Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Regularization of orbital integrals.
Yiannis Sakellaridis
Rutgers–Newark and National Technical University of Athens
Simons Symposium onGeometric Aspects of the Trace formula
Schloss Elmau, Germany, 10-16 April 2016.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Outline
Introduction
Linear reduction
Asymptotics along a torus
Equivariant toroidal compactifications
Regularization of orbital integrals
Details on the construction of rHsF
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Question: What is the (Relative) Trace Formula?Setup: Xi “ HizG , i “ 1, 2.“Naive” isomorphisms: X1 ˆ X2{G
diag » H1zG{H2.If X1 “ X2 “ H, G “ H ˆ H, this is also: » H{H-conj.At a more sophisticated level: Need these isomorphisms tobe true for F -points. OK if we think of the quotients asstacks.Deeper reason: Need all Langlands parameters to appear onthe spectral side. (E.g., for Gross-Prasad it is not enough toconsider one orthogonal/hermitian space.)
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Question: What is the (Relative) Trace Formula?Setup: Xi “ HizG , i “ 1, 2.“Naive” isomorphisms: X1 ˆ X2{G
diag » H1zG{H2.If X1 “ X2 “ H, G “ H ˆ H, this is also: » H{H-conj.At a more sophisticated level: Need these isomorphisms tobe true for F -points. OK if we think of the quotients asstacks.Deeper reason: Need all Langlands parameters to appear onthe spectral side. (E.g., for Gross-Prasad it is not enough toconsider one orthogonal/hermitian space.)
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Question: What is the (Relative) Trace Formula?Setup: Xi “ HizG , i “ 1, 2.“Naive” isomorphisms: X1 ˆ X2{G
diag » H1zG{H2.If X1 “ X2 “ H, G “ H ˆ H, this is also: » H{H-conj.At a more sophisticated level: Need these isomorphisms tobe true for F -points. OK if we think of the quotients asstacks.Deeper reason: Need all Langlands parameters to appear onthe spectral side. (E.g., for Gross-Prasad it is not enough toconsider one orthogonal/hermitian space.)
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Question: What is the (Relative) Trace Formula?Setup: Xi “ HizG , i “ 1, 2.“Naive” isomorphisms: X1 ˆ X2{G
diag » H1zG{H2.If X1 “ X2 “ H, G “ H ˆ H, this is also: » H{H-conj.At a more sophisticated level: Need these isomorphisms tobe true for F -points. OK if we think of the quotients asstacks.Deeper reason: Need all Langlands parameters to appear onthe spectral side. (E.g., for Gross-Prasad it is not enough toconsider one orthogonal/hermitian space.)
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
F -local field, X-smooth stack, now have notion ofSpXpF qq-Schwartz space of XpF q.Naively: if X “ X {G , SpXpF qq “ SpX pF qqGpF q(coinvariants),so that distributions DpXpF qq “ DpX pF qqGpF q.Global Schwartz space: SpXpAkqq “
Â1v SpXpkv qq.
Problem: Define a distribution RTF : SpXpAkqq Ñ C.From now on: X “ X {G , X : smooth affine, G : reductive.Pretend SpXpF qq “ SpX pF qqGpF q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
F -local field, X-smooth stack, now have notion ofSpXpF qq-Schwartz space of XpF q.Naively: if X “ X {G , SpXpF qq “ SpX pF qqGpF q(coinvariants),so that distributions DpXpF qq “ DpX pF qqGpF q.Global Schwartz space: SpXpAkqq “
Â1v SpXpkv qq.
Problem: Define a distribution RTF : SpXpAkqq Ñ C.From now on: X “ X {G , X : smooth affine, G : reductive.Pretend SpXpF qq “ SpX pF qqGpF q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
F -local field, X-smooth stack, now have notion ofSpXpF qq-Schwartz space of XpF q.Naively: if X “ X {G , SpXpF qq “ SpX pF qqGpF q(coinvariants),so that distributions DpXpF qq “ DpX pF qqGpF q.Global Schwartz space: SpXpAkqq “
Â1v SpXpkv qq.
Problem: Define a distribution RTF : SpXpAkqq Ñ C.From now on: X “ X {G , X : smooth affine, G : reductive.Pretend SpXpF qq “ SpX pF qqGpF q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
F -local field, X-smooth stack, now have notion ofSpXpF qq-Schwartz space of XpF q.Naively: if X “ X {G , SpXpF qq “ SpX pF qqGpF q(coinvariants),so that distributions DpXpF qq “ DpX pF qqGpF q.Global Schwartz space: SpXpAkqq “
Â1v SpXpkv qq.
Problem: Define a distribution RTF : SpXpAkqq Ñ C.From now on: X “ X {G , X : smooth affine, G : reductive.Pretend SpXpF qq “ SpX pF qqGpF q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
F -local field, X-smooth stack, now have notion ofSpXpF qq-Schwartz space of XpF q.Naively: if X “ X {G , SpXpF qq “ SpX pF qqGpF q(coinvariants),so that distributions DpXpF qq “ DpX pF qqGpF q.Global Schwartz space: SpXpAkqq “
Â1v SpXpkv qq.
Problem: Define a distribution RTF : SpXpAkqq Ñ C.From now on: X “ X {G , X : smooth affine, G : reductive.Pretend SpXpF qq “ SpX pF qqGpF q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
F -local field, X-smooth stack, now have notion ofSpXpF qq-Schwartz space of XpF q.Naively: if X “ X {G , SpXpF qq “ SpX pF qqGpF q(coinvariants),so that distributions DpXpF qq “ DpX pF qqGpF q.Global Schwartz space: SpXpAkqq “
Â1v SpXpkv qq.
