eisenstein series and l–functions...eisenstein series and l{functions freydoon shahidi purdue...

Eisenstein Series and L–functions

Freydoon Shahidi

Purdue University

November 14, 2018

Abel Conference in honor of Robert Langlands

IMA - University of Minnesota

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 1 / 38

Timeline

Langlands wrote his mimeographed notes on Eisenstein series overthe period 1962–64.

Announced at Boulder’s AMS School: Algebraic Groups andDiscontinuous Subgroups in 1965 (Volume published in 1966).

Langlands’ notes were published as SLN544 in 1976.

Related work: Harish–Chandra SLN62, 1968,· · · ,Moeglin–Waldspurger, CUP, 1995, all mainly based on Langlands.


Eisenstein Series

k : Q, A : adeles of QG : connected reductive group/QX(G)Q = Q – rational characters of G

Define:

aG = Hom(X(G)Q,R), and

HG : G(A)→ aG by

exp 〈HG(g), χ〉 =∏v

|χ(gv)|v, g = (gv)v, χ ∈ X(G)Q


Eisenstein Series

Set

G(A)1 = {x ∈ G(A)|HG(x) = 0}, and

a∗G = X(G)Q ⊗Z R a∗G,C = a∗G ⊗R CFor an irreducible unitary representation π of G(A) and ν ∈ ia∗G, set

πν(x) = π(x) exp〈ν,HG(x)〉ia∗G - orbits of irreducible uni-tary representations of G(A)

←→ irreducible unitary rep-resentations of G(A)1

Here,

AG = split component of G.

= maximal split tours in the connected component

of identity of the center.


Eisenstein Series

P0 = M0N0, minimal parabolic

P = MN ⊇ P0, parabolic subgroup

AM = split component of M

DefineHM : M(A)→ aM = Hom(X(M)Q,R)

similarly, define exp〈ν,HM (x)〉, with ν ∈ a∗P,C = a∗M,C, x ∈M(A).

Let VP be the space of measurable functions

φ : N(A)M(Q)AM (R)0 \G(A)→ C,

such thatφx : m 7→ φ(mx) m ∈M(Q) \M(A)1

is in L2disc(M(Q) \M(A)1) for ∀x ∈ G(A), satisfying

‖φ‖2 =

∫K

∫M(Q)\M(A)1

|φ(mk)|2dmdk <∞.


Eisenstein Series

Then G(A) acts on VP by IP (ν, y) :

(IP (ν, y)φ)(x) = φ(xy) exp〈ν+ρP , HP (xy)〉·exp〈−(ν+ρP ), HP (x)〉.

The space VP is independent of ν ∈ a∗M,C, but ν appears in theaction IP (ν, ·) of G(A) on VP . In fact right regular action byy ∈ G(A) will take φ(x) exp〈ν + ρP , HP (x)〉 toφ(xy) exp〈ν + ρP , HP (xy)〉. To realize it again in VP , we need tomultiply with exp〈−(ν + ρP ), HP (x)〉.One can choose φ so that φx lands in specific cuspidal space oreven K–finite members of the cuspidal space. (See Langlands1966–Proc. of AMS, Algebraic Groups and DiscontinuousSubgroups.)


Eisenstein Series

For x ∈ G(A), φ ∈ VP and ν ∈ a∗P,C, define the Eisenstein seriesattached to φ as:

E(x, φ, ν) =∑

P (Q)\G(Q)

φ(δx) exp〈ν + ρP , HP (δx)〉

=∑

P (Q)\G(Q)

φν(δx)


Eisenstein Series

Langlands chose φ ∈ V 0P , the space of K–finite functions in VP , i.e.

dim 〈IP (ν, k)φ|k ∈ K〉 <∞

and the span is a subspace of a finite sum of irreduciblesubrepesentations of VP under IP (ν).

They are independent conditions and don’t depend on ν.

They must also be Z∞–finite.

The set V 0P is dense in VP . Let

(a∗P )+ = {ν ∈ a∗P |〈ν, α∨〉 > 0}

α = roots of aP in Lie(N).


Weyl Sets and Intertwining Operators

Suppose P, P ′ ⊃ P0, aP , aP ′ ⊂ a0.

W (aP , aP ′) = linear isomorphisms between aP and aP ′ obtainedby restricting elements in W (G,A0).

P and P ′ are associates if W (aP , aP ′) 6= φ.

Take s ∈W (aP , aP ′), ws ∈ G(Q) representative. Define the globalintertwining operator as follows.

M(s, ν) : VP → VP ′ ,

(M(s, ν)φ)(x) =

∫φ(w−1s nx) exp〈ν + ρP , HP (w−1s nx)〉

exp〈−(sν + ρP ′), HP ′(x)〉dn

where the integral is over N ′(A) ∩ wsN(A)w−1s \N ′(A).


