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A user’s guide to the trace formula for covering groups
Wen-Wei Li
Pan Asian Number Theory Conference, 2012
24 July, 2012
Wen-Wei Li
Trace formula for covers
July 24, 2012
1 / 39
Introduction
.
References
.
1. Arthur’s papers.
2. Moeglin and Waldspurger, Décomposition spectrale et séries
d’Eisenstein, Progress in Math. 113 (1994).
.3 La formule des traces pour les revêtements de groupes réductifs
connexes. I.
Le développement géométrique fin (arXiv:1004.4011)
. La formule des traces pour les revêtements de groupes réductifs
connexes. II.
Analyse harmonique locale (arXiv:1107.1865)
5. La formule des traces pour les revêtements de groupes réductifs
connexes. III.
Le développement spectral fin (arXiv:1107.2220)
6. La formule des traces pour les revêtements de groupes réductifs
4
.
connexes. IV.
Distributions invariantes (in prepartation)
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Trace formula
Arthur-Selberg trace formula
F : number field, A its ring of adèles,
G: a connected reductive group over F ,
G(A)1 := Ker(HG ) where HG : G(A) → aG is the Harish-Chandra
homomorphism,
˜ on L2 (G(F )\G(A)1 ),
R: right regular representation of G
f ∈ Cc∞ (G(A)),
KG : the kernel of R(f ), k(x) := KG (x, x) for x ∈ G(A).
The Arthur-Selberg trace formula calculates a truncated integral of k(x)
over G(F )\G(A)1 :
(geometric expansion) = J(f ) = (spectral expansion).
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Trace formula
Roughly speaking:
the geometric side: distributions on G(A)1 indexed by rational
conjugacy classes (eg. orbital integrals),
the spectral side: distributions on G(A)1 indexed by automorphic
representations (eg. characters).
.
Some of the applications
.
Base change and Jacquet-Langlands correspondence for GL(n).
Endoscopic classification of representations of classical groups.
Formula for the trace of Hecke operators.
Various results in local harmonic analysis (character identities, etc.)
.
Applications in analytic number theory.
In each case, it is crucial to have some refined versions of this trace
formula. Examples: invariant trace formula, stable trace formula.
Wen-Wei Li
Trace formula for covers
July 24, 2012
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A class of covers
Covers of connected reductive groups: the local case
Consider central extensions of locally compact groups as follows.
.
The local setting
.
F : local field, G: connected reductive F -group.
˜ → G(F ) → 1,
1→N→G
.where N is finite abelian.
.
The global setting
.
F : global field, A its ring of adèles, G: connected reductive F -group.
˜ → G(A) → 1.
1→N→G
.
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Trace formula for covers
July 24, 2012
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A class of covers
Examples
G = Sp(2n): A. Weil (1964) ⇒ representation-theoretic interpretation
of Siegel modular forms of half-integral weight.
G = SL(2) or GL(2): Shimura (1973), Kubota.
G split, simple and simply connected: R. Steinberg (1962), H.
Matsumoto (1969) constructed the universal central extension of
G(F ) – related to algebraic K-theory.
G = GLn : metaplectic correspondence (Flicker, Kazhdan,
Patterson,..., ≥ 1980).
G arbitrary: Deligne and Brylinski (2001) classified their
K2 -extensions.
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Trace formula for covers
July 24, 2012
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A class of covers
Genuine representations
In harmonic analysis, on may assume N = µm := {z ∈ C× : z m = 1} for
some m.
˜ which are genuine, i.e.
It suffices to study the representations π of G
π(ϵ˜
x) = ϵπ(˜
x) for all ϵ ∈ µm .
Test functions: it suffices to consider π(f ) with antigenuine
˜ i.e. f (ϵ˜
f ∈ Cc∞ (G),
x) = ϵ−1 f (˜
x).
˜ let ω : N → C× be its central
Justification: given a smooth irrep π of G,
character on N. Then it suffices to study the push-forward of
˜ → G(F ) → 1 by ω.
