ${}^{t}gJg=\ulambda J$, by its center
$\{(\ulambda ,\ulambda ^{2})\}\simeq \G _{m}$.
Since $\ulambda $ is uniquely determined by $g$
(we write $\ulambda =\ulambda (g)$),
we view GSp(2) as a subgroup of GL(4) and PGSp(2) of PGL(4).

The group $\wh {C}$ is the dual group of the special orthogonal group
$\uC =$``SO(4)'' of pairs $(g_{1},g_{2})\in (\GL (2)\times \GL (2))/\G _{m}$ with
$\det g_{1}=\det g_{2}$. Here $z\in \G _{m}$ embeds as the central element $(z,z)$.
Also we write $((\GL (2)\times \GL (2))/\GL (1))'$ for $\uC $, where the prime
indicates that the two factors in GL(2) have equal determinants, and
$(\dots )''$ if their product is 1.

The principle of functoriality suggests that automorphic
discrete spectrum representations of $\uH (\A )$ and $\uC (\A )$
parametrize (or lift to) the $\theta $-invariant automorphic
discrete spectrum representations of the group $\uG (\A )$ of
$\A $-valued points of $\uG $. Our main purpose is to describe
this parametrization, in particular define tensor products of
two automorphic forms of $\GL (2,\mathbb{A})$ the product of
whose central characters is 1, and especially describe the
automorphic representations of the projective symplectic
group of similitudes of rank two, $\PGSp (2,\A )$, in terms of
$\theta $-invariant representations of $\PGL (4,\A )$.

\section*{2. Unramified lifting}

We proceed to explain how the liftings are defined, first for
unramified representations.

An irreducible admissible representation $\pi $ of an ad\`{e}le group $\uG (\A )$
is the restricted tensor product $\otimes \pi _{v}$ of irreducible
admissible (\cite{BZ1}) representations $\pi _{v}$ of the groups $\uG (F_{v})$
of $F_{v}$-points of $\uG $, where $F_{v}$ is the completion of $F$
at the place $v$ of $F$. Almost all the local components $\pi _{v}$
are unramified, that is contain a (unique up to a scalar
multiple) nonzero $K_{v}$-fixed vector. Here $K_{v}$ is the standard maximal
compact subgroup of $\uG (F_{v})$, namely the group $\uG (R_{v})$ of
$R_{v}$-points, $R_{v}$ being the ring of integers of the nonarchimedean
local field $F_{v}$; $\uG $ is defined over $R_{v}$ at almost all $v$. For
such $v$, an irreducible unramified $\uG (F_{v})$-module $\pi _{v}$ is the
unique unramified irreducible constituent in an unramified principal
series representation $I(\eta _{v})$, normalizedly induced (\cite{BZ2})
from an unramified character $\eta _{v}$ of the maximal torus $\uT (F_{v})$
of a Borel subgroup $\uB (F_{v})$ of $\uG (F_{v})$ (extended trivially
to the unipotent radical $\uN (F_{v})$ of $\uB (F_{v}))$. The space of
$I(\eta _{v})$ consists of the smooth functions $\phi :\uG (F_{v})\to \C $ with
$\phi (ank)=(\delta ^{1/2}_{v}\eta _{v})(a)\phi (k)$, $k\in K_{v}$,
$n\in \uN (F_{v})$, $a\in \uT (F_{v}),$
$\delta _{v}(a)=\det [\Ad (a)\vert \Lie \uN (F_{v})]$, and the $\uG (F_{v})$-action
is $(g\cdot \phi )(h)=\phi (hg)$, $g, h\in \uG (F_{v})$.

The character $\eta _{v}$ is unramified, thus it factorizes as
$\eta _{v}:\uT (F_{v})/\uT (R_{v})\to \C ^{\times }$. As $\uT (F_{v})/\uT (R_{v})
\simeq X_{\ast }(\uT )=\Hom (\G _{m},\uT )$, $\eta _{v}$ lies in $\Hom (X_{\ast }(\uT ),
\C ^{\times })=\Hom (X^{\ast }(\wh {T}),\C ^{\times })$, where $\wh {T}$ is the
maximal torus in the Borel subgroup $\wh {B}$ of $\wh {G}$, both fixed
in the definition of the (complex) dual group $\wh {G}$ (\cite{Bo}, \cite{Ko}).
Now $\Hom (X^{\ast }(\wh {T}),\C ^{\times })=X_{\ast }(\wh {T})\otimes \C ^{\times }=\wh {T}\subset \wh {G}$, thus the unramified irreducible
$\uG (F_{v})$-module $\pi _{v}$ determines a conjugacy class $t(\pi _{v})
=t(I(\eta _{v}))$ (the ``Langlands parameter'') in $\wh {G}$, represented
by the image of $\eta _{v}$ in $\wh {T}$.

For lack of space here we shall describe elsewhere (see \cite{F6}) our
results on our secondary lifting $\lambda _{1}$, to $\uG =\PGL (4)$
from $\uC =\SO (4)=((\GL (2)\times \GL (2))/\GL (1))'$.

\section*{3. The lifting $\lambda $ from PGSp(2) to PGL(4)}

We now turn to the study of our main lifting $\lambda $, and of
the automorphic representations of the $F$-group $\uH =\PGSp (2)=\GSp (2)/\G _{m}$.
The center $\G _{m}$ of
$\GSp (2)=\{g\in \GL (4);\,{}^{t}gJg=\ulambda J,
\,\exists \ulambda =\ulambda (g)\in \G _{m}\}$
consists of the scalar matrices. Its dual group is
$\wh {H}=\mySp (2,\C )=Z_{\wh {G}}(\wh {\theta })\subset \wh {G}=\SL (4,\C )$,
where $\wh {\theta }(g)=J^{-1}{}^{t}g^{-1}J$. It has a single
elliptic endoscopic group $\uC _{0}$ different from $\uH $ itself.
Thus
\begin{equation*}\wh {C}_{0}=Z_{\wh {H}}(\wh {s}_{0})=\left
\{\left(\begin{matrix}a&0&0&b\\0&\alpha &\beta &0\\0&\gamma &\delta
&0\\c&0&0&d\end{matrix}\right)
\in \wh {H}\right \}\simeq \SL (2,\C )\times \SL (2,\C ),\end{equation*}
where $\wh {s}_{0}=\diag (-1,1,1,-1)$, and $\uC _{0}=\PGL (2)\times \PGL (2)$.
Write $\lambda _{0}$ for the embedding $\wh {C}_{0}\into \wh {H}$,
and $\lambda $ for the embedding $\wh {H}\into \wh {G}$.

