Intertwining Operators, L–Functions,

and Representation Theory*

by Freydoon Shahidi** Purdue University

*Lecture Notes of the Eleventh KAIST Mathematics Workshop 1996, Taejon, Korea; Edited by Ja Kyung Koo

**Partially supported by NSF Grant DMS9622585

Table of Contents

Introduction

1. Notation

2. Modular forms as forms for GL_{2}

3. Maass forms and Ramanujan–Petersson’s Conjecture 4. Automorphic cuspidal representations

5. L–functions for GL_{2}

6. Cuspidal representations for GL_{n}

7. Rankin–Selberg L–functions for GL_{n}

8. Applications of properties of L(s, π × π′)

9. The method 9.1 L–groups 9.2 Parabolic subgroups 9.3 Cusp forms and Eisenstein series 9.4 Constant term and intertwining operators 9.5 L–functions in the constant term 9.6 Generic representations 9.7 Local coefficients 9.8 Main global results 9.9 Local factors and functional equations

10. Some local applications

Introduction

These notes are based on a series of twelve lectures delivered at the Korea Advanced Institute of Science and Technology (KAIST), Taejeon, during their biannual workshop in algebra that took place over the period of July 29–August 4, 1996.

As its main goal, I have chosen to sketch a proof of the estimate

q_{v}^{−1/5} < |α_{v}| < q_{v}^{1/5}

for Hecke eigenvalues of a cusp form on GL_{2}(𝔸_{F}), where F is an arbitrary number field, using the method initiated by Langlands and developed by myself. The first half of the notes (Sections 1–8) is devoted to developing the problem and introducing some important L–functions (symmetric power L–functions for GL(2) and Rankin– Selberg product L–functions for GL(m) × GL(n)). The second half (Section 9) is spent to explain the method in the split case and finally prove the estimate. The notes are concluded by describing a number of local results in harmonic analysis and representation theory of local groups that can be proved using this method (Section 10).

It is a pleasure to thank Professor Ja Kyung Koo, chairman of the Department of Mathematics at KAIST, for his wonderful and memorable hospitality, as well as his mathematical interest.

Many of the initial efforts in arranging this visit was done by Dr. Hi–joon Chae of KAIST to whom I would like to extend my thanks and appreciation. I should also thank the Department of Mathematics, especially the algebra group, at KAIST for their hospitality.

Last but not least, I should thank my audience. It was a pleasure to deliver these talks to such a pleasant and enthusiastic audience, a number of whom traveled long distances to participate in this workshop.

§1. Notation

Throughout these lectures F will be either a local or global field of characteristic zero. More precisely, F is a global field of characteristic zero if it is a number field, i.e. a finite extension of ℚ. A local field of characteristic zero is either ℝ, ℂ, or a finite extension of ℚ_{p}, the field of p–adic numbers, i.e. the completion of ℚ with respect to p–adic absolute value | |_{p}, |rp^{m}/s|_{p} = p^{−m}, p ∤ rs. Then | |_{p} extends to an absolute value | | = | |_{v}, on F where v is the place lying over p.

Suppose F is a p–adic local field (non–archimedean). Then

O = {x ∈ F ||x| ≤ 1}

is a ring, called ring of integers. The set

P = {x ∈ O||x| < 1}

is the maximal ideal of O and O^{∗} = O\P is the group of units. The field \overline{F} = O/P is called the residue field of F whose cardinality which is finite is denoted by q = p^{f}. If e is the ramification index of p in F, i.e. pO = P^{e}, then [F : ℚ_{p}] = ef. The ideal P is principal and one usually fixes a uniformizing parameter ϖ ∈ P for P. Then |ϖ|_{v} = q^{−1}.

Now let F be a number field. For each place v of F, let F_{v} be its completion with respect to v. A place is an equivalence class of absolute values for F. We will say v < ∞ if v|p for some rational prime p. Otherwise F_{v} ≅ ℝ or ℂ and we say v = ∞. In either case F ⊗_{ℚ} ℚ_{p} = ⊕_{v|p}F_{v}, F ⊗_{ℚ} ℝ = ⊕_{v|∞}F_{v}. If v < ∞, let O_{v}, P_{v}, \overline{F}_{v} = O_{v}/P_{v}, q_{v}, ϖ_{v}, and O_{v}^{∗} be as before.

Let 𝔸 = 𝔸_{F} ⊂ ∏_{v} F_{v} be the subring of ∏_{v} F_{v} defined by

𝔸 = {(x_{v})_{v}|x_{v} ∈ O_{v}, ∀′v},

i.e. the direct limit of ∏_{v} F_{v} with respect to ∏_{v<∞} O_{v}. With direct limit topology, 𝔸 becomes a locally compact ring, called the ring of adeles of F. The group of units in 𝔸, 𝔸^{∗} = 𝕀 will be

𝔸^{∗} = {(x_{v})_{v}|x_{v} ∈ F_{v}^{∗}, x_{v} ∈ O_{v}^{∗}, ∀′v}.

Considered as the direct limit of ∏_{v} F_{v}^{∗} with respect to ∏_{v<∞} O_{v}^{∗} and with direct limit topology, 𝔸^{∗} is called the group of ideles of F. Then one can define

∥(x_{v})_{v}∥ = ∏_{v} |x_{v}|_{v}

whose kernel I^{1} = (𝔸^{∗})^{1} contains F^{∗} by diagonal imbedding. Then F^{∗} is a lattice in I^{1}, i.e. F^{∗} is discrete in I^{1} and I^{1}/F^{∗} is compact. So is F in 𝔸. The finiteness of class number, the number of ideals (fractional) modulo the principal ones, is equivalent to the compactness of I^{1}/F^{∗}.

Fix an integer n > 0 and let GL_{n}(R) be the group of invertible matrices with R–entries, where R is a commutative ring with 1.

Let R = 𝔸. Then GL_{n}(𝔸) is called the adelization of GL_{n} over F. If g ∈ GL_{n}(𝔸), then g = (g_{v})_{v}, g_{v} ∈ GL_{n}(F_{v}) with g_{v} ∈ GL_{n}(O_{v}), ∀′v < ∞.

Let G be a Zariski–closed subgroup of a GL_{n}(F_{a}), where F_{a} is an algebraic closure of F. Assume G is defined over F. Then it is defined over every F_{v}. Let G_{v} = G(F_{v}). Then g ∈ GL_{n}(𝔸) belongs to G(𝔸), adelization of G/F, if and only if each g_{v} ∈ G_{v}. For almost all v < ∞, G is defined over O_{v} and therefore G(O_{v}) makes sense and G_{v} ∩GL_{n}(O_{v}) = G(O_{v}). Consequently (g_{v})_{v} ∈ G(𝔸) implies that g_{v} ∈ G(O_{v}) for almost all v < ∞.

§2. Modular forms as forms for GL_{2}

In this section we shall briefly review how classical modular forms and automorphic forms on GL_{2}(𝔸_{ℚ}) are related.

Let Γ be a congruence subgroup for SL_{2}, i.e. a subgroup of SL_{2}(ℤ), containing a principal congruence subgroup

Γ_{N} = \{g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ)|g ≡ I(mod N)\}

for some N. The group SL_{2}(ℤ) and therefore every congruence subgroup Γ acts on upper half plane 𝔥

𝔥 = {z ∈ ℂ|Im(z) > 0}

by

( \begin{matrix}a & b \\ c & d \end{matrix} ) · z = \frac{az + b}{cz + d}.

In fact Im(g · z) = Im(z)/|cz + d|^{2} as long as g ∈ SL_{2}(ℝ). Let 𝔥^{∗} be the compactification of 𝔥 by adding cusps of Γ to 𝔥. Roughly speaking, a modular form of weight k > 0, k an integer, is a complex function on 𝔥^{∗} which is holomorphic there and satisfies

(2.1)  f(γ · z) = (cz + d)^{k}f(z)  (γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ).

We may fix an integer N > 0 and consider the Hecke subgroup

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) |c ≡ 0(N)\} ⊂ SL_{2}(ℤ).

Given a character χ mod N of (ℤ/Nℤ)^{∗}, extended to all of ℤ, we may consider holomorphic forms f such that

f(γ · z) = χ(a)^{−1}(cz + d)^{k}f(z), γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N).

Then given any congruence subgroup Γ and a holomorphic form with respect to Γ, one can find a Γ_{0}(N) and a χ such that f becomes a χ–modular form with respect to Γ_{0}(N). Thus one may only consider forms with respect to Γ_{0}(N) but with respect to arbitrary characters of (ℤ/Nℤ)^{∗}. A modular form f is a cusp form if it vanishes on every cusp of Γ (or Γ_{0}(N)). The best reference for this is [Shi1].

Let us show how modular cusp forms on 𝔥 are in fact functions on GL_{2}(ℚ)\GL_{2}(𝔸_{ℚ}), i.e. an automorphic form. We refer to [B,G] for details.

First observe that every g ∈ SL_{2}(ℝ) can be written as

g = [ \begin{matrix}y^{1/2} & xy^{−1/2} \\ 0 & y^{−1/2} \end{matrix} ] k(θ) (y > 0)

with

k(θ) = ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) .

Thus

g ↦ (z = x + iy ∈ 𝔥, θ)

and g · i = z since k(θ) · i = i. Given a modular cusp form f, define:

ϕ_{f}(g) = f(g · i)(ci + d)^{−k}
= f(g · i)y^{\frac{k}{2}} e^{−iθk}.

If j(g, z) = cz + d for g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ), then

j(g_{1}g_{2}, z) = j(g_{1}, g_{2} · z)j(g_{2}, z).

Thus

j(γg, i) = j(γ, g · i)j(g, i)

from which we get

ϕ_{f}(γg) = f(γg · i)j(γg, i)^{−k}
= f(g · i)j(γ, g · i)^{k}j(γg, i)^{−k}
= ϕ_{f}(g)

It is then clear that ϕ_{f} is bounded as a function of Γ\G, G = SL_{2}(ℝ), and by the finiteness of the volume of Γ\G, ϕ_{f} ∈ L^{2}(Γ\G). Thus the space of modular cusp forms with respect to Γ can be embedded into L^{2}(Γ\G). It can be easily seen that

∫∫_{F} |f(z)|^{2}y^{k} \frac{dxdy}{y^{2}} = ∫_{Γ\G} |ϕ_{f}(g)|^{2}dg < ∞

where F is a fundamental domain.

One likes to find a larger space so that it will cover for all Γ’s. It is here that we use GL_{2}(𝔸_{ℚ}). We may assume Γ = Γ_{0}(N) for some N and assume f is attached to a character χ.

It can be shown, using approximation, that given Γ_{0}(N)

GL_{2}(𝔸_{ℚ}) = GL_{2}(ℚ)GL_{2}^{+}(ℝ) ∏_{p∤N} GL_{2}(ℤ_{p}) ∏_{p|N} K_{p,N}

K_{p,N} = GL_{2}(ℤ_{p}), for p ∤ N, while K_{p,N} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ_{p})|c ≡ 0(N)\} , p|N

and GL_{2}^{+}(ℝ) is the subgroup of elements with positive determinant, i.e. the connected component of GL_{2}(ℝ). Moreover

GL_{2}(ℚ) ∩ GL_{2}^{+}(ℝ) ∏_{p∤N} GL_{2}(ℤ_{p}) ∏_{p|N} K_{p,N} = Γ_{0}(N).

For g ∈ GL_{2}^{+}(ℝ), define

j(g, z) = (cz + d)(det g)^{−1/2}.

The character χ of (ℤ/Nℤ)^{∗} then defines a character of ℚ^{∗}\𝔸_{ℚ}^{∗}, write χ = ⊗_{p}χ_{p}. Each χ_{p} defines a character

[ \begin{matrix}a & b \\ c & d \end{matrix} ] ↦ χ_{p}(a)

of K_{p,N} and thus we get a character of

K_{0}^{N} = ∏_{p∤N} GL_{2}(ℤ_{p}) ∏_{p|N} K_{p,N}.

Then we let

ϕ̃_{f}(γg_{∞}k_{0}) = f(g_{∞} · i)j(g_{∞}, i)^{−k}χ(k_{0}).

It can then easily be checked that this is a well defined function and

ϕ̃_{f} ∈ L_{0}^{2}(Z(𝔸_{ℚ})GL_{2}(ℚ)\GL_{2}(𝔸_{ℚ}), χ),

the space of L^{2}–functions φ on Z(𝔸_{ℚ})GL_{2}(ℚ)\GL_{2}(𝔸_{ℚ}) which satisfy φ(zg) = χ(z)φ(g),

z ∈ Z(𝔸_{ℚ})

∫_{ℚ\𝔸_{ℚ}} φ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) dx = 0, ∀′g.