1→N→G
Wen-Wei Li
Trace formula for covers
July 24, 2012
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A class of covers
Constraints on covers
The class of such extensions under consideration should be:
stable under push-forward by any homomorphism µm → µm′ ;
stable under passage to Levi subgroups (philosophy of cusp forms);
when F is global,
˜ (⇒ spectral decomposition, see [MW]),
∃ splitting G(F ) ,→ G
∃
splittings
over
hyperspecial
subgroups G(ov ) at almost all v such that
∏
˜
G(o
)
,→
G
is
continuous;
here we fix an integral model of G;
v
v
(continued) the corresponding
antigenuine
spherical Hecke algebra at v
⊗
must be commutative (⇒ -decomposition of smooth irreps)
Existence of canonical splittings over unipotent subgroups: automatic. ⇒
notions of constant terms and Jacquet functors.
These conditions are satisfied by the K2 -extensions of Brylinski-Deligne.
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
Desiderata
Goal: establish the Arthur-Selberg trace formula for a large class of covers.
Fix a minimal Levi M0 and set L(M0 ) := {Levi containing M0 }
The coarse trace formula
∑
∑
Jo (f ) = J(f ) =
Jχ (f ).
χ
o
Refined trace formula, in terms of weighted characters and weighted
orbital integrals:
∑
L∈L(M0 )
|W0L |
|W0G |
∑
˜
aL (˜
γ )JL˜ (˜
γ , fV ) = J(f )
˜ 1 ,V )
γ∈Γ(L
=
∑
L∈L(M0 )
Wen-Wei Li
|W0L |
|W0G |
∫
˜ 1 ,V )
Π− (L
Trace formula for covers
˜
aL (π)JL˜ (π, 0, f )dπ.
July 24, 2012
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Non-invariant trace formula
The invariant trace formula
∑
L∈L(M0 )
|W0L |
|W0G |
∑
˜
aL (˜
γ )IL˜ (˜
γ , fV ) = I(f )
˜ 1 ,V )
γ∈Γ(L
=
∑
L∈L(M0 )
|W0L |
|W0G |
∫
˜ 1 ,V )
Π− (L
˜
aL (π)IL˜ (π, 0, f )dπ.
where the IL˜ (· · · ) are invariant distributions. For L = G, we get the
usual orbital integrals and characters.
Simple trace formula: for suitable choice of f , only the terms with
L = G survive.
˜ stabilization ⇒ rewrite
Long-term goal (for some special G):
everything in terms of stable distributions on certain linear reductive
groups.
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Non-invariant trace formula
Coarse trace formula
Coarse trace formula
˜ → G(A) be a cover, Ker(p) = µm .
Let p : G
˜ 1 := Ker(HG ◦ p) where HG : G(A) → aG is the Harish-Chandra
G
homomorphism.
˜ on L2 (G(F )\G
˜ 1 ),
R: right regular representation of G
˜ antigenuine,
f ∈ Cc∞ (G)
KG : the kernel of R(f ), k(x) := KG (˜
x, x
˜) for x ∈ G(A), x
˜ ∈ p−1 (x),
for any parabolic P = M U , RP the right regular representation on
˜ 1 ) and KP its kernel.
L2 (U (A)M (F )\G
Fix minimal Levi M0 and maximal compact K ⊂ G(A) in good relative
˜ := p−1 (K), a0 := aM .
position. Set K
0
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
Coarse trace formula
Fix P0 ∈ P(M0 ). For T ∈ a0 , define the truncated kernel à la Arthur
k T (x) :=
∑
(−1)dim AP /AG
P ⊃P0
∑
KP (δ x
˜, δ x
˜)ˆ
τP (HP (δx) − T ).
δ∈P (F )\G(F )
.
Theorem
.
˜1.
