On certain zeta integrals · homogeneous spaces, i.e. 9 open Borel orbit. Let X+ be spherical homogeneous under Gand ˇ: irrep. Main issues in harmonic analysis include decomposition

On certainzeta integrals

Wen-Wei Li ‖ 李文威Chinese Academy of Sciences

HangzhouJuly 26-31, 2015

The cover picture is taken from http://lvtu.qunar.com

Added on 2015-8-25: see arXiv:1508.05594 for full details.

http://lvtu.qunar.com/mobile_ugc/web/album.htm?albumId=26979

http://arxiv.org/abs/1508.05594

Integral representations of L-functions

Common features:Use tractable integrals involving automorphic forms + other stuff tostudy L-functions, etc.Adélic formulation.Meromorphic continuation, bounded in vertical strips, etc.Functional equation(s).Euler product; theory of local integrals.

Usually called zeta integrals — the core of the L-function machine.

Wen-Wei Li (AMSS-CAS) Zeta integrals July 28, 2015 3 / 34

Godement-Jacquet theory (< 1972)

Goal: study the standard L-functions of GLn, both locally and globally. Webegin with the case over a local field F

GLn × GLn-equivariant embedding GLn ↪→ Matn.ξ ∈ S(Matn×n): Schwartz-Bruhat functions.For an irrep π of GLn(F ), integrate against ξ the matrix coefficients⟨v, π(·)v⟩ (= images under equivariant π ⊠ π → C∞(GLn), satisfyingmultiplicity one):

Z(s, π, v ⊗ v, ξ) :=

∫GL(n,F )

ξ(x)⟨v, π(x)v⟩|det(x)|s+n−12 d×x.

Facts: convergence for Re(s) ≫ 0, meromorphic/rational continuation.


Local functional equation: there exists a meromorphic/rationalfunction γ(s, π) such that

Z(1− s, π, v ⊗ v,Fξ) = γ(s, π)Z(s, π, v ⊗ v, ξ)

where F is the Fourier transform on Matn×n(F ) relative to(X,Y ) 7→ ψ(trXY ), for a chosen additive character ψ.Relation to L-functions: taking greatest common divisor

Z(s, π, ξ)⇝ L(s, π).

Shift in s: the −12 is nice, but where does n/2 come from?

Note: |det |nd×x = dx


Global case: let A = AF , π be cuspidal automorphic, ϕ ∈ π and ϕ ∈ π becusp forms, ∫

GL(n,A)β(x)ξ(x)|det(x)|s+

n−12 d×x

where ℜ(s) ≫ 0 andxϕ(·) = ϕ(· x), β(x) = ⟨ϕ, xϕ⟩Pet (the Petersson pairing) is the globalmatrix coefficient: it factorizes,ξ ∈ S(Matn×n(A)).

It serves as a prototype for many other integral representations ofL-functions.Exercise: rewrite it as a convergent integral involving ϕ and ϕ ondiag(a)GL(n, F )2\GL(n,A)2, for a suitable central connected subgroupa ⊂ GL(n, F∞).


Doubling integrals – a sketch (Piatetski-Shapiro and Rallis)

A typical set-up: let W be a symplectic F -vector space, dimW ≥ 2 andW□ := (W, ⟨, ⟩)

⊥⊕ (W,−⟨, ⟩).

G = Sp(W ), H = Sp(W□);P ⊂ H is the stabilizer of the Lagrangian x0 := diag(W ) in W□,Pab ≃ Gm, and we have a natural algebraic characterdetP : Pab → Gm;G×G ↪→ H naturally;since (G×G) ∩ P = diag(G), one obtains a G×G-equivariantembedding G ↪→ P\H with open dense image;the boundaries are negligible: every γ ∈ ∂H is stabilized by theunipotent radical of some proper parabolic. Morally, this means ∂Hdoes not interact with cusp forms.


Global integrals. Let F be a number field. Let σ ⊠ π ⊠ π be a cuspidalautomorphic representation of (Pab ×G×G)(AF ). Form∫

[G×G]Ef (σ, s)(x, x

′)ϕ(x)ϕ′(x′) dx dx′

ϕ ∈ π, ϕ′ ∈ π;f a “good section” for IHP (σ ⊗ detsP ) (unitary parabolic induction),parametrized by s ∈ C;Ef (σ, s) : H(F )\H(AF ) → C the Eisenstein series made from f .

Properties of Eisenstein series (eg. intertwining operators) =⇒meromorphic continuation, functional equation...


