regularization of orbital...

Yiannis Sakellaridis

Introduction

Linear reduction

Asymptotics alonga torus

Equivarianttoroidalcompactifications

Regularization oforbital integrals

Details on theconstruction ofrHsF

Regularization of orbital integrals.

Rutgers–Newark and National Technical University of Athens

Simons Symposium onGeometric Aspects of the Trace formula

Schloss Elmau, Germany, 10-16 April 2016.

Introduction

Linear reduction

Outline

Introduction

Linear reduction

Asymptotics along a torus

Equivariant toroidal compactifications

Regularization of orbital integrals

Details on the construction of rHsF

Introduction

Linear reduction

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.)

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

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.

Introduction

Linear reduction

Â1v SpXpkv qq.

Introduction

Linear reduction

Â1v SpXpkv qq.

Introduction

Linear reduction

Â1v SpXpkv qq.

Introduction

Linear reduction

Â1v SpXpkv qq.

Introduction

Linear reduction

Â1v SpXpkv qq.

Introduction

Linear reduction

Outline

Introduction

Linear reduction

Introduction

Linear reduction

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 “

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”:

γPopkq

f pγgqdg .

Introduction

Linear reduction

RTFpf q “ÿ

ξPX pkq{Gpkq

Oξpf q “

γPX pkq

f pγgqdg , (1)

γPopkq

f pγgqdg .

Introduction

Linear reduction

RTFpf q “ÿ

ξPX pkq{Gpkq

Oξpf q “

γPX pkq

f pγgqdg , (1)

γPopkq

f pγgqdg .

Introduction

Linear reduction

RTFpf q “ÿ

ξPX pkq{Gpkq

Oξpf q “

γPX pkq

f pγgqdg , (1)

γPopkq

f pγgqdg .

Introduction

Linear reduction

RTFpf q “ÿ

ξPX pkq{Gpkq

Oξpf q “

γPX pkq

f pγgqdg , (1)

γPopkq

f pγgqdg .

Introduction

Linear reduction

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

γPNpkq f pγhqdh.References: Arthur, Jacquet, Lapid, Rogawski, Ichino,Yamana, Zydor, Jason Levy.

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

Outline

Introduction

Linear reduction

Introduction

Linear reduction

By an easy argument, enough to regularize

γPV pkq

f pγhqdh.

Behavior along a 1-dimensional torus λ : Gm Ñ H:Decompose: V “ V`

λ ‘ V 0λ ‘ V´

(positive, zero and negative weights).

γ´PV´

γ0PV 0

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.

Introduction

Linear reduction

γPV pkq

f pγhqdh.

λ ‘ V 0λ ‘ V´

γ´PV´

γ0PV 0

γ`PV`

Introduction

Linear reduction

γPV pkq

f pγhqdh.

λ ‘ V 0λ ‘ V´

γ´PV´

γ0PV 0

γ`PV`

Introduction

Linear reduction

γPV pkq

f pγhqdh.

λ ‘ V 0λ ‘ V´

γ´PV´

γ0PV 0

γ`PV`

Introduction

Linear reduction

γPV pkq

f pγhqdh.

λ ‘ V 0λ ‘ V´

γ´PV´

γ0PV 0

γ`PV`

Introduction

Linear reduction

Hence,ÿ

γ0PV 0

γ`PV`

Poisson summation on V` (f : partial Fourier transform):

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 “

V`pAk qf pnqdn.

In the end:

f pγλptqq|t|Ñ0„

γ0PV 0

V`pAk q

f ppγ0 ` nqλptqqdn.

Introduction

Linear reduction

Hence,ÿ

γ0PV 0

γ`PV`

f pγ`q “ÿ

δPpV`q˚

f pδq,

V`pAk qf pnqdn.

In the end:

γ0PV 0

V`pAk q

Introduction

Linear reduction

Hence,ÿ

γ0PV 0

γ`PV`

f pγ`q “ÿ

δPpV`q˚

f pδq,

V`pAk qf pnqdn.

In the end:

γ0PV 0

V`pAk q

Introduction

Linear reduction

Hence,ÿ

γ0PV 0

γ`PV`

f pγ`q “ÿ

δPpV`q˚

f pδq,

V`pAk qf pnqdn.

