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98m:11125 11R39 11F27 11F67 11F70 22E50 22E55
Gelbart, Stephen (IL-WEIZ) ; Rogawski, Jonathan (1-UCLA) ;
Soudry, David (IL-TLAV)
Endoscopy, theta-liftings, and period integrals for the unitary
group in three variables.
Ann. of Math. (2) 145 (1997), no. 3, 419–476.
FEATURED REVIEW.
This article describes the structure of local and global L-packets for
the quasisplit unitary group U(3) from the points of view of endoscopy,
theta lifting and period integrals. It provides the clearest evidence
to date of the connections one should expect to find for general
reductive groups among these three notions which lie at the heart
of the theory of automorphic representations. It is also of interest
because of its connections to the theory of algebraic cycles on Picard
modular surfaces. The research represents the culmination of a series
of papers by the authors.
Understanding this paper requires substantial preparation. Some
of this is provided by Gelbart’s survey article [in Theta functions:
from the classical to the modern, 129–174, Amer. Math. Soc., Providence, RI, 1993; MR 94f:22023]. On the specific topics of endoscopy,
theta liftings and period integrals, we also mention the following introductory articles by J. G. Arthur [in Representation theory and
automorphic forms (Edinburgh, 1996), 433–442, Proc. Sympos. Pure
Math., 61, Amer. Math. Soc., Providence, RI, 1997; MR 98j:22029],
D. Prasad [in Theta functions: from the classical to the modern, 105–
127, Amer. Math. Soc., Providence, RI, 1993; MR 94e:11043], and H.
M. Jacquet [in Representation theory and automorphic forms (Edinburgh, 1996), 443–455, Proc. Sympos. Pure Math., 61, Amer. Math.
Soc., Providence, RI, 1997; MR 98m:22024].
To state the results of the present paper, it is necessary to recall
some basic concepts. According to Langlands’ theory, one expects to
attach a stable trace formula to a general connected reductive group
defined over a global field. The spectral side of the trace formula
involves a sum over L-packets of representations of the given group,
as well as its elliptic endoscopic subgroups. In the case of interest, G =
U(3), the L-packets are understood, as is the stable trace formula,
thanks to Rogawski’s work [Automorphic representations of unitary
groups in three variables, Ann. of Math. Stud., 123, Princeton Univ.
Press, Princeton, NJ, 1990; MR 91k:22037]. In this case, there is
a unique elliptic endoscopic group, namely, H = U(2) × U(1), where
U(2) denotes the quasisplit unitary group in two variables. All of
these unitary groups are defined with respect to some fixed quadratic
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extension E/F of global fields. For each idele class character γ of E
which extends the class field theory character of E/F , there is an
associated embedding of L-groups L H → L G which is used to map
each L-packet ρ of H to an L-packet Π(ρ) of G.
The L-packets of primary interest are referred to as (cuspidal) endoscopic L-packets. These are the cuspidal L-packets Π(ρ), for which
ρ = ρ2 × ρ1 is a cuspidal L-packet (consisting of infinite-dimensional
representations). It is shown that each endoscopic L-packet contains
a unique generic representation. The analogous local result is also
proven.
b under endoscopic
If Π is an endoscopic L-packet, then its fiber Π
transfer is a set of at most three L-packets for H. In fact, Rogawski
b∪
has shown that there is a natural way to identify the set SΠ = Π
{1} with one of the groups Z/2 or Z/2 ⊕ Z/2 so that P
the cuspidal
multiplicity of π ∈ Π is given by a formula m(π) = |SΠ |−1 ρ∈SΠ hρ, πi,
for some character ρ 7→ hρ, πi of SΠ . This is compatible with the
multiplicity formulas outlined by Arthur for general groups. The
multiplicity-one property of U(3) implies that m(π) = 0 or 1.