Problem: Define a distribution RTF : SpXpAkqq Ñ C.From now on: X “ X {G , X : smooth affine, G : reductive.Pretend SpXpF qq “ SpX pF qqGpF q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Outline
Introduction
Linear reduction
Asymptotics along a torus
Equivariant toroidal compactifications
Regularization of orbital integrals
Details on the construction of rHsF
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: X “ smooth affine variety. Then RTF “ř
Xpkq.Special case of this: X “ X {G with G acting freely on X , sothat X is a variety.Then
RTFpf q “ÿ
ξPX pkq{Gpkq
Oξpf q “
ż
rG s
ÿ
γPX pkq
f pγgqdg , (1)
rG s “ G pkqzG pAkq.
We will try to generalize (1). Can group terms intoequivalence classes according to “the semisimple part of theJordan decomposition”:
ÿ
o
ż ˚
rG s
ÿ
γPopkq
f pγgqdg .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: X “ smooth affine variety. Then RTF “ř
Xpkq.Special case of this: X “ X {G with G acting freely on X , sothat X is a variety.Then
RTFpf q “ÿ
ξPX pkq{Gpkq
Oξpf q “
ż
rG s
ÿ
γPX pkq
f pγgqdg , (1)
rG s “ G pkqzG pAkq.
We will try to generalize (1). Can group terms intoequivalence classes according to “the semisimple part of theJordan decomposition”:
ÿ
o
ż ˚
rG s
ÿ
γPopkq
f pγgqdg .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: X “ smooth affine variety. Then RTF “ř
Xpkq.Special case of this: X “ X {G with G acting freely on X , sothat X is a variety.Then
RTFpf q “ÿ
ξPX pkq{Gpkq
Oξpf q “
ż
rG s
ÿ
γPX pkq
f pγgqdg , (1)
rG s “ G pkqzG pAkq.
We will try to generalize (1). Can group terms intoequivalence classes according to “the semisimple part of theJordan decomposition”:
ÿ
o
ż ˚
rG s
ÿ
γPopkq
f pγgqdg .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: X “ smooth affine variety. Then RTF “ř
Xpkq.Special case of this: X “ X {G with G acting freely on X , sothat X is a variety.Then
RTFpf q “ÿ
ξPX pkq{Gpkq
Oξpf q “
ż
rG s
ÿ
γPX pkq
f pγgqdg , (1)
rG s “ G pkqzG pAkq.
We will try to generalize (1). Can group terms intoequivalence classes according to “the semisimple part of theJordan decomposition”:
ÿ
o
ż ˚
rG s
ÿ
γPopkq
f pγgqdg .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: X “ smooth affine variety. Then RTF “ř
Xpkq.Special case of this: X “ X {G with G acting freely on X , sothat X is a variety.Then
RTFpf q “ÿ
ξPX pkq{Gpkq
Oξpf q “
ż
rG s
ÿ
γPX pkq
f pγgqdg , (1)
rG s “ G pkqzG pAkq.
We will try to generalize (1). Can group terms intoequivalence classes according to “the semisimple part of theJordan decomposition”:
ÿ
o
ż ˚
rG s
ÿ
γPopkq
f pγgqdg .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Each o contains a unique semisimple (= geometricallyclosed) G pkq-orbit. Let x0 P X pkq be a representative,H “ Gx0 , V “ the fiber of the normal bundle to x0G at x0.Luna slice theorem: the stack X {G is etale-locallyisomorphic to V {H.This reduces the problem to the “Lie algebra version” V {H,with o replaced by the k-points of N “ tv P V |0 P v ¨ hu.The isomorphism is not canonical, but one can show thatthe definitions we will give do not depend on choices.Goal: regularize
ş˚
rHs
ř
γPNpkq f pγhqdh.References: Arthur, Jacquet, Lapid, Rogawski, Ichino,Yamana, Zydor, Jason Levy.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Each o contains a unique semisimple (= geometricallyclosed) G pkq-orbit. Let x0 P X pkq be a representative,H “ Gx0 , V “ the fiber of the normal bundle to x0G at x0.Luna slice theorem: the stack X {G is etale-locallyisomorphic to V {H.This reduces the problem to the “Lie algebra version” V {H,with o replaced by the k-points of N “ tv P V |0 P v ¨ hu.The isomorphism is not canonical, but one can show thatthe definitions we will give do not depend on choices.Goal: regularize
ş˚
rHs
ř
γPNpkq f pγhqdh.References: Arthur, Jacquet, Lapid, Rogawski, Ichino,Yamana, Zydor, Jason Levy.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Each o contains a unique semisimple (= geometricallyclosed) G pkq-orbit. Let x0 P X pkq be a representative,H “ Gx0 , V “ the fiber of the normal bundle to x0G at x0.Luna slice theorem: the stack X {G is etale-locallyisomorphic to V {H.This reduces the problem to the “Lie algebra version” V {H,with o replaced by the k-points of N “ tv P V |0 P v ¨ hu.The isomorphism is not canonical, but one can show thatthe definitions we will give do not depend on choices.Goal: regularize
ş˚
rHs
ř
γPNpkq f pγhqdh.References: Arthur, Jacquet, Lapid, Rogawski, Ichino,Yamana, Zydor, Jason Levy.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Each o contains a unique semisimple (= geometricallyclosed) G pkq-orbit. Let x0 P X pkq be a representative,H “ Gx0 , V “ the fiber of the normal bundle to x0G at x0.Luna slice theorem: the stack X {G is etale-locallyisomorphic to V {H.This reduces the problem to the “Lie algebra version” V {H,with o replaced by the k-points of N “ tv P V |0 P v ¨ hu.The isomorphism is not canonical, but one can show thatthe definitions we will give do not depend on choices.Goal: regularize
ş˚
rHs
ř
γPNpkq f pγhqdh.References: Arthur, Jacquet, Lapid, Rogawski, Ichino,Yamana, Zydor, Jason Levy.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Each o contains a unique semisimple (= geometricallyclosed) G pkq-orbit. Let x0 P X pkq be a representative,H “ Gx0 , V “ the fiber of the normal bundle to x0G at x0.Luna slice theorem: the stack X {G is etale-locallyisomorphic to V {H.This reduces the problem to the “Lie algebra version” V {H,with o replaced by the k-points of N “ tv P V |0 P v ¨ hu.The isomorphism is not canonical, but one can show thatthe definitions we will give do not depend on choices.Goal: regularize
ş˚
rHs
ř
γPNpkq f pγhqdh.References: Arthur, Jacquet, Lapid, Rogawski, Ichino,Yamana, Zydor, Jason Levy.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Outline
Introduction
Linear reduction
Asymptotics along a torus
Equivariant toroidal compactifications
Regularization of orbital integrals
Details on the construction of rHsF
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
By an easy argument, enough to regularize
ż ˚
rHs
ÿ
γPV pkq
f pγhqdh.