The map f ↦ ϕ̃_{f} can now be extended to all of modular cusp forms, holomorphic or non–holomorphic.

Now let GL_{2}(𝔸_{ℚ}) act by right regular action on the Hilbert space

L^{2}(Z(𝔸_{ℚ})GL_{2}(ℚ)\GL_{2}(𝔸_{ℚ}), ω),

ω a unitary character of ℚ^{∗}\𝔸_{ℚ}^{∗}. It decomposes to a continuous part and a discrete part.

¿From now on assume all the modular forms f are eigenvalues of Hecke operators for all p. Recall that if f is of weight k, then the p–th Hecke operator T_{k}(p) is defined by

T_{k}(p)f(z) = p^{k−1} ∑_{{a>0 \atop ad=p}} ∑_{b=0}^{d−1} f (\frac{az + b}{d}) d^{−k}.

For a modular cusp form f, ϕ̃_{f} will belong to the discrete part. Moreover

∫_{ℚ\𝔸} ϕ̃_{f} (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} )) dx = 0 (∀g ∈ GL_{2}(𝔸_{ℚ}))

and in fact the space of cusp forms are exactly those L^{2}–functions which satisfy the above condition for ∀′g ∈ GL_{2}(𝔸_{ℚ}).

In conclusion, the set of normalized cuspidal eigenfunctions of all weights and for all Hecke operators, holomorphic or non, are in one–one correspondence with irreducible constituents of

L_{0}^{2}(Z(𝔸_{ℚ})GL_{2}(ℚ)\GL_{2}(𝔸_{ℚ}), ω)

for all ω.

One can then translate problems from classical theory to representation theory and try to solve them. One important one will be discussed next.

§3. Maass forms and Ramanujan–Petersson’s Conjecture

Holomorphic modular forms are defined to satisfy the property that the differential f(z)(dz)^{\frac{k}{2}} , k = even, remains invariant under the action of congruence subgroup Γ, i.e.

f(γ · z)(d(γ · z))^{\frac{k}{2}} = f(z)(dz)^{\frac{k}{2}} .

The only holomorphic modular functions (i.e. k = 0) are constants and therefore no non–zero cuspidal holomorphic modular function exists. On the other hand if we allow the Laplace–Beltrami operator

Δ = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} )

to have a non–zero eigenfunction (a real analytic function) say

Δf = \frac{1}{4}(1 − s^{2})f,

we can then pick up non–holomorphic modular functions i.e. forms of weight zero, which are cusp forms, the so called Maass forms. More precisely, one needs functions f : 𝔥^{∗} ⟶ ℂ for which

1. f(γ · z) = f(z) γ ∈ Γ,

2. Δf = \frac{1}{4}(1 − s^{2})f, s ∈ iℝ or −1 < s < 1,

3. f is bounded, 4. f is cuspidal,

5. f is an eigenfunction for all Hecke operators.

Then it is an easy exercise to show that

f(x + iy) = ∑_{n≠0}(|n|y)^{1/2}a_{n}K_{s/2}(2π|n|y)e^{2πinx},

where K_{ν} satisfies

z^{2} \frac{d^{2}K_{ν}}{dz^{2}} + z \frac{dK_{ν}}{dz} − (z^{2} + ν^{2})K_{ν} = 0

with

K_{ν}(z) ∼ \sqrt{\frac{π}{2z}} e^{−z}

as z ⟶ +∞. The numbers a_{n} are called the Fourier coefficients of f. Let us say f is normalized if a_{1} = 1.

There is an analogue of classical Ramanujan–Petersson’s conjecture. More precisely:

Suppose f is a normalized Maass form which is an eigenfunction for every Hecke operator. Then

|a_{p}| ≤ 2p^{−1/2}.

Define α_{p} by

a_{p} = p^{−1/2}(α_{p} + α_{p}^{−1}).

Then by R–P conjecture

(α_{p} − α_{p}^{−1})^{2} ≤ 0,

since a_{p} is real. This implies that α_{p} − α_{p}^{−1} is pure imaginary or

Re(α_{p}) − \frac{Re(α_{p})}{|α_{p}|^{2}} = 0.

Since Re(α_{p}) ≠ 0, |α_{p}| = 1, and therefore

|a_{p}| ≤ 2p^{−1/2} ⇔ |α_{p}| = 1,

i.e. Ramanujan–Petersson’s conjecture demands |α_{p}| = 1.

§4. Automorphic cuspidal representations

We now assume F is an arbitrary number field and consider L_{0}^{2}(Z(𝔸)GL_{2}(F)\GL_{2}(𝔸), ω), where ω is a (unitary) character of Z(F)\Z(𝔸) ≅ F^{∗}\𝔸^{∗} and L_{0}^{2} is the space of all the square integrable functions φ on the quotient space which satisfy:

φ(zg) = ω(z) φ(g) (z ∈ Z(𝔸)) 1)

∫_{F \𝔸} φ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) dx = 0 (∀′g ∈ GL_{2}(𝔸)). 2)

Under the right regular action of GL_{2}(𝔸) this space decomposes to a direct sum of irreducible subrepresentations each of which appear with multiplicity one [JL, Go].

Let π be an irreducible constituent as above. Then there is a non–unique decomposition of π into an infinite restricted tensor product of irreducible representation π_{v} of each local group GL_{2}(F_{v}) for every place v of F. Almost all of π_{v}’s have a vector fixed by the action of GL_{2}(O_{v}) and it’s the choices made for these vectors which leads to the decomposition π = ⊗_{v}π_{v} in a non–unique way. But the classes of representations π_{v} are all unique.

A representation π_{v} which has a vector fixed by GL_{2}(O_{v}) is called a class one, spherical, or unramified representation. It can be realized on the space V (μ_{1,v}, μ_{2,v}) of smooth functions

f : GL_{2}(F_{v}) ⟶ ℂ

which satisfy

f (( \begin{matrix}a_{1} & x \\ 0 & a_{2} \end{matrix} ) g) = μ_{1,v}(a_{1})μ_{2,v}(a_{2})|a_{1}/a_{2}|_{v}^{1/2}f(g)

where μ_{v,1} and μ_{v,2} are a pair of unramified quasicharacters of F_{v}^{∗}, i.e. μ_{v,i}|O_{v}^{∗} = 1, a_{1}, a_{2} ∈ F_{v}^{∗}, and x ∈ F_{v}. Moreover μ_{1,v}/μ_{2,v} ≠ | |_{v}^{±1} as quasicharacters. If we denote the right regular action of GL_{2}(F_{v}) on V (μ_{1,v}, μ_{2,v}) by I(μ_{1,v}, μ_{2,v}), the representation unitarily induced from μ_{1,v} and μ_{2,v}, then I(μ_{1,v}, μ_{2,v}) and I(μ_{2,v}, μ_{1,v}) are equivalent and if I(μ_{1,v}, μ_{2,v}) ≅ I(μ′_{1,v}, μ′_{2,v}), then (μ_{1,v}, μ_{2,v}) = (μ′_{1,v}, μ′_{2,v}) or (μ_{1,v}, μ_{2,v}) = (μ′_{2,v}, μ′_{1,v}).

Now assume π = ⊗_{p}π_{p} is generated by one of our ϕ̃_{f}, where f is a classical modular cusp form with respect to a Hecke subgroup Γ_{0}(N). Then for each p∤N, π_{p} is of class one and if π_{p} = I(μ_{1,p}, μ_{2,p}), then μ_{1,p}(p) = α_{p} and μ_{2,p}(p) = α_{p}^{−1} or vice– versa. Thus R–P conjecture is equivalent to characters μ_{1,p} and μ_{2,p} being unitary or said in the language of representation theory I(μ_{1,p}, μ_{2,p}) must be tempered. One can then formulate a generalized R–P conjecture for an arbitrary number field.

If π = ⊗_{v}π_{v} is an irreducible cuspidal representation of GL_{2}(𝔸), then each π_{v} is tempered.

Not every tempered representation of GL_{2}(F_{v}) is of the form of above induced representation. The so called discrete series representations, i.e. those which appear discretely in L^{2}(GL_{2}(F_{v})) will provide the rest of them.

This is a very difficult conjecture and due to Deligne’s work we know its validity, but only for holomorphic cusp forms, i.e. those for which every π_{v}, v archimedean, is in the discrete series.

If π_{v} is of class one, then π_{v} ≅ I(μ_{1,v}, μ_{2,v}) ≅ I(μ_{2,v}, μ_{1,v}) and the choices of μ_{1,v} and μ_{2,v} are uniquely given by π_{v} up to permutation of the two characters. Moreover the two characters are determined uniquely by their evaluations at ϖ_{v}. Let A_{v} denote the conjugacy class of

( \begin{matrix}α_{v} & 0 \\ 0 & β_{v} \end{matrix} )

in GL_{2}(ℂ), where α_{v} = μ_{1,v}(ϖ_{v}) and β_{v} = μ_{2,v}(ϖ_{v}). Thus class of π_{v} is uniquely determined by A_{v}. Moreover, π = ⊗_{v}π_{v} is uniquely determined by {A_{v}|v ∉ S}, where S is a finite set of places of F such that π_{v} is of class one, whenever v ∉ S. By strong multiplicity one theorem, the choice of S is irrelevant.

The Ramanujan–Petersson’s conjecture is then equivalent to |α_{v}| = |β_{v}| = 1 for ∀ v ∉ S.

When [F : ℚ] = 1, the best estimate is due to Luo–Rudnick–Sarnak [LRSa] and Bump–Duke–Hoffstein–Iwaniec [BDHI] and is

p^{−5/28} ≤ |α_{p}| and |β_{p}| ≤ p^{5/28}.

For an arbitrary number field, the best estimate is

q_{v}^{−1/5} < |α_{v}| and |β_{v}| < q_{v}^{1/5}

and is due to Shahidi [Sh1,Sh2]. We refer to [Ra] for a density result on the conjecture.

Proof are all based on this theory of automorphic L–functions which we will discuss next.

§5. L–functions for GL_{2}

The group GL_{2}(ℂ) has already showed up in the study of automorphic forms for GL_{2}(𝔸), where 𝔸 = 𝔸_{F} is the ring of adeles of a number field F. More precisely, if π = ⊗_{v}π_{v} is a cuspidal automorphic representation of GL_{2}(𝔸), then almost all components π_{v} of π are determined, up to isomorphism, by semisimple conjugacy classes A_{v} in GL_{2}(ℂ). Fix a complex number s. If r is a complex analytic (finite dimensional) representation of GL_{2}(ℂ), then det(I − r(A_{v})q_{v}^{−s})^{−1} is the inverse of the characteristic polynomial of r(A_{v}) at λ = q_{v}^{−s} and is therefore independent of all the choices except classes of π_{v} and r. This is what one calls the local Langlands L–function attached to π_{v} and r and is denoted by

L(s, π_{v}, r) = det(I − r(A_{v})q_{v}^{−s})^{−1}.

In fact, Langlands defined his local L–functions in the generality of any quasisplit reductive group in [L1], using exactly the same kind of definition.

The derived group SL_{2}(ℂ) of GL_{2}(ℂ) has precisely one irreducible finite dimensional representation of any degree. They can be naturally extended to GL_{2}(ℂ) and they are called symmetric powers of the standard representation of GL_{2}(ℂ) which we shall now explain.

Fix a pair of positive integers m and n. The group GL_{n}(ℂ) acts on ℂ^{n} by natural matrix multiplication which we call the standard representation ρ_{n} of GL_{n}(ℂ). It also acts on (ℂ^{n})^{⊗^{m}}, the tensor product of m–copies of ℂ^{n} by acting naturally on each component of pure tensors of rank m. Let S_{m} be the symmetric group in m letters. Then the symmetrization operator

Sym^{m} = \frac{1}{m!} ∑_{σ∈S_{m}} σ

acts on (ℂ^{n})^{⊗^{m}}. In fact, let {e_{1}, …, e_{n}} be a basis for ℂ^{n} and define the action of S_{m} on the basis by σ(e_{i}) = e_{σ(i)}. Moreover {e_{i_{1}} ⊗ e_{i_{2}} … ⊗ e_{i_{m}}} makes a basis for (ℂ^{n})^{⊗^{m}} and S_{m} acts by acting at each component. Then the subspace of symmetric tensors is by definition the image Sym^{m}(ℂ^{n}) of (ℂ^{n})^{⊗^{m}} under Sym^{m}. The natural action of GL_{n}(ℂ) on (ℂ^{n})^{⊗^{m}} commutes with Sym^{m}, making its image invariant under GL_{n}(ℂ). The resulting representation of GL_{n}(ℂ) on Sym^{m}(ℂ)^{n} denoted by Sym^{m}(ρ_{n}) is irreducible and is called the m–th symmetric power of ρ_{n}.