The embedding $\lambda _{0}:\wh {C}_{0}=\SL (2,\C )\times \SL (2,\C )\into \wh {H}
=\mySp (2,\C )$ defines the ``endoscopic'' lifting
$\lambda _{0}:\pi _{2}(\mu _{1},\mu _{1}^{-1})\times \pi _{2}
(\mu _{2},\mu _{2}^{-1})\mapsto \pi _{\PGSp (2)}(\mu _{1},\mu _{2}).$
Here $\pi _{2}(\mu _{i},\mu _{i}^{-1})$ is the unramified irreducible
constituent of the normalizedly induced $\PGL (2,F_{v})$-module
$I(\mu _{i},\mu _{i}^{-1})$ ($\mu _{i}$ are unramified characters of
$F_{v}^{\times }$, $i=1,2$); $\pi _{\PGSp (2)}(\mu _{1},\mu _{2})$ is the
unramified irreducible constituent of the $\PGSp (2,F_{v})$-module
$I_{\PGSp (2)}(\mu _{1},\mu _{2})$ normalizedly induced from the
character $n\cdot \diag (\alpha ,\beta ,\gamma ,\delta )\mapsto \mu _{1}
(\alpha /\gamma )\mu _{2}(\alpha /\beta )$ of the upper triangular
subgroup of $\PGSp (2,F_{v})$ ($n$ is in the unipotent radical,
$\alpha \delta =\beta \gamma )$.

The embedding $\lambda :\wh {H}=\mySp (2,\C )\into \SL (4,\C )=\wh {G}$
defines the lifting $\lambda $ which maps the unramified irreducible
$\PGSp (2,F_{v})$-module $\pi _{\PGSp (2)}(\mu _{1},\mu _{2})$ to the unramified
irreducible $\PGL (4,F_{v})$-module
$\pi _{4}(\mu _{1},\mu _{2},\mu _{2}^{-1},\mu _{1}^{-1})$.

The composition $\lambda \circ \lambda _{0}:\wh {C}_{0}=\SL (2,\C )\times \SL (2,\C )\to \wh {G}=\SL (4,\C )$ takes
$\pi _{2}(\mu _{1},\mu _{1}^{-1})\times \pi _{2}(\mu _{2},\mu _{2}^{-1})$ to
$\pi _{4}(\mu _{1},\mu _{2},\mu _{2}^{-1},\mu _{1}^{-1})
=\pi _{4}(\mu _{1},\mu _{1}^{-1},\mu _{2},\mu _{2}^{-1}),$
namely the unramified irreducible $\PGL (2,F_{v})\times \PGL (2,F_{v})$-module
$\pi _{2}\times \pi '_{2}$ to the unramified irreducible constituent
$\pi _{4}(\pi _{2},\pi '_{2})$ of the $\PGL (4,F_{v})$-module $I_{4}(\pi _{2},\pi '_{2})$
normalizedly induced from the representation $\pi _{2}\otimes \pi '_{2}$
of the parabolic of type $(2,2)$ of $\PGL (4,F_{v})$ (extended
trivially to the unipotent radical). For example
$\lambda \circ \lambda _{0}$ takes the trivial $\PGL (2,F_{v})\times \PGL (2,F_{v})$-module $\1_{2}\times \1_{2}$ to the unramified
irreducible constituent $\pi _{4}(\1_{2},\1_{2})$ of
$I_{4}(\1_{2},\1_{2})$, and $\1_{2}\times \pi _{2}$ to
$\pi _{4}(\1_{2},\pi _{2})=\pi _{4}(\nu ^{1/2}\pi _{2},\nu ^{-1/2}\pi _{2})$.
This last $\pi _{4}$ is traditionally denoted by $J$.

The definition of lifting is extended from the case of
unramified representations to that of any admissible
representations. For this purpose we defined in \cite{F5} norm
maps from the $\theta $-stable $\theta $-regular conjugacy
classes in $G=\uG (F)$ to stable conjugacy classes in $H=\uH (F)$,
and from these to conjugacy classes in $\uC _{0}(F)$. These
norm maps extend the norm maps on the split tori in these groups. 
The latter maps are dual
to the homomorphisms $\lambda $ and $\lambda _{0}$ of the dual groups.
This is used to define a relation of matching functions $f$,
$f_{H}$ and $f_{C_{0}}$ (they have suitably defined matching
orbital integrals) and a dual relation of liftings of representations.

To express the lifting results we use the following notations
for induced representations of $H=\PGSp (2,F)$. For characters
$\mu _{1}$, $\mu _{2}$, $\sigma $ of $F^{\times }$ with $\mu _{1}\mu _{2}
\sigma ^{2}=1$ we write $\mu _{1}\times \mu _{2}\rtimes \sigma $
for the $H$-module normalizedly induced from the character
$mu\mapsto \mu _{1}(a)\mu _{2}(b)\sigma (\ulambda )$, $u\in U$,
$m=\diag (a,b,\ulambda /b,\ulambda /a)$, $a,b,\ulambda \in F^{\times },$
of the upper triangular minimal parabolic of $H$.

For a $\GL (2,F)$-module $\pi _{2}$ and character $\mu $ we write
$\pi _{2}\rtimes \mu $ for the $\PGSp (2,F)$-module normalizedly
induced from the representation
$p=mu\mapsto \pi _{2}(g)\mu (\ulambda )$, $m=\diag (g,\ulambda w^{t}g^{-1}w)$,
$u\in U_{(2)}$, $\ulambda \in F^{\times }$
(here the product of the central character $\omega $ of
$\pi _{2}$ with $\mu ^{2}$ is 1) of the Siegel parabolic subgroup
(whose unipotent radical $U_{(2)}$ is abelian).