Behavior along a 1-dimensional torus λ : Gm Ñ H:Decompose: V “ V`
λ ‘ V 0λ ‘ V´
λ
(positive, zero and negative weights).
ÿ
γPV
“ÿ
γ´PV´
ÿ
γ0PV 0
ÿ
γ`PV`
Terms with γ´ ‰ 0 contribute something of rapid decayalong λptq as |t| Ñ 0:
f ppγ´ ` γ0 ` γ`qλptqq “ f pγ´λptq ` pγ0 ` γ`qλptqq
Ñ 0 very rapidly as |t| Ñ 0, because γ´λptq Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
By an easy argument, enough to regularize
ż ˚
rHs
ÿ
γPV pkq
f pγhqdh.
Behavior along a 1-dimensional torus λ : Gm Ñ H:Decompose: V “ V`
λ ‘ V 0λ ‘ V´
λ
(positive, zero and negative weights).
ÿ
γPV
“ÿ
γ´PV´
ÿ
γ0PV 0
ÿ
γ`PV`
Terms with γ´ ‰ 0 contribute something of rapid decayalong λptq as |t| Ñ 0:
f ppγ´ ` γ0 ` γ`qλptqq “ f pγ´λptq ` pγ0 ` γ`qλptqq
Ñ 0 very rapidly as |t| Ñ 0, because γ´λptq Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
By an easy argument, enough to regularize
ż ˚
rHs
ÿ
γPV pkq
f pγhqdh.
Behavior along a 1-dimensional torus λ : Gm Ñ H:Decompose: V “ V`
λ ‘ V 0λ ‘ V´
λ
(positive, zero and negative weights).
ÿ
γPV
“ÿ
γ´PV´
ÿ
γ0PV 0
ÿ
γ`PV`
Terms with γ´ ‰ 0 contribute something of rapid decayalong λptq as |t| Ñ 0:
f ppγ´ ` γ0 ` γ`qλptqq “ f pγ´λptq ` pγ0 ` γ`qλptqq
Ñ 0 very rapidly as |t| Ñ 0, because γ´λptq Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
By an easy argument, enough to regularize
ż ˚
rHs
ÿ
γPV pkq
f pγhqdh.
Behavior along a 1-dimensional torus λ : Gm Ñ H:Decompose: V “ V`
λ ‘ V 0λ ‘ V´
λ
(positive, zero and negative weights).
ÿ
γPV
“ÿ
γ´PV´
ÿ
γ0PV 0
ÿ
γ`PV`
Terms with γ´ ‰ 0 contribute something of rapid decayalong λptq as |t| Ñ 0:
f ppγ´ ` γ0 ` γ`qλptqq “ f pγ´λptq ` pγ0 ` γ`qλptqq
Ñ 0 very rapidly as |t| Ñ 0, because γ´λptq Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
By an easy argument, enough to regularize
ż ˚
rHs
ÿ
γPV pkq
f pγhqdh.
Behavior along a 1-dimensional torus λ : Gm Ñ H:Decompose: V “ V`
λ ‘ V 0λ ‘ V´
λ
(positive, zero and negative weights).
ÿ
γPV
“ÿ
γ´PV´
ÿ
γ0PV 0
ÿ
γ`PV`
Terms with γ´ ‰ 0 contribute something of rapid decayalong λptq as |t| Ñ 0:
f ppγ´ ` γ0 ` γ`qλptqq “ f pγ´λptq ` pγ0 ` γ`qλptqq
Ñ 0 very rapidly as |t| Ñ 0, because γ´λptq Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Hence,ÿ
γPV
„ÿ
γ0PV 0
ÿ
γ`PV`
Poisson summation on V` (f : partial Fourier transform):
ÿ
γ`PV`
f pγ`q “ÿ
δPpV`q˚
f pδq,
But now Gm acts through λ with negative weights onpV`q˚. Hence, only the term with δ “ 0 remains, up tomoderate growth.Notice: f pδ “ 0q “
ş
V`pAk qf pnqdn.
In the end:
ÿ
γPV
f pγλptqq|t|Ñ0„
ÿ
γ0PV 0
ż
V`pAk q
f ppγ0 ` nqλptqqdn.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Hence,ÿ
γPV
„ÿ
γ0PV 0
ÿ
γ`PV`
Poisson summation on V` (f : partial Fourier transform):
ÿ
γ`PV`
f pγ`q “ÿ
δPpV`q˚
f pδq,
But now Gm acts through λ with negative weights onpV`q˚. Hence, only the term with δ “ 0 remains, up tomoderate growth.Notice: f pδ “ 0q “
ş
V`pAk qf pnqdn.
In the end:
ÿ
γPV
f pγλptqq|t|Ñ0„
ÿ
γ0PV 0
ż
V`pAk q
f ppγ0 ` nqλptqqdn.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Hence,ÿ
γPV
„ÿ
γ0PV 0
ÿ
γ`PV`
Poisson summation on V` (f : partial Fourier transform):
ÿ
γ`PV`
f pγ`q “ÿ
δPpV`q˚
f pδq,
But now Gm acts through λ with negative weights onpV`q˚. Hence, only the term with δ “ 0 remains, up tomoderate growth.Notice: f pδ “ 0q “
ş
V`pAk qf pnqdn.