In this section, we are interested in the case n = 2. Then for each m, dim Sym^{m}(ρ_{n})) = m + 1. If t_{v} = diag(α_{v}, β_{v}) ∈ GL_{2}(ℂ) is a representative for A_{v}, then fixing the standard basis e_{1} = (1, 0) and e_{2} = (0, 1) for ℂ^{2}, so that

{Sym^{m}(e_{1}^{⊗^{i}} ⊗ e_{2}^{⊗^{j}})|i + j = m}

is a basis for Sym^{m}(ℂ^{2}), implies that

Sym^{m}(ρ_{2})(t_{v}) = diag(α_{v}^{m}, α_{v}^{m−1}β_{v}, …, α_{v}β_{v}^{m−1}, β_{v}^{m}),

an element in GL_{m+1}(ℂ).

The restriction of each Sym^{m}(ρ_{2}) to SL_{2}(ℂ) is irreducible and, up to isomorphism, this is the only irreducible representation of SL_{2}(ℂ) of dimension m + 1. Moreover all the finite dimensional irreducible representations of SL_{2}(ℂ) are so obtained.

For each unramified place v, let

L(s, π_{v},Sym^{m}(ρ_{2})) = det(I − Sym^{m}(ρ_{2})(A_{v})q_{v}^{−s})^{−1}

= ∏_{j=0}^{m}(1 − α_{v}^{j}β_{v}^{m−j}q_{v}^{−s})^{−1}

and if S is a finite set of places of F (including the archimedean ones) such that π_{v} is unramified for every v ∉ S, let

L_{S}(s, π, Sym^{m}(ρ_{2})) = ∏_{v∉S} L(s, π_{v}, Sym^{m}(ρ_{2}))

= ∏_{v∉S} ∏_{j=0}^{m}(1 − α_{v}^{j}β_{v}^{m−j}q_{v}^{−s})^{−1}.

Langlands [L1]: Suppose L_{S}(s, π, Sym^{m}(ρ_{2})) is absolutely convergent for Re(s) > 1 for every m. Then Ramanujan–Petersson’s conjecture is valid for π.

In fact, |α_{v}^{m}| < q_{v} and |β_{v}^{m}| < q_{v} if the partial L–function converges for Re(s) > 1. Then |α_{v}| < q_{v}^{1/m} and |β_{v}| < q_{v}^{1/m}. Letting m ⟶ +∞, then implies the conjecture since |α_{v}|^{−1} = |β_{v}|.

The estimate

q_{v}^{−1/5} < |α_{v}| and |β_{v}| < q_{v}^{1/5}

is a consequence of the following result which is a special case of a general theorem proved about automorphic L–functions.

For each v ∉ S, the local L–function L(s, π_{v}, Sym^{5}(ρ_{2})) is holomorphic for Re(s) ≥ 1.

This is clearly a global result. In fact for an arbitrary unramified unitary representation π whose attached semisimple conjugacy class in GL_{2}(ℂ) is represented by t = diag(α, β), q^{−1/2} < |α| and |β| < q^{1/2} and there are unramified unitary representations for which |α| = q^{1/2−ε} for arbitrarily small ε > 0.

§6. Cuspidal representations for GL_{n}

Using the language of adeles and the theory of group representations, it is now quite easy to define cusp forms for GL_{n}. We rather define the corresponding representations.

Let ω be a unitary character of F^{∗}\𝔸^{∗}. Identifying the center Z(𝔸) of GL_{n}(𝔸) with 𝔸^{∗}, ω is then a unitary character of Z(𝔸). By a parabolic subgroup of GL_{n}, we shall mean any conjugate (under GL_{n}) of a subgroup P of the form

{( g_{1} ∗ ; g_{2}  ∗ ; 0  ... ; g_{r} )} ,

where g_{i} ∈ GL_{n_{i}}, 1 ≤ i ≤ r, n_{1} + n_{2} + … + n_{r} = n. Then

M = {( g_{1} ; 0 ; g_{2} ; 0 ... ; g_{r} )}

is called a Levi subgroup of P and P = MN, where

N = ( \begin{matrix}I_{n_{1}} & & & & ∗ \\ & . & . & . \\ 0 & & & & I_{n_{r}} \end{matrix} ) ,

a subgroup of unipotent upper triangular elements, is called the unipotent radical of P.

An L^{2}–function φ is called cuspidal if

∫_{N(F)\N(𝔸)} φ(ng)dn = 0 (∀′g ∈ GL_{n}(𝔸))

for every possible unipotent radical. It can be shown that it is enough to consider only those P which are maximal, i.e. r = 2.

Let L_{0}^{2}(Z(𝔸)GL_{n}(F)\GL_{n}(𝔸), ω) be the Hilbert space of square integrable cuspidal functions satisfying φ(zg) = ω(z)φ(g), z ∈ ℤ(𝔸), g ∈ GL_{n}(𝔸). Under right regular translation by elements in GL_{n}(𝔸), L_{0}^{2} becomes a unitary representation which is a direct sum of irreducible unitary representation of GL_{n}(𝔸). Let π be an irreducible constituent of this space. Then again π = ⊗_{v}π_{v} with almost all π_{v} of class one, i.e. having a vector fixed by GL_{n}(O_{v}). The vector will then be unique up to scalar multiplication. Moreover, the class of a class one π_{v} can be uniquely determined by a unique conjugacy class A_{v} in GL_{n}(ℂ).

Again by strong multiplicity one, π is determined by almost all A_{v}’s, i.e. if π = ⊗_{v}π_{v} and π′ = ⊗_{v}π′_{v}, and A_{v} = A′_{v} or equivalently π_{v} ≅ π′_{v} for almost all v, then π ≅ π′ and in fact π = π′ (due to Shalika [S]).

It is important to produce interesting cuspidal representations. When n = 2 the theory is equivalent to that of modular forms, holomorphic or non–holomorphic, as explained. Here is an important example for n = 3.

Gelbart–Jacquet lift. Let π = ⊗_{v}π_{v} be a cuspidal representation of GL_{2}(𝔸). Choose a finite set S of places of F such that if v ∉ S, then π_{v} is unramified. Let A_{v} be the corresponding conjugacy class for π_{v} for each v ∉ S. The group P GL_{2}(ℂ) acts on the 3–dimensional Lie algebra sl_{2}(ℂ) of SL_{2}(ℂ) (and itself) by adjoint representation Ad, giving a 3–dimensional irreducible and faithful representation of P GL_{2}(ℂ). Let Ad^{2} be the corresponding 3–dimensional representation of GL_{2}(ℂ), i.e. the one obtained by combining Ad with the natural projection of GL_{2}(ℂ) onto P GL_{2}(ℂ). It is this representation which is called the adjoint square representation of GL_{2}(ℂ). If t_{v} = diag(α_{v}, β_{v}) represents A_{v}, then Ad^{2}(t_{v}) = diag(α_{v}β_{v}^{−1}, 1, α_{v}^{−1}β_{v}). Set t′_{v} = Ad^{2}(t_{v}) and let A′_{v} be its conjugacy class in GL_{3}(ℂ). Each A′_{v}, v ∉ S, determines a unique (up to isomorphism) unramified representation π_{v} of GL_{3}(F_{v}). The question is whether there exists an irreducible automorphic representation ∏ = ⊗_{v} ∏_{v} of GL_{3}(𝔸) such that ∏_{v} = π′_{v} for v ∉ S and the answer is yes and is due to Gelbart and Jacquet ([GJ], [Shi2]). Before, we explain how this is proved, let us formulate the problem in the language of functoriality.

The map

Ad^{2} : GL_{2}(ℂ) ⟶ GL_{3}(ℂ)

is a homomorphism from GL_{2}(ℂ), the L–group of GL_{2}, into the L–group GL_{3}(ℂ) of GL_{3}. In fact for every reductive group over a local or global field, Langlands [L1] defined a complex group by transposing the Cartan matrix of the original group, called its “L–group”. When the group is GL_{n}, the L–group ^{L}GL_{n} = GL_{n}(ℂ). Langlands functoriality conjecture then requires that every homomorphism between two L–groups must give rise to a “map” between automorphic forms of the original groups. Thus there must exist a map Ad_{∗}^{2} which sends automorphic representations of GL_{2}(𝔸) to those of GL_{3}(𝔸) in such a way that if Ad_{∗}^{2}(π) = ∏ = ⊗_{v} ∏_{v}, then Ad^{2}(t_{v}) represents the conjugacy class in GL_{3}(ℂ) attached to ∏_{v} for all unramified places v.

The proof is based on the theory of L–functions. Gelbart and Jacquet showed that the partial L–function L_{S}(s, π, Ad^{2}) can be completed to an L–function which is entire (as a function of s) unless π = π ⊗ χ for some non–trivial character χ of F^{∗}\𝔸^{∗}, is bounded in vertical strips, and satisfies a functional equation. More generally, given any character ρ = ⊗ρ_{v} of F^{∗}\𝔸^{∗}, one can define a local L–function

L(s, π_{v}, ρ_{v}, Ad^{2}) = det(I − Ad^{2}(t_{v})ρ_{v}(ϖ_{v})q_{v}^{−s})^{−1}

whenever π_{v} and ρ_{v} are unramified with t_{v} ∈ GL_{2}(ℂ) representing π_{v}. Similar statements are then proven in [Sh3] for the completed form of

L_{S}(s, π, ρ, Ad^{2}) = ∏_{v∉S} L(s, π_{v}, ρ_{v}, Ad^{2}).

Converse theorem for GL_{3}(𝔸) due to Jacquet, Piatetski–Shapiro, and Shalika now applies, generalizing that of Hecke, Weil, and Jacquet–Langlands, proving the existence of an automorphic representation Π of GL_{3}(𝔸) such that

L(s, π, ρ, Ad^{2}) = L(s, Π ⊗ ρ, ρ_{3}),

for every ρ, where ρ_{3} is the standard representation of GL_{3}(ℂ) and (Π ⊗ ρ)(g) = Π(g)ρ(det g). Moreover Π is cuspidal if and only if π is cuspidal and π = π ⊗ χ implies χ ≡ 1.

Experts familiar with classical theory must realize that the introduction of L– functions for each ρ ∈ F^{∗}\A^{∗} is equivalent to twisting the original Dirichlet series with primitive characters as needed by the converse theorem. It is very special of GL_{3} that only twisting with characters of GL_{1}(𝔸) is enough for applying the converse theorem.

Let ω be the central character of π. Then setting Sym_{∗}^{2}(π) = Ad_{∗}^{2}(π) ⊗ ω we see that Sym_{∗}^{2} is the dual “map” to

Sym^{2}(ρ_{2}) = Sym^{2} : GL_{2}(ℂ) ⟶ GL_{3}(ℂ)

as defined before, i.e. Sym^{2}(diag(α_{v}, β_{v})) = diag(α_{v}^{2}, α_{v}β_{v}, β_{v}^{2}).

It is natural to ask whether Sym_{∗}^{m} is defined for any other m > 2. At present, our best chance is m = 3. In fact, lots of information is available about

L_{S}(s, π, Sym^{3}(ρ_{2})).

Local L–functions are canonically defined for all v and therefore the L–function is completed [Sh3]. The completed L–function satisfies a functional equation sending s to 1 − s. The fact that it is entire unless π is monomial, i.e. π = π ⊗ χ for some non–trivial χ ∈ F^{∗}\^{∧}A^{∗}, is still incomplete and combining the work in [BGiH] and [Sh3], one only knows that poles are real and only between −3/4 and 3/4.

To apply the converse theorem one needs to twist with ρ ∈ F^{∗}\^{∧}𝔸^{∗} as well as cusp forms on GL_{2}(𝔸). With respect to the first one, one needs another irreducible four dimensional representation of GL_{2}(ℂ) which we have called in [Sh3], adjoint cube, denoted by Ad^{3}. It is defined by

Ad^{3} (( \begin{matrix}α_{v} & 0 \\ 0 & β_{v} \end{matrix} )) = diag(α_{v}^{2}β_{v}^{−1}, α_{v}, β_{v}, α_{v}^{−1}β_{v}^{2}).