For T highly regular, k T (x) is integrable over G(F )\G
There is an identity of absolutely convergent integrals
∑
∑
JoT (f ) = J T (f ) =
JχT (f ).
χ
o
.Everything in sight is polynomial in T .
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Non-invariant trace formula
Coarse trace formula
Spectral side: χ ranges over cuspidal data (M, σ), where M ⊃ M0 is
˜
a Levi subgroup, σ is a cuspidal automorphic representation of M
2
1
˜ ).
inside L (M (F )\M
Geometric side: o ranges over semisimple classes in G(F ). The
T (f ) corresponds to 1.
unipotent term Junip
.
About the proof
.
Combinatorics: the same as the case of reductive groups (Arthur),
Spectral decomposition: included in Moeglin-Waldspurger,
.
Geometric side: the same as in the case of reductive groups – we only
look at conjugacy classes in G(F ).
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
Coarse trace formula
Refinement
Let T0 ∈ a0 be the canonical element (depending on K) defined by Arthur.
Then
J(f ) := J T0 (f )
Jχ (f ) := JχT0 (f )
Jo (f ) := JoT0 (f )
The problem is to find explicit formulas for them.
.
Goal
.
1. Express J (f ), J (f ) in terms of weighted orbital integrals and
χ
o
weighted characters (local objects).
2. Isolate global and local information.
.
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined geometric side
Descend to the unipotent case
Idea: get rid of the cover on the geometric side.
˜ set
For x, y ∈ G(A) with liftings x
˜, y˜ ∈ G,
[x, y] := x
˜−1 y˜−1 x
˜y˜.
Let σ ∈ G(F ) be semisimple, Gσ := ZG (σ)◦ . Then [·, σ] defines a
homomorphism Gσ (A) → µm .
.
Principle
.
˜
Gσ ,[·,σ]
Let o be the G(F )-orbit containing σ. Reduce JoG (f ) to Junip
, the
unipotent term of the trace formula of Gσ twisted by the character [·, σ].
.(More precisely, some Levi subgroups of Gσ may appear...)
Remark: we have Jordan decomposition on covers!
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined geometric side
Example: descent of orbital integrals
The same formalism about [·, σ] applies to the local case. Let F be a local
˜ → G(F ) a cover and f ∈ Cc∞ (G)
˜ antigenuine.
field, p : G
.
Theorem
.
γ = σ exp(X) with X ∈ gσ (F ) sufficiently
∃f ♭ ∈ Cc∞ (gσ (F )) such that ∀˜
close to 0, we have
1
˜
1
G ,[·,σ]
Gσ
|DG (γ)| 2 OγG
(X)| 2 OXσ
˜ (f ) = |D
(f ♭ )
where
DG and DGσ are the Weyl discriminants on G and gσ , respectively.
.
It’s tempting to remove [·, σ] and replace Gσ (F ) by Ker([·, σ])|Gσ (F ) .
However the latter group is less manageable.
Wen-Wei Li
Trace formula for covers
July 24, 2012
16 / 39
Non-invariant trace formula
The refined geometric side
Basic ideas for the refined geometric expansion
. Reduce to the study of J Gσ ,[·,σ] (f ).
unip
1
. Express J Gσ ,[·,σ] (f ) in terms of weighted unipotent orbital integrals
unip
twisted by [·, σ] (adapt Arthur’s arguments).
˜ and deduce analogous descent
3. Define weighted orbital integrals on G
2
formulas.
. Compare the descent formulas to express ∑ Jo (f ) in terms of
o
˜
weighted orbital integrals on G.
4
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined geometric side
The result will be expressed in terms of good weighted orbital integrals
JM˜ S (˜
γS , fS ), where
S: a sufficiently large finite set of places containing the archimedean
and the ramified ones, depending on Supp(f );
˜ S := p−1 (M (FS ));
M
f = fS fK S , where fK S is the
∏unit in the antigenuine spherical Hecke
S
S
˜
algebra of G w.r.t. K := v∈S
/ Kv ;
˜
γ˜S : conjugacy class in MS which is good, i.e. x
˜γ˜S = γ˜S x
˜ iff
xγS = γS x, where p(˜∗) = ∗.