The endoscopic lifting of ρ = ρ2 × ρ1 to Π(ρ) is characterized by an
L-function identity which relates the L-function of a twist of π ∈ Π(ρ)
to the product of L-functions of corresponding twists of the standard
L-functions associated to the base change lifts, with respect to E/F ,
of ρ2 and ρ1 to representations of GL(2) and GL(1), respectively.
In addition to endoscopic lifting from H to G, one may construct
representations of G via theta lifting from unitary
groups
in two
0
variables. If d ∈ F × , we define a matrix Φd = d0 −1
and let Hd
denote the associated unitary group. Then the isomorphism class of
Hd is determined by the class of d modulo norms from E × . The
choice of a nontrivial character ψ of A/F together with the choice of
γ determines a theta correspondence between cuspidal representations
σ of Hd and certain representations Θψ (σ, γ) of G. When σ does not
correspond to a representation of G, then the theta lift is considered
to be zero.
The group G lies in a tower of unitary groups U(2n + 1) and one
may consider theta lifting from Hd to any of the groups in the tower.
The philosophy of towers suggests that, given σ, the first occurrence
of a nonzero theta lift of σ in the tower must be cuspidal. Indeed, the
authors showed in a previous paper that if the theta lift of σ to U(1)
is zero then the theta lift to G must be cuspidal, though possibly
N zero.
In the present paper, they show that if the theta lift of σ = v σv to
U(1) is zero then the theta lift Θψ (σ, γ) is nonzero precisely when the
local Howe lift of each component σv is nonzero.
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Much of the above discussion has an obvious adaptation to the local
case, which we will not provide here. In short, if E/F is local, we are
interested in L-packets of irreducible, admissible representations of
the quasisplit group G(F ) = U(3, E/F ) which are obtained from the
elliptic, endoscopic group H by functorial transfer. Such L-packets
will be called endoscopic L-packets. There are local “multiplicity
pairings” hρ, πi which are related to the global pairing by a product
formula. Given an irreducible, admissible representation σ of the local
group Hd (F ), we will let Hψ (σ, γ) denote the Howe lift of σ to G(F ).
We now describe the connection between endoscopic lifting and
Howe lifting. Suppose that π = Hψ (σ, γ) belongs to a local endoscopic
b such that ρ1 =
L-packet Π. Then it is shown that there exists ρ ∈ Π
1
1
γ|E , where E = U(1). On the other hand, if π is any representation
b satisfies ρ1 =
which belongs to an endoscopic L-packet Π and if ρ ∈ Π
1
γ|E then there exist d, σ, ψ such that π = Hψ (σ, γ). Moreover, d is
unique modulo norms. This leads to the definition ερ (π) = 1 if d is
a norm, and ερ (π) = −1 if d is not a norm. It is proven that, in
fact, ερ (π) = hρ, πi. A more general statement of these results which
applies to twists of Howe lifts by characters of E 1 is actually proven.
The p-adic endoscopic L-packets may be parametrized in terms of
Howe lifts as follows. Recall that both the functorial lifting and the
theta lifting depend on the choice of the character γ. For simplicity,
we are using the same choice of γ for both purposes. (In the notation
of the paper, we are assuming µ = γ −1 .) Fix a nonnorm d0 . Then there
is an analogue of the Jacquet-Langlands correspondence between the
quasisplit group H1 and the non-quasisplit group Hd0 . Assume ρ2 is
an L-packet on H1 . Let ρ02 be the corresponding L-packet on Hd0 ,
if it exists, and let ρ02 = ∅ otherwise. The authors show that the
endoscopic L-packet associated to ρ = ρ2 × ρ1 , with ρ1 = γ|E 1 , is
identical to the collection of all nonzero Howe lifts Hψ (σ, γ) such that
σ ∈ ρ2 ∪ ρ02 . Moreover, the set of all such σ (with nonzero Howe lift)
is precisely described. Once again, the actual theorem proven applies
more generally to twists of Howe lifts.