Langlands’ Theorem on Eisenstein Series

Theorem:

(a) Suppose φ ∈ V 0P and Re(ν) ∈ ρP + (a∗P )+. Then both the series for

E(x, φ, ν) and M(s, ν)φ converge absolutely.

(b) Suppose φ ∈ V 0P . Then E(x, φ, ν) and M(s, ν)φ can be extended to

meromorphic functions of ν ∈ a∗P,C satisfying:

E(x,M(s, ν)φ, sν) = E(x, φ, ν)

andM(ts, ν) = M(t, sν)M(s, ν) (t ∈W (aP ′ , aP ′′ )).

If ν ∈ ia∗P , then both E(x, φ, ν) and M(s, ν)φ are analytic andM(s, ν) extends to a unitary operator from VP to VP ′ .


Constant term : Langlands

Let P, P ′ ⊃ P0, φ ∈ V 0P , ν ∈ a∗P,C, with P = MN and P ′ = M ′N ′.

Define the constant term of E(x, φ, ν) along P ′ by

EP ′(x, φ, ν) =

∫N ′(Q)\N ′(A)

E(nx, φ, ν)dn.

Assume ∀y ∈ G(A), φy : m 7→ φ(my),m ∈M(Q) \M(A)1, is inL2cusp(M(Q)\ M(A)1). Then for Re(ν) ∈ ρP + (a∗P )+

EP ′(x, φ, ν) =∑

s∈W (aP ,aP ′ )

(M(s, ν)φ)(x) exp(〈sν + ρP ′ , HP ′(x)〉).

Thus, no constant term unless P and P ′ are associate.


Constant term calculations

Maximal case: Assume P, P ′ are maximal, i.e., dim aP /aG = 1.Depending on whether W (aP , aP ) = {1} or {±1}, we have thefollowing cases.

(a) P ′ = P and P = self associate, i.e., W (aP , aP ) = {±1}

EP (x, φ, ν) = φν(x) +

∫N(A)

φν(w0nx)dn.

(b) P ′ = P but P is not self-associate, , i.e., W (aP , aP ) = {1}

EP (x, φ, ν) = φν(x)

(c) P ′ 6= P, (P ′)− = w−10 Pw0. Then

EP ′(x, φ, ν) =

∫N ′(A)

φν(w0nx)dn.


Applications of the Eisenstein Series

Continuous and residual spectrum

Trace formula

Applications in number theory, including

Integral representations of L-functions.


L–functions (Fall 1966):

(See “Messages to Mueller and Volpato, December 22 and 27, 2013” inpublications.ias.edu/rpl)Langlands calculated the constant term in the maximal parabolic cases.Recall: k = p-adic G = unramified group over kH(G(k), G(O)) = spherical Hecke algebra,

A0 ⊂ T ⊂ G W0 = W (G,A0)

T : maximally split maximal torus /k

A0 : maximal split torus of T/k.

G, A0, T = dual groups, LH = H o Γ


publications.ias.edu/rpl

Satake Isomorphism

Satake isomorphism: S : H = H(G(k), G(O)) ' C[A0]W0

C[A0] = polynomial ring of A0 = group algebra of X∗(A0).

Here is where Langlands comes in:Rep(LG) = subalgebra of C[LG] generated by characters of finitedimensional complex-analytic representations of LG.

U = Algebra of restrictions of elements of Rep(LG) to(Go σ)ss/IntG, σ = Frobenius.


Frobenius-Hecke parameters

Langlands establishes an explicit isomorphism

β : U ' C[A0]W0 ,

given for a representation ρ of LG by

trace ρ(to σ) =∑λ∈M

cλλ(t) (M := X∗(A0)),

where cλ is the trace of ρ(σ) on the λ-eigenspace of A0; (Langlandsnotation, proved in “Problems in the Theory of AutomorphicForms”.)

This is beyond what Satake did which basically proves thatspherical Hecke algebra H is commutative, a matter ofrepresentation theory · · ·


Frobenius-Hecke parameters

What Langlands does is to put this in the context of L–functionsand number theory by relating it to representation ring of LG;something that Satake would not have thought of.

Given an irreducible unramified representation π of G(k), letA(π) ∈ A0/W0 (recall Xun(T (k)) ' A0) be the corresponding orbitunder W0. There exists an isomorphism

α : A0/W0 ' (Go σ)ss/IntG

which induces β.

We then set c(π) := α(A(π)), the Frobenius–Hecke conjugacy class(not Satake parameter) attached to π.


L-functions

Thus Langlands showed that

Unramified representation theory of G(k)

←→ C− algebra homomorphisms of H(G(k), G(O))

←→ semi-simple conjugacy classes in Go σ

←→ number theory / L–functions

Langlands: H(G(k), G(O))←→ Representation ring of LG.Then as is well–known, we have Langlands unramifiedL–function attached to π and ρ defined by

L(s, π, ρ) = det(I − ρ(c(π))q−s)−1, q = |O/P |.