When M = G, we get the usual orbital integrals.
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined geometric side
Regular semisimple weighted orbital integrals
˜ v be semisimple, regular and good; in G
˜ v (M ⊂ G:
Let S = {v}, γ˜ ∈ M
Levi). Then
∫
1
M
2
γ , fv ) = |D (γ)|
JM˜ v (˜
fv (x−1 γ˜ x)vM (x)dx.
Mγ (Fv )\G(Fv )
vM : M (F )\G(F )/Kv → R≥0 is the volume for some convex
polytope in aG
M;
JM˜ v (˜
γ , fv ) depends on
the choice of Kv ,
a Haar measure on the R-v.s. aG
M,
the ratio of the Haar measures on G(F ) and Gγ (F ).
For general S: can be reduced to S = {v} via Arthur’s splitting
formula.
For general γ˜ : defined by a limiting process.
Wen-Wei Li
Trace formula for covers
July 24, 2012
19 / 39
Non-invariant trace formula
The refined geometric side
Let us return to the refined geometric expansion. We have:
J(f ) =
∑
M ∈L(M0 )
|W0M |
|W0G |
∑
˜
˙S )J ˜ (γf
˙
aM (S, γf
MS S , f ).
K,good
γ∈(M (F ))M,S
γ⇝f
γS
S: depending on Supp(f ).
(M (F ))M,S : the set of (M, S)-equivalence classes defined by Arthur.
(· · · )K,good
M,S : the classes γ admitting a representative whose
components outside S are in K S , and whose local component γf
S in
−1
S
S
˜
˜
MS is good (using p (M (FS ) × K ) = MS × K ).
The correspondence γ ⇝ γf
S : described as above.
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined geometric side
One difficulty
Arthur’s proof can be adapted to prove the refined geometric expansion.
But it requires new ideas as well.
˜
˙S )J ˜ (γf
˙S , f )
In the course of proof, one has to show that aM (S, γf
MS
behaves well with respect to the correspondence γ ⇝ γf
S.
Unlike the case treated by Arthur, the character [·, σ] intervenes and
we need a somewhat technical result of “transport of structure” for
˜
˙S ) and J ˜ (γf
˙
aM (S, γf
MS S , f ).
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined geometric side
Compression of coefficients
Note: the set of places S depends on the support of f .
Goal: we want an expression depending only on a set V containing
{v : v|∞} and the ramified places, and we use test functions of the form
f = fV fK V , where fK V is the unit of the antigenuine spherical Hecke
˜ V , K V ).
algebra w.r.t. (G
.
Theorem
.
For V as above, there are
˜ 1 , V ) of good conjugacy classes in L
˜V ;
a set Γ(L
˜
˜ 1 , V ),
γ ) for γ˜ ∈ Γ(L
coefficients aL (˜
such that
J(f ) =
∑
L∈L(M0 )
.
Wen-Wei Li
|W0L |
|W0G |
∑
˜
γ )JL˜ (˜
aL (˜
γ , fV ).
˜ 1 ,V )
γ∈Γ(L
Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined geometric side
˜
The compressed coefficients aL (˜
γ ) are defined in terms of
˜
1. the global coefficients aM (S, ·), for various M ⊂ L and S ⊃ V large
enough;
.2 the unramified weighted orbital integrals rL (·), i.e. the weighted
M
orbital integrals of the unit in the genuine unramified spherical Hecke
˜ V w.r.t. K V .
algebra G
S
S
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined spectral side
The refined spectral side
˜
Suppose f to be left and right K-finite
from now on. Then
Jχ (f ) =
∑
∑
∑
∑
˜ 1 ) L∈L(M ) s∈W L (M )reg
M ∈L(M0 ) π∈Π(M
· | det(s −
−1
1|aL
M )|
∫
∗
i(aG
L)
|W0M |
·
|W0G |
˜
tr(ML (P˜ , λ)MP |P (s, 0)IPG
˜ (λ, f )χ,π )dλ.
