Limiting form of the stable trace formula for
special odd orthogonal groups
Chung Pang Mok
Purdue University
November 9th, 2016
Overview
• Generalities on Arthur’s endoscopic classification for special
odd orthogonal groups
Overview
• Generalities on Arthur’s endoscopic classification for special
odd orthogonal groups
• A weak form of beyond endoscopic decomposition of the
stable trace formula
Overview
• Generalities on Arthur’s endoscopic classification for special
odd orthogonal groups
• A weak form of beyond endoscopic decomposition of the
stable trace formula
• Limiting forms of the stable trace formula
Overview
• Generalities on Arthur’s endoscopic classification for special
odd orthogonal groups
• A weak form of beyond endoscopic decomposition of the
stable trace formula
• Limiting forms of the stable trace formula
• Concluding remarks
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
• Firstly, G will be a connected semi-simple group over a
number field F , which we assume to be split over F .
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
• Firstly, G will be a connected semi-simple group over a
number field F , which we assume to be split over F .
• L2 (G (F )\G (AF )), equipped with the usual right regular
action of G (AF ).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
• Firstly, G will be a connected semi-simple group over a
number field F , which we assume to be split over F .
• L2 (G (F )\G (AF )), equipped with the usual right regular
action of G (AF ).
• The discrete spectrum
L2disc (G (F )\G (AF )) ⊂ L2 (G (F )\G (AF ))
b π mπ · π
L2disc (G (F )\G (AF )) = ⊕
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
• Firstly, G will be a connected semi-simple group over a
number field F , which we assume to be split over F .
• L2 (G (F )\G (AF )), equipped with the usual right regular
action of G (AF ).
• The discrete spectrum
L2disc (G (F )\G (AF )) ⊂ L2 (G (F )\G (AF ))
b π mπ · π
L2disc (G (F )\G (AF )) = ⊕
• here π ranges over the irreducible unitary representations of
G (AF ), and mπ ∈ Z≥0 is the multiplicity of π in L2disc .
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
Arthur’s general conjecture: the decomposition of L2disc is
parametrized by the global Arthur parameters ψ for G :
ψ : LF × SL2 (C) → Gb
b for any
that are square-integrable, i.e. do not factor through M
Levi subgroup M of G over F .
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
Arthur’s general conjecture: the decomposition of L2disc is
parametrized by the global Arthur parameters ψ for G :
ψ : LF × SL2 (C) → Gb
b for any
that are square-integrable, i.e. do not factor through M
Levi subgroup M of G over F .
The set of global Arthur parameters for G is noted as Ψ(G ), and
the subset of square integrable parameters is noted as Ψ2 (G ).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
Here LF is the automorphic Galois group (also called the Langlands
group) over F .
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
Here LF is the automorphic Galois group (also called the Langlands
group) over F . It is conjectured to be a locally compact topological
group, having a surjection to the global Weil group over F :
LF → WF
with compact connected kernel,
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
Here LF is the automorphic Galois group (also called the Langlands
group) over F . It is conjectured to be a locally compact topological
group, having a surjection to the global Weil group over F :
LF → WF
with compact connected kernel, and having a Tannakian type
characterization: for any positive integer N, an N-dimensional
irreducible unitary representation of LF , say
φ : LF → GLN (C)
corresponds to an unitary cuspidal automorphic representation Πφ
of GLN (AF ).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
For each place v of F , there ought to have injections of the local
Weil-Deligne group of Fv :
WF0 v ,→ LF
that is compatible with the local Langlands correspondence for
GLN (Fv ) for any N.
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In more details, given a global Arthur parameter ψ ∈ Ψ(G ), one
would want a packet Πψ consisting of irreducible unitary
representations of G (AF );
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In more details, given a global Arthur parameter ψ ∈ Ψ(G ), one
would want a packet Πψ consisting of irreducible unitary
representations of G (AF ); representations occuring in a packet
ought to be nearly equivalent.