L-functions

Langlands was led to this definition after he calculated theconstant term of the Eisenstein series in the maximal cases:

P = MN = maximal parabolic

A = split component of M

a = real Lie alg. of A

= Hom(X(A)Q,R).

There exists a unique α ∈ ∆, α 6∈ ∆M . Defineα = 〈ρP , α〉−1ρP ∈ a∗, and ν = sα ∈ a∗C, s ∈ C.

r = adjoint action of LM on Ln = Lie algebra of LN

Let Vi = {Xβ∨ ∈Ln|(α, β∨) = 〈α, β〉 = i}. Then ri := r|Vi isirreducible for 1 ≤ i ≤ m and r = ⊕mi=1ri, where m = nilpotencedegree of LN .


L-functions

Langlands calculated (Published in Euler Products)∫N ′(A)

φsα(w0nx)dn,

choosing φ to take values in the space of π = ⊗vπv, a cuspidalrepresentation of M(A):

Let S be a finite set of places such that for each v ∈ S, πv isunramified. Then, up to a product of local intertwining operatorsfor all v ∈ S,

(contd.)


L-functions

(M(w0, sα)φ)(e) =

∫N ′(A)

φsα(won)dn ∼m∏i=1

LS(is, π, ri)

LS(1 + is, π, ri),

whereLS(s, π, ri) =

∏v 6∈S

L(s, πv, ri · ηv)

withLMv

ηv−→LM.

Langlands notices the immediate meromorphy of LS(s, π, r1) ifm = 1 and more generally if one knows meromorphy of all others.(See Borel, Corvallis, Automorphic L– functions §14.5.)Otherwise, there is an induction which gives the meromorphy ofeach LS(s, π, ri).


L-functions

For each i, ∃Gi with M as a Levi and ri appears as r′1 for action ofLM on Lni as ⊕m′

j=1r′j with m′ < m. (Observed later.)

This is where new Euler products appeared, showing the way forLanglands’ definition of an automorphic L–function which he wassearching for quite sometime.


L-functions

Theorem:Each LS(s, π, ri) is meromorphic on all of C.

A remark on L–groups. Langlands noticed the need forintroducing a dual group (L–group) after he considered the pair(G = SO2n+3,M = SO2n+1 ×GL1) (similarly for(G = Sp2n+2,M = Sp2n ×GL1)) for which the dimension of r1was not the standard representation of SO2n+1, but rather that ofSp2n. This led him to the definition of the dual group and laterthe L–group in “Problems in the theory of automorphic forms”.


Langlands-Shahidi Method

Langlands “Euler products” manuscript led to a body of work,now called Langlands–Shahidi method with many consequences. Itstarted while I was at IAS during 1975–1976 academic year. Hepointed me to the example of (G2, GL2) with GL2 generated bythe long root β which gives L(s, π,Sym3) for a cuspidalrepresentation π of GL2(A), which he called “particularly striking”in E.P. He gave me a copy of his letter to Godement where heconjectured the Casselman–Shalika’s formula for unramifiedWhittaker functions and sketched a way to prove the functionalequation of these L–functions in a fully unramified setting usingthat of E.S. I proved the functional equation for L(s, π,Sym3) inmy paper in Compositio Math. 1978, which he kindly solicited foras one of its editors.

And that’s where things started!


IAS - 1975


Non-constant Terms

Briefly one needs to calculates:

Eψ(x, φ, ν) =

∫U(k)\U(A)

E(ux, φ, ν)ψ(u)du,

where ψ is a generic character of U(k) \ U(A), k a number field,B = TU , B = P0, minimal parabolic. This requires G to bequasisplit. Then one shows using Casselman-Shalika formula,

Eψ(e, φ, sα) =∏v∈S

Wv(ev) ·∏v 6∈S

m∏i=1

L(1 + is, πv, ηv · ri)−1

∼m∏i=1

LS(1 + is, π, ri)−1.


Non-vanishing of L-functions

Since E and thus Eψ are holomorphic on√−1a∗, this implies the

non-vanishing of∏mi=1 L

S(1 + is, π, ri) on the line “Re(s) = 1”.

More precisely, the non-vanishing of

m∏i=1

LS(1 + it√−1, π, ri) 6= 0 (t ∈ R).

This has many consequences, e.g., classification of automorphicforms on GLn due to Jacquet–Shalika.

Functional equation for each L(s, π, ri) follows from that of theEisenstein series after the introduction of “local coefficients” todefine local factors at ramified places:


Globally Generic Forms

Let π be a cuspidal automorphic representation of M(A). Assumethere exists ϕ ∈ H(π) such that∫

UM (Q)\UM (A)

ϕ(u)ψ(u)du 6= 0;

thus π is ψ–generic–(globally). We may assume ϕ is KM–finiteand ZM,∞–finite.