˜ 1 ): the set of unitary irreps of M
˜ 1 up to equivalence,
Π(M
P ∈ P(M ) arbitrary,
MP |P (s, 0): global intertwining operators,
ML (P˜ , λ): an operator defined by a (G, L)-family arising from
intertwining operators,
˜
IPG
˜ (· · · ): unitary parabolic induction.
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined spectral side
.
Local ingredients of the proof
.
Mostly Harish-Chandra’s theory:
local intertwining operators,
c-functions, µ-functions,
Plancherel formula,
normalization of local intertwining operators:
.
Archimedean case: juggling with Γ-functions,
non-archimedean case: adapt Langlands’ proof,
“unramified” case: need some theory of unramified genuine principal
series, cf. [McNamara].
.
Global ingredients of the proof
.
In
. view of Moeglin-Waldspurger, one can simply copy Arthur’s arguments.
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined spectral side
Global weighted characters
One can define
˜ 1 ) of “genuine discrete parameters” for each Levi M ;
a set Πdisc,− (M
˜
˜ 1 );
the discrete coefficients aM (π) for each π ∈ Πdisc,− (M
disc
∗
for λ ∈ i(aG
M ) , the global weighted character
(
)
˜
JM˜ (πλ , f ) = tr JM (πλ , P˜ )IPG
(π
,
f
)
,
λ
˜
where
πλ := π ⊗ exp⟨λ,HM˜ (·)⟩ ,
P is a parabolic with Levi component M ,
JM (πλ , P˜ ) is an intertwining operator coming from MM (P˜ , λ).
˜
IPG
˜ (−) is the normalized parabolic induction.
˜ 1 ) is not necessarily contained in the discrete spectrum.
Warning: Πdisc,− (M
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined spectral side
.
Theorem
.
We have
∑
J(f ) =
.
M ∈L(M0 )
|W0M |
|W0G |
∑
∫
G ∗
˜ 1 ) i(aM )
π∈Πdisc,− (M
˜
aM
˜ (πλ , f )dλ.
disc (πλ )JM
.
Remark
.
To make this integral absolutely convergent, Arthur put some restriction
on the infinitesimal characters of the ∞-part of π. This seems to be
avoidable
by the work of Finis-Lapid-Müller.
.
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Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined spectral side
Local unitary weighted characters
Let
V : a finite set of places such that
either V contains some v|∞, or
the places in V have the same residual characteristic > 0;
P = M U : a parabolic subgroup with Levi component M ⊃ M0 .
˜ V ) unitary and genuine, there is an operator MM (π, P˜ )
For π ∈ Π(M
˜
G
acting on the space of IM
˜ (π) such that
(
)
˜
G
JM˜ V (π, fV ) := tr MM (π, P˜ )IM
(π,
f
)
V
˜
is well-defined. It does not depend on P .
Remark: MM (π, P˜ ) is defined in terms of local intertwining operators and
Harish-Chandra’s µ-functions; they are canonical objects attached to
(M, π).
When M = G, we get the usual characters.
Wen-Wei Li
Trace formula for covers
July 24, 2012
28 / 39
Non-invariant trace formula
The refined spectral side
Compression of coefficients
˜ V ).
Goal: re-index the fine spectral expansion by the local objects π ∈ Π(M
.
Theorem
.
For any Levi L ⊃ M0 , one can define
˜ 1 , V ) of genuine representations of L
˜ V , endowed with a
a space Π− (L
measure,
˜
˜ 1 , V ),
coefficients aL (π) for π ∈ Π− (L
such that for f = fV fK V where fK V is the unit of the antigenuine
˜ V , K V ), we have
spherical Hecke algebra w.r.t. (G
J(f ) =
∑
L∈L(M0 )
.
|W0L |
|W0G |
∫
˜ 1 ,V )
Π− (L
˜
aL (π)JL˜ (π, 0, fV ).