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In more details, given a global Arthur parameter ψ ∈ Ψ(G ), one
would want a packet Πψ consisting of irreducible unitary
representations of G (AF ); representations occuring in a packet
ought to be nearly equivalent. When ψ ∈ Ψ2 (G ), one also wants a
formula for the multiplicity of a given π ∈ Πψ to occur in
L2disc (G (F )\G (AF )), and in addition, showing that any π occuring
in L2disc belongs to Πψ for a unique ψ ∈ Ψ2 (G ).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In more details, given a global Arthur parameter ψ ∈ Ψ(G ), one
would want a packet Πψ consisting of irreducible unitary
representations of G (AF ); representations occuring in a packet
ought to be nearly equivalent. When ψ ∈ Ψ2 (G ), one also wants a
formula for the multiplicity of a given π ∈ Πψ to occur in
L2disc (G (F )\G (AF )), and in addition, showing that any π occuring
in L2disc belongs to Πψ for a unique ψ ∈ Ψ2 (G ).
Langlands’ theory of Eisenstein series then allows the complete
spectral description of L2 (G (F )\G (AF )), in terms of
L2disc (G (F )\G (AF )), and also L2disc (M(F )\M(AF )) for the Levi
subgroup M of G over F .
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In more details, given a global Arthur parameter ψ ∈ Ψ(G ), one
would want a packet Πψ consisting of irreducible unitary
representations of G (AF ); representations occuring in a packet
ought to be nearly equivalent. When ψ ∈ Ψ2 (G ), one also wants a
formula for the multiplicity of a given π ∈ Πψ to occur in
L2disc (G (F )\G (AF )), and in addition, showing that any π occuring
in L2disc belongs to Πψ for a unique ψ ∈ Ψ2 (G ).
Langlands’ theory of Eisenstein series then allows the complete
spectral description of L2 (G (F )\G (AF )), in terms of
L2disc (G (F )\G (AF )), and also L2disc (M(F )\M(AF )) for the Levi
subgroup M of G over F .
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A global parameter ψ ∈ Ψ(G ) is called generic if it is trivial on
SL2 (C); such a parameter is usually notes as φ:
φ : LF → Gb
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A global parameter ψ ∈ Ψ(G ) is called generic if it is trivial on
SL2 (C); such a parameter is usually notes as φ:
φ : LF → Gb
The set of generic parameters for G is noted as Φ(G ).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A global parameter ψ ∈ Ψ(G ) is called generic if it is trivial on
SL2 (C); such a parameter is usually notes as φ:
φ : LF → Gb
The set of generic parameters for G is noted as Φ(G ). Given
φ ∈ Φ(G ), the localization
φv = φ|WF0 : WF0 v → Gb
v
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A global parameter ψ ∈ Ψ(G ) is called generic if it is trivial on
SL2 (C); such a parameter is usually notes as φ:
φ : LF → Gb
The set of generic parameters for G is noted as Φ(G ). Given
φ ∈ Φ(G ), the localization
φv = φ|WF0 : WF0 v → Gb
v
is a local Langlands pramater for G over Fv , and it would
correspond to a local L-packet Πφv for G over Fv (the local
Langlands correspondence for G over Fv ).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
An extension of the Generalized Ramanujan Conjecture: if π
occurs in L2disc (G (F )\G (AF )), then π is of Ramanujan type (i.e.
tempered at every place of F ), if and only if, π belongs to Πφ for
some φ ∈ Φ(G ) ∩ Ψ2 (G ).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
An extension of the Generalized Ramanujan Conjecture: if π
occurs in L2disc (G (F )\G (AF )), then π is of Ramanujan type (i.e.
tempered at every place of F ), if and only if, π belongs to Πφ for
some φ ∈ Φ(G ) ∩ Ψ2 (G ).
Put Φ2 (G ) := Φ(G ) ∩ Ψ2 (G ), referred to as the set of cuspidal
parameters.
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In the following, we will often fix S, a finite set of places of F that
includes all the archimedean places of F .
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In the following, we will often fix S, a finite set of places of F that
includes all the archimedean places of F .
Suppose π ∈ Πφ for a certain φ ∈ Φ(G ), with π being unramified
outside S.
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In the following, we will often fix S, a finite set of places of F that
includes all the archimedean places of F .
Suppose π ∈ Πφ for a certain φ ∈ Φ(G ), with π being unramified
outside S. Then for any finite dimensional comlex representation
r : Gb → GL(V ), we have the (partial) Langlands L-function of π
with respect to r :
Y
LS (s, π, r ) :=
det(IV − r (c(πv ))Nv −s )−1
v ∈S
/
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
In the following, we will often fix S, a finite set of places of F that
includes all the archimedean places of F .