In the end:
ÿ
γPV
f pγλptqq|t|Ñ0„
ÿ
γ0PV 0
ż
V`pAk q
f ppγ0 ` nqλptqqdn.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Hence,ÿ
γPV
„ÿ
γ0PV 0
ÿ
γ`PV`
Poisson summation on V` (f : partial Fourier transform):
ÿ
γ`PV`
f pγ`q “ÿ
δPpV`q˚
f pδq,
But now Gm acts through λ with negative weights onpV`q˚. Hence, only the term with δ “ 0 remains, up tomoderate growth.Notice: f pδ “ 0q “
ş
V`pAk qf pnqdn.
In the end:
ÿ
γPV
f pγλptqq|t|Ñ0„
ÿ
γ0PV 0
ż
V`pAk q
f ppγ0 ` nqλptqqdn.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
ÿ
γPV
f pγλptqq|t|Ñ0„
ÿ
γ0PV 0
ż
V`pAk q
f pγ0 ` nλptqqdn.
The integralş
V`pAk qis a λpAk
ˆq-eigendistribution witheigencharacter equal to the absolute value of:
ź
χ
χ´mχ ,
where χ ranges over the characters of Gm on V`pAkq (viaλ) and mχ is the multiplicity of χ.If we write the characters additively:
´ÿ
χ
mχ ¨ χ.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
ÿ
γPV
f pγλptqq|t|Ñ0„
ÿ
γ0PV 0
ż
V`pAk q
f pγ0 ` nλptqqdn.
The integralş
V`pAk qis a λpAk
ˆq-eigendistribution witheigencharacter equal to the absolute value of:
ź
χ
χ´mχ ,
where χ ranges over the characters of Gm on V`pAkq (viaλ) and mχ is the multiplicity of χ.If we write the characters additively:
´ÿ
χ
mχ ¨ χ.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
ÿ
γPV
f pγλptqq|t|Ñ0„
ÿ
γ0PV 0
ż
V`pAk q
f pγ0 ` nλptqqdn.
The integralş
V`pAk qis a λpAk
ˆq-eigendistribution witheigencharacter equal to the absolute value of:
ź
χ
χ´mχ ,
where χ ranges over the characters of Gm on V`pAkq (viaλ) and mχ is the multiplicity of χ.If we write the characters additively:
´ÿ
χ
mχ ¨ χ.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
To sum up: to describe the asymptotic behavior of thefunction
Σf phq :“ÿ
γPV pkq
f pγhq
on rHs, need a way to encode the fact that
for every λ : Gm Ñ H, as |t| Ñ 0,
Σf pλptqq „ź
χ|λě0
χ´mχptq.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
To sum up: to describe the asymptotic behavior of thefunction
Σf phq :“ÿ
γPV pkq
f pγhq
on rHs, need a way to encode the fact that
for every λ : Gm Ñ H, as |t| Ñ 0,
Σf pλptqq „ź
χ|λě0
χ´mχptq.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Outline
Introduction
Linear reduction
Asymptotics along a torus
Equivariant toroidal compactifications
Regularization of orbital integrals
Details on the construction of rHsF
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Let A “ maximal (central) split torus in P0{N0 “ universalCartan of H.a “ HompGm,Aq bZ Q.Recall: a strictly convex polyhedral cone C in a describes anaffine toric embedding: A ãÑ AC , characterized by:
limtÑ0
λptq P AC ðñ λ P C.
A more general toric variety is described by a fan F on a (=subdivision of a subset into cones C).We are interested in a subdivision F of a` “anti-dominantcone.Definition: the subdivision F of a` is determined by its walls,and the hyperplanes perpendicular to χ, χ a weight of V .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Let A “ maximal (central) split torus in P0{N0 “ universalCartan of H.a “ HompGm,Aq bZ Q.Recall: a strictly convex polyhedral cone C in a describes anaffine toric embedding: A ãÑ AC , characterized by:
limtÑ0
λptq P AC ðñ λ P C.
A more general toric variety is described by a fan F on a (=subdivision of a subset into cones C).We are interested in a subdivision F of a` “anti-dominantcone.Definition: the subdivision F of a` is determined by its walls,and the hyperplanes perpendicular to χ, χ a weight of V .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Let A “ maximal (central) split torus in P0{N0 “ universalCartan of H.a “ HompGm,Aq bZ Q.Recall: a strictly convex polyhedral cone C in a describes anaffine toric embedding: A ãÑ AC , characterized by:
limtÑ0
λptq P AC ðñ λ P C.
A more general toric variety is described by a fan F on a (=subdivision of a subset into cones C).We are interested in a subdivision F of a` “anti-dominantcone.Definition: the subdivision F of a` is determined by its walls,and the hyperplanes perpendicular to χ, χ a weight of V .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Let A “ maximal (central) split torus in P0{N0 “ universalCartan of H.a “ HompGm,Aq bZ Q.Recall: a strictly convex polyhedral cone C in a describes anaffine toric embedding: A ãÑ AC , characterized by:
limtÑ0
λptq P AC ðñ λ P C.
A more general toric variety is described by a fan F on a (=subdivision of a subset into cones C).We are interested in a subdivision F of a` “anti-dominantcone.Definition: the subdivision F of a` is determined by its walls,and the hyperplanes perpendicular to χ, χ a weight of V .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Let A “ maximal (central) split torus in P0{N0 “ universalCartan of H.a “ HompGm,Aq bZ Q.Recall: a strictly convex polyhedral cone C in a describes anaffine toric embedding: A ãÑ AC , characterized by:
limtÑ0
λptq P AC ðñ λ P C.
A more general toric variety is described by a fan F on a (=subdivision of a subset into cones C).We are interested in a subdivision F of a` “anti-dominantcone.Definition: the subdivision F of a` is determined by its walls,and the hyperplanes perpendicular to χ, χ a weight of V .
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: In the group case, V “ h, weights of V are theroots, and a` is not divided further.For each cone C (including “faces”) of the subdivision, wehave a character:
χC “ ´ÿ
χ
mχχ,
where mχ is the multiplicity of χ in V` “ the positivesubspace of all cocharacters in C.Example: In the group case, this is just the modularcharacter 2ρ.