If Ad_{∗}^{3} exists, then Sym_{∗}^{3} = Ad_{∗}^{3} ⊗ ω, where ω is the central character of π, also exists. To twist Ad_{∗}^{3}(π) with a ρ ∈ F^{∗}\^{∧}A^{∗}, it is enough to consider

L_{S}(s, π ⊗ ω, Ad^{3}).

It is proved in [Sh3], that L_{S}(s, π ⊗ ω, Ad^{3}) can be completed, the completed one satisfies a functional equation, and has only real poles, which using [BGiH], must lie between −3/4 and 3/4.

Twisting with a cusp form π′ on GL_{2}(𝔸) is much harder. We know that [Sh4]

L_{S}(s, Ad_{∗}^{3}(π) × π′)

is meromorphic and satisfies a functional equation. But not much more. When full properties of these L–functions are established, we will have the existence of Ad_{∗}^{3} and therefore Sym_{∗}^{3}. Methods of analytic number theory will then apply, implying

p^{−3/22} ≤ |α_{p}| ≤ p^{3/22},

an estimate even better than p^{1/7}. But until the necessary properties of L–functions are proved, these estimates remain out of our reach.

We conclude by pointing out that establishing Langlands functoriality is one of the most important goals of modern theory of automorphic forms. Besides the converse theorem which may be called the method of L–functions, the trace formula as developed by Arthur in remarkable generality [A1,A2], can be used to establish certain cases of functoriality. The most notable is that of endoscopy, a major work in progress, which when established, proves one case of functoriality, but in generality of every reductive group.

§7. Rankin–Selberg L–functions for GL_{n}

Fix two positive integers m and n. Let π = ⊗_{v}π_{v} and π′ = ⊗_{v}π′_{v} be irreducible cuspidal automorphic representations of GL_{m}(𝔸) and GL_{n}(𝔸), respectively. Fix a finite set of places S of F such that for v ∉ S, both π_{v} and π′_{v} are unramified. Given v ∉ S, let A_{v} and A′_{v} denote the corresponding semisimple conjugacy classes of GL_{m}(ℂ) and GL_{n}(ℂ), attached to π_{v} and π′_{v}, respectively. Let

L(s, π_{v} × π′_{v}) = L(s, π_{v} ⊗ π′_{v}, ρ_{m} ⊗ ρ_{n})

= det(I − A_{v} ⊗ A′_{v}q_{v}^{−s})^{−1}

= ∏_{{1≤i≤m \atop 1≤j≤n}}(1 − α_{v,i}α′_{v,j}q_{v}^{−s})^{−1},

when t_{v} = diag(α_{v,1}, …, α_{v,m}) ∈ A_{v} and t′_{v} = diag(α′_{v,1}, …, α′_{v,n}) ∈ A′_{v}. Finally, let

L_{S}(s, π × π′) = ∏_{v∉S} L(s, π_{v} × π′_{v}).

This is what is usually called the (partial) Rankin–Selberg product L–function for π and π′. Each L(s, π_{v} ×π′_{v}) is called the corresponding local Rankin–Selberg product L–function. We have

Theorem 7.1 (Jacquet–Shalika [JS1]). The partial L–function L_{S}(s, π × π′) converges absolutely for Re (s) > 1.

The fact that any partial Langlands L–function converges absolutely for Re(s) sufficiently large is a general fact due to Langlands [L1]. But a bound as little as 1 is usually hard to get and proof of 7.1 is fairly hard. Let us sketch it very roughly here.

Sketch of the proof of 7.1. One first represents the L–function by means of an integral representation which is holomorphic for Re (s) > 1, showing that it is holomorphic as a function of s for Re (s) > 1. This is quite involved and requires lots of work. Now, techniques of Dirichlet series with positive coefficients apply, showing that the L–function must be absolutely convergent to the right of its first pole which will happen at s = 1, if m = n and π′ = \overline{π} for which A′_{v} = \overline{A}_{v} for almost all v. Observe that in this case L_{S}(s, π × \overline{π}) gives a Dirichlet series with non–negative coefficients. The general case follows from this and Schwarz Lemma.

Theorem 7.2 (Langlands [L2]). The partial L–function L_{S}(s, π × π′) extends to a meromorphic function of s on all of ℂ.

Sketch. One realizes GL_{m}×GL_{n} as a Levi subgroup of GL_{m+n} and considers π̃⊗π′ as a cuspidal representation of GL_{m}(𝔸) × GL_{n}(𝔸). For definition and detail see proof of Theorem 7.6 here. The ratio of partial L–functions

L_{S}(s, π × π′)/L_{S}(1 + s, π × π′)

appears in the constant term of the corresponding Eisenstein series which is meromorphic on all of ℂ. Starting with s with Re (s) large and using induction, one concludes the meromorphy of L_{S}(s, π × π′) for all s.

Remark. This work of Langlands [L2] is the origin of what is now called the Langlands–Shahidi method.

The theory of L–functions as predicted by Langlands [L1] requires local factors to be defined at all places and not only at unramified ones. There is one type of L–functions which are defined at all places. They are the so called Artin L– functions and Langlands philosophy is that Artin L–functions are automorphic and that is how Langlands proved their holomorphy, the Artin’s conjecture on Artin L– functions, for a large number of two dimensional representations of Galois groups.

As usual, Artin L–functions are defined by a product of local L–functions. Let F be a local field. The Weil group W_{E/F} of F is an extension of Gal (E/F) by E^{∗} by means of a non–trivial 2–cocycle. Here E/F is a finite Galois extension. The Weil group of F, denoted by W_{F}, as W_{\overline{F} /F}, is defined as a projective limit for all W_{E/F}. There is a further thickening of these groups by a unipotent group, the Deligne– Weil group W′_{E/F} or W′_{F}, which will be necessary when F is non–archimedean. What follows is expected in the generality of every reductive algebraic group. But let us restrict ourselves to the case of GL_{n}.

First, one should mention that given any continuous representation ρ of W′_{F} on a complex vector space V, the Artin L–function L(s, ρ) is defined by

L(s, ρ) = det(I − ρ(ϕ)q^{−s}|V^{I})^{−1},

where ϕ denotes an inverse Frobenius and V^{I} is the subspace of all vectors fixed in V by the inertia subgroup I. We refer to [T] for definitions of ϕ and I. Also given a non–trivial additive character ψ of F, one can attach an Artin root number ε(s, ρ, ψ) which is a monomial in q^{−s}, and we again refer to [L3] for its definition.

To make matters more precise and clear, let us consider the easier case of archimedean fields. Thus let F = ℂ or ℝ. Then W_{ℂ} = ℂ^{∗} and W_{ℝ} consist of pairs (z, τ) ∈ ℂ^{∗}× Gal(ℂ/ℝ) with multiplication rule

(z_{1}, τ_{1})(z_{2}, τ_{2}) = (z_{1}τ_{1}(z_{2})a_{τ_{1},τ_{2}}, τ_{1}τ_{2}),

with a_{τ_{1},τ_{2}} = 1 unless τ_{1} = τ_{2} = 1 in which case a_{τ_{1},τ_{2}} = −1 (cf. [L4]).

Irreducible representations of W_{ℂ} are 1–dimensional and if ρ is one such, then ρ(z) = |z|_{ℂ}^{t}(z/\overline{z})^{n/2}, |z|_{ℂ} = z\overline{z}, for some t ∈ ℂ and integer n. The corresponding Artin L–function is

L(s, ρ) = 2(2π)^{−(s+t+|n|/2)}Γ(s + t + |n|/2)

with

Γ(s) = ∫_{0}^{∞} e^{−t}t^{s}dt/t,

the ordinary Γ–function.

Now assume F = ℝ and let ρ be an irreducible representation of W_{ℝ}. If dim ρ = 2, then ρ = Ind_{ℂ^{∗}}^{W_{ℝ}}θ, where θ is a character of ℂ^{∗}. Then

L(s, ρ) = L(s, θ)

with L(s, θ) defined before.

On the other hand, if dim ρ = 1, then using W_{ℝ}/[W_{ℝ}, W_{ℝ}] = ℝ^{∗}, ρ becomes a character of ℝ^{∗}. Here [W_{ℝ}, W_{ℝ}] is the commutator group of W_{ℝ}. Then

ρ(x) = (x/|x|)^{ε}|x|^{t}

with t ∈ ℂ and ε = 0 or 1. The corresponding Artin L–function is

L(s, ρ) = π^{−1/2(ε+t+s)}Γ(1/2(ε + t + s)).

Let F be a local field. The parametrization problem for GL_{n}(F) requires that every irreducible admissible representation π of GL_{n}(F) be parametrized by a continuous representation φ of W′_{F}, i.e. a continuous homomorphism of W′_{F} into GL_{n}(ℂ) =^{L} GL_{n}. If r is a representation of GL_{n}(ℂ), then the corresponding Langlands L–function L(s, π, r) will be defined to equal the Artin L–function L(s, r · φ). Similarly

ε(s, π, r, ψ) = ε(s, r · φ, ψ)

with the one on the right, the Artin root number.

The parametrization problem is quite general and for any reductive group and in the generality of all reductive groups it is only solved when F is archimedean, i.e. F = ℝ or ℂ. This is due to Langlands [L4]. For non–archimedean F, the results are quite fragmented and we will make no effort to explain them.

In practice L(s, π, r) and ε(s, π, r, ψ) are defined by other means and one task is to show

L(s, π, r) = L(s, r · φ)

and

ε(s, π, r, ψ) = ε(s, r, φ · ψ).

In the case in hand, i.e. for G = GL_{m} × GL_{n} and r = ρ_{m} ⊗ ρ_{n}, both

L(s, π_{v} × π′_{v}) = L(s, π_{v} ⊗ π′_{v}, ρ_{m} ⊗ ρ_{n})

and

ε(s, π_{v} × π′_{v}, ψ_{v}) = ε(s, π_{v} ⊗ π′_{v}, ρ_{m} ⊗ ρ_{n}, ψ_{v})

are defined using the two different methods, Rankin–Selberg in [JPSS] and Langlands– Shahidi in [Sh5] and were proved to be equal to each other in [Sh6], when F_{v} is non–archimedean. When F_{v} = ℝ, ℂ, the factors defined from Langlands–Shahidi methods and in the full generality of the method are proved to be those of Artin in [Sh7], in particular so are L(s, π_{v} × π′_{v}) and ε(s, π_{v} × π′_{v}, ψ_{v}). The method of Rankin–Selberg for archimedean fields is addressed in [JS3]. Here ψ_{v} is a non–trivial additive character of F_{v}.

Now, let π = ⊗_{v}π_{v} and π′ = ⊗_{v}π′_{v} be cuspidal representations of GL_{m}(𝔸) and GL_{n}(𝔸) as in the beginning of the section. Fix a non–trivial additive character ψ = ⊗_{v}ψ_{v} of F \𝔸. Let

L(s, π × π′) = ∏_{v} L(s, π_{v} × π′_{v})

and

ε(s, π × π′) = ∏_{v} ε(s, π_{v} × π_{v}, ψ_{v}),

where the factors are as in the previous paragraph. Let π̃ and π′ be contragredients of π and π′, i.e. their ∏_{v} GL_{m}(O_{v})–finite (respectively ∏_{v} GL_{n}(O_{v})–finite) duals.

Theorem 7.3 [Sh5]. The functional equation

L(s, π × π′) = ε(s, π × π′)L(1 − s, π̃ × π′)

is valid.

The fact that a non–trivial additive character is necessary to define local root numbers which will have no appearance in the global root number is one of the mysteries of the subject.

Sketch of the proof. In the setting of the proof of Theorem 7.2, one considers a non–constant Fourier coefficient of the Eisenstein series built on π̃ ⊗ π′. Up to a finite number of factors, in fact attached to those in S, the Fourier coefficient is equal to L_{S}(1 + s, π × π′)^{−1}. One then applies the functional equation satisfied by the Eisenstein series and combines with lots of local work. We refer to [Sh1,Sh8,CS] for details.

The method of the proof is quite general and applies to a large number of L– functions. We will discuss this later.

Just as it is the case with classical Dirichlet series, one can also prove a non– vanishing for these L–functions on the line Re(s) = 1 and, as its celebrated classical application to the infiniteness of number of primes in an arithmetic progression has shown, many applications are expected. Some has already been established and are truly significant such as classification of automorphic forms for GL_{n}(𝔸) proved in [JS2]. Many more will be proved in future, especially since the results are true in a much more general setting than L_{S}(s, π × π′).