Suppose π ∈ Πφ for a certain φ ∈ Φ(G ), with π being unramified
outside S. Then for any finite dimensional comlex representation
r : Gb → GL(V ), we have the (partial) Langlands L-function of π
with respect to r :
Y
LS (s, π, r ) :=
det(IV − r (c(πv ))Nv −s )−1
v ∈S
/
which converges absolutely for Re(s) 0. It ought to depend only
on φ, so we will denote this as LS (s, φ, r ).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
We now take G to be the split special odd orthogonal group over
F ; thus G = SO(2N + 1) for some N ≥ 1. One has
Gb = Sp(2N, C).
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
We now take G to be the split special odd orthogonal group over
F ; thus G = SO(2N + 1) for some N ≥ 1. One has
Gb = Sp(2N, C).
The general conjectures of Arthur (with appropriate
understanding) were established in Arthur’s book; see chapter one
of Arthur’s book for an introduction.
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A cuspidal parameter φ for G can be described as:
φ = φ1 ⊕ · · · ⊕ φk
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A cuspidal parameter φ for G can be described as:
φ = φ1 ⊕ · · · ⊕ φk
where
• Each φi : LF → GL(mi , C) is irreducible.
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A cuspidal parameter φ for G can be described as:
φ = φ1 ⊕ · · · ⊕ φk
where
• Each φi : LF → GL(mi , C) is irreducible.
• Each φi is of symplectic type, in particular mi is even.
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A cuspidal parameter φ for G can be described as:
φ = φ1 ⊕ · · · ⊕ φk
where
• Each φi : LF → GL(mi , C) is irreducible.
• Each φi is of symplectic type, in particular mi is even.
• φ and φj are non-equivalent for i 6= j.
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A cuspidal parameter φ for G can be described as:
φ = φ1 ⊕ · · · ⊕ φk
where
• Each φi : LF → GL(mi , C) is irreducible.
• Each φi is of symplectic type, in particular mi is even.
• φ and φj are non-equivalent for i 6= j.
• 2N = m1 + · · · + mk
Generalities on Arthur’s endoscopic classification for
special odd orthogonal groups
A cuspidal parameter φ for G can be described as:
φ = φ1 ⊕ · · · ⊕ φk
where
• Each φi : LF → GL(mi , C) is irreducible.
• Each φi is of symplectic type, in particular mi is even.
• φ and φj are non-equivalent for i 6= j.
• 2N = m1 + · · · + mk
When k = 1, we refer to such a parameter as simple. The set of
simple parameters for G is referred to as Φsim (G ).
A weak form of beyond endoscopic decomposition of the
stable trace formula
Question: How is the endoscopic classification reflected in the
decomposition of the (stable) trace formula?
A weak form of beyond endoscopic decomposition of the
stable trace formula
Question: How is the endoscopic classification reflected in the
decomposition of the (stable) trace formula?
A more sophisticated form of the question: formulate the generic
portion of the endoscopic classification, in terms of beyond
endoscopy.
A weak form of beyond endoscopic decomposition of the
stable trace formula
Question: How is the endoscopic classification reflected in the
decomposition of the (stable) trace formula?
A more sophisticated form of the question: formulate the generic
portion of the endoscopic classification, in terms of beyond
endoscopy.
Hecke space of test functions: H(G (AF )) =
Q0
v
H(G (Fv )).
A weak form of beyond endoscopic decomposition of the
stable trace formula
Question: How is the endoscopic classification reflected in the
decomposition of the (stable) trace formula?
A more sophisticated form of the question: formulate the generic
portion of the endoscopic classification, in terms of beyond
endoscopy.
Hecke space of test functions: H(G (AF )) =
Q0
v
H(G (Fv )).
The Invariant trace formula: for f ∈ H(G (AF )),
X
G
mπ · TπG (f ) + · · · (spectral expansion)
Idisc
(f ) =
π
=
X
γ
vol(Gγ ) · OγG (f ) + · · · (geometric expansion)
A weak form of beyond endoscopic decomposition of the
stable trace formula
For our purpose, we need the stable version of the trace formula:
X
G
Sdisc
(f ) =
mψst · STψG (f ) + · · · (spectral expansion)
ψ∈Ψ2 (G )
=
X
δ
vol(G ) · SOδG (f ) + · · · (geometric expansion)
A weak form of beyond endoscopic decomposition of the
stable trace formula
For our purpose, we need the stable version of the trace formula:
X
G
Sdisc
(f ) =
mψst · STψG (f ) + · · · (spectral expansion)
ψ∈Ψ2 (G )
=
X
vol(G ) · SOδG (f ) + · · · (geometric expansion)
δ
G on H(G (A )) is stable.