Here JL˜ (π, 0, fV ) is the “Fourier coefficient at 0” of JL˜ (πλ , fV ).
Wen-Wei Li
Trace formula for covers
July 24, 2012
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Non-invariant trace formula
The refined spectral side
˜
The compressed coefficients aL (π) is defined using
˜
1. the global discrete coefficients aM (σ), for various M ⊂ L and
disc
˜ 1 );
σ ∈ Πdisc,− (M
2. the normalizing factors r L (c) (for the intertwining operators) of
M
˜ V w.r.t. K V .
various genuine unramified representations c of G
G (c) is expected to be
In view of the works of Langlands and Shahidi, rM
related to L-functions of some linear group. Cf. [McNamara].
.
Remark
.
Note the formal resemblance between the refined geometric and spectral
expansions
with compressed coefficients.
.
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Trace formula for covers
July 24, 2012
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Invariant trace formula
Invariant trace formula for covers
Let V be a set of places containing the archimedean and ramified ones.
.
Motivation
.
In harmonic analysis, we usually choose test functions fV that are defined
only through their
orbital integrals JG˜ V (˜
γ , fV ), or
characters JG˜ V (π, fV ) (eg. the Euler-Poincaré functions).
These distributions fV 7→ JG˜ V (π, fV ) (resp. JG˜ V (˜
γ , fV )) are weakly dense
˜V .
.in the space of invariant distributions on G
.
Goal
.
Express the distributions JL˜ (· · · ) in the refined trace formula in terms of
invariant
distributions on Levi subgroups.
.
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Trace formula for covers
July 24, 2012
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Invariant trace formula
Arthur’s idea
Let J be the distribution of the refined trace formula, viewed as a
˜ V by
distribution on G
˜ V ) ∋ fV 7−→ f := fV fK V ∈ Cc∞ (G).
˜
Cc∞ (G
. For each Levi L, choose a suitable space of test functions H ˜ together
L
with a surjection HL˜ → IHL˜ .
2. Find a good linear map ϕ
˜ : HG
˜ → IHL
˜ , satisfying similar identities
L
under conjugation as the weighted characters/orbital integrals.
˜
3. By induction, define a distribution I = I G so that
1
I is invariant (follows from the conditions above on ϕL˜ );
I factors through IHG˜ ;
we have the decomposition
J(f ) =
∑
L∈L(M0 )
Wen-Wei Li
|W0L | L˜
I (ϕL˜ (fV )),
|W0G |
Trace formula for covers
f ∈ HG˜ ,
July 24, 2012
32 / 39
Invariant trace formula
We shall apply a similar procedure to the distributions JM˜ (· · · ) in the
refined expansions, using Levi subgroups L ⊃ M . This will yield invariant
distributions IL˜ V (˜
γ , ·), IL˜ V (π, ·).
.
Invariant trace formula
.
∑
L∈L(M0 )
|W0L |
|W0G |
∑
˜
aL (˜
γ )IL˜ (˜
γ , fV ) = I(f )
˜ 1 ,V )
γ∈Γ(L
=
∑
L∈L(M0 )
.
|W0L |
|W0G |
∫
˜ 1 ,V )
Π− (L
˜
aL (π)IL˜ (π, 0, fV ).
This is almost a formal consequence of the recipe above.
.
Remark
.
For L = G, the distributions IG˜ (· · · ) = JG˜ (· · · ) are the usual characters
.and orbital integrals.