Theorem 7.4. L(s, π × π′) ≠ 0 for Re(s) = 1.

Sketch of the proof. The Fourier expansion discussed in the proof of Theorem 7.3 is equal (up to a finite number of local Whittaker functions) to L_{S}(1 + it, π × π′)^{−1}.

It must also be holomorphic for all t ∈ ℝ, i.e. on the unitary axis by the general theory of Eisenstein series [L5,MW1]. One must then apply some deep local theory at archimedean places, due to Casselman–Wallach, to show that local Whittaker functions are not identically zero. (This was a serious problem about the time when the theory was being developed). Details are given in [Sh8]. The theorem follows.

Next, one must address the question of poles. The first result is due to Jacquet– Shalika [JS2].

Theorem 7.5 (Jacquet–Shalika [JS2]). The only poles for L_{S}(s, π × π′) on the half plane Re(s) ≥ 1 are on the line Re(s) = 1. They happen if and only if m = n and π′ ≅ π̃ ⊗ | det( )|^{t}, t ∈ ℂ, i.e. when π′ is an unramified twist of π̃. The partial L–function L_{S}(s, π × π̃) has a simple pole at s = 1.

Sketch of the proof. This is accomplished using an integral representation which itself involves an Eisenstein series if m = n. Again good amount of local result for Re(s) > 1 is necessary. For more details see [JS1,JS2].

The continuation to the whole complex plane is due to Moeglin and Waldspurger [MW2]. It is an ingenious application of results of Jacquet–Piatetski– Shapiro–Shalika and Shahidi.

Theorem 7.6 (Moeglin–Waldspurger [MW2]). The complete L–function L(s, π × π′) is entire unless m = n and π′ ≅ π̃ ⊗ | det( )|^{t}, t ∈ ℂ. The only poles of L(s, π × π̃) are at s = 0 and 1 and are both simple.

Sketch of the proof. Using Theorems 7.1–7.5, it is enough to prove L(s, π̃ × π′) has no poles for 0 < Re(s) ≤ 1/2. Assume s with 0 < Re(s) ≤ 1/2 is a pole of L(·, π̃ × π′) of order r.

Let us first say a few words about Eisenstein series in this case which is the main tool in this whole program. Let Q and Q′ denote standard parabolic subgroups of G = GL_{m+n} whose Levi subgroups are M = GL_{m} × GL_{n} and M′ = GL_{n} × GL_{m}, respectively. Let U and U′ be their unipotent radicals. Set \underline{π} = π ⊗ π′ and \underline{s} = (s, s′), s, s′ ∈ ℂ and denote by I(\underline{π}) and I(\underline{π}, \underline{s}), representations induced from \underline{π} and \underline{π}[\underline{s}] = (π ⊗ | det( )|^{s}) ⊗ (π′ ⊗ | det( )|^{s′}). If φ ∈ I(\underline{π}), define φ(\underline{s}) ∈ I(\underline{π}, \underline{s}) by

φ(\underline{s})(mug) = | det m|\underline{s}φ(mg).

Let

G_{Q} = {f : Q(F)U(𝔸)\G(𝔸) ⟶ ℂ}

and define i : I(\underline{π}, \underline{s}) ⟶ G_{Q} by

(if)(g) = f(g)(1, 1).

The Eisenstein series attached to φ and \underline{s} is defined by

E(φ, \underline{s}, g) = ∑_{γ∈Q(F)\G(F)}[iφ(\underline{s})](γg).

Consider

E(φ, g) = lim_{s′⟶s}(s′ − s)^{r}E(φ, \underline{s}′, g)

with \underline{s}′ = (s′/2, −s′/2), where E(φ, \underline{s}′, g) is the Eisenstein series built on representation (π ⊗ | det( )|^{s′/2}) ⊗ (π′ ⊗ | det( )|^{−s′/2}) of the Levi subgroup M(𝔸) = GL_{m}(𝔸) × GL_{n}(𝔸) of GL_{m+n}(𝔸) discussed above. Since the poles of E(φ, \underline{s}′, g) in this interval are those of L(s′, π̃ × π′), E(φ, g) is not identically zero. Let U be the unipotent radical of the parabolic subgroup Q = MU of GL_{m+n} on which the Eisenstein series is built. Next they show that if E^{U}(φ, g) is the limit

E^{U}(φ, g) = lim_{s′⟶s}(s′ − s)^{r}E^{U}(φ, \underline{s}′, g),

where E^{U}(φ, \underline{s}′, g) is the constant term of E(φ, \underline{s}′, g) along Q, then

(7.6.1)  E(φ, g) = ∑_{γ∈(P ∩Q)(F)\P (F)} E^{U}(φ, γg),

where P is the standard parabolic subgroup of GL_{m+n} whose Levi is isomorphic to GL_{m+n−1} × GL_{1}.

Using density of GL_{m+n}(F)B(𝔸) in GL_{m+n}(𝔸), where B is the Borel subgroup of upper triangular elements in GL_{m+n}, one concludes that if E^{U}(φ, ) ≠ 0, then there exist γ ∈ P (F) and b ∈ B(𝔸) such that E^{U}(φ, γb) ≠ 0.

Now, up to a constant, E^{U}(φ, g) is given by a normalized intertwining operator N(\underline{s})φ(\underline{s})(g). Moreover

\{N(\underline{s})φ(\underline{s})|φ ∈ I(\underline{π})\} = ⊗_{v}J_{v},

where each J_{v} is a space of functions on GL_{m+n}(F_{v}) with values in the space of π′_{v} ⊗ π_{v}. More precisely, J_{v} is a subspace of the space of representation induced from a twist of π′_{v} ⊗ π_{v}. They then show that if v ∉ S, then there exists a function j_{v} ∈ J_{v}, j_{v} ≠ 0, such that j_{v}|Q(F_{v})P (F_{v}) ≡ 0.

Choose φ such that N(\underline{s})φ(\underline{s}) is not identically zero and its v–th component is j_{v} for some v ∉ S. Then E^{U}(φ, g) ≠ 0 and consequently E(φ, ) is not identically zero. But then by (7.6.1) and the density argument below it E^{U}(φ, γb) ≠ 0 for some γ ∈ P (F) and b ∈ B(𝔸). On the other hand N(\underline{s})φ(\underline{s})(γb) = 0 since N(\underline{s})φ(\underline{s}) has a j_{v}–component. This contradicts E^{U}(φ, γb) ≠ 0.

The proof is quite remarkable and mixes the results of the two different methods to prove the full holomorphy which is usually hardest step. It has the potential of being applied to other cases which have remained unresolved.

§8. Applications of properties of L(s, π × π′)

8.1 Classification of automorphic forms for GL_{n}. Let G = GL_{r}(𝔸). Fix a cusp form σ = σ_{1} ⊗ … ⊗ σ_{u} of M = GL_{r_{1}}(𝔸) × … × GL_{r_{u}}(𝔸), not necessarily unitary, r_{1}+…+r_{u} = r. Here M is considered as the standard Levi subgroup of the standard parabolic subgroup P = MN of G. Let ξ = ⊗_{v}ξ_{v} be the representation induced from σ ⊗ 1 of MN = P. Similarly let Q be another standard parabolic subgroup of G with a cuspidal representation τ of the standard Levi subgroup of Q. Denote by η the representation of G induced from τ ⊗ 1. Write η = ⊗_{v}η_{v}. The following theorem is proved in [JS2].

Theorem 8.1 (Jacquet–Shalika [JS2]). Let S be a finite set of places of F such that for v ∉ S, σ_{v} and τ_{v} are both unramified. Then ξ_{v} and η_{v} have the same unramified components if and only if (σ, P) and (τ, Q) are conjugate, i.e. up to a permutation they are equivalent.

When M = G, this is the strong Multiplicity one Theorem alluded to before.

The connection with classification is explained in [JS2]. Its significance and connection with global parametrization problem which requires introduction of certain Tanakian categories is well explained in an important article of Langlands [L6].

Proof of 8.1 is a clever application of Theorems 7.1, 7.4, and 7.5.

8.2 Base change for GL_{n}. Let E/F be a cyclic extension of F. Set r = [E : F] and let Gal(E/F) = ⟨τ⟩. There is a group denoted by Res_{E/F} GL_{n}. It is defined by

Res_{E/F} GL_{n} = \overbrace{GL_{n}(E) × … × GL_{n}(E)}^{r} ⋊Gal(E/F)

in which τ(g_{1}, …, g_{r}) = (τ(g_{r}), τ(g_{1}), τ(g_{2}), …, τ(g_{r−1})). Thus τ(g_{1}, …, g_{r}) = (g_{1}, …, g_{r}) implies g_{1} = τ(g_{r}), g_{2} = τ(g_{1}), g_{3} = τ(g_{2}), …, g_{r} = τ(g_{r−1}), and therefore (g_{1}, …, g_{r}) = (g_{1}, τ(g_{1}), τ^{2}(g_{1}), …, τ^{r−2}(g_{1}), τ^{r−1}(g_{1})). Consequently (Res_{E/F} GL_{n})(F) = GL_{n}(E). Its L–group is (GL_{n}(ℂ)×…×GL_{n}(ℂ))⋊Gal(E/F) in which

τ(g) = ( \begin{matrix}& 1 \\ 1 \end{matrix} )^{t}g^{−1} ( \begin{matrix}& 1 \\ 1 \end{matrix} )

for every g ∈ GL_{n}(ℂ).

Let

θ : GL_{n}(ℂ) × Gal(E/F) ⟶ \overbrace{GL_{n}(ℂ) × … × GL_{n}(ℂ)}^{r} ⋊Gal(E/F)

be defined by

θ(g, τ) = (g, …, g) ⋊ τ.

Then it is a major work of Arthur and Clozel [ACl] that θ_{∗} from the space of automorphic representations of GL_{n}(𝔸_{F}) into those of GL_{n}(𝔸_{E}) exists, generalizing the fundamental work of Langlands [L7] from n = 2, in which he proves Artin’s conjecture for a large number of irreducible two dimensional continuous representations of Gal(\overline{F} /F). In fact for all those whose image in P GL_{2}(ℂ) are solvable.

Proof makes a fundamental use of trace formula and its twisted version. Again L–functions L(s, π × π′) play a role both through the normalizing factors for intertwining operators, as well as by means of their analytic properties and their defining role in the theory of local base change.

8.3 Residual spectrum for GL_{n}. Langlands theory of Eisenstein series [L5] gives a recipe to how to determine non–cuspidal discrete spectrum of reductive groups. One must determine residues of Eisenstein series coming from different conjugacy classes of parabolic subgroups. This is a hard and difficult process. One must determine residues of constant terms which are given, up to a finite number of factors, as ratios of partial L–functions. For GL_{r}(𝔸) and the parabolic subgroup Q = MU in which M ≅ GL_{m} × GL_{n}, r = m + n, with inducing cuspidal representations π ⊗ π′ on M(𝔸), the ratio is

L_{S}(s, π̃ × π′)/L_{S}(1 + s, π̃ × π′),

and clearly analytic properties of L_{S}(s, π̃ × π′) play an important role in determination of these residues. This whole project has been masterfully executed by Moeglin and Waldspurger in [MW2], completely classifying the residual spectrum for GL_{n}(𝔸).

8.4 Ramanujan–Petersson’s and Selberg’s Conjectures. Let Γ be a congruence subgroup of SL_{2}(ℤ). The Laplace operator

Δ = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} )

acts on L^{2}(Γ\𝔥). Let λ_{1}(Γ) be the smallest non–zero eigenvalue of Δ. The corresponding eigenfunctions are called Maass forms.

Selberg’s Conjecture [Se]. λ_{1}(Γ) ≥ 1/4 = 0.25.

Recall that

λ_{1}(Γ) = \frac{1}{4}(1 − s^{2})

in which s must be either pure imaginary or −1 ≤ s ≤ 1. If the Maass form π = ⊗_{p}π_{p} has π_{∞} as its archimedean component, then π_{∞} = I(μ_{1,∞}, μ_{2,∞}), where μ_{1,∞} = | |^{s/2} and μ_{2,∞} = | |^{−s/2} with | | = | |_{ℝ}. Thus

\frac{1}{4}(1 − s^{2}) = \frac{1}{4}

implies s = 0 and consequently π_{∞} is tempered and therefore Selberg’s conjecture may be considered as the archimedean version of Ramanujan–Petersson’s conjecture.

Selberg himself proved λ_{1}(Γ) ≥ 3/16 or −1/4 ≤ s/2 ≤ 1/4 which was sharpened to −1/4 < s/2 < 1/4 by Gelbart and Jacquet [GJ]. We again refer to [I], Section 11.3, for further discussion of this subject and its history.