We write $\mu \rtimes \pi _{2}$, if $\omega \mu =1$, for the representation
of $\PGSp (2,F)$ normalizedly induced from the representation
$p=mu\mapsto \mu (a)\pi _{2}(g)$, $m=\diag (a,g,\ulambda (g)/a)$,
$u\in U_{(1)}$, $\ulambda (g)=\det g,$ of the Heisenberg parabolic
subgroup (whose unipotent radical $U_{(1)}$ is a Heisenberg group).

These inductions are normalized by multiplying the inducing
representation by the character $p\mapsto \vert \det (\Ad (p))\vert \Lie U\vert ^{1/2}$, as usual. For example, $I_{H}(\mu _{1},\mu _{2})=\mu _{1}\mu _{2}
\times \mu _{1}/\mu _{2}\rtimes \mu _{1}^{-1}$. Note that $\pi \rtimes \sigma \simeq \check \pi \rtimes \omega \sigma $ and $\mu (\pi \rtimes \sigma )
=\pi \rtimes \mu \sigma $. Complete results describing reducibility
of these induced representations are recorded in \cite{ST}---whose results
and notations we use---following earlier work of \cite{Ro2}, \cite{Sh2}, 
\cite{Sh3}
and \cite{W}. Our lifting results explain the results of \cite{ST}.

For properly induced representations, defining $\lambda $- and
$\lambda _{0}$-liftings by character relations $(\lambda (\pi _{H})=\pi _{4}$
if $\tr \pi _{4}(f\times \theta )=\tr \pi _{H}(f_{H})$ for all matching $f$,
$f_{H}$, and $\lambda _{0}(\pi _{1}\times \pi _{2})=\pi _{H}$ if $\tr \pi _{H}(f_{H})
=\tr (\pi _{1}\times \pi _{2})(f_{C_{0}})$ for all matching $f_{H}$, $f_{C_{0}}$),
our preliminary results (obtained by local character evaluations)
are that $\omega ^{-1}\rtimes \pi _{2}$ $\lambda $-lifts to $\pi _{4}=I_{G}
(\pi _{2},\check \pi _{2})$, that $\mu \pi _{2}\rtimes \mu ^{-1}$ (here
$\omega =1$) $\lambda $-lifts to $\pi _{4}=I_{G}(\mu ,\pi _{2},\mu ^{-1})$,
and that $I_{2}(\mu ,\mu ^{-1})\times \pi _{2}$ $\lambda _{0}$-lifts from
$C_{0}$ to $\mu \pi _{2}\rtimes \mu ^{-1}$ on $H=\PGSp (2,F)$.

Let $\chi $ be a character of $F^{\times }/F^{\times 2}$. It defines
a one-dimensional representation $\chi _{H}(h)=\chi (\ulambda (h))$ of
$H$, which $\lambda $-lifts to the one-dimensional representation
$\chi (g)=\chi (\det g)$ of $G$ (if $h=\!Ng$ then $\ulambda (h)=\det g$;
on diagonal matrices $N(\diag (a,b,c,d))\!=\diag (ab,ac,db,dc))$.
The Steinberg representation of $H$ $\lambda $-lifts to the
Steinberg representation of $G$, and for any character $\chi $ of
$F^{\times }$ with $\chi ^{2}=1$ we have $\lambda (\chi _{H}\St _{H})=\chi \St _{G}$.

\section*{4. Elliptic representations}

Our local liftings, $\lambda $, $\lambda _{0}$ and $\lambda _{1}$, are relations
of representations, defined by means of character relations.
Thus our finer local lifting results concern elliptic representations (whose
characters are nonzero on the set of elliptic elements). They follow on
using global techniques. Elliptic representations include the cuspidal ones
(terminology of \cite{BZ1}, \cite{BZ2}; these are called ``supercuspidal'' 
by Harish-Chandra,
who used the word ``cuspidal'' for what is currently named ``discrete series''
or ``square integrable'' representations).

\begin{theorem1} For any unordered
pair $\pi _{1}$, $\pi _{2}$ of square integrable irreducible representations
of $\PGL (2,F)$ there exists a unique pair $\pi ^{+}_{H}$, $\pi ^{-}_{H}$ of
tempered $($square integrable if $\pi _{1}\neq \pi _{2}$, cuspidal if
$\pi _{1}\neq \pi _{2}$ are cuspidal$)$ representations of $H$ with
\begin{equation*}\begin{split}\tr (\pi _{1}\times \pi _{2})(f_{C_{0}})&=\tr \pi ^{+}_{H}(f_{H})-\tr \pi ^{-}_{H}(f_{H}),\\
\tr I_{G}(\pi _{1},\pi _{2}; f\times \theta )&=\tr \pi ^{+}_{H}(f_{H})+\tr \pi ^{-}_{H}(f_{H})\end{split}\end{equation*}
for all triples $(f,f_{H},f_{C_{0}})$ of matching functions.

If $\pi _{1}=\pi _{2}$ is cuspidal, $\pi ^{+}_{H}$ and $\pi ^{-}_{H}$ are the two
inequivalent constituents of $1\rtimes \pi _{1}$.

If $\pi _{1}=\pi _{2}=\sigma \mysp _{2}$ where $\sigma $ is a character of $F^{\times }$
with $\sigma ^{2}=1$, then $\pi ^{+}_{H}$ and $\pi ^{-}_{H}$ are the two tempered
inequivalent constituents $\tau (\nu ^{1/2}\mysp _{2},\sigma \nu ^{-1/2})$ and
$\tau (\nu ^{1/2}\1_{2},\sigma \nu ^{-1/2})$ of $1\rtimes \sigma \mysp _{2}$.