The linear form Sdisc
F
A weak form of beyond endoscopic decomposition of the
stable trace formula
For our purpose, we need the stable version of the trace formula:
X
G
Sdisc
(f ) =
mψst · STψG (f ) + · · · (spectral expansion)
ψ∈Ψ2 (G )
=
X
vol(G ) · SOδG (f ) + · · · (geometric expansion)
δ
G on H(G (A )) is stable.
The linear form Sdisc
F
We will focus on the case where ψ = φ ∈ Φ2 (G ). We have the
1
stable multiplicity formula: mφst = 2k−1
, for
φ = φ1 ⊕ · · · ⊕ φk
A weak form of beyond endoscopic decomposition of the
stable trace formula
Define:
G
Scusp
(f ) =
X
φ∈Φ2 (G )
mφst · STφG (f )
A weak form of beyond endoscopic decomposition of the
stable trace formula
Define:
G
Scusp
(f ) =
X
mφst · STφG (f )
φ∈Φ2 (G )
G
The linear form Scusp
on H(G (AF )) is again stable, and will be
referred to as the cuspidal component of the stable trace formula.
G (f ) according to the shape of
We want to decompose Scusp
φ ∈ Φcusp (G ).
A weak form of beyond endoscopic decomposition of the
stable trace formula
Define:
G
Scusp
(f ) =
X
mφst · STφG (f )
φ∈Φ2 (G )
G
The linear form Scusp
on H(G (AF )) is again stable, and will be
referred to as the cuspidal component of the stable trace formula.
G (f ) according to the shape of
We want to decompose Scusp
φ ∈ Φcusp (G ).
Given a partition
2N = m1 + · · · + mk , mi positive even integers
define
H = SO(2m1 + 1) × · · · × SO(2mk + 1)
A weak form of beyond endoscopic decomposition of the
stable trace formula
So
Hb = Sp(2m1 , C) × · · · × Sp(2mk , C)
and one has the embedding of dual groups:
ρ : Hb ,→ Gb
which is uniquely determined up to conjugacy by Gb.
A weak form of beyond endoscopic decomposition of the
stable trace formula
So
Hb = Sp(2m1 , C) × · · · × Sp(2mk , C)
and one has the embedding of dual groups:
ρ : Hb ,→ Gb
which is uniquely determined up to conjugacy by Gb.
The pair (H, ρ) is an example of an elliptic beyond endoscopic
datum.
A weak form of beyond endoscopic decomposition of the
stable trace formula
H is just a product a special odd orthogonal groups of smaller size.
In particular the endoscopic classification is valid for H.
A weak form of beyond endoscopic decomposition of the
stable trace formula
H is just a product a special odd orthogonal groups of smaller size.
In particular the endoscopic classification is valid for H. For
instance:
Φ2 (H) = Φ2 (SO(2m1 + 1)) × · · · × Φ2 (SO(2mk + 1))
A weak form of beyond endoscopic decomposition of the
stable trace formula
H is just a product a special odd orthogonal groups of smaller size.
In particular the endoscopic classification is valid for H. For
instance:
Φ2 (H) = Φ2 (SO(2m1 + 1)) × · · · × Φ2 (SO(2mk + 1))
Define Φprim (H) ⊂ Φ2 (H) to consist of parameters
φ0 = φ1 × · · · × φk
with φi ∈ Φsim (SO(2mi + 1)) and φi , φj are non-equivalent for
i 6= j.
A weak form of beyond endoscopic decomposition of the
stable trace formula
Every φ ∈ Φ2 (G ) is of the form ρ∗ φ0 for a unique pair (H, ρ) and
φ0 ∈ Φprim (H).
A weak form of beyond endoscopic decomposition of the
stable trace formula
Every φ ∈ Φ2 (G ) is of the form ρ∗ φ0 for a unique pair (H, ρ) and
φ0 ∈ Φprim (H).