In the end:

γ0PV 0

V`pAk q

Introduction

Linear reduction

γ0PV 0

V`pAk q

f pγ0 ` nλptqqdn.

The integralş

V`pAk qis a λpAk

ˆq-eigendistribution witheigencharacter equal to the absolute value of:

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:

mχ ¨ χ.

Introduction

Linear reduction

γ0PV 0

V`pAk q

f pγ0 ` nλptqqdn.

The integralş

V`pAk qis a λpAk

χ´mχ ,

mχ ¨ χ.

Introduction

Linear reduction

γ0PV 0

V`pAk q

f pγ0 ` nλptqqdn.

The integralş

V`pAk qis a λpAk

χ´mχ ,

mχ ¨ χ.

Introduction

Linear reduction

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.

Introduction

Linear reduction

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.

Introduction

Linear reduction

Outline

Introduction

Linear reduction

Introduction

Linear reduction

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 .

Introduction

Linear reduction

limtÑ0

Introduction

Linear reduction

limtÑ0

Introduction

Linear reduction

limtÑ0

Introduction

Linear reduction

limtÑ0

Introduction

Linear reduction

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

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.

Introduction

Linear reduction

χC “ ´ÿ

mχχ,

Introduction

Linear reduction

χC “ ´ÿ

mχχ,

Introduction

Linear reduction

χC “ ´ÿ

mχχ,

Introduction

Linear reduction

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.

Introduction

Linear reduction

Σf phq “ÿ

γPV pkq

f pγhq

Example

Introduction

Linear reduction

Σf phq “ÿ

γPV pkq

f pγhq

Example

Introduction

Linear reduction

Σf phq “ÿ

γPV pkq

f pγhq

Example

Introduction

Linear reduction

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.

Introduction

Linear reduction

Example

H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)

Introduction

Linear reduction

Example

H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)

Introduction

Linear reduction

Example

H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)

Introduction

Linear reduction

Example

H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)

Introduction

Linear reduction

Example

H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)

Introduction

Linear reduction

Example

H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)

Introduction

Linear reduction

Example

H :“ Hn :“ SOn, V “ hn`1, V “ h‘ std .

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ` εn (on εn ě 0)

χC “ 2ρ` ε1 ` ¨ ¨ ¨ ` εn´1 ´ εn (on εn ď 0)

Introduction

Linear reduction

Outline

Introduction

Linear reduction

Introduction

Linear reduction

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:

ΓzHf px ` iyq

dxdˆy

(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.

Introduction

Linear reduction

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:

ΓzHf px ` iyq

dxdˆy

(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.

Introduction

Linear reduction

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.

Introduction

Linear reduction

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.

Introduction

Linear reduction

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.

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

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

γPV pkq

f pγhq “ÿ

γPV˚pkq

f pγhq,

so Poisson summation automatically holds for theregularized integrals

Introduction

Linear reduction

Example

makes sense.

Example

Remark

γPV pkq

f pγhq “ÿ

γPV˚pkq

f pγhq,

Introduction

Linear reduction

Example

makes sense.

Example

Remark

γPV pkq

f pγhq “ÿ

γPV˚pkq

f pγhq,

Introduction

Linear reduction

Example

makes sense.

Example

Remark

γPV pkq

f pγhq “ÿ

γPV˚pkq

f pγhq,

Introduction

Linear reduction

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 “

i 2pm ´ iqεi .On vi : 2ρpvi q “ 2xipm ´ i`1

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

Example: Kudla-Rallis regularized period for theSiegel-Weil formula (cont.).

χCpvi q “ nix

2ρpvi q “ 2xipm ´i ` 1

Condition:

nix ‰ 2xipm ´i ` 1

2q for all 1 ď i ď m

ðñ n R rm ´ 1, 2pm ´ 1qs

Introduction

Linear reduction

Outline

Introduction

Linear reduction

Introduction

Linear reduction

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.

Introduction

Linear reduction

APpRq AFPP pRq.

Introduction

Linear reduction

APpRq AFPP pRq.

Introduction

Linear reduction

PpkqzHpAkq

&&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.

Introduction

Linear reduction

PpkqzHpAkq

&&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.

Introduction

Linear reduction

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.

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

Introduction

Linear reduction

regularization of orbital...

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