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Trace formula for covers
July 24, 2012
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Invariant trace formula
Choice of ϕL˜ : HL˜ → IHL˜
Following Arthur, we take
˜ V -finite (left and right), C ∞ antigenuine
HL˜ to be the space of K
c
1
˜ ;
functions on L
V
IHL˜ to be a space of C-valued functions on the tempered genuine
˜ 1 , characterized by trace Paley-Wiener theorems, so that for
dual of L
V
any fV ∈ HL˜ ,
π 7→ trace(π(fV ))
lies in IHL˜ ;
ϕL˜ to be the map sending fV to
π 7→ JL˜ (π, 0, fV ).
Recall: JL˜ (π, 0, fV ) is the 0-th Fourier coefficient of the weighted
character λ 7→ JL˜ (πλ , fV ).
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Trace formula for covers
July 24, 2012
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Invariant trace formula
. To show ϕ ˜ has image in IH ˜ , a detailed analysis of weighted
L
L
characters is needed.
˜ V -finite
2. We also need to assume the trace Paley-Wiener theorem of K
1
functions.
. To show that the invariant distribution
3
I(fV ) := J(f ) −
∑ |W L | ˜
0
I L (ϕL˜ (fV ))
G|
|W
0
L̸=G
factors through IHG˜ , Arthur uses a global argument à la Kazhdan,
via the trace formula. For some technical reason, this does not work
f
˜ = Sp(2n)
for covers except for G = GL(n) or G
(the twofold cover of
Sp(2n)).
. We use a purely local argument via the invariant local trace formula
for covering groups.
4
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Trace formula for covers
July 24, 2012
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Invariant trace formula
Trace Paley-Wiener theorem
.
Assumption
.
For all Levi L ⊂ G, the trace Paley-Wiener theorem characterizing the
image of
fV 7−→ [π 7→ trace(π(fV ))]
˜1
.for fV ∈ HL˜ holds, where π is a genuine tempered irrep of LV .
. This can be reduced to the case V = {v}, for genuine tempered irreps
˜v.
of L
2. When v is nonarchimedean: simply copy the arguments of
1
Bernstein-Deligne-Kazhdan.
. When v is archimedean: some arguments of Clozel-Delorme seem
problematic for covers!
3
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Trace formula for covers
July 24, 2012
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Invariant trace formula
For archimedean v, some special cases can be check by ad hoc arguments:
G = GL(n) (used in Mezo’s work on metaplectic correspondence),
G = SL(2) (Hiraga-Ikeda),
f
˜ = Sp(2n)
G
the twofold cover of Sp(2n) (Weil...),
G unitary group.
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Trace formula for covers
July 24, 2012
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Invariant trace formula
Simple trace formulas
Cuspidal test functions
˜ v ) be antigenuine.
Let v be a place of F , fv ∈ Cc∞ (G
.
Definition
.
We say fv is cuspidal if trace(π(f )) = 0 for any π of the form
˜
π = IPG
˜ (σ)
˜
.where P ̸= G and σ is a tempered genuine irrep of Mv .
∏
Let fV = v∈V fv ∈ HG˜ , we say fV is cuspidal at v if fv is cuspidal.
Recall: the invariant distribution fV 7→ I(fV ) has two expansions: spectral
and geometric.
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Trace formula for covers
July 24, 2012
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Invariant trace formula
Simple trace formulas
.
Theorem
.
∏
˜ 1)
Let fV = v∈V fv ∈ HG˜ . Put f := fV fK V ∈ Cc∞ (G
1. If f
V is cuspidal at one place, then the spectral expansion of I(fV )
becomes
∑
˜
aG (πV )JG˜ (πV , fV )
˜ 1 ,V )
πV ∈Πdisc,− (G
or in global terms:
∑
˜
aG
˜ (π, f ).
disc (π)JG
˜1)
π∈Πdisc,− (G
. If f is cuspidal at two places, then the geometric expansion of I(f )
becomes
∑
˜
aG (˜
γ )JG˜ (˜
γ , f ).
2
˜ 1 ,V )
γ
˜ ∈Γ(G
.
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Trace formula for covers
July 24, 2012
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