Define, for f 0 ∈ H(H(AF ))
H
Pcusp
(f 0 ) =
X
φ0 ∈Φprim (H)
STφH0 (f 0 )
A weak form of beyond endoscopic decomposition of the
stable trace formula
Every φ ∈ Φ2 (G ) is of the form ρ∗ φ0 for a unique pair (H, ρ) and
φ0 ∈ Φprim (H).
Define, for f 0 ∈ H(H(AF ))
H
Pcusp
(f 0 ) =
X
STφH0 (f 0 )
φ0 ∈Φprim (H)
so in the particular case where H = G , we have
X
G
Pcusp
(f ) =
STφG (f )
φ∈Φsim (G )
A weak form of beyond endoscopic decomposition of the
stable trace formula
Every φ ∈ Φ2 (G ) is of the form ρ∗ φ0 for a unique pair (H, ρ) and
φ0 ∈ Φprim (H).
Define, for f 0 ∈ H(H(AF ))
H
Pcusp
(f 0 ) =
X
STφH0 (f 0 )
φ0 ∈Φprim (H)
so in the particular case where H = G , we have
X
G
Pcusp
(f ) =
STφG (f )
φ∈Φsim (G )
H
The linear form Pcusp
on H(H(AF )) is stable.
A weak form of beyond endoscopic decomposition of the
stable trace formula
Theorem
Q
We have the the decomposition, for f = v fv ∈ H(G (AF )):
X
G
H
Scusp
(f ) =
ι(G , H) · Pcusp
(f H )
(H,ρ)
A weak form of beyond endoscopic decomposition of the
stable trace formula
Theorem
Q
We have the the decomposition, for f = v fv ∈ H(G (AF )):
X
G
H
Scusp
(f ) =
ι(G , H) · Pcusp
(f H )
(H,ρ)
Here ι(G , H) =
1
2k−1
for H = SO(2m1 + 1) × · · · × SO(2mk + 1).
A weak form of beyond endoscopic decomposition of the
stable trace formula
Theorem
Q
We have the the decomposition, for f = v fv ∈ H(G (AF )):
X
G
H
Scusp
(f ) =
ι(G , H) · Pcusp
(f H )
(H,ρ)
Here ι(G , H) =
1
2k−1
for H = SO(2m1 + 1) × · · · × SO(2mk + 1).
Q
f H = v fvH ∈ H(H(AF )); the stable orbital integral of fvH is
uniquely determined by fv by the condition:
A weak form of beyond endoscopic decomposition of the
stable trace formula
Theorem
Q
We have the the decomposition, for f = v fv ∈ H(G (AF )):
X
G
H
Scusp
(f ) =
ι(G , H) · Pcusp
(f H )
(H,ρ)
Here ι(G , H) =
1
2k−1
for H = SO(2m1 + 1) × · · · × SO(2mk + 1).
Q
f H = v fvH ∈ H(H(AF )); the stable orbital integral of fvH is
uniquely determined by fv by the condition:
STφH0v (fvH ) = STρG∗ φ0v (fv )
for every (bounded) local Langlands parameter φ0v for H over Fv .
Limiting forms of the stable trace formula
Langlands’ beyond endoscopy proposal: for each finite dimensional
complex representation r ∈ Rep(Gb), insert the the order of poles at
s = 1 of Langlands L-functions, into the the cuspidal component
of the stable trace formula:
Limiting forms of the stable trace formula
Langlands’ beyond endoscopy proposal: for each finite dimensional
complex representation r ∈ Rep(Gb), insert the the order of poles at
s = 1 of Langlands L-functions, into the the cuspidal component
of the stable trace formula:
r ,G
Scusp
(f ) =
X
φ∈Φ2 (G )
Ress=1 −
d
log LS (s, φ, r ) · mφst · STφG (f )
ds
Limiting forms of the stable trace formula
Langlands’ beyond endoscopy proposal: for each finite dimensional
complex representation r ∈ Rep(Gb), insert the the order of poles at
s = 1 of Langlands L-functions, into the the cuspidal component
of the stable trace formula:
r ,G
Scusp
(f ) =
X
φ∈Φ2 (G )
Ress=1 −
d
log LS (s, φ, r ) · mφst · STφG (f )
ds
1,G
G (f ), for 1 being the trivial representation
We have Scusp
(f ) = Scusp
of Gb.