If $\pi _{1}=\sigma \mysp _{2}$, $\sigma ^{2}=1$, and $\pi _{2}$ is cuspidal,
then $\pi ^{+}_{H}$ is the square integrable constituent
$\delta (\sigma \nu ^{1/2}\pi _{2},\sigma \nu ^{-1/2})$ of the induced
$\sigma \nu ^{1/2}\pi _{2}\rtimes \sigma \nu ^{-1/2};$ $\pi ^{-}_{H}$ is
cuspidal, denoted here by
$\delta ^{-}(\sigma \nu ^{1/2}\pi _{2},\sigma \nu ^{-1/2}).$

If $\pi _{1}=\sigma \mysp _{2}$ and $\pi _{2}=\xi \sigma \mysp _{2}$, $\xi $ $(\neq 1=\xi ^{2})$
and $\sigma $ $(\sigma ^{2}=1)$ are characters of $F^{\times }$, then $\pi ^{+}_{H}$
is the square integrable constituent
$\delta (\xi \nu ^{1/2}\mysp _{2},\sigma \nu ^{-1/2})$
of the induced $\xi \nu ^{1/2}\mysp _{2}\rtimes \sigma \nu ^{-1/2};$ $\pi ^{-}_{H}$ is
cuspidal, denoted here by $\delta ^{-}(\xi \nu ^{1/2}\mysp _{2},\sigma \nu ^{-1/2})$.

For every character $\sigma $ of $F^{\times }/F^{\times 2}$ and square
integrable $\pi _{2}$ there exists a nontempered representation
$\pi ^{\times }_{H}$ of $H$ such that for all matching $f$, $f_{H}$, $f_{C_{0}}$
\begin{equation*}\begin{split}\tr (\sigma \1_{2}\times \pi _{2})(f_{C_{0}})&=\tr \pi ^{\times }_{H}(f_{H})
+\tr \pi ^{-}_{H}(f_{H}),\\
\tr I_{G}(\sigma \1_{2},\pi _{2}; f\times \theta )&=\tr \pi ^{\times }_{H}(f_{H})
-\tr \pi ^{-}_{H}(f_{H}).\end{split}\end{equation*}
Here $\pi ^{-}_{H}=\pi ^{-}_{H}(\sigma \mysp _{2}\times \pi _{2})$ and
$\pi ^{\times }_{H}=L(\sigma \nu ^{1/2}\pi _{2},\sigma \nu ^{-1/2}).$

For any characters $\xi $, $\sigma $ of $F^{\times }/F^{\times 2}$
and matching $f$, $f_{H}$, $f_{C_{0}}$ we have
\begin{equation*}\begin{split}\tr (\sigma \xi \1_{2}\times \sigma \1_{2})(f_{C_{0}})
&=\tr L(\nu \xi ,\xi \rtimes \sigma \nu ^{-1/2})(f_{H})\!-\tr X(\xi \nu ^{1/2}\mysp _{2},
\xi \sigma \nu ^{-1/2})(f_{H}),\\
\tr I_{G}(\sigma \xi \1_{2},\sigma \1_{2}; f\times \theta )&=\tr L(\nu \xi ,\xi \rtimes \sigma \nu ^{-1/2})(f_{H})\!+\tr X(\xi \nu ^{1/2}\mysp _{2},\xi \sigma \nu ^{-1/2})
(f_{H}).\end{split}\end{equation*}
Here $X=\delta ^{-}$ if $\xi \neq 1$ and $X=L$ if $\xi =1$.

Any $\theta $-invariant irreducible square integrable representation
$\pi $ of $G$ which is not a $\lambda _{1}$-lift is a $\lambda $-lift
of an irreducible square integrable representation $\pi _{H}$ of $H$,
thus $\tr \pi (f\times \theta )=\tr \pi _{H}(f_{H})$ for all matching $f$,
$f_{H}$. In particular, the square integrable $($resp. nontempered$)$
constituent $\delta (\xi \nu ,\nu ^{-1/2}\pi _{2})$ $($resp. $L(\xi \nu ,
\nu ^{-1/2}\pi _{2}))$ of the induced representation $\xi \nu \rtimes \nu ^{-1/2}\pi _{2}$ of $H$, where $\pi _{2}$ is a cuspidal $($irreducible$)$
representation of $\GL (2,F)$ with central character $\xi \neq 1=\xi ^{2}$
and $\xi \pi _{2}=\pi _{2}$, $\lambda $-lifts to the square integrable
$($resp. nontempered$)$ constituent $S(\nu ^{1/2}\pi _{2},\nu ^{-1/2}\pi _{2})$
$($resp. $J(\nu ^{1/2}\pi _{2},\nu ^{-1/2}\pi _{2}))$ of the induced representation
$I_{G}(\nu ^{1/2}\pi _{2},\nu ^{-1/2}\pi _{2})$ of $G=\PGL (4,F)$.
\hfill $\square $\end{theorem1}


These character relations permit us to introduce the notion of a
packet of an irreducible representation, and of a quasi-packet,
over a local field. Thus we say that the {\em {packet}} of a
representation $\pi _{H}$ of $H$ consists of $\pi _{H}$ alone unless it
is tempered of the form $\pi ^{+}_{H}$ or $\pi ^{-}_{H}$ for some pair
$\pi _{1}$, $\pi _{2}$ of (irreducible) square integrable representations
of $\PGL (2,F)$, in which case the packet $\{\pi _{H}\}$ is defined to be
$\{\pi ^{+}_{H}$, $\pi ^{-}_{H}\}$, and we write $\lambda _{0}(\pi _{1}\times \pi _{2})
=\{\pi _{H}^{+},\pi ^{-}_{H}\}$ and $\lambda (\{\pi ^{+}_{H},\pi ^{-}_{H}\})=
I_{G}(\pi _{1},\pi _{2})$. Further, we define a {\em {quasi-packet}}
only for the nontempered (irreducible) representations $\pi ^{\times }_{H}$
and $L=L(\nu \xi , \xi \rtimes \sigma \nu ^{-1/2})$, to consist of
$\{\pi ^{\times }_{H}, \pi ^{-}_{H}\}$ and $\{L, X\}$,
$X=X(\xi \nu ^{1/2}\mysp _{2},\xi \sigma \nu ^{-1/2})$.
We say that $\sigma \1_{2}\times \pi _{2}$ $\lambda _{0}$-lifts to the quasi-packet
$\lambda _{0}(\sigma \1_{2}\times \pi _{2})=\{\pi ^{\times }_{H}$, $\pi ^{-}_{H}\}$,
which in turn $\lambda $-lifts to $I_{G}(\sigma \1_{2},\pi _{2})$, and
similarly, $\sigma \xi \1_{2}\times \sigma \1_{2}$ $\lambda _{0}$-lifts to
$\lambda _{0}(\sigma \xi \1_{2}\times \sigma \1_{2})=\{L,X\}$ which
$\lambda $-lifts to $I_{G}(\sigma \xi \1_{2},\sigma \1_{2})$.