Local integrals. Let σ ⊠ π ⊠ π be a smooth irreducible representation of(Pab ×G×G)(F ). Form∫

G(F )f(x0(x, 1))cπ(x) dx

wheref : Ru(P )\H(F ) → C is a “good section” for IHP (σ⊗detsP ) as before;cπ ∈ C∞(G(F )) is a matrix coefficient of π.

By taking gcd, these integrals represents the Rankin-Selberg L-functionL(s+ 1

2 , χ⊗ π).Furthermore, the global integral factorizes into local ones.


The doubling construction can be adapted to G = GL(n), H = GL(2n) toyield the Godement-Jacquet integrals. Piatetski-Shapiro and Rallis claimedthat the doubling method is the correct generalization ofGodement-Jacquet theory.Things can go the other way around.

Braverman-Kazhdan (2002)Use the affine embedding Pab ×G ↪→ Pder\H =: X. Replace good sectionf by suitable test function on X(F ). Normalized intertwining operators forIPH(· · · ) get replaced by “Fourier transforms”. The resulting theoryresembles the original Godement-Jacquet.

Furthermore: in the unramified setting, there is a distinguishedPab ×H(oF )-invariant test function ξ◦ (denoted by cP,0 in loc. cit.) whosevalues are closely related to the trace-of-Frobenius of the IC sheaf onDrinfeld’s compactification BunP of BunP , the moduli stack of P -bundlesover a smooth projective Fq-curve. Why?


Igusa zeta integrals

One of the interesting integrals in the pre-Langlands era.It originates from ideas of Gelfand et al. on complex powers. Let F be alocal field. We want to study integrals∫

X(F )|f |sξ, Re(s) ≫ 0

forappropriate varieties X (often: affine n-space),ξ ∈ C∞

c (X(F )),f ∈ F [X]: “interesting” function.

Seek for meromorphic continuation in s, location of poles, evaluation atspecial ξ, etc.It is even more interesting if symmetries enter into this picture, by letting agroup G act on X so that f is an eigenfunction.


Prehomogeneous zeta integrals

Let X be an F -vector space, on which a reductive group G acts with denseopen orbit X+. Assume that

1 ∂X = X ∖X+ is a hypersurface (f = 0); we assume f ∈ F [X]transforms with eigencharacter ω : G→ Gm;

2 X+(F ) is a single G(F )-orbit (for simplicity).Define

Z(s, ξ) :=

∫X(F )

|f |sξ, ξ ∈ S(X).

Local functional equation (M. Sato, T. Shintani, F. Sato, et al)Assume X “regular” (hence X is prehomogeneous) and X+(F ) is a singleG(F )-orbit. Z(∗ − s,F(·)) and Z(s, ·) can be related up to “γ-factor”.

Multiple G(F )-orbits in X+(F ) leads to functional equation with“γ-matrices”, by considering integrals over each orbit separately.


In comparison with the local Godement-Jacquet and doublingconstructions, one may consider

π: irrep of G(F ),φ ∈ Nπ := HomG(F )(π,C

∞(X+)), andstudy the zeta integral Z(s, φ, v, ξ) =

∫X(F ) φ(v)|f |

sξ.An attempt in this direction is made by Bopp and Rubenthaler (2005),under various conditions (eg. assuming X+ is a symmetric space, specificseries of π). One indispensable premise seems to be

dimNπ <∞.


Braverman-Kazhdan (1999, 2000)

For split reductive G, they formulated a vast conjectural generalization ofGodement-Jacquet integrals, using

G×G-equivariant embedding of G into some algebraic monoid Mwith unit group G together with detM :M → Ga, restricting to ahomomorphism G→ Gm;conjectural Schwartz space S;conjectural Fourier transform, satisfying an even more conjecturalPoisson formula.

Conjecturally, integration of matrix coefficients against ξ ∈ S, twisted by|detM |s, represents a large family of L-functions attached toρ : G→ GL(N,C).

ρ↔ highest weight Vinberg theory−−−−−−−−→M =Mρ

Related works: L. Lafforgue, Bouthier-Ngô-Sakellaridis......


Analysis on homogeneous spaces

G under G×G is just a special case of homogeneous spaces — thegroup case.A generally accepted framework for doing harmonic analysis: sphericalhomogeneous spaces, i.e. ∃ open Borel orbit.Let X+ be spherical homogeneous under G and π: irrep. Main issuesin harmonic analysis include

decomposition of L2(X+),intertwining operators π → C∞(X+) yielding the “coefficients” of π,intertwining operators relating different X+.

Must also consider spherical embeddings: G-equivariant morphismsX+ ↪→ X with open dense image and normal X.

Important works are being done by Sakellaridis-Venkatesh.