Then for ν ∈ a∗M,C

I(ν, π) = IndG(A)P (A)π ⊗ exp〈ν,HM ( )〉 ⊗ 1 ↪→ VP exp〈ν + ρP , HM ( )〉.


Intertwining Operators

We can then restrict M(w0, ν) to I(ν, π). SinceI(ν, π) = ⊗vI(ν, πv), define

M(w0, ν, π) := M(w0, ν)|I(ν, π) = ⊗vA(ν, πv, w0),

identifyingX(M)k ⊗ C ↪→ X(M)kv ⊗ C

anda∗M,C ↪→ a∗M/kv ,C.

ThenA(ν, πv, w0) : I(ν, πv) −→ I(w0(ν), w0(πv))

is called the standard intertwining operator attached to w0.


Local Coefficients

To each I(ν, πv) there is attached a canonical inducedJacquet–Whittaker functional λψv(ν, πv).

By multiplicity one (locally at v), there exists Cψv(ν, πv, w0) ∈ C,a local coefficient, such that

Cψv(ν, πv, w0)λψv(w0(ν), w0(πv)) ·A(ν, πv, w0) = λψv(ν, πv)

Crude Functional Equation:

m∏i=1

LS(is, π, ri) =∏v

Cψv(sα, πv, w0)

m∏i=1

LS(1− is, π, ri).


γ-factors

Theorem: [Sh.]Assume P is maximal. There exists a family of γ–factors

γ(s, πv, ri,v, ψv) and a product of Langlands λ–functions λG(ψv, w0) foreach v such that

Cψ(sα, πv, w0) = λG(ψv, w0)−1

m∏i=1

γ(is, πv, ri,v, ψv).

We thus get a true functional equation for each L(s, π, ri). Thefactors are Artin factors whenever k is archimedean for all ri throughLanglands correspondence.


Artin factors

For p-adic places, the γ–factors are Artin factors through localLanglands correspondence for GLn in the cases:

L(s, π,Λ2) and L(s, π,Sym2) for π = irreducible representation ofGLn(k) – k local field (Cogdell–Sh–Tsai); the pair is:G = GSp2n, M = GLn = Siegel Levi.

L(s, π,Λ2 ⊗ χ) and L(s, π,Sym2 ⊗ χ) (Dongming She); the pair isG = GSpin2n+1, M = GLn ×GL1

L(s, π,Asai), π = irreducible representation of GLn(K),[K : k] = 2 (Daniel Shankman); Asai= Asai representation ofGLn(C)×GLn(C) o ΓK/k and realizing GLn(K) asResK/kGLn(k). The pair is G = U(n, n) and M = ResK/kGL(n).

Also equal in the cases of classical groups as Rankin–productsmatching for GLn ×G.


Normalization of Intertwining Operators

These factors also normalize intertwining operators A(sα, πv, w0)in the form

m∏i=1

ε(is, πv, ri,v, ψv)L(1 + is, πv, ri)

L(is, πv, ri)A(sα, πv, w0),

as conjectured by Langlands. They play a central role inintertwining relations of Arthur in his trace formula, as well as,reducibility questions for induced representations (Goldberg, Sh,. . . ).


Normalization of Intertwining Operators

The local coefficients defining γ–factors also normalizeintertwining operators and certain singular versions of themproduce normalized operators which give the “Fourier transforms”in the case of doubling method of (PS–Rallis / Lapid) for standardL–functions of classical groups in the context of Braverman –Kazhdan / Ngo program generalizing Godement – Jacquet. Manymore applications exist and are expected. (SeeCai–Frienderg–Kaplan.)


Functoriality

One of the striking consequences of the method has beenSymmetric Power (Symm) transfers of cuspidal representationsfrom GL2(A) to GLm+1(A) for m = 3 (Kim–Sh) and m = 4(Kim); two surprising and completely unaccessible cases offunctoriality before, as well as that of generic transfer fromclassical groups to GL(n) (Cogdell–Kim–PS–Sh). (Arthur has nowestablished the general case using trace formula.)


Progress towards Ramanujan Conjecture

Remarkable sharpening on bounds for Fourier coefficients(Frobenius–Hecke parameters), both at p-adic and archimedeanplaces of Maass forms have followed (Kim–Sarnak), again pointingto Langlands idea of using Symmetric power L–functions to provethe Ramanujan’s conjecture: Holomorphy of L(s, π,Symm), forRe(s) > 1 and for all m implies the conjecture. We now have quitea bit of information about these L–functions for m up to 9,including invertibility at Re(s) = 1, when m ≤ 8.


Congratulations on the Abel Prize!


eisenstein series and l–functions...eisenstein series and l{functions freydoon shahidi purdue...

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