\section*{5. Automorphic representations}

With these local definitions we can state our global results.
These global results are partial, since we work with test
functions whose components are elliptic at least at two places.
This constraint will be removed once the trace formulae identity is established
for general test functions. With further work, known techniques can reduce
the constraint to a single place. 

With this reservation, emphasized by a $\ast $-superscript in
the following Global Theorem, the discrete spectrum representations of
$\uH (\A )=\PGSp (2,\A )$ can now be described by means of the liftings.
They consist of two types, stable and unstable. Global packets and
quasi-packets define a partition of the spectrum. To define a (global)
[quasi-]packet $\{\pi _{H}\}$, fix a local [quasi-]packet $\{\pi _{Hv}\}$
at each place $v$ of $F$, such that $\{\pi _{Hv}\}$ contains an unramified
member $\pi ^{0}_{Hv}$ (and then $\{\pi _{Hv}\}$ consists only of $\pi ^{0}_{Hv}$
in case it is a packet) for almost all $v$. The [quasi-]{\em {packet}}
$\{\pi _{H}\}$ is then defined to consist of all products $\otimes _{v}\pi '_{Hv}$
with $\pi '_{Hv}$ in $\{\pi _{Hv}\}$ for all $v$, and $\pi '_{Hv}=\pi ^{0}_{Hv}$
for almost all $v$. The [quasi-]packet $\{\pi _{H}\}$ of an automorphic
representation $\pi _{H}$ is defined by the local [quasi-]packets
$\{\pi _{Hv}\}$ of the components $\pi _{Hv}$ of $\pi _{H}$ at almost all places.

The discrete spectrum of $\PGSp (2,\A )$ will be described by means of the
$\lambda _{0}$- and $\lambda $-liftings. We say that the discrete spectrum
representation
$\pi _{1}\times \pi _{2}$ $\lambda _{0}$-{\em {lifts}} to a packet $\{\pi _{H}\}$ (or to a
member thereof) if $\{\pi _{Hv}\}=\lambda _{0}(\pi _{1v}\times \pi _{2v})$ for
almost all $v$, and that a packet $\{\pi _{H}\}$ (or a member of it)
$\lambda $-{\em {lifts}} to an irreducible self-contragredient automorphic
representation $\pi $ if $\lambda (\{\pi _{Hv}\})=\pi _{v}$ for almost all $v$.
The {\em {unstable}} spectrum of $\PGSp (2,\A )$ is defined to be the set of
discrete spectrum representations which are $\lambda _{0}$-lifts; its complement
is the {\em {stable}} spectrum.

\begin{theorem2} The packets and
quasi-packets partition the discrete spectrum of the group $\PGSp (2,\A )$,
namely they satisfy the rigidity theorem$:$ if $\pi _{H}$ and $\pi '_{H}$ are
discrete spectrum representations locally equivalent at almost all places,
then their packets or quasi-packets are equal.

The $\lambda $-lifting is a bijection between the set of packets $($resp.
quasi-packets$)$ of discrete spectrum representations in the stable spectrum
$($of $\PGSp (2,\A ))$ and the set of self-contragredient discrete spectrum
representations of $\PGL (4,\A )$ which are one dimensional, or cuspidal
and not a $\lambda _{1}$-lift from $\uC (\A )$ $($resp.~residual
$J(\nu ^{1/2}\pi _{2},\nu ^{-1/2}\pi _{2})$ where $\pi _{2}$ is a cuspidal
representation of $\GL (2,\A )$ with central character $\xi \neq 1=\xi ^{2}$ and
$\xi \pi _{2}=\pi _{2})$.

The $\lambda _{0}$-lifting is a bijection between the set of pairs of
discrete spectrum representations $\{\pi _{1}\times \pi _{2},\pi _{2}\times \pi _{1};\pi _{1}\neq \pi _{2}\}$ of $\PGL (2,\A )\times \PGL (2,\A )$, and the
set of packets and quasi-packets in the unstable spectrum of
$\PGSp (2,\A )$. The $\lambda $-lifting is a bijection from this last
set to the set of automorphic representations $I_{G}(\pi _{1},\pi _{2})$ of
$\PGL (4,\A )$, normalizedly induced from the discrete spectrum representation
$\pi _{1}\times \pi _{2}$ $(\pi _{1}\neq \pi _{2})$ on the parabolic subgroup with Levi factor
of type $(2,2)$. If
$\pi _{1}\times \pi _{2}$ is cuspidal, its $\lambda _{0}$-lift is a 
packet\textup{;}
otherwise, a quasi-packet.

Each member of a stable packet occurs in the discrete spectrum of
$\PGSp (2,\A )$ with multiplicity one. The multiplicity $m(\pi _{H})$ of a
member $\pi _{H}=\otimes \pi _{Hv}$ of an unstable $[$quasi-$]$packet
$\lambda _{0}(\pi _{1}\times \pi _{2})$ $(\pi _{1}\neq \pi _{2})$ is not $($``stable", or$)$
constant over the $[$quasi-$]$packet. If $\pi _{1}\times \pi _{2}$ is cuspidal,
it is $m(\pi _{H})={\frac{1}{2}}(1+(-1)^{n(\pi _{H})})\ (\in \{0,1\}),$
where $n(\pi _{H})$ is the number of components $\pi ^{-}_{Hv}$ of $\pi _{H}$
and each $\pi _{H}$ with $m(\pi _{H})=1$ is cuspidal $(n(\pi _{H})$ is bounded
by the number of places $v$ where both $\pi _{1v}$ and $\pi _{2v}$
are square integrable$)$.