N
Returning to the global case, assume that π = v πvQbelongs to an
endoscopic L-packet Π(ρ) and γ|E 1 = ρ1 . Let ερ (π) = v ερv (πv ) and
assume that ερ (π) = 1. Then we may choose d ∈ F × which is a norm
at precisely those places v at which
N ερv (πv ) = 1. We may then form an
admissible representation σ = v σv of Hd (A) by choosing σv such
that πv = Hψv (σv , γ). It is shown that m(π) = 1 if and only if m(σ) =
1, where m(σ) is the cuspidal multiplicity of σ with respect to the
spectrum of Hd . On the other hand, if ερ (π) = −1 then m(π) = 0.
We now turn to the relation between the theta lifts and endoscopic
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lifts considered above and the nonvanishing of certain period integrals.
The period integrals of interest arise as period integrals on Picard
modular surfaces over algebraic cycles corresponding to unitary groups
in two variables. They are defined by
Z
P (ϕ, η, d) =
ϕ(h)η −1 (det h)dh,
Hd (F )\Hd (A)
where ϕ is an automorphic form in the space of the cuspidal automor1
phic representation π of G, the character η is defined on E 1 \EA
, and
we have fixed an embedding of Hd in G. These integrals are analogous to the period integrals used by Harder, Langlands and Rapoport
to prove Tate’s conjectures for Hilbert modular surfaces.
If P (ϕ, η, d) is nonzero for some d and ϕ, we say π is η-distinguished.
An endoscopic L-packet which contains a (cuspidal) η-distinguished
element is also said to be η-distinguished. It is proven that if Π is such
b must contain an element ρ such that η = ρ1 . If
an L-packet, then Π
π ∈ Π is cuspidal, then the global condition of π being ρ1 -distinguished
is shown to be equivalent to the local conditions hρv , πv i = 1, for all
v. In this case, the L-function L(s, π ⊗ η −1 ) must have a pole at s =
1.
NGiven an η-distinguished representation π, we may choose ϕ =
v ϕv such that P (ϕ, η, d) 6= 0 and then vary one of the components
ϕv . This yields a local “period functional” on the space of πv , that is,
a nonzero linear functional λ on the space of πv such that λ(πv (h)v) =
ηv (det(h))λ(v), for all h ∈ Hd (Fv ). An irreducible, admissible, infinitedimensional representation of G(Fv ) is said to be ηv -distinguished if
it admits a period functional, for some Hd . It is shown in the p-adic
case that such a representation must lie in an endoscopic L-packet
Π(ρv ) such that ηv = ρv1 and hρv , πv i = 1, and, conversely, that these
conditions characterize the ηv -distinguished representations. It is also
shown that a cuspidal representation π is η-distinguished if and only
if it is ηv -distinguished for all v. On the other hand, if one fixes d
and only considers periods relative to this choice, then it is possible
to find π such that P (ϕ, η, d) ≡ 0 even though there exist (nonzero)
period functionals relative to Hd at each place of F .
Putting together some of the results mentioned above, one deduces
that the cuspidal representations π with nonzero periods are those
cuspidal representations which are theta lifts of cuspidal representations σ of the quasisplit group H1 . Suppose, in this case, that σE is
the base change lift of σ to GL(2)/E . The last result of the paper is
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the formula
1
P (ϕ, γ|EA
, d) = LS (1, σE ⊗ γ −3 )
Y
Cv (ϕv ),
v∈S
in which S is a finite set of places of F , the Cv ’sN
are period functionals,
and the components of the test function ϕ = v ϕv outside of S are
held fixed.
Besides the work of Harder, Langlands and Rapoport on Hilbert
modular surfaces, one of the most obvious precursors of this paper
is Waldspurger’s work on the Shimura correspondence. Many of the
connections among theta lifting, special values of L-functions and
period integrals were already evident in the work of Waldspurger,
as well as in many subsequent applications of Jacquet’s theory of
relative trace formulas on GL(2). The present paper fully integrates
the theory of endoscopy into this mix and it greatly increases our
understanding of the theory of automorphic representations for U(3).
Jeff Hakim (1-AMER)