RemarkWe are only interested in the restriction of χC to thesubtorus AC spanned by cocharacters in C. From now on χCis considered as a character of AC only.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: In the group case, V “ h, weights of V are theroots, and a` is not divided further.For each cone C (including “faces”) of the subdivision, wehave a character:
χC “ ´ÿ
χ
mχχ,
where mχ is the multiplicity of χ in V` “ the positivesubspace of all cocharacters in C.Example: In the group case, this is just the modularcharacter 2ρ.
RemarkWe are only interested in the restriction of χC to thesubtorus AC spanned by cocharacters in C. From now on χCis considered as a character of AC only.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: In the group case, V “ h, weights of V are theroots, and a` is not divided further.For each cone C (including “faces”) of the subdivision, wehave a character:
χC “ ´ÿ
χ
mχχ,
where mχ is the multiplicity of χ in V` “ the positivesubspace of all cocharacters in C.Example: In the group case, this is just the modularcharacter 2ρ.
RemarkWe are only interested in the restriction of χC to thesubtorus AC spanned by cocharacters in C. From now on χCis considered as a character of AC only.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: In the group case, V “ h, weights of V are theroots, and a` is not divided further.For each cone C (including “faces”) of the subdivision, wehave a character:
χC “ ´ÿ
χ
mχχ,
where mχ is the multiplicity of χ in V` “ the positivesubspace of all cocharacters in C.Example: In the group case, this is just the modularcharacter 2ρ.
RemarkWe are only interested in the restriction of χC to thesubtorus AC spanned by cocharacters in C. From now on χCis considered as a character of AC only.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
TheoremThere is an HpAkq-equivariant compactification rHsF of rHs,such that for every f P SpV pAkqq, the function
Σf phq “ÿ
γPV pkq
f pγhq
on rHs is a section of the sheaf over rHsF of χC-equivariantfunctions.
Example
(Tate’s thesis)H “ Gm, V “ Ga, a “ Q, F “ C` Y C0 Y C´, C˘ “ Q˘,C0 “ t0u.χC` “ ´1, χC´ “ 0.Σf ptq „ |t|´1 when |t| Ñ 0, „ 1 when |t| Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
TheoremThere is an HpAkq-equivariant compactification rHsF of rHs,such that for every f P SpV pAkqq, the function
Σf phq “ÿ
γPV pkq
f pγhq
on rHs is a section of the sheaf over rHsF of χC-equivariantfunctions.
Example
(Tate’s thesis)H “ Gm, V “ Ga, a “ Q, F “ C` Y C0 Y C´, C˘ “ Q˘,C0 “ t0u.χC` “ ´1, χC´ “ 0.Σf ptq „ |t|´1 when |t| Ñ 0, „ 1 when |t| Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
TheoremThere is an HpAkq-equivariant compactification rHsF of rHs,such that for every f P SpV pAkqq, the function
Σf phq “ÿ
γPV pkq
f pγhq
on rHs is a section of the sheaf over rHsF of χC-equivariantfunctions.
Example
(Tate’s thesis)H “ Gm, V “ Ga, a “ Q, F “ C` Y C0 Y C´, C˘ “ Q˘,C0 “ t0u.χC` “ ´1, χC´ “ 0.Σf ptq „ |t|´1 when |t| Ñ 0, „ 1 when |t| Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
TheoremThere is an HpAkq-equivariant compactification rHsF of rHs,such that for every f P SpV pAkqq, the function
Σf phq “ÿ
γPV pkq
f pγhq
on rHs is a section of the sheaf over rHsF of χC-equivariantfunctions.
Example
(Tate’s thesis)H “ Gm, V “ Ga, a “ Q, F “ C` Y C0 Y C´, C˘ “ Q˘,C0 “ t0u.χC` “ ´1, χC´ “ 0.Σf ptq „ |t|´1 when |t| Ñ 0, „ 1 when |t| Ñ 8.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
Orthogonal Gross-Prasad: X “ SOn`1{SOn-conj
H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .
Say n even, split SOn. (Odd or non-split case is similar.)GLn ãÑ H (into a Levi subgroup)ε1, . . . , εn: the standard characters of the torus of diagonalelements in GLn,a` : ε1 ď ¨ ¨ ¨ ď εn´1 ď εn ď ´εn´1,Non-zero weights of h: dual to the walls of a`.Non-zero weights of the std : ˘ε1,˘ε2, . . . ,˘εn.New wall: εn “ 0.Thus, the fan F: subdivision of a` into two subcones.
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)
Notice: in all cases, χC ‰ 2ρ, except for C “ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
Orthogonal Gross-Prasad: X “ SOn`1{SOn-conj
H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .
Say n even, split SOn. (Odd or non-split case is similar.)GLn ãÑ H (into a Levi subgroup)ε1, . . . , εn: the standard characters of the torus of diagonalelements in GLn,a` : ε1 ď ¨ ¨ ¨ ď εn´1 ď εn ď ´εn´1,Non-zero weights of h: dual to the walls of a`.Non-zero weights of the std : ˘ε1,˘ε2, . . . ,˘εn.New wall: εn “ 0.Thus, the fan F: subdivision of a` into two subcones.
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)
Notice: in all cases, χC ‰ 2ρ, except for C “ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
Orthogonal Gross-Prasad: X “ SOn`1{SOn-conj
H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .
Say n even, split SOn. (Odd or non-split case is similar.)GLn ãÑ H (into a Levi subgroup)ε1, . . . , εn: the standard characters of the torus of diagonalelements in GLn,a` : ε1 ď ¨ ¨ ¨ ď εn´1 ď εn ď ´εn´1,Non-zero weights of h: dual to the walls of a`.Non-zero weights of the std : ˘ε1,˘ε2, . . . ,˘εn.New wall: εn “ 0.Thus, the fan F: subdivision of a` into two subcones.