Let H be a split reductive group. In the global case, Sakellaridis (2012)proposes to generate many integral representations using

affine spherical embedding X+ ↪→ X,∀v, conjectural Schwartz space Sv ⊂ C∞(X+(Fv))) from these data.for almost all v ∤ ∞, there should be a distinguished G(ov)-invariantelement ξ◦v ∈ Sv (the basic function).

The expected behavior of ξ ∈ Sv is complicated by the singularities of X.

Example: Bouthier-Ngô-SakellaridisIn the Braverman-Kazhdan case X+ = G ↪→ X =Mρ, suppose everythingunramified.They interpreted ξ◦v ∈ Sv as the trace of Frobenius of an appropriatelydefined IC sheaf on L+X, the formal arc space attached to X+, by passingto the equal-characteristic case.


Generalizations?

It seems that a correct way of viewing zeta integrals is crucial.We try to refine Sakellaridis’ proposal as follows. G: split connectedreductive group over a local field F , char(F ) = 0, common patternsinclude

1 an affine G-spherical embedding X+ ↪→ X;2 a Schwartz space S of “test functions” on a X+(F );3 integrate the coefficients of an irrep π (smooth admissible, SAF ifF ⊃ R) against test functions from S;

4 a mechanism to twist coefficients by |f |s for appropriatef ∈ F [X]G−eigen;

5 meromorphic/rational continuations of the zeta-integrals/zetadistributions so obtained;

6 functional equation of zeta integrals, which is a manifestation of some“Fourier transform” of Schwartz functions.


Reality checkAny such formalism must give a simple, conceptual explanation of theGodement-Jacquet theory.

Other issues:Can we prove anything under such a general framework?Clarify the functional-analytic underpinnings.Local-global compatibility.Try to find manageable examples, for which Schwartz space andFourier transform are already available.It should make zeta integrals appear more natural to an outsider.


Towards a broader frameworkG: split connected reductive group

1 X+ ↪→ X: affine spherical embedding.2 The boundary X ∖X+ is the union of prime divisors fi = 0

(eigencharacter =: ωi), i = 1, . . . , r.3 Assume the eigencharacters ω1, . . . are linearly independent and

generate a lattice Λ. Write |ω|λ =∏

i |ωi|λi , |f |λ :=∏

i |fi|λi forλ =

∑i λiωi ∈ Λ⊗ C.

4 Assume Nπ := HomG(F )(π,C∞(X+)) is finite-dimensional. Known

for F = R (Kobayashi-Oshima) or F non-archimedean and X+

wavefront (Sakellaridis-Venkatesh). Here we should take continuousHom.

5 Work with a given Schwartz space C∞c (X+) ⊂ S ⊂ C∞(X+), a

continuous smooth G(F )-representation.

Zλ(ξ, φ(v)) :=

∫X+

ξφ(v)|f |λ, φ ∈ Nπ, ξ ∈ S, v ∈ π.


Digression: What can we integrate?Integration applies to any density, locally written as |ω| (ω: volumeform). Density bundle = L .The L2-ness makes sense for a 1

2 -density, locally as |ω|1/2.Integration of densities is canonical if we fix a Haar measure on F .The L2-pairing is canonical and invariant under all symmetries ofX+(F ), giving rise to unitary representations.

Hence: L2(X+) stands for the L2-sections of the bundle L 1/2 of12 -densities. Re-define

S ⊂ L2(X+) is L 1/2-valued,C∞(X+) := C∞(X+(F ),L 1/2),φ ∈ Nπ := HomG(F )(π,C

∞(X+)).More generally, we should allow values in some line bundles, eg. theWhittaker-induced case.


What is this good for? Let’s return to the Godement-Jacquet scenario

S := {Schwartz-Bruhat half-densities on Matn×n(F )}= {ξ0|dx|1/2 : ξ0 ∈ Susual},

where dx is a translation-invariant volume form. Therefore

ξ0|dx|1/2 = ξ0|detx|n/2|detx|−n/2|dx|1/2 = ξ0|det |n/2|d×x|1/2

where |d×x| stands for the Haar measure on GL(n, F ).This explains the n/2-shift.Fix ψ, the Fourier transform for S is canonically defined andGL(n, F )-equivariant (already known to E. Stein ≤ 1967).

Lesson: Don’t trivialize L even though you can.


Partial results

Assume F non-archimedean and ∀ξ ∈ S, the support of ξ has compactclosure in X(F ).