The multiplicity $m(\pi _{H})$ $($in the discrete spectrum of
$\PGSp (2,\A ))$ of $\pi _{H}=\otimes \pi _{Hv}$ of a quasi-packet
$\lambda _{0}(\sigma \1_{2}\times \pi _{2})$, where $\pi _{2}$ is a cuspidal
representation of $\PGL (2,\A )$ and $\sigma $ is a character of
$\A ^{\times }/F^{\times }\A ^{\times 2}$, is ${\frac{1}{2}}(1+\varepsilon (\sigma \pi _{2},{\frac{1}{2}})(-1)^{n(\pi _{H})})$ $(=0$ or $1)$,
where $n(\pi _{H})$ is the number of components $\pi ^{-}_{Hv}$ of $\pi _{H}$,
and $\varepsilon (\sigma \pi _{2},s)$ is the usual $\varepsilon $-factor
which appears in the functional equation of the $L$-function
$L(\pi _{2},s)$. In particular $\pi ^{\times }_{H}=\otimes \pi ^{\times }_{Hv}$
$(n(\pi _{H})=0)$ is in the discrete spectrum if and only if
$\varepsilon (\sigma \pi _{2},{\frac{1}{2}})=1$.

Finally we have $m(\pi _{H})={\frac{1}{2}}(1+(-1)^{n(\pi _{H})})$ for
$\pi _{H}=\otimes \pi _{Hv}$ in $\lambda _{0}(\sigma \xi \1_{2}\times \sigma \1_{2})$
with $n(\pi _{H})$ components $\pi _{Hv}=X_{v}$. Here $\pi _{H}=\otimes L_{v}$
$(n(\pi _{H})=0)$ is residual.\hfill $\square $\end{theorem2}


\section*{6. Unstable spectrum}

Note that the quasi-packet $\lambda _{0}(\sigma \xi \1\times \sigma \1_{2})$ is
defined by the local quasi-packets $\{L_{v}=L(\nu _{v}\xi _{v},\xi _{v}\rtimes \sigma _{v}
\nu _{v}^{-1/2})$, $X_{v}=X(\xi _{v}\nu _{v}^{1/2}\mysp _{2v},\xi _{v}\sigma _{v}\nu _{v}^{-1/2})\}$
for every $v$, where $\xi $ $(\neq 1)$, $\sigma $ are characters of
$\A ^{\times }/F^{\times }$ with $\xi ^{2}=1=\sigma ^{2}$ and $\xi _{v}$, $\sigma _{v}$
are their components. When $\xi _{v}$, $\sigma _{v}$ are unramified, this
quasi-packet contains the unramified representation $\pi ^{0}_{Hv}=L_{v}$.
Members of this quasi-packet have been studied by means of the theta
correspondence by Howe and Piatetski-Shapiro; see, e.g., \cite{PS1}, Theorem 2.5.
They attracted interest since they violate the naive generalization of
the Ramanujan conjecture, which expects the components of a cuspidal
representation to be tempered. (The form of the Ramanujan conjecture
which is expected to be true asserts that the components of a cuspidal
representation of PGSp(2,$\A $) which $\lambda $-lifts to a cuspidal
representation of PGL(4,$\A $) are tempered.) Members of this quasi-packet
are equivalent at almost all places to the quotient of the properly
induced representation $\nu \xi \times \xi \rtimes \sigma \nu ^{-1/2}$.

Let $\pi _{2}$ be a cuspidal representation of $\PGL (2,\A )$ and $\sigma $
a character of $\A ^{\times }\!/F^{\times }\A ^{\times 2}\!$. The packet
$\lambda _{0}(\sigma \1_{2}\times \pi _{2})$ contains the constituent $\pi ^{\times }_{H}
=\otimes _{v}\pi ^{\times }_{Hv}$ of the representation
$\sigma \nu ^{1/2}\pi _{2}\rtimes \sigma \nu ^{-1/2}\simeq \sigma \nu ^{-1/2}\pi _{2}\rtimes \sigma \nu ^{1/2}$ properly induced from an
automorphic representation, hence it is automorphic by \cite{L}. It is known
that $\pi ^{\times }_{H}$ is residual precisely when
$L(\sigma \pi _{2},{\frac{1}{2}})\neq 0$;
hence $\varepsilon (\sigma \pi _{2},{\frac{1}{2}})=1$ in this case.

Let $n(\pi _{2})$ denote the number of square integrable components of $\pi _{2}$.
The quasi-packet $\lambda _{0}(\sigma \1_{2}\times \pi _{2})$ thus consists of
$2^{n(\pi _{2})}$ (irreducible) representations. If $n(\pi _{2})\ge 1$, half of
them are in the discrete spectrum, all cuspidal if $L(\sigma \pi _{2},{\frac{1}{2}})=0$,
and all but one: $\pi ^{\times }_{H}=\otimes _{v}\pi ^{\times }_{Hv}$, are cuspidal if
$L(\sigma \pi _{2},{\frac{1}{2}})\not =0$. If $n(\pi _{2})\ge 1$ and
$L(\sigma \pi _{2},{\frac{1}{2}})=0$, the automorphic nonresidual
$\pi ^{\times }_{H}$ is cuspidal when $\varepsilon (\sigma \pi _{2},{\frac{1}{2}})=1$.