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)
Notice: in all cases, χC ‰ 2ρ, except for C “ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
Orthogonal Gross-Prasad: X “ SOn`1{SOn-conj
H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .
Say n even, split SOn. (Odd or non-split case is similar.)GLn ãÑ H (into a Levi subgroup)ε1, . . . , εn: the standard characters of the torus of diagonalelements in GLn,a` : ε1 ď ¨ ¨ ¨ ď εn´1 ď εn ď ´εn´1,Non-zero weights of h: dual to the walls of a`.Non-zero weights of the std : ˘ε1,˘ε2, . . . ,˘εn.New wall: εn “ 0.Thus, the fan F: subdivision of a` into two subcones.
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)
Notice: in all cases, χC ‰ 2ρ, except for C “ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
Orthogonal Gross-Prasad: X “ SOn`1{SOn-conj
H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .
Say n even, split SOn. (Odd or non-split case is similar.)GLn ãÑ H (into a Levi subgroup)ε1, . . . , εn: the standard characters of the torus of diagonalelements in GLn,a` : ε1 ď ¨ ¨ ¨ ď εn´1 ď εn ď ´εn´1,Non-zero weights of h: dual to the walls of a`.Non-zero weights of the std : ˘ε1,˘ε2, . . . ,˘εn.New wall: εn “ 0.Thus, the fan F: subdivision of a` into two subcones.
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)
Notice: in all cases, χC ‰ 2ρ, except for C “ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
Orthogonal Gross-Prasad: X “ SOn`1{SOn-conj
H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .
Say n even, split SOn. (Odd or non-split case is similar.)GLn ãÑ H (into a Levi subgroup)ε1, . . . , εn: the standard characters of the torus of diagonalelements in GLn,a` : ε1 ď ¨ ¨ ¨ ď εn´1 ď εn ď ´εn´1,Non-zero weights of h: dual to the walls of a`.Non-zero weights of the std : ˘ε1,˘ε2, . . . ,˘εn.New wall: εn “ 0.Thus, the fan F: subdivision of a` into two subcones.
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)
Notice: in all cases, χC ‰ 2ρ, except for C “ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
Orthogonal Gross-Prasad: X “ SOn`1{SOn-conj
H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .
Say n even, split SOn. (Odd or non-split case is similar.)GLn ãÑ H (into a Levi subgroup)ε1, . . . , εn: the standard characters of the torus of diagonalelements in GLn,a` : ε1 ď ¨ ¨ ¨ ď εn´1 ď εn ď ´εn´1,Non-zero weights of h: dual to the walls of a`.Non-zero weights of the std : ˘ε1,˘ε2, . . . ,˘εn.New wall: εn “ 0.Thus, the fan F: subdivision of a` into two subcones.
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)
Notice: in all cases, χC ‰ 2ρ, except for C “ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
Orthogonal Gross-Prasad: X “ SOn`1{SOn-conj
H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .
Say n even, split SOn. (Odd or non-split case is similar.)GLn ãÑ H (into a Levi subgroup)ε1, . . . , εn: the standard characters of the torus of diagonalelements in GLn,a` : ε1 ď ¨ ¨ ¨ ď εn´1 ď εn ď ´εn´1,Non-zero weights of h: dual to the walls of a`.Non-zero weights of the std : ˘ε1,˘ε2, . . . ,˘εn.New wall: εn “ 0.Thus, the fan F: subdivision of a` into two subcones.
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)
χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)
Notice: in all cases, χC ‰ 2ρ, except for C “ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Outline
Introduction
Linear reduction
Asymptotics along a torus
Equivariant toroidal compactifications
Regularization of orbital integrals
Details on the construction of rHsF
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Regularization: baby case
Let s P C. Consider the space SpΓzHqs of functions on ΓzH,where H is the complex upper half-plane and Γ “ SL2pZq,which are smooth and have the property that:
f px ` iyq „ y s
for y " 0, where „ means that the difference is a functionwhich, together with all its polynomial derivatives, is of rapiddecay. Then, the regularized integral:
ż ˚
ΓzHf px ` iyq
dxdˆy
y
(where dˆy denotes multiplicative measure dyy ) is
well-defined unless s “ 1, i.e. unless the growth of thefunction is inverse to that of volume. The exponent(multiplicative character) y ÞÑ y 1 is what I call a criticalexponent.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Regularization: baby case
Let s P C. Consider the space SpΓzHqs of functions on ΓzH,where H is the complex upper half-plane and Γ “ SL2pZq,which are smooth and have the property that:
f px ` iyq „ y s
for y " 0, where „ means that the difference is a functionwhich, together with all its polynomial derivatives, is of rapiddecay. Then, the regularized integral:
ż ˚
ΓzHf px ` iyq
dxdˆy
y
(where dˆy denotes multiplicative measure dyy ) is
well-defined unless s “ 1, i.e. unless the growth of thefunction is inverse to that of volume. The exponent(multiplicative character) y ÞÑ y 1 is what I call a criticalexponent.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Regularization: baby case (cont.)
The definition of the regularized integral is as follows: fixany large T ą 0, and define:
ftpx ` iyq “
#
f px ` iyq, if y ď T
f px ` iyq|y |´t , if y ą T .
Thenş
ΓzH ftpx ` iyqdxdˆy|y | is convergent for <ptq " 0, and
admits meromorphic continuation with only a simple pole att ` 1 “ s. Thus, if s ‰ 1 we can define the regularizedintegral as the analytic continuation of the above integral tot “ 0.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Regularization: baby case (cont.)
The definition of the regularized integral is as follows: fixany large T ą 0, and define:
ftpx ` iyq “
#
f px ` iyq, if y ď T
f px ` iyq|y |´t , if y ą T .
Thenş
ΓzH ftpx ` iyqdxdˆy|y | is convergent for <ptq " 0, and
admits meromorphic continuation with only a simple pole att ` 1 “ s. Thus, if s ‰ 1 we can define the regularizedintegral as the analytic continuation of the above integral tot “ 0.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Local picture on rHsF:
ş˚f px , yq|x |s1 |y |s2dˆxdˆy .