Limiting forms of the stable trace formula
Langlands’ beyond endoscopy proposal: for each finite dimensional
complex representation r ∈ Rep(Gb), insert the the order of poles at
s = 1 of Langlands L-functions, into the the cuspidal component
of the stable trace formula:
r ,G
Scusp
(f ) =
X
φ∈Φ2 (G )
Ress=1 −
d
log LS (s, φ, r ) · mφst · STφG (f )
ds
1,G
G (f ), for 1 being the trivial representation
We have Scusp
(f ) = Scusp
r ,G
of Gb. One would like to establish decomposition of Scusp
(f ) similar
G
to that of Scusp (f ). These would provide the new tools for
establishing the Principle of Functoriality.
Limiting forms of the stable trace formula
In Langlands’ Beyond Endoscopy proposal, one hopes that the data
given by the order of poles of LS (s, φ, r ) at s = 1, for varying r ,
would reflect the functorial origin of φ.
Limiting forms of the stable trace formula
In Langlands’ Beyond Endoscopy proposal, one hopes that the data
given by the order of poles of LS (s, φ, r ) at s = 1, for varying r ,
would reflect the functorial origin of φ.
For instance, consider a parameter
φ = φ1 ⊕ · · · ⊕ φk
with H = SO(2m1 + 1) × · · · × SO(2mk + 1) as before.
Limiting forms of the stable trace formula
In Langlands’ Beyond Endoscopy proposal, one hopes that the data
given by the order of poles of LS (s, φ, r ) at s = 1, for varying r ,
would reflect the functorial origin of φ.
For instance, consider a parameter
φ = φ1 ⊕ · · · ⊕ φk
with H = SO(2m1 + 1) × · · · × SO(2mk + 1) as before. By a
result due to Dihua Jiang, the set of dimension data for H:
mH (r ) := dim HomHb (1, r )
Limiting forms of the stable trace formula
In Langlands’ Beyond Endoscopy proposal, one hopes that the data
given by the order of poles of LS (s, φ, r ) at s = 1, for varying r ,
would reflect the functorial origin of φ.
For instance, consider a parameter
φ = φ1 ⊕ · · · ⊕ φk
with H = SO(2m1 + 1) × · · · × SO(2mk + 1) as before. By a
result due to Dihua Jiang, the set of dimension data for H:
mH (r ) := dim HomHb (1, r )
when r ranges over the fundamental representations r1 , r2 , · · · , rN
of Gb = Sp(2N, C), determines the partition 2N = m1 + · · · + mk ,
b
and hence H.
Limiting forms of the stable trace formula
Here for a = 1, · · · , N, the a-th fundamental representation ra of
Sp(2N, C), fits into the following shot exact sequence:
0 → ra → Λa std → Λa−2 std → 0
Limiting forms of the stable trace formula
Here for a = 1, · · · , N, the a-th fundamental representation ra of
Sp(2N, C), fits into the following shot exact sequence:
0 → ra → Λa std → Λa−2 std → 0
so for example
r1 = std
r2 ⊕ 1 = Λ2 std
Limiting forms of the stable trace formula
Here for a = 1, · · · , N, the a-th fundamental representation ra of
Sp(2N, C), fits into the following shot exact sequence:
0 → ra → Λa std → Λa−2 std → 0
so for example
r1 = std
r2 ⊕ 1 = Λ2 std
and for H as above,
mH (r1 ) = 0, mH (r2 ) = k − 1
mH (ra ) = 0 for a odd, 1 ≤ a ≤ N
Limiting forms of the stable trace formula
With φ = φ1 ⊕ · · · ⊕ φk as above, one expects the equality
−ords=1 LS (s, φ, r ) = mH (r )
Limiting forms of the stable trace formula
With φ = φ1 ⊕ · · · ⊕ φk as above, one expects the equality
−ords=1 LS (s, φ, r ) = mH (r )
to hold for any finite dimensional representation r of
Gb = Sp(2N, C), if φ does not factor through any smaller
b
L-subgroup of H.
Limiting forms of the stable trace formula
With φ = φ1 ⊕ · · · ⊕ φk as above, one expects the equality
−ords=1 LS (s, φ, r ) = mH (r )
to hold for any finite dimensional representation r of
Gb = Sp(2N, C), if φ does not factor through any smaller
b
L-subgroup of H.
