1.2. ε-INVARIANT AUTOMORPHIC REPRESENTATIONS 3
1.1.2. Local Lifting. Langlands Functoriality as stated above has a local
analogue. Namely, for every reductive group H over a local p-adic field F , the ho-
momorphisms from the Weil group WF to LH is conjectured to correspond canon-
ically to packets of admissible representations of H(F ). Consequently, analogous
to the global case, an L-homomorphism between L-groups should give rise to the
lifting of admissible representations from one p-adic group to another.
Since each square-integrable admissible representation is a local component of
an automorphic representation, it is a general philosophy that one can derive a
large class of local liftings from global liftings. Works in the past which establish
global lifting using trace formulas typically also derive local lifting. More precisely,
they express the local lifts in the form of trace identities among admissible repre-
sentations. For example, for each admissible representation π of GL(n, F ) which
is invariant under tensor product with a character ε of F
×
of order n, Kazhdan
establishes in [K] an expression of the ε-twisted Harish-Chandra character of π
in terms of the Galois orbit of a character of
E×,
where E is the degree n cyclic
extension of F determined by ε via local class field theory.
The second half of this work is occupied with deriving local twisted character
identities between admissible representations of GSp(2) and those of its twisted
endoscopic groups. We obtain these identities using our global lifting results.
1.2. ε-Invariant Automorphic Representations
Let F be a number field. Let AF denote the ring of ad` eles of F , and CF the
id` ele class group F
×\AF ×.
Let V denote the set of places of F . For each finite place
v of F , let Ov denote the ring of integers of Fv.
Let H be a reductive algebraic group over F . Put H := H(F ), Hv := H(Fv)
for any place v of F . Let Z be the center of H. Let Z0 be the maximal F -split
component of Z. Put Z := Z(F ), Z0 := Z0(F ). For the groups GSp(2) and GL(4),
which are F -split, there is no distinction between Z and Z0.
Let ω be a unitary character of Z0\Z0(AF ). Let L(H(AF ),ω) be the space
of measurable functions φ on H\H(AF ) such that φ(zg) = ω(z)φ(g) for all z
Z0(AF ), and:
Z0(AF )H\H(AF )
|φ(h)|2
dh ∞.
Let ρω be the right-regular representation of H(AF ) on L(H(AF ),ω), that is:
(ρω(h)φ) (g) = φ(gh), h, g H(AF ), φ L(H(AF ),ω).
We say that a representation of H(AF ) is automorphic if it is equivalent to a
subquotient of ρω for some character ω of Z0\Z0(AF ). Let A(H,ω) denote the set
of equivalence classes of irreducible automorphic representations π of H(AF ) with
the property that their central characters restrict to ω on Z0(AF ).
By [Fla], an irreducible automorphic representation π of H(AF ) is equivalent
to a tensor product ⊗v∈V πv of local representations πv of Hv. At almost every
finite place v, the local representation πv is unramified , i.e. it contains a nonzero
vector fixed by the hyperspecial maximal compact subgroup H(Ov).
As an H(AF )-module, L(H(AF ),ω) decomposes into a direct sum:
Ld(H(AF ),ω) Lc(H(AF ),ω),
where Ld(H(AF ),ω) is the closed span of the irreducible, closed, invariant subspaces
of L(H(AF ),ω), and Lc(H(AF ),ω) is its orthogonal complement in L(H(AF ),ω).
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