While R−P conjecture was making progress at all the finite places, its archimedean version, the Selberg’s conjecture remained resilient to any progress. Except for some partial results by Iwaniec, no real progress took place until the recent paper of Luo, Rudnick, and Sarnak [LRSa] in which they proved

Theorem 8.2 (Luo–Rudnick–Sarnak) [LRSa]. λ_{1}(Γ) ≥ 21/100 or equivalently

−1/5 ≤ s/2 ≤ 1/5.

The proof relies in a fundamental way on the theory of Rankin–Selberg L– functions for GL_{m} × GL_{n} when m = n = 3. Their proof also gives the estimate

p^{−1/5} ≤ |α_{p}| & |β_{p}| ≤ p^{1/5}

discussed earlier. Using GL_{2}–theory alone Iwaniec has also obtained the slightly weaker bound λ_{1}(Γ) ≥ 10/49.

Sketch of the proof. Let Π = ⊗_{p} ∏_{p} be the Gelbart–Jacquet lift (§6) of π = ⊗_{p}π_{p}. Fix a place p = p_{0}, finite or infinite. We need the following theorem whose proof is in [LRSa] and becomes unavailable as we replace ℚ by a number field.

Theorem 8.3 [LRSa]. Given s with Re(s) > 4/5, there exists a character χ = ⊗_{p}χ_{p} of ℚ^{∗}\𝔸_{ℚ}^{∗} with χ_{p_{0}} = 1, such that

L_{{p_{0}}}(s, (Π ⊗ χ) × Π) = ∏_{p≠p_{0}} L(s, (Π_{p} ⊗ χ_{p}) × Π_{p})

is non–zero.

Proof. See [LRSa].

Fix a s = s_{0} with 4/5 < Re(s_{0}) < 1 and choose χ as in Theorem 8.3. By Theorem 7.6

L(s_{0}, (Π ⊗ χ) × Π)

is holomorphic. But

L(s, (Π ⊗ χ) × Π) = L(s, Π_{p_{0}} × Π_{p_{0}})L_{{p_{0}}}(s, (Π ⊗ χ) × Π).

Since L_{{p_{0}}}(s, (Π ⊗ χ) × Π) is non–zero at s = s_{0}, L(s, Π_{p_{0}} × Π_{p_{0}}) must be holomorphic at s = s_{0}. Let us first assume p_{0} < ∞ and π_{p_{0}} (and thus Π_{p_{0}}) is unramified. Recall that if

t_{p_{0}} = ( \begin{matrix}α_{p_{0}} & 0 \\ 0 & β_{p_{0}} \end{matrix} )

is attached to π_{p_{0}}, then

Ad^{2}(t_{p_{0}}) = ( \begin{matrix}α_{p_{0}}β_{p_{0}}^{−1} & & 0 \\ & 1 \\ 0 & & α_{ρ_{0}}^{−1}β_{p_{0}} \end{matrix} )

represents the conjugacy class of GL_{3}(ℂ) attached to Π_{p_{0}} (cf. §6). For a Maass form β_{p_{0}}^{−1} = α_{p_{0}} and

(1 − p^{−s}α_{p_{0}}^{4})

divides L(s, Π_{p_{0}} × Π_{p_{0}})^{−1}. Consequently (1 − p^{−s}α_{p_{0}}^{4}) must have no zeros for any s with 4/5 < Re(s) < 1. Thus

p^{−(4/5)+ε}|α_{p_{0}}|^{4} < 1

for every ε > 0. Taking the limit as ε ⟶ 0 one gets

|α_{p_{0}}| ≤ p_{0}^{1/5}.

Now suppose p_{0} = ∞ and π_{∞} = I(| |^{s′/2}, | |^{−s′/2}), s′ ∈ ℝ. In fact this is always the case for a Maass form. If

φ : W_{ℝ} ⟶ GL_{2}(ℂ)

parametrizes π_{∞}, then

φ(x) = ( \begin{matrix}|x|^{s′/2} & 0 \\ 0 & |x|^{−s′/2} \end{matrix} )

for ∀ x ∈ ℝ^{∗} ⊂ W_{ℝ}. Moreover Π_{∞} = I(| |^{s′}, 1, | |^{−s′}) and L(s, Π_{∞} × Π_{∞}) will have Γ(s − 2s′) as a factor. By the same argument Γ(s − 2s′) must be holomorphic for Re(s) > 4/5, or

4/5 + ε − 2s′ > 0

for every ε > 0. Then s′ ≤ 2/5 and s′/2 ≤ 1/5.

The estimate 5/28 which for a classical Maass form π = ⊗_{p}π_{p} is the best presently proved, uses a different L–function. Again one takes Π = Ad_{∗}^{2}(π) which is cuspidal unless π is monomial in which case R − P is valid anyway, π being parametrized by a representation of the global Weil group. One then uses the following theorem of Bump and Ginzburg.

Theorem 8.4 (Bump–Ginzburg [BGi]). Let Π = ⊗_{v}Π_{v} be a cuspidal automorphic representation of GL_{3}(𝔸) and choose a finite set S of places of F such that Π_{v} is unramified for every v ∉ S. Then

L_{S}(s, Π, Sym^{2}(ρ_{3})) = ∏_{v∉S} L(s, Π_{v}, Sym^{2}(ρ_{3}))

is entire with possibly only a pole at s = 1.

The 5/28 estimate of Bump–Duke–Hoffstein–Iwaniec [BDHI]

p^{−5/28} ≤ |α_{p}| ≤ p^{5/28}

follows from Theorem 8.4, the functional equation satisfied by this L–function, and an ingenious idea of Duke and Iwaniec [DI].

Although it may be possible to extend the 5/28 estimate to quadratic extensions of ℚ (private communications with Sarnak), still the best estimate for an arbitrary number field is (cf. [Sh1, Sh2])

q_{v}^{−1/5} < |α_{v}| & |β_{v}| < q_{v}^{1/5}.

This follows from the theory developed in [L2, Sh1, Sh8] which is now being called the Langlands–Shahidi method which we will discuss next.

§9. The Method

In 1967 Langlands gave a series of lectures at Yale which were published in a book titled “Euler Products” by Yale University Press [L2]. The book is the origin for his notion of an L-group. This was later pursued by the author and is now being called the Langlands–Shahidi method and is one of the two methods (the Rankin–Selberg being the other one) to study L–functions directly. Efforts are now being made to generalize the method to non–generic representations [FGol]. For the purpose of these lectures, let us consider only the case of split groups.

9.1 L–groups. Let F be a number field and denote by G a Zariski–closed subgroup of GL_{n}(F_{a}), for some n, where F_{a} denotes an algebraic closure of F. This means that G is given as common zeros of a finite number of polynomials (in several variables). If the polynomials are with coefficients in F, then G will be said to be defined over F. We refer to §1 for adelization of G. Radical of G is the largest connected solvable normal subgroup of G. G will be called reductive, if its radical consists of only semisimple (diagonalizable) elements. The radical is then equal to the connected component of the center of G. A maximal torus of G is one which is maximal among its closed connected abelian subgroups whose elements are semisimple. Let T be one. Denote by X^{∗}(T) the group of rational characters of T into GL_{1}. Let X_{∗}(T) be the group of all rational homomorphism of GL_{1} into T. Let X^{∗}(T)_{F} be the group of F –rational character of T into GL_{1}, i.e. those defined over F which means that they send T(F) into GL_{1}(F). We will say T is split if

X^{∗}(T) = X^{∗}(T)_{F} .

We shall say G is split over F if it has maximal tori which are split over F.

For simplicity from now on we shall assume G is split. Let T be a maximal torus and fix a Borel subgroup B, i.e. a maximal connected solvable subgroup, containing T. Let U be the unipotent radical of B, i.e. the subgroup of all its unipotent elements. Then B = T · U. The torus T acts by adjoint action on U, defining root characters of T in U. Let Σ^{∗} ⊂ X^{∗}(T) be their subset. One can identify the coroots Σ_{∗} of T with a subset of X_{∗}(T). Observe that the choice of B, already determines a set of positive roots.

Theorem 9.1. The root datum (X^{∗}(T), Σ^{∗}, X_{∗}(T), Σ_{∗}) determines the class of G uniquely.

Definition 9.2. The complex group whose root datum is dual of that of G, i.e. is (X_{∗}(T), Σ_{∗}, X^{∗}(T), Σ^{∗}) is called the L–group of G and is denoted by ^{L}G.

Remark. All of these are valid if F is local.

Examples. One can quickly notice that the roots of G are coroots of ^{L}G. If α and β are roots of G and α^{∨} and β^{∨}, their corresponding coroots, i.e. so that ⟨α, α^{∨}⟩ = ⟨β, β^{∨}⟩ = 2. Then

⟨α, β⟩ = ⟨β^{∨}, α^{∨}⟩.

The Cartan matrix of G is

C = (⟨α_{i}, α_{j}⟩),

where α_{i} and α_{j} denote simple roots of T in U. Let ^{L}C be the Cartan matrix of LG, then

LC = (⟨α_{i}^{∨}, α_{j}^{∨}⟩)
= (⟨α_{j}, α_{i}⟩)
= ^{t}C,

from which one concludes that G and ^{L}G have transposed Cartan matrices.

Suppose G is semisimple, i.e. it is reductive and its center is finite, thus its radical is trivial, then G is called simply connected, if

X_{∗}(T) = ℤ–span of Σ_{∗}

and adjoint, if

X^{∗}(T) = ℤ–span of Σ^{∗}.

Thus G is simply connected if and only if ^{L}G is adjoint and conversely.

A simple way of characterizing an adjoint group is that it has no center. Examples are P GL_{n}, P Sp_{2n}, P SO_{2n}, SO_{2n+1}, exceptional groups of type G_{2}, F_{4}, and E_{8}. On the other hand SL_{n}, Sp_{2n}, Spin_{n}, G_{2}, F_{4}, and E_{8} are all simply connected. The groups SO_{2n} are neither simply connected nor adjoint. The following table for Chevalley groups is then clear:

G  ^{L}G
SL_{n}  P GL_{n}(ℂ)
Sp_{2n}  SO_{2n+1}(ℂ)
SO_{2n+1}  Sp_{2n}(ℂ)
SO_{2n}  SO_{2n}(ℂ)
Spin_{2n}  P SO_{2n}(ℂ)
Spin_{2n+1}  P Sp_{2n}(ℂ)
G_{2}  G_{2}(ℂ)
F_{4}  F_{4}(ℂ)
simply connected E_{6} & E_{7} adjoint E_{6}(ℂ) and E_{7}(ℂ), resp.
E_{8}  E_{8}(ℂ)

Examples of non–semisimple reductive groups are GL_{n} and similitude groups, such as GSp_{2n}. Their L–groups require a little more work to figure out. We refer to [L1] and [Bo] for details. The result is ^{L}GL_{n} = GL_{n}(ℂ), as mentioned before, and ^{L}GSp_{4} = GSp_{4}(ℂ). The L–group of GSp_{2n}, n > 2, is again a reductive group whose derived group is Sp_{2n}(ℂ). But one needs to build in the extra dimension in the maximal torus of GSp_{2n} as opposed to Sp_{2n} (cf. [Bo]).

9.2 Parabolic subgroups. Fix a maximal split torus T for G which lies in our fixed Borel subgroup B = TU. Let Δ be the subset of simple roots for roots of T in U, spanning Σ_{+} ⊂ Σ^{∗} the set of (positive) roots in U. Given a set θ ⊂ Δ, let A = A_{θ} = ( ⋂_{α∈θ} ker α)^{0}, the connected component of ⋂_{α∈θ} ker α. Let M = M_{θ} be the centralizer of A in G. Clearly T ⊂ M. Let N = N_{θ} be the subgroup of U spanned by roots in Σ_{+}\⟨θ⟩, where ⟨θ⟩ is the subset of Σ_{+} spanned by the roots in θ. Then P = P_{θ} = MN is called the standard parabolic subgroup of G attached to θ with a Levi subgroup M and unipotent radical N. Being standard means that N ⊂ U for a fixed B. The decomposition P = MN is called a Levi decomposition. A parabolic subgroup of G is then a conjugate of a P_{θ} for some θ. Every parabolic subgroup is its own normalizer. P is called maximal if A/Z(G) is one–dimensional.

If F is a local non–archimedean field, then

G = G(F) = P(F)G(O),

where O is the ring of integers of F. The subgroup K = G(O) is a maximal compact subgroup of G and G = P K is called an Iwasawa decomposition of G.