Thus in this case, the set of data −ords=1 LS (s, φ, ra ), for
b
a = 2, 4, · · · , 2[N/2], determines H.
Limiting forms of the stable trace formula
Problem: we do not yet have analytic continuation of the
r ,G
L-functions LS (s, φ, r ), so the definition for the linear form Scusp
as
given previously, only serve as a motivation.
Limiting forms of the stable trace formula
Problem: we do not yet have analytic continuation of the
r ,G
L-functions LS (s, φ, r ), so the definition for the linear form Scusp
as
given previously, only serve as a motivation.
r ,G
Langlands suggests one should construct Scusp
as a certain limiting
G , by using modified form of test functions, as follows.
form of Scusp
Limiting forms of the stable trace formula
For a fixed set of valuations S of F as before,
Limiting forms of the stable trace formula
For a fixed set of valuations S of F as before, define, for each
w∈
/ S, n ≥ 0, and r ∈ Rep(Gb), the following element in the
spherical Hecke algebra of G (Fw ):
n r
hw
∈ Hsph (G (Fw ))
Limiting forms of the stable trace formula
For a fixed set of valuations S of F as before, define, for each
w∈
/ S, n ≥ 0, and r ∈ Rep(Gb), the following element in the
spherical Hecke algebra of G (Fw ):
n r
hw
∈ Hsph (G (Fw ))
whose Satake transform satisfies:
nd
r (c)
hw
= tr (r (c)n )
for semi-simple conjugacy class c in Gb.
Limiting forms of the stable trace formula
Now for a general test function f ∈ H(G (AF )), that is spherical at
valuations outside S, define the modified test function, for each
w∈
/ S, n ≥ 0, and r ∈ Rep(Gb):
n r ,w
f
∈ H(G (AF ))
Limiting forms of the stable trace formula
Now for a general test function f ∈ H(G (AF )), that is spherical at
valuations outside S, define the modified test function, for each
w∈
/ S, n ≥ 0, and r ∈ Rep(Gb):
n r ,w
f
∈ H(G (AF ))
by the rule:
(n f r ,w )w
r
= fw ? n hw
(n f r ,w )v
= fv if v 6= w
Limiting forms of the stable trace formula
Then form the Dirichlet series:
r ,G
Scusp
(f , s) :=
X X log Nw
w ∈S
/ n≥1
Nw ns
G
· Scusp
(n f r ,w )
Limiting forms of the stable trace formula
Then form the Dirichlet series:
r ,G
Scusp
(f , s) :=
X X log Nw
w ∈S
/ n≥1
Nw ns
G
· Scusp
(n f r ,w )
which converges absolutely for Re(s) 0.
Limiting forms of the stable trace formula
Then form the Dirichlet series:
r ,G
Scusp
(f , s) :=
X X log Nw
w ∈S
/ n≥1
Nw ns
G
· Scusp
(n f r ,w )
which converges absolutely for Re(s) 0.
Langlands’ beyond endoscopic proposal: use the trace formula to
r ,G
show that Scusp
(f , s) has analytic continuation to the region
Re(s) > 1,
Limiting forms of the stable trace formula
Then form the Dirichlet series:
r ,G
Scusp
(f , s) :=
X X log Nw
w ∈S
/ n≥1
Nw ns
G
· Scusp
(n f r ,w )
which converges absolutely for Re(s) 0.
Langlands’ beyond endoscopic proposal: use the trace formula to
r ,G
show that Scusp
(f , s) has analytic continuation to the region
Re(s) > 1, and show that the limit:
r ,G
r ,G
(f ) := lim (s − 1) · Scusp
(f , s)
Scusp
s→1
exists.