1 Using the asymptotics of coefficients φ(v) ∈ C∞(X+) fromSakellaridis-Venkatesh, one can show the convergence for Re(λ) ≫ 0when X+ is wavefront. Idea: use a smooth toroidal compactificationof X+ that is compatible with X+ ↪→ X.

2 In the prehomogeneous case with the usual S, Igusa’s theory impliesthe rationality of zeta integrals.

This framework accommodates:the aforementioned generalization of prehomogeneous zeta integralswith X+ spherical and wavefront,Godement-Jacquet ⊂ Braverman-Kazhdan,doubling integrals interpreted via the Pab ×G×G-equivariantembedding X+ := Pab ×G ↪→ Pder\H ↪→ Pder\H

aff= X.


In the symplectic case at least, the doubling method can be subsumed intoBraverman-Kazhdan. The reason is as follows.

Theorem (Rittatore)View G as a G×G-variety (the “group case”). Its affine sphericalembeddings are the same as normal affine algebraic monoids with unitgroup G.

Moreover, the monoid/embedding X+ ↪→ X in the doubling method forG = Sp(2n) matches Ngô’s recipe with the expected representation

ρ := id⊠ Std : C× × SO(2n+ 1,C) → GL(2n+ 1,C)

of (Pab ×G)∧ = C× × SO(2n+ 1,C). The verification usesa closer look at the stratification of X = Pder\H intoPab ×G×G-orbits;Luna-Vust classification.


Obvious drawbacks

Currently, the formalism does not include Rankin-Selberg integrals onGL(m)× GL(n) with n < m. Doing this will require:

allowing homogeneous G-spaces that are “Whittaker-induced” from aLevi,understanding the “unfolding” process.

The most accessible case: X smooth. Such G-varieties tend to beprehomogeneous (Luna); according to Sakellaridis (2012), theL-functions coming from prehomogeneous zeta integrals have beenknown by other methods.


Relation to spectral decomposition

Generally, every X+ has an abstract Plancherel decomposition

L2(X+) =

∫ ⊕

Πunit(G)Hτ dµ(τ), Hτ = τ⊗Mτ .

In the group case: X+ = G under G×G-action, Harish-Chandra gives afar-reaching refinement of the decomposition above.In the Godement-Jacquet case G = GL(n, F ), zeta integrals are related tothe spectral decomposition of L2(GL(n, F )):

1 Plancherel formula describes the image of ξ 7→ π(ξ) ∈ EndC(Vπ),ξ ∈ C∞

c (X+), π in the tempered dual of G(F );2 for ξ ∈ S(Matn×n) and Re(λ) ≫ 0, the image ofZλ(ξ, · · · ) ↔ (π ⊗ | det |λ)(ξ) has a similar description involvingL(λ, π).


Lafforgue turned this approach over to study Braverman-Kazhdan program:

Given local factors L(s, π), one defines S by spectral means.

This leads the notion of functions of L-type, natural formulations ofFourier transforms, etc.

Heuristically very useful.In general: have no L-factor at hand, except when π is in the principalseries, etc.Putting the cart before the horse?

Issue of spectral decompositionClarify the relation between zeta integrals and spectral decomposition ofL2(X+) for general X+. Deduce this from properties of S (not conversely)


Gelfand-Kostyuchenko method

L2(X+) =∫ ⊕Πunit(G)Hτ dµ(τ), assuming the multiplicity space Mτ is

finite-dimensional.In general: the decomposition is not defined by operatorsL2(X+) → Hτ (think of L2(R) — classical Fourier analysis!)Rigged Hilbert spaces: S ↪→ L2(X+) dense continuous, requirement:it is pointwise defined by a family ατ : S → Hτ with dense images.OK for the minimalist choice S = C∞

c (X+) := C∞c (X+(F ),L 1/2).

Sufficient condition for the existence of (ατ )τ : S is nuclear separable;this appeared in Generalized Functions, Vol. IV.

Can embed

M∨τ ≃ HomG(F )(τ ⊗Mτ , τ) = HomG(F )(Hτ , τ) ↪→ HomG(F )(S, τ)

since Hτ is the completion of S w.r.t. the continuous semi-norm∥ατ (·)∥Hτ .


When S = C∞c (X+), Bernstein (1988) showed

HomG(F )(S, τ) ≃ HomG(F )(π,C∞(X+)) = Nπ

where π := τ∞. Summing up, we obtain a relation

Spectral decomposition (L2-theory) ↔ distinction (smooth theory).

S = C∞c (X+) is easier to work with.

Expectation: choice of S is closely related to affine embeddingsX+ ↪→ X.Different homogeneous spaces X+

1 , X+2 may have isometric L2

induced from some equivariant F : S2 ≃ S1, which is difficult to seefrom C∞

c test functions. Again, one may motivate by consideringL2(R) and the usual Schwartz space.