Let P = MN be a parabolic subgroup of G standard with respect to a Borel subgroup B = TU of G. Let ^{L}G be the L–group of G, and denote by ^{L}M the Levi subgroup of ^{L}G defined by θ^{∨} the coroots of T defined by roots in θ. It should now be clear that ^{L}G has a maximal torus ^{L}T such that

X^{∗}(^{L}T) = X_{∗}(T)

and

X_{∗}(^{L}T) = X^{∗}(T).

The subset of roots of ^{L}T is Σ_{∗}, the coroots of T. Fix a Borel subgroup ^{L}B of ^{L}G such that ^{L}T ⊂ ^{L}B. Let ^{L}U be its unipotent radical. Then the set of coroots of T defined by positive roots of T can be identified with the roots of ^{L}T in ^{L}U. Let LN be the subgroup of ^{L}U spanned by positive roots in ^{L}U which are not spanned by θ^{∨}. Then the subgroup ^{L}P = ^{L}M^{L}N of ^{L}G is called the L–group of P. It is standard with respect to ^{L}B.

Example. The group Sp_{4} has two maximal parabolic subgroups. To explain, let α and β be the short and the long simple roots of Sp_{4}. Then M_{1} = M_{⟨α⟩} ≅ GL_{2} and M_{2} = M_{⟨β⟩} ≅ SL_{2} × GL_{1}. Moreover ^{L}Sp_{4} = SO_{5}(ℂ). Since

⟨α, β⟩ = ⟨β^{∨}, α^{∨}⟩,

then α^{∨} and β^{∨} are the long and the short simple roots of SO_{5}(ℂ), respectively. It can be easily checked that M_{⟨α^{∨}⟩} ⊆ SO_{5}(ℂ) is in fact GL_{2}(ℂ) which is ^{L}M_{1}, the L–group of the reductive group M_{1}. On the other hand

LM_{2} = P GL_{2}(ℂ) × GL_{1}(ℂ)
≅ SO_{3}(ℂ) × GL_{1}(ℂ)

for the reductive group M_{2}. But this is exactly M_{⟨β^{∨}⟩} ⊂ SO_{5}(ℂ).

Now assume F is either local or global. Let G be as before a split reductive group over F. Fix B, T, U, … as before. Let P = MN be a standard parabolic subgroup of G, N ⊆ U, and let A be its split component, i.e. the connected component of the center of M. Let X(M)_{F} and X(A)_{F} be the groups of F –rational character of M and A, respectively. Set

𝔞 = Hom(X(M)_{F} , ℝ)
= Hom(X(A)_{F} , ℝ),

since X(M)_{F} is of finite index in X(A)_{F}. The ℝ–vector space 𝔞 is called the real Lie algebra of A. Then its ℝ–dual is

𝔞^{∗} = X(M)_{F} ⊗_{ℤ} ℝ
= X(A)_{F} ⊗_{ℤ} ℝ.

In fact, if χ ⊗ r ∈ 𝔞^{∗} and λ ∈ 𝔞, then

⟨λ, χ ⊗ r⟩ = λ(χ)r.

Let 𝔞_{ℂ}^{∗} = 𝔞^{∗} ⊗_{ℝ} ℂ.

If F is global, then for every place v of F, X(M)_{F} ↪ X(M)_{F_{v}} will then induce an embedding 𝔞_{v} = Hom(X(M)_{F_{v}}, ℝ) ↪ 𝔞 = Hom(X(M)_{F} , ℝ).

Suppose first that F is local. Define the homomorphism

H_{M} : M = M(F) ⟶ 𝔞

by

exp⟨χ, H_{M}(m)⟩ = |χ(m)|

where χ ∈ X(M)_{F} and | | is that of F. Extend H_{M} to H_{P} on all of G = MNK by extending it trivially on NK.

Next suppose F is global. Let v be a place of F with F_{v} the completion of F at v. The embedding X(M)_{F} ↪ X(M)_{F_{v}} induces a map 𝔞_{v} → 𝔞, where 𝔞_{v} = Hom(X(M)_{F_{v}}, ℝ). We again define a homomorphism

H_{M} : M = M(𝔸) ⟶ 𝔞

by

exp⟨χ, H_{M}(m)⟩ = ∏_{v} |χ(m_{v})|_{v},

where χ ∈ X(M)_{F} and m = (m_{v}) ∈ M(𝔸). We extend H_{M} to H_{P} on all of G = G(𝔸) in the same manner. Observe that

exp⟨χ, H_{M}(m)⟩ = ∏_{v} exp⟨χ, H_{M_{v}}(m_{v})⟩.

Since m_{v} ∈ M(O_{v}) for almost all v < ∞, the product is a finite product.

When F is non–archimedean, it is natural and is usual to use q rather than exp to define H_{M} locally, i.e.

q^{⟨χ,H_{M} (m)⟩} = |χ(m)| (∀χ ∈ X(M)_{F}).

Now assume F is global and define

H_{M} : M = M(𝔸) ⟶ 𝔞

as before by

exp⟨χ, H_{M}(m)⟩ = ∏_{v} |χ(m_{v})|_{v} (∀χ ∈ X(M)_{F})

and therefore

exp⟨χ, H_{M}(m_{v})⟩ = |χ(m_{v})|_{v}.

If H′_{M_{v}} is the one defined by

q_{v}^{⟨χ,H′_{Mv} (m)⟩} = |χ(m)|_{v} (m ∈ M_{v}),

then

H_{M}(m_{v}) = (log q_{v})H′_{M_{v}}(m_{v}),

where q_{v} = exp if F is archimedean. Therefore if we define

H′_{M} : M ⟶ 𝔞

by

H′_{M}|M_{v} = (log q_{v})^{−1}H_{M}|M_{v},

then

exp⟨χ, H_{m}(m)⟩ = ∏_{v=∞} exp⟨χ, H′_{M}(m_{v})⟩ · ∏_{v<∞} q_{v}^{⟨χ,H′_{M} (m_{v})⟩}.

Suppose χ = 2ρ_{P}, sum of positive roots in N, and t ∈ ℂ. Then

exp⟨tρ_{P}, H_{M}(m)⟩ = exp \frac{t}{2}⟨2ρ_{P}, H_{M}(m)⟩

= ∏_{v} |(2ρ_{P})(m_{v})|_{v}^{t/2}

= ∏_{v} δ_{P_{v}}(m_{v})^{t},

where δ_{P_{v}} is the modular character of P_{v}. In fact 2ρ_{P} is the determinant of the adjoint action of M on the Lie algebra of N, a rational character of M. The modulus character δ_{P_{v}}, the ratio of the right and the left invariant measures on N_{v}, when evaluated at m_{v} ∈ M_{v} is just |(2ρ_{P})(m_{v})|_{v}^{1/2}.

9.3. Cusp forms and Eisenstein series. We will be interested in cusp forms on M = M(𝔸). Let us fix a character χ of A = A(𝔸), where A is the split component of M. Let L_{0}^{2}(AM(F)\M, χ) be the Hilbert space of all L^{2}–functions φ on AM(F)\M which transform under A according to χ and moreover

∫_{N_{M}(F)\N_{M}(𝔸)} φ(nm)dn = 0 (∀′m ∈ M),

where N_{M} is the unipotent radical of a parabolic subgroup P_{M} of M for all such parabolic subgroups. The space L_{0}^{2}(AM(F)\M, χ) is called the space of χ–cusp forms on M and M acts on it by right translations. It decomposes discretely to a direct sum of irreducible subspaces each called a cuspidal representation of M. Let π be one such representation. Then again π = ⊗_{v}π_{v} with almost all π_{v}, v < ∞, having a vector fixed by M_{v}(O_{v}), where M_{v} denotes M considered as a group over F_{v}. The representation π_{v} is then uniquely determined by a unique semisimple conjugacy class in ^{L}M_{v} where ^{L}M_{v} = ^{L}M since M is split over F.

Choose a function φ ∈ L_{0}^{2}(AM(F)\M, χ), belonging to π such that

V_{φ} = ⟨π(k)φ|k ∈ K_{M} = ∏_{v} K_{M_{v}}⟩

is finite dimensional. Such functions exist by density arguments and are dense. There are natural ways of extending φ to a function Φ on G = G(𝔸), for example via a finite dimensional representation of K, a maximal compact subgroup of G with K_{M} = K ∩ M(𝔸), which contains the representation of K_{M} on V_{φ} upon restriction [L5, MW1, HC1].

Given ν ∈ 𝔞_{ℂ}^{∗}, let

Φ_{ν}(g) = Φ(g) exp⟨ν + ρ_{P}, H_{P} (g)⟩

where ρ_{P} is half the sum of roots in N, an element in 𝔞^{∗}. Let

E(ν, φ, g) = ∑_{γ∈P(F)\G(F)} Φ_{ν}(γg),

where g ∈ G = G(𝔸), the Eisenstein series attached to ν and ϕ. It converges for ν in a cone inside the positive cone, i.e. the one defined by

⟨ν, H_{α}⟩ > 0

for every simple root α in N. Here

H_{α} = 2α/(α, α)

is the corresponding coroot. It extends to a meromorphic function of ν on all of 𝔞_{ℂ}^{∗} (cf. [L5]).

Now assume P is maximal. Let s be a complex number. Denote by α the unique simple root in N. Set

α̃ = ⟨ρ_{P}, α⟩^{−1}ρ_{P} ∈ 𝔞^{∗}.

Then sα̃ ∈ 𝔞_{ℂ}^{∗}.

9.4 Constant term and intertwining operators. Recall that A ⊂ T, a maximal split torus of G. Let W be the Weyl group of T in G, i.e. the quotient of its normalizer by its centralizer. Assume M = M_{θ}, θ ⊂ Δ and θ ∪ {α} = Δ. There exists a unique element w̃_{0} ∈ W such that w̃_{0}(θ) ⊂ Δ while w̃_{0}(α) ∈ Σ_{−} = −Σ_{+}. Fix a representative w_{0} ∈ K ∩ G(F) for w̃_{0}.

For every place v of F, let

I(sα̃, π_{v}) = Ind_{M_{v}N_{v}↑G_{v}}π_{v} ⊗ q_{v}^{⟨s_{α},H_{Pv} ( )⟩} ⊗ 1.

More precisely, the space V (sα̃, π_{v}) of I(sα̃, π_{v}) is the vector space of all smooth functions f_{v} from G_{v} = G(F_{v}) into the space H(π_{v}) of π_{v} satisfying

f_{v}(mng) = π_{v}(m)q_{v}^{⟨s_{α}+ρ_{P},H_{Pv} (m)⟩}f_{v}(g),

for all m ∈ M_{v}, n ∈ N_{v}, and g ∈ G_{v}. We use exp instead of q_{v} if v = ∞. Finally given f_{v} ∈ V (sα̃, π_{v}), define the corresponding intertwining operator by

A(sα̃, π_{v}, w_{0})f_{v}(g) = ∫_{N′_{v}} f_{v}(w_{0}^{−1}ng)dn,

for all g ∈ G_{v}. Here N′ is the unipotent radical of the standard parabolic subgroup P′ of G which has M′ = M_{_{w_{0}}(θ)} as its Levi subgroup.

Let

M(sα̃, π) = ⊗_{v}A(s, π_{v}, w_{0}).

It acts on

I(sα̃, π) = ⊗_{v}I(sα̃, π_{v}),

which itself is a restricted tensor product, to be explained below, as follows. Given a function f ∈ V (sα̃, π), there exists a finite set of places, including the archimedean ones, such that

f ∈ ⊗_{v∈S}V (sα̃, π_{v}) ⊗ ⊗_{v∉S}{f_{v}^{0}},

where each f_{v}^{0}, v ∉ S, satisfies, f_{v}^{0}(k_{v}) = 1, ∀k_{v} ∈ G(O_{v}). Clearly for such v ∉ S, π_{v} would have to be unramified. The operator M(sα̃, π) then acts by linearity on each component of f. It is therefore important to explain how the resulting infinite product behaves. The operator M(sα̃, π) is called, by abuse of notation in the maximal case, the constant term of E(sα̃, −, −), attached to π. The poles of M(sα̃, π) are exactly those of E(sα̃, −, −).