Makes sense as long as |x |s1
or |y |s2 is not inverse to vol-ume growth (equal to themodular character).
TheoremThe regularized integral makes sense, as long as χC ‰ 2ρ forevery C ‰ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Local picture on rHsF:
ş˚f px , yq|x |s1 |y |s2dˆxdˆy .
Makes sense as long as |x |s1
or |y |s2 is not inverse to vol-ume growth (equal to themodular character).
TheoremThe regularized integral makes sense, as long as χC ‰ 2ρ forevery C ‰ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Local picture on rHsF:
ş˚f px , yq|x |s1 |y |s2dˆxdˆy .
Makes sense as long as |x |s1
or |y |s2 is not inverse to vol-ume growth (equal to themodular character).
TheoremThe regularized integral makes sense, as long as χC ‰ 2ρ forevery C ‰ t0u.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
In the Gross-Prasad case, χC ‰ 2ρ. Thus,ş˚
makes sense.
Example
In the group case, V “ h, χC “ 2ρ for every C. Thus,ş˚
does not make sense.There is no purely geometric regularization of orbitalintegrals.
Remark
ÿ
γPV pkq
f pγhq “ÿ
γPV˚pkq
f pγhq,
so Poisson summation automatically holds for theregularized integrals
ş˚.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
In the Gross-Prasad case, χC ‰ 2ρ. Thus,ş˚
makes sense.
Example
In the group case, V “ h, χC “ 2ρ for every C. Thus,ş˚
does not make sense.There is no purely geometric regularization of orbitalintegrals.
Remark
ÿ
γPV pkq
f pγhq “ÿ
γPV˚pkq
f pγhq,
so Poisson summation automatically holds for theregularized integrals
ş˚.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
In the Gross-Prasad case, χC ‰ 2ρ. Thus,ş˚
makes sense.
Example
In the group case, V “ h, χC “ 2ρ for every C. Thus,ş˚
does not make sense.There is no purely geometric regularization of orbitalintegrals.
Remark
ÿ
γPV pkq
f pγhq “ÿ
γPV˚pkq
f pγhq,
so Poisson summation automatically holds for theregularized integrals
ş˚.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example
In the Gross-Prasad case, χC ‰ 2ρ. Thus,ş˚
makes sense.
Example
In the group case, V “ h, χC “ 2ρ for every C. Thus,ş˚
does not make sense.There is no purely geometric regularization of orbitalintegrals.
Remark
ÿ
γPV pkq
f pγhq “ÿ
γPV˚pkq
f pγhq,
so Poisson summation automatically holds for theregularized integrals
ş˚.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: Kudla-Rallis regularized period for theSiegel-Weil formula.
H “ SOpW q, dim W “ 2m, acting on V “ W b X ,dim X “ n.Assume split, a` : ε1 ď ¨ ¨ ¨ ď εm´1 ď εm ď ´εm´1 as before.Weights: n times the weights of the standard representation:˘ε1, . . . ,˘εm. Decomposition of a` into two subcones asbefore.Extremal rays of the cones:
vi “ px , x , . . . , xpi-th positionq, 0, . . . , 0q, x ď 0
and v 1m “ px , x , . . . , x ,´xq.Character χC for C “ R`vi : vi ÞÑ nix .Modular character: 2ρ “
ř
i 2pm ´ iqεi .On vi : 2ρpvi q “ 2xipm ´ i`1
2 q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: Kudla-Rallis regularized period for theSiegel-Weil formula.
H “ SOpW q, dim W “ 2m, acting on V “ W b X ,dim X “ n.Assume split, a` : ε1 ď ¨ ¨ ¨ ď εm´1 ď εm ď ´εm´1 as before.Weights: n times the weights of the standard representation:˘ε1, . . . ,˘εm. Decomposition of a` into two subcones asbefore.Extremal rays of the cones:
vi “ px , x , . . . , xpi-th positionq, 0, . . . , 0q, x ď 0
and v 1m “ px , x , . . . , x ,´xq.Character χC for C “ R`vi : vi ÞÑ nix .Modular character: 2ρ “
ř
i 2pm ´ iqεi .On vi : 2ρpvi q “ 2xipm ´ i`1
2 q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: Kudla-Rallis regularized period for theSiegel-Weil formula.
H “ SOpW q, dim W “ 2m, acting on V “ W b X ,dim X “ n.Assume split, a` : ε1 ď ¨ ¨ ¨ ď εm´1 ď εm ď ´εm´1 as before.Weights: n times the weights of the standard representation:˘ε1, . . . ,˘εm. Decomposition of a` into two subcones asbefore.Extremal rays of the cones:
vi “ px , x , . . . , xpi-th positionq, 0, . . . , 0q, x ď 0
and v 1m “ px , x , . . . , x ,´xq.Character χC for C “ R`vi : vi ÞÑ nix .Modular character: 2ρ “
ř
i 2pm ´ iqεi .On vi : 2ρpvi q “ 2xipm ´ i`1
2 q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: Kudla-Rallis regularized period for theSiegel-Weil formula.
H “ SOpW q, dim W “ 2m, acting on V “ W b X ,dim X “ n.Assume split, a` : ε1 ď ¨ ¨ ¨ ď εm´1 ď εm ď ´εm´1 as before.Weights: n times the weights of the standard representation:˘ε1, . . . ,˘εm. Decomposition of a` into two subcones asbefore.Extremal rays of the cones:
vi “ px , x , . . . , xpi-th positionq, 0, . . . , 0q, x ď 0
and v 1m “ px , x , . . . , x ,´xq.Character χC for C “ R`vi : vi ÞÑ nix .Modular character: 2ρ “
ř
i 2pm ´ iqεi .On vi : 2ρpvi q “ 2xipm ´ i`1
2 q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: Kudla-Rallis regularized period for theSiegel-Weil formula.