If $\pi _{2}$ has no square integrable components ($n(\pi _{2})=0$), the packet
$\lambda _{0}(\sigma \1_{2}\times \pi _{2})$ consists only of $\pi ^{\times }_{H}$. This
$\pi ^{\times }_{H}$ is residual if $L(\sigma \pi _{2},{\frac{1}{2}})\not =0$; cuspidal
(by \cite{PS1}, Theorem 2.6, and \cite{PS2}, Theorem A.2) if
$L(\sigma \pi _{2},{\frac{1}{2}})=0$ and $\varepsilon (\sigma \pi _{2},{\frac{1}{2}})=1$;
or (automorphic but) not in the discrete spectrum otherwise:
$L(\sigma \pi _{2},{\frac{1}{2}})=0$ and $\varepsilon (\sigma \pi _{2},{\frac{1}{2}})=-1$.
In this last case the $\lambda _{0}$-lift of $\sigma \1_{2}\times \pi _{2}$ 
is not in the
discrete spectrum, and there is no discrete spectrum representation
$\lambda $-lifting to $I_{G}(\sigma \1_{2},\pi _{2})$.

At a place $v$ where $\pi _{2v}$ is induced $I(\mu _{v},\mu _{v}^{-1})$, the
packet $\pi _{Hv}=\lambda _{0}(\sigma _{v}\1_{2}\times \pi _{2v})$ is the
irreducible induced $\mu _{v}\sigma _{v}\1_{2}\rtimes \mu ^{-1}_{v}$, which
$\lambda $-lifts to $I_{G}(\mu _{v},\sigma _{v}\1_{2},\mu _{v}^{-1})$, and {\em not}
the irreducible induced
\begin{equation*}\sigma _{v}\mu _{v}\nu _{v}^{1/2}\times \sigma _{v}\mu _{v}^{-1}\nu _{v}^{1/2}\rtimes \sigma _{v}\nu _{v}^{-1/2}=\sigma _{v}\mu _{v}\nu _{v}^{1/2}\rtimes I(\mu _{v}^{-1},\sigma _{v}\nu _{v}^{-1/2}),\end{equation*}
which $\lambda $-lifts to the reducible induced
$I_{G}(\mu _{v},\sigma _{v} I(\nu _{v}^{1/2},\nu _{v}^{-1/2}),\mu _{v}^{-1})$, which has
the constituent $I_{G}(\mu _{v},\sigma _{v}\1_{2},\mu _{v}^{-1})$.

Members of the quasi-packet $\lambda _{0}(\sigma \1_{2}\times \pi _{2})$ were
studied numerically by H. Saito and N. Kurokawa, and using the theta
correspondence by Piatetski-Shapiro and others; see \cite{PS1}, Theorem 2.6.
They attracted interest since they violate the naive generalization of
the Ramanujan conjecture. They are equivalent at almost all places to
the quotient of the properly induced representation
$\sigma \nu ^{1/2}\pi _{2}\rtimes \sigma \nu ^{-1/2}$.

A discrete spectrum representation $\pi _{H}$ with component $L(\nu _{v}\xi _{v},
\nu _{v}^{-1/2}\pi _{2v})$ (whose packet consists of itself), where
$\pi _{2v}$ is a cuspidal representation with central character
$\xi _{v}\neq 1=\xi _{v}^{2}$ and $\xi _{v}\pi _{2v}=\pi _{2v}$, is in the packet
of $L(\nu \xi ,\nu ^{-1/2}\pi _{2})$. Here $\pi _{2}$ is cuspidal with
central character $\xi \neq 1=\xi ^{2}$, hence $\xi \pi _{2}=\pi _{2}$,
whose components at $v$ are $\pi _{2v}$ and $\xi _{v}$. It $\lambda $-lifts
to $J_{G}(\nu ^{1/2}\pi _{2},\nu ^{-1/2}\pi _{2})$. At $v$ with $\xi _{v}=1$
the component $\pi _{2v}$ is induced. If $\pi _{2v}=I(\mu _{v},\mu _{v}\xi _{v})$,
$\xi ^{2}_{v}=1$ and $\mu ^{2}_{v}=1$ (in particular whenever $\xi _{v}\neq 1$
and $\pi _{2v}$ is not cuspidal), then $L(\nu _{v}\xi _{v},\nu _{v}^{-1/2}
\pi _{2v})$ is $L_{v}=L(\nu _{v}\xi _{v},\xi _{v}\rtimes \mu _{v}\nu _{v}^{-1/2})$,
which $\lambda $-lifts to $I_{G}(\mu _{v}\1_{2},\mu _{v}\xi _{v}\1_{2})$. Its
packet contains also $X_{v}=X(\nu _{v}^{1/2}\xi _{v}\mysp _{2v},\xi _{v}
\mu _{v}\nu _{v}^{-1/2})$. Thus the packet of $\pi _{H}$ is determined by
$\{L_{v}, X_{v}\}$ at all $v$ where $\pi _{2v}=I(\mu _{v},\mu _{v}\xi _{v})$,
$\mu _{v}^{2}=1=\xi ^{2}_{v}$, and by the singleton $\{L_{v}=L(\nu _{v}\xi _{v},
\nu _{v}^{-1/2}\pi _{2v})\}$ at all other $v$, where $\pi _{2v}$ is cuspidal,
or $\xi _{v}=1$ and $\pi _{2v}=I(\mu _{v},\mu _{v}^{-1})$, $\mu ^{2}_{v}\neq 1$.
Each member of this infinite packet occurs in the discrete spectrum
with multiplicity one, and is cuspidal, with the exception of
$L(\nu \xi ,\nu ^{-1/2}\pi _{2})=\otimes _{v}L(\nu _{v}\xi _{v},\nu _{v}^{-1/2}
\pi _{2v})$, which is residual (\cite{Kim}, Theorem 7.2). Members of the
packet $L(\nu \xi ,\nu ^{-1/2}\pi _{2})$ are considered in the Appendix
of \cite{PS1} and its corrigendum.