Functional equations: L2-aspectQuestion: why should we expect functional equations?

1 Disintegrate F : L2(X+2 )

∼−→ L2(X+1 ) using the abstract Plancherel

decomposition: get η(τ) : M(2)τ → M(1)

τ on multiplicity spaces. Howto handle η(τ)?

2 Suppose that F restricts to S2∼−→ S1. By Gelfand-Kostyuchenko,

M(i),∨τ ⊂ HomG(F )(Si, τ) and η(τ)∨ is induced from a transport of

structure via F . How to describe the image of M(i),∨τ ?

3 It is relatively easier to identify M(i),∨τ as a subspace of N (i)

π withπ = τ∞. How to compare the relevant embeddings of π intoC∞(X+) and S∨

i ?4 My proposal: use zeta integrals + meromorphic continuation. This

can be justified if for X+ = X+1 , X

+2 , C∞

c (X+) ↪→ S inducesHomG(F )(π|ω|λ,S∨) ↪→ HomG(F )(π|ω|λ, C∞

c (X+)∨)

for |ω|λ =∏

i |ωi|λi and λ ∈ ΛC in general position. Plus sometechnical assumptions...


Functional equations: smooth aspectAssuming the meromorphic continuation, etc. for zeta integrals, we deduce

Nπ ↪→ Lπ := {nice meromorphic invariant Bλ : πλ ⊗ S → C}

with πλ := π ⊗ |ω|λ. It maps φ to Bλ(v ⊗ ξ) = Zλ(ξ, φ(v)).Let K be the field of meromorphic functions on T := {|ω|λ : λ ∈ ΛC}.Obtain a K-linear map Nπ ⊗K ↪→ Lπ.Now work with X+

i ↪→ Xi for i = 1, 2. For ease of notations, assumeΛ1 = Λ2, i.e. allow the same twists by characters |ω|λ.

DefinitionThe local functional equation for F : S2

∼−→ S1 and π holds if:

L(1)π L(2)

π

N (1)π ⊗K N (1)

π ⊗K∃γ(π)

γ(π) ↔ γ(π, λ) : N (1)π → N (2)

π

meromorphic in λ


In comparison of the L2-aspect, one may infer (under some hypotheses aswe have seen) that γ(π, 0) restricts to η(τ)∨ for almost all τ and π := τ∞.This is compatible with some known properties of γ-factors: unitarity alongthe critical line for tempered irreps, etc.

QuestionHow to obtain local functional equations in this sense?

In the non-archimedean case, one approach is to establish multiplicity-oneof invariant bilinear forms Bλ : πλ ⊗ S → C for general λ.

This is closely related to the geometry of ∂X.It is possible to formulate a sufficient condition that impliesGodement-Jacquet case, and probably some other prehomogeneousvector spaces.


Speculations on the global case

Let F be a number field, A = AF .Same conditions on X+ ↪→ X, assuming the Schwartz spaces Sv

defined at each place v.Assume that “basic functions” ξ◦v ∈ Sv are chosen for almost all v ∤ ∞so that S =

⊗′v Sv makes sense.

Consider a continuous G(F )-invariant functional ϑ : S → C, eg.

ϑ(ξ) =∑

x∈X+(F )

evγ(ξ)

for appropriate evaluation maps evγ , with due care on thehalf-densities, etc.Frobenius reciprocity yields ξ 7→ ϑξ ∈ C∞(G(F )\G(A)).


It is natural to consider integrals

Zλ =

∫Xϕλϑξ, ℜ(λ) ≫ 0,

for ϕ: cusp form, ϕλ = ϕ|ω|λ, X = G(F )\G(A) divided by some suitablecentral subgroup of G(F∞). Some issues:

Convergence for ℜ(λ) ≫ 0 (look for sufficient conditions).Try to prove meromorphic continuation (hard).In principle, global functional equations should come from Poissonformulas relating Schwartz spaces of different spaces, as in the localcase. Idealistically: F : S2

∼−→ S1, ϑ(1)(Fξ) = ϑ(2)(ξ) (hard).One can relate Zλ to period integrals in its range of convergence;when periods factorize, Zλ also factorizes into local zeta integralstreated before.


Warning:All statements above are subject to revision.

Work in progress...


Documents

On certain zeta integrals · homogeneous spaces, i.e. 9 open Borel orbit. Let X+ be spherical homogeneous under Gand ˇ: irrep. Main issues in harmonic analysis include decomposition