H “ SOpW q, dim W “ 2m, acting on V “ W b X ,dim X “ n.Assume split, a` : ε1 ď ¨ ¨ ¨ ď εm´1 ď εm ď ´εm´1 as before.Weights: n times the weights of the standard representation:˘ε1, . . . ,˘εm. Decomposition of a` into two subcones asbefore.Extremal rays of the cones:
vi “ px , x , . . . , xpi-th positionq, 0, . . . , 0q, x ď 0
and v 1m “ px , x , . . . , x ,´xq.Character χC for C “ R`vi : vi ÞÑ nix .Modular character: 2ρ “
ř
i 2pm ´ iqεi .On vi : 2ρpvi q “ 2xipm ´ i`1
2 q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Example: Kudla-Rallis regularized period for theSiegel-Weil formula (cont.).
χCpvi q “ nix
2ρpvi q “ 2xipm ´i ` 1
2q
Condition:
nix ‰ 2xipm ´i ` 1
2q for all 1 ď i ď m
ðñ n R rm ´ 1, 2pm ´ 1qs
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Outline
Introduction
Linear reduction
Asymptotics along a torus
Equivariant toroidal compactifications
Regularization of orbital integrals
Details on the construction of rHsF
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
C ù a`P (face) Ø P “ MN Ă HFP “ restriction of F to a`P .
AFPP “ toric variety for AP .
rHsP “ MpkqNpAkqzHpAkq
DefinitionrHsFP :“ rHsP ˆ
APpRq AFPP pRq.
HpAkq-orbits Ø subcones P FP .ZC Ø C.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
C ù a`P (face) Ø P “ MN Ă HFP “ restriction of F to a`P .
AFPP “ toric variety for AP .
rHsP “ MpkqNpAkqzHpAkq
DefinitionrHsFP :“ rHsP ˆ
APpRq AFPP pRq.
HpAkq-orbits Ø subcones P FP .ZC Ø C.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
C ù a`P (face) Ø P “ MN Ă HFP “ restriction of F to a`P .
AFPP “ toric variety for AP .
rHsP “ MpkqNpAkqzHpAkq
DefinitionrHsFP :“ rHsP ˆ
APpRq AFPP pRq.
HpAkq-orbits Ø subcones P FP .ZC Ø C.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
PpkqzHpAkq
α
yy
β
&&rHs rHsP
α is an isomorphism “close to P-cusp”.
DefinitionFor all C P F, C ù P, glue ZC to rHs “in the P-cusp” byglueing it to PpkqzHpAkq: a sequence zn P PpkqzHpAkq
converges to z P ZC iff βpznq Ñ z in rHsFP .
RemarkWhen F “ the faces of a`, this is the reductive Borel-Serrecompactification.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
PpkqzHpAkq
α
yy
β
&&rHs rHsP
α is an isomorphism “close to P-cusp”.
DefinitionFor all C P F, C ù P, glue ZC to rHs “in the P-cusp” byglueing it to PpkqzHpAkq: a sequence zn P PpkqzHpAkq
converges to z P ZC iff βpznq Ñ z in rHsFP .
RemarkWhen F “ the faces of a`, this is the reductive Borel-Serrecompactification.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Suppose now that we are given a set E of characters χC ofAC , for each cone C P F.Define the space of “asymptotically pF,E q-finite” functionson rHs inductively:SpZC , χCq “ Schwartz pAC , χCq-sections over ZC .SpZC ,E q “ the space of sections which are “asymptoticallypF,E q-finite”.It is a cosheaf over ZC . In a neighborhood of ZD Ă ZC , itssections are equal to a section of SpZD,E q, up to rapiddecay (i.e. up to a section of SpZC , χCq).For C “ t0u ù SprHsF,E q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Suppose now that we are given a set E of characters χC ofAC , for each cone C P F.Define the space of “asymptotically pF,E q-finite” functionson rHs inductively:SpZC , χCq “ Schwartz pAC , χCq-sections over ZC .SpZC ,E q “ the space of sections which are “asymptoticallypF,E q-finite”.It is a cosheaf over ZC . In a neighborhood of ZD Ă ZC , itssections are equal to a section of SpZD,E q, up to rapiddecay (i.e. up to a section of SpZC , χCq).For C “ t0u ù SprHsF,E q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Suppose now that we are given a set E of characters χC ofAC , for each cone C P F.Define the space of “asymptotically pF,E q-finite” functionson rHs inductively:SpZC , χCq “ Schwartz pAC , χCq-sections over ZC .SpZC ,E q “ the space of sections which are “asymptoticallypF,E q-finite”.It is a cosheaf over ZC . In a neighborhood of ZD Ă ZC , itssections are equal to a section of SpZD,E q, up to rapiddecay (i.e. up to a section of SpZC , χCq).For C “ t0u ù SprHsF,E q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Suppose now that we are given a set E of characters χC ofAC , for each cone C P F.Define the space of “asymptotically pF,E q-finite” functionson rHs inductively:SpZC , χCq “ Schwartz pAC , χCq-sections over ZC .SpZC ,E q “ the space of sections which are “asymptoticallypF,E q-finite”.It is a cosheaf over ZC . In a neighborhood of ZD Ă ZC , itssections are equal to a section of SpZD,E q, up to rapiddecay (i.e. up to a section of SpZC , χCq).For C “ t0u ù SprHsF,E q.
Yiannis Sakellaridis
Introduction
Linear reduction
Asymptotics alonga torus
Equivarianttoroidalcompactifications
Regularization oforbital integrals
Details on theconstruction ofrHsF
Suppose now that we are given a set E of characters χC ofAC , for each cone C P F.Define the space of “asymptotically pF,E q-finite” functionson rHs inductively:SpZC , χCq “ Schwartz pAC , χCq-sections over ZC .SpZC ,E q “ the space of sections which are “asymptoticallypF,E q-finite”.It is a cosheaf over ZC . In a neighborhood of ZD Ă ZC , itssections are equal to a section of SpZD,E q, up to rapiddecay (i.e. up to a section of SpZC , χCq).For C “ t0u ù SprHsF,E q.