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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 over the period 1962–64.

Announced at Boulder’s AMS School: Algebraic Groups and Discontinuous 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.

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 2 / 38

Eisenstein Series

k : Q, A : adeles of Q G : connected reductive group/Q X(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

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 3 / 38

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 C

For 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.

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 4 / 38

Eisenstein Series

P0 = M0N0, minimal parabolic

P = MN ⊇ P0, parabolic subgroup AM = split component of M

Define HM : 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 that φx : 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 the action IP (ν, ·) of G(A) on VP . In fact right regular action by y ∈ G(A) will take φ(x) exp〈ν + ρP , HP (x)〉 to φ(xy) exp〈ν + ρP , HP (xy)〉. To realize it again in VP , we need to multiply with exp〈−(ν + ρP ), HP (x)〉. One can choose φ so that φx lands in specific cuspidal space or even K–finite members of the cuspidal space. (See Langlands 1966–Proc. of AMS, Algebraic Groups and Discontinuous Subgroups.)

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 6 / 38

Eisenstein Series

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

E(x, φ, ν) = ∑

P (Q)\G(Q)

φ(δx) exp〈ν + ρP , HP (δx)〉

= ∑

P (Q)\G(Q)

φν(δx)

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 7 / 38

Eisenstein Series

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

dim 〈IP (ν, k)φ|k ∈ K〉 0}

α = roots of aP in Lie(N).

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 8 / 38

Weyl Sets and Intertwining Operators

Suppose P, P ′ ⊃ P0, aP , aP ′ ⊂ a0. W (aP , aP ′) = linear isomorphisms between aP and aP ′ obtained by 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 global intertwining 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).

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 9 / 38

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)

and M(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 and M(s, ν) extends to a unitary operator from VP to VP ′ .

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 10 / 38

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 in L2cusp(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.

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 11 / 38

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 the following 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.

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 12 / 38

Applications of the Eisenstein Series

Continuous and residual spectrum

Trace formula

Applications in number theory, including

Integral representations of L-functions.

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 13 / 38

L–functions (Fall 1966):

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

Ĝ, Â0, T̂ = dual groups, LH = Ĥ o Γ

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

publications.ias.edu/rpl

Satake Isomorphism

Satake isomorphism: S : H = H(G(k), G(O)) ' C[Â0]W0 C[Â0] = polynomial ring of Â0 = group algebra of X∗(Â0). Here is where Langlands comes in: Rep(LG) = subalgebra of C[LG] generated by characters of finite dimensional complex-analytic representations of LG.

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

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 15 / 38

Frobenius-Hecke parameters

Langlands establishes an explicit isomorphism

β : U ' C[Â0]W0 ,

given for a representation ρ of LG by

trace ρ(to σ) = ∑ λ∈M̂

cλλ(t) (M̂ := X ∗(Â0)),

where cλ is the trace of ρ(σ) on the λ-eigenspace of Â0; (Langlands notation, proved in “Problems in the Theory of Automorphic Forms”.)

This is beyond what Satake did which basically proves that spherical Hecke algebra H is commutative, a matter of representation theory · · ·

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 16 / 38

Frobenius-Hecke parameters

What Langlands does is to put this in the context of L–functions and 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), let A(π) ∈ Â0/W0 (recall Xun(T (k)) ' Â0) be the corresponding orbit under W0. There exists an isomorphism

α : Â0/W0 ' (Ĝo σ)ss/IntĜ

which induces β.

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

Freydoon Shahidi (Purdue University)Eisenstein Series and L–functions November 14, 2018 17 / 38

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 Ĝo σ ←→ number theory / L–functions

Langlands: H(G(k), G(O)

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