Limiting forms of the stable trace formula
This still seems to be a very difficult problem. To conclude this
talk, we would like to illustrate how (the generic portion of)
Arthur’s endoscopic classification could be rephrased in this setup:
Limiting forms of the stable trace formula
This still seems to be a very difficult problem. To conclude this
talk, we would like to illustrate how (the generic portion of)
Arthur’s endoscopic classification could be rephrased in this setup:
Theorem
r ,G
For r = r1 , r2 , the Dirichlet series Scusp
(f , s) has holomorphic
continuation to the region Re(s) > 1;
Limiting forms of the stable trace formula
This still seems to be a very difficult problem. To conclude this
talk, we would like to illustrate how (the generic portion of)
Arthur’s endoscopic classification could be rephrased in this setup:
Theorem
r ,G
For r = r1 , r2 , the Dirichlet series Scusp
(f , s) has holomorphic
r ,G
(f ) exists,
continuation to the region Re(s) > 1; the limit Scusp
Limiting forms of the stable trace formula
This still seems to be a very difficult problem. To conclude this
talk, we would like to illustrate how (the generic portion of)
Arthur’s endoscopic classification could be rephrased in this setup:
Theorem
r ,G
For r = r1 , r2 , the Dirichlet series Scusp
(f , s) has holomorphic
r ,G
(f ) exists, and
continuation to the region Re(s) > 1; the limit Scusp
we have the decomposition:
X
r ,G
H
Scusp
(f ) =
ι(r , H) · Pcusp
(f H )
(H,ρ)
Limiting forms of the stable trace formula
This still seems to be a very difficult problem. To conclude this
talk, we would like to illustrate how (the generic portion of)
Arthur’s endoscopic classification could be rephrased in this setup:
Theorem
r ,G
For r = r1 , r2 , the Dirichlet series Scusp
(f , s) has holomorphic
r ,G
(f ) exists, and
continuation to the region Re(s) > 1; the limit Scusp
we have the decomposition:
X
r ,G
H
Scusp
(f ) =
ι(r , H) · Pcusp
(f H )
(H,ρ)
where ι(r , H) := mH (r ) · ι(G , H).
Limiting forms of the stable trace formula
This still seems to be a very difficult problem. To conclude this
talk, we would like to illustrate how (the generic portion of)
Arthur’s endoscopic classification could be rephrased in this setup:
Theorem
r ,G
For r = r1 , r2 , the Dirichlet series Scusp
(f , s) has holomorphic
r ,G
(f ) exists, and
continuation to the region Re(s) > 1; the limit Scusp
we have the decomposition:
X
r ,G
H
Scusp
(f ) =
ι(r , H) · Pcusp
(f H )
(H,ρ)
where ι(r , H) := mH (r ) · ι(G , H).
r1 ,G
In particular Scusp
(f ) = 0 (since mH (r1 ) = 0).
Limiting forms of the stable trace formula
For the proof, besides the input from Arthur’s work on endoscopic
classification, we also used the results on Langlands L-function
with respect to , Sym2 , Λ2 , due to Shahidi, S. Takeda, D. Belt.
Concluding remarks
For the case of G = SO(3) ∼
= PGL(2) over Q, and r = r1 , the
r1 ,G
vanishing of Scusp
(f ) (under certain condition on f ) is established
in the thesis of A. Altug, by working with the trace formula,
without using the known results on automorphic L-functions.
Concluding remarks
For the case of G = SO(3) ∼
= PGL(2) over Q, and r = r1 , the
r1 ,G
vanishing of Scusp
(f ) (under certain condition on f ) is established
in the thesis of A. Altug, by working with the trace formula,
without using the known results on automorphic L-functions.
Can one generalize Altug’s results to the case of
G = SO(5) ∼
= PGSp(4)?
Concluding remarks
For the case of G = SO(3) ∼
= PGL(2) over Q, and r = r1 , the
r1 ,G
vanishing of Scusp
(f ) (under certain condition on f ) is established
in the thesis of A. Altug, by working with the trace formula,
without using the known results on automorphic L-functions.
Can one generalize Altug’s results to the case of
G = SO(5) ∼
= PGSp(4)? Note that for r = r2 , one has, according
to the theorem:
1 H
r2 ,G
Scusp
(f ) = Pcusp
(f H )
2
Concluding remarks
For the case of G = SO(3) ∼
= PGL(2) over Q, and r = r1 , the
r1 ,G
vanishing of Scusp
(f ) (under certain condition on f ) is established
in the thesis of A. Altug, by working with the trace formula,
without using the known results on automorphic L-functions.
Can one generalize Altug’s results to the case of
G = SO(5) ∼
= PGSp(4)? Note that for r = r2 , one has, according
to the theorem:
1 H
r2 ,G
Scusp
(f ) = Pcusp
(f H )
2
where H = SO(3) × SO(3) ∼
= PGL(2) × PGL(2). Can one prove
this by generalizing Altug’s techniques?