If $\pi _{1}$ and $\pi _{2}$ are cuspidal but there is no place $v$ where
both are square integrable, $\lambda _{0}(\pi _{1}\times \pi _{2})$ consists
of a single irreducible cuspidal representation. This instance of
the lifting $\lambda _{0}$---where $\pi _{i}$ are cuspidal---can also be
studied (\cite{Rb}) using the theta correspondence for suitable dual reductive
pairs (SO(4), PGSp(2)) for the isotropic and anisotropic forms of the
orthogonal group, to describe further properties of the packets, such
as their periods.

\section*{7. Generic representations}

Our proof of the existence of the lifting $\lambda $ uses only
the trace formula, orbital integrals and character relations.
To compute the multiplicities in the discrete spectrum we use
a global proof. It relies on results of \cite{GRS}, \cite{KRS},
and \cite{Sh1} from the theory of generic representations.

This global proof in our case of PGSp(2)
is similar to the second proof of \cite[II]{F4}, Proposition 3.5, p. 48,
in the case of U(3). The 2nd proof of \cite[II]{F4} is also based on the
theory of generic representations. But it is not complete. Indeed,
the claim in Proposition 2.4(i) in reference [GP] to \cite[II]{F4} that
``$L^{2}_{0,1}$ has multiplicity 1'', is interpreted in \cite[II]{F4} as
asserting that generic representations of U(3) occur in the discrete
spectrum with multiplicity one. But it should be interpreted as asserting
that irreducible $\pi $ in $L^{2}_{0,1}$ have multiplicity one {\em only in
the subspace} $L^{2}_{0,1}$ of the discrete spectrum. This assertion does not
exclude the possibility of having a cuspidal $\pi '$ perpendicular and
equivalent to $\pi \subset L^{2}_{0,1}$. Multiplicity one for the generic
spectrum of U(3) could be deduced from Proposition 2.4(i) in [GP] of 
\cite[II]{F4} 
using the statement that a locally generic representation is globally generic.
(In fact this statement is equivalent to multiplicity one.)

In our case of PGSp(2) a similar statement follows from \cite{KRS}, \cite{GRS},
\cite{Sh1}.

A proof for U(3) (independent of the local proof of \cite[II]{F4}, p. 47)
still needs to be given.

The first proof of Proposition 3.5 in \cite[II]{F4}, p. 47, namely that the
multiplicity one theorem holds for the discrete spectrum of U(3), is
purely local. It uses only character relations. It is based on a twisted
analogue of Rodier's result \cite{Ro1} that the number of Whittaker models (0
or 1) is encoded in the behavior of the character near the origin.
For further details see \cite[IV]{F4}. Such a technique was first used in \cite{FK},
to prove the multiplicity one theorem for the metaplectic group of GL($n$).

There is some overlap between our results on the existence of the
$\lambda $-lifting and the work of \cite{GRS}. However, the methods of \cite{GRS} apply only to generic representations,
while our methods apply to all representations of PGSp(2). Thus
we can define packets, describe their structure, establish the multiplicity one
theorem and rigidity theorem for packets of PGSp(2), specify which member
in a packet or a quasi-packet is in the discrete spectrum, and we can also
$\lambda $-lift the nongeneric nontempered (at almost all places) packets
to residual self-contragredient representations of $\PGL (4,\A )$. Our
\pagebreak liftings are proven in terms of all places, not only almost all places.
In addition we establish the lifting $\lambda _{1}$ from SO(4) to PGL(4),
determine its fibers (that is, prove the multiplicity one theorem for SO(4)
and rigidity in the sense explained in \cite{F6} and \cite{F1}), and show that any
self-contragredient discrete spectrum representation of $\PGL (4,\A )$
which is not a $\lambda $-lift from $\PGSp (2,\A )$ is a $\lambda _{1}$-lift
from $\SO (4,\A )$.

Our work is an analogue for (SO(4), PGSp(2), PGL(4)) of \cite{F3}, which dealt
with the symmetric square lifting, thus with (PGL(2), SL(2), PGL(3)), and
of \cite{F4}, which dealt with quadratic base change for the unitary group
U(3,$E/F$), thus with (U(2,$E/F$), U(3,$E/F$), GL(3,$E$)). These works
use the twisted---by transpose-inverse (and the Galois action in the
unitary groups case)---trace formulae on PGL(4), PGL(3), GL(3,$E$).
They are based on the fundamental lemma: \cite{F5} in our case, 
\cite[V]{F3} and
\cite[I]{F4} in the other cases. The technique employed in these last papers
benefited from the work of Weissauer \cite{We} and Kazhdan \cite{K}.

The present
work, which deals with the applications of the fundamental lemma and
the trace formula to character relations, liftings and the definition
of packets, is analogous to \cite[IV]{F3} and \cite[II]{F4}. The trace formulae
identity is proven in \cite[VI]{F3} and \cite[III]{F4} for all test functions.
Here we deal only with test functions which have at least two elliptic
components. The trace formulae identity for a general test function has
not yet been proven in our case. Arthur is working on this. It
would be interesting to pursue the elementary techniques of \cite[VI]{F3}
and
\cite[III]{F4}, and \cite{F2}, which establishes the trace formulae identity for
base change for GL(2) by elementary means, based on the usage of regular,
Iwahori test functions. In particular our work does not develop the trace
formula. It only uses a form of it.

Our approach to the lifting uses the trace formula developed by Arthur,
as envisaged by Langlands e.g. in his work on base change for GL(2). It is
compatible with the conjectures of Arthur \cite{A}. 

Of course Siegel modular
forms have been extensively studied by many authors (e.g. Siegel, Maass,
Shimura, Andrianov, and others) over a long period of time.

An important representation theoretic approach alternative to the trace
formula, based on the theta correspondence, Weil representation, Howe's
dual reductive pairs, $L$-functions and converse theorems, has been
fruitfully developed in our context of the symplectic group by many authors,
see, e.g., \cite{PS1}, \cite{PS2}, \cite{KRS}, \cite{GRS}.

Another---purely local---approach is developed in \cite{FZ}.

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\pagebreak

\bibitem[F3]{F3}
Y. Flicker, On the symmetric square. IV. Applications of
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research monograph, 2003.
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\end{document}