9.5 L–functions in the constant term. The L–group ^{L}M of M acts on the Lie algebra ^{L}𝔫 of the complex Lie group ^{L}N by adjoint action r (conjugation). Given a positive integer i, let V_{i} be the subspace of ^{L}𝔫 generated by those dual roots β^{∨} for which ⟨α̃, β⟩ = i. There is a positive integer m, such that V_{i} = ϕ for every i > m, while V_{i} ≠ ϕ for every 1 ≤ i ≤ m. Moreover each V_{i} is invariant under ^{L}M. Let r_{i} = r|V_{i}.

Examples.

1) Let G = GL_{m+n}, M = GL_{m}×GL_{n}, and N = M_{m×n}. Then^{L}M = GL_{m}(ℂ)× GL_{n}(ℂ) acts on M_{m×n}(ℂ) by

r(g_{1}, g_{2})X = g_{1}Xg_{2}^{−1},

g_{1} ∈ GL_{m}(ℂ), g_{2} ∈ GL_{n}(ℂ), and X ∈ M_{m×n}(ℂ). The representation r is irreducible and m = 1. This is the case of Rankin–Selberg L–functions discussed in Section 7.

2) Let G be the exceptional group of type G_{2}. It is a rank 2 split group. Let α and β be its simple roots. We assume α is short and β is long. That simply means ∥α∥^{2} = 1 and ∥β∥^{2} = 3. Moreover (α, β) = (β, α) = −\frac{3}{2}. Here ∥α∥^{2} = (α, α), where ( , ) is the standard inner product on ℝ^{2}. The Cartan matrix is

( \begin{matrix}2 & −1 \\ −3 & 2 \end{matrix} ) .

Assume θ = {β}. Then M_{θ} ≅ GL_{2}. (So is M_{θ} if θ = {α}.) The Lie algebra L𝔫 is five dimensional. The L–group ^{L}M = ^{L}M_{θ} ≅ GL_{2}(ℂ) is the Levi subgroup generated by β^{∨} which is the short root of G_{2}(ℂ). The integer m = 2, dim V_{1} = 4, and dim V_{2} = 1. Moreover r_{1} = Ad^{3}, the adjoint cube discussed in Section 6, and r_{2} = det. Recall that

Ad^{3}(( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )) = diag(a^{2}b^{−1}, a, b, a^{−1}b^{2}),

a, b ∈ ℂ^{∗}. This example was called by Langlands in [L2] “extremely striking.”

One can quickly determine the subspaces V_{1} and V_{2}. We will use α and β to also denote the short and the long simple roots of G_{2}(ℂ). Then ^{L}𝔫 is generated by root vectors of β, α + β, 2α + β, 3α + β, and 3α + 2β, since ^{L}M is generated by the short simple root α. The simple root in ^{L}N is β and the numbers i are coefficients of β in the above roots. Then V_{1} is spanned by β, α + β, 2α + β, 3α + β, while V_{2} is spanned by 3α + 2β. Their dimensions are obviously 4 and 1.

On the other hand if θ = {α}, then m = 3, dim V_{2} = 1, while dim V_{1} = dimV_{3} = 2.

A good part of progress on the symmetric cube L–functions for GL_{2}(𝔸) has come from this example (cf. [Sh3, Sh4, Sh9]).

Choose a finite set S of places of F such that π_{v} is unramified. Then π_{v} is uniquely determined by a semisimple conjugacy class A_{v} in ^{L}M_{v} = ^{L}M, M being split. For each complex analytic (finite dimensional) representation r of ^{L}M, there exists a Langlands L–function L(s, π_{v}, r) defined by

L(s, π_{v}, r) = det(I − r(A_{v})q_{v}^{−s})^{−1}.

The following theorem is due to Langlands [L2].

Theorem 9.1. Fix v ∉ S. Then there exists a choice of w such that:

A(sα̃, π_{v}, w)f_{v}^{0} = ∏_{i=1}^{m} L(is, π_{v}, r̃_{i})/L(1 + is, π_{v}, r̃_{i})f̃_{v}^{0}

where f̃_{v}^{0} ∈ V (w(sα̃), w(π_{v})) satisfies f̃_{v}^{0}(k) = 1, ∀k ∈ G(O_{v}).

We should remark that the functions f_{v}^{0} and f̃_{v}^{0} are realized by complex valued functions that one obtains by identifying π_{v} on a constituent of an unramified principal series, i.e. one induced from an unramified character of T ⊂ M and trivial on U ∩ M to M.

Corollary (Langlands). There exists a s_{0} ∈ ℝ such that M(sα̃, π) is defined by an absolutely convergent product for Re(s) > s_{0}.

We refer to [GPSR] for another important application of Theorem 9.1.

Finally let us point out that intertwining operators can be defined for any parabolic subgroup P_{θ} = M_{θ}N_{θ} of G, where G is any (split) group over a local field F. Again let w̃ ∈ W be such that w̃(θ) ⊂ Δ. Let N^{−} = N_{−θ} and

N_{w} = U ∩ wN^{−}w^{−1}.

Given ν ∈ 𝔞_{ℂ}^{∗}, where 𝔞 is the real Lie algebra of A_{θ}, a representative w of w̃, and an irreducible admissible representation σ of M = M_{θ}(F), let

A(ν, σ, w)f(g) = ∫_{N_{w}} f(w^{−1}ng)dn (g ∈ G),

for every f ∈ V (ν, σ), the space of the representation I(ν, σ) induced from

σ ⊗ q^{⟨ν,H_{M} ( )⟩} ⊗ 1.

It converges absolutely for ν in a cone inside the positive Weyl chamber.

There is a product formula giving A(ν, σ, w) as a product of operators for maximal parabolics as we defined using elements w̃_{0} before ([L5, Sh8]) and therefore one needs to study those defined earlier more closely as they are building blocks of the general ones.

9.6 Generic representations.

We will first assume F is a local field whose ring of integer is O. Let G be a split reductive group over F and fix B, T, and U as before. Let Δ be the set of simple roots in U. Then

U/[U, U] ≅ ∏_{α∈Δ} U_{α}(F),

where each U_{α} = U_{α}(F) is the one dimensional subgroup generated by

exp(tX_{α}) = ∑_{n=0}^{∞} \frac{t^{n}X_{α}^{n}}{n!} , (t ∈ F)

where X_{α} is the corresponding root vector. Let ψ be a character of U. Then

ψ = ∏_{α} ψ_{α}

with each ψ_{α} a character of U_{α}. The character ψ is called generic or non–degenerate if each ψ_{α} is non–trivial. Having fixed X_{α}, α ∈ Δ, then each ψ_{α} is a character of F which we still denote by ψ_{α}. A character of F is called unramified if O is the largest ideal on which the character is trivial. The character ψ is called unramified if and only if each ψ_{α} is unramified.

An irreducible admissible representation (σ, H(σ)) of G = G(F) is called ψ– generic, where ψ is a non–degenerate character of U, if there exists a functional λ ∈ H(σ)′, the full dual of H(σ), such that

λ(σ(u)v) = ψ(u)λ(v)

for all u ∈ U and v ∈ H(σ). When F is archimedean one would require λ to be continuous with respect to semi–norm topology defined by elements of universal enveloping algebra of the Lie algebra of G, or more simply by right invariant differential operators on 𝔤. A non–zero such λ is called a Whittaker functional.

Theorem 9.2 (Shalika [S], Gelfand–Kazhdan). The dimension of the space of Whittaker functionals for σ is at most 1.

Now suppose F is global. A character χ = ⊗_{v}χ_{v} of U(F)\U(𝔸) is generic if and only if each χ_{v} is generic. Let φ be a cusp form. Set:

W_{φ}(g) = ∫_{U(F)\U(𝔸)} φ(ug)\overline{χ(u)}du,

g ∈ G = G(𝔸). φ is called globally χ–generic if there exists a g ∈ G such that W_{φ}(g) ≠ 0. The function W_{φ} is called the Whittaker function attached to φ. Finally if φ belongs to the space of a cuspidal representation π = ⊗_{v}π_{v}, then the vector space of all W_{φ} is called a Whittaker model for π.

Let F be local and assume σ is an irreducible admissible ψ–generic representation of G. Fix a Whittaker functional λ and define a function W_{ν} for each v ∈ H(σ) by

W_{v}(g) = λ(σ(g)v).

The space of all W_{v}’s is called a Whittaker model for σ.

9.7. Local coefficients. Let F be local and σ be an irreducible admissible ψ_{M} generic representation of M = M(F), F –points of a Levi subgroup for the parabolic subgroup P = MN with N ⊂ U. As usual B = TU is a Borel subgroup of G. The character ψ_{M} is defined by means of restriction from a character ψ of U = U(F) which is assumed to be generic. For the sake of simplicity from now on we shall assume all the ψ_{α}’s appearing in ψ, α ∈ Δ, are equal.

Fix ν ∈ 𝔞_{ℂ}^{∗} and consider the representation I(ν, σ) of G induced from

σ ⊗ q^{⟨ν,H_{M} ( )⟩} ⊗ 1.

Let w̃_{0} ∈ W be such that w̃_{0}(θ) ⊂ Δ, P = P_{θ}, while w̃_{0}(α) is negative for every α in N. Let M′ = M_{_{w_{0}}(θ)} and P′ = M′N′, P′ = P_{_{w_{0}}(θ)}, N′ = N_{_{w_{0}}(θ)} ⊂ U. Let λ_{M} be a ψ_{M}–Whittaker functional for the space H(σ) of σ. For each f ∈ V (ν, σ), define

λ_{ψ}(ν, σ)(f) = ∫_{N′=N′(F)} \overline{ψ(n′)}λ_{M}(f(w_{0}^{−1}n′))dn′,

where w_{0} is a representative for w̃_{0}. Clearly

λ(I(u)f) = ψ(u)λ(f),

where λ = λ_{ψ}(ν, σ) and I = I(ν, σ), i.e. λ is a ψ–generic Whittaker functional for I(ν, σ). It is easy to see λ ≠ 0. Moreover, as a function of ν, λ_{ψ}(ν, σ) is holomorphic for all ν (cf. [CS]).

Now fix a w̃ ∈ W and choose a representative w for w̃. Let A(ν, σ, w) be the intertwining operator from I(ν, σ) into I(w(ν), w(σ)). Denote by λ_{ψ}(w(ν), w(σ)) the Whittaker functional defined as above for I(w(ν), w(σ)). Then

λ_{ψ}(w(ν), w(σ))A(ν, σ, w)

is another non–zero (generically) Whittaker functional for I(ν, σ). There is a theorem of Rodier [Ro] which states that such functionals for the induced representation I(ν, σ) are unique up to scalars. Thus there exists a scalar C_{ψ}(ν, σ, w), called the local coefficient [Sh8] attached to ν, σ, and w such that

λ_{ψ}(ν, σ) = C_{ψ}(ν, σ, w)λ_{ψ}(w(ν), w(σ))A(ν, σ, w).

We shall now explain the most important property of local coefficients.

Assume F is global with 𝔸 as its ring of adeles. Let G = G(A), M = M(𝔸), N, P, B, T, U be as before. We will assume P is maximal. Let ψ_{M} be a generic character of U_{M}(F)\U_{M}(𝔸), where U_{M} = U ∩ M. Let φ be a globally ψ_{M}–generic cusp form on M = M(𝔸), belonging to the space of the irreducible cuspidal representation π = ⊗_{v}π_{v} of M. Write ψ_{M} = ⊗_{v}ψ_{M,v}. Let s ∈ ℂ and if α is the unique simple root in N, let α̃ = ⟨ρ, α⟩^{−1}ρ be as in 9.3. Finally let r_{i}, 1 ≤ i ≤ m, be representations of ^{L}M as defined in 9.5.

Theorem 9.3 (Crude functional equation [Sh8]). Let S be a finite set of places such that π_{v} and ψ_{v} are both unramified for every v ∉ S. Then

∏_{i=1}^{m} L_{S}(is, π, r_{i}) = ∏_{v∈S} C_{\overline{ψ}_{v}}(sα̃, π̃_{v}, w_{0}) ∏_{i=1}^{m} L_{S}(1 − is, π, r̃_{i}).

Here ψ = ⊗_{v}ψ_{v} is any generic character of U(F)\U which restricts to ψ_{M} and L_{S} is as usual the product of corresponding local L–functions at all v ∉ S.

Sketch of the proof. Let E(sα̃, Φ, g) be the Eisenstein series defined by extension Φ of φ as in 9.3. Consider the non–constant Fourier coefficient

E_{ψ}(sα̃, Φ, g) = ∫_{U(F)\U} E(sα̃, Φ, ug)\overline{ψ(u)}du

of E(sα̃, Φ, −). Then it can be shown [Sh1, Sh8] that

E_{ψ}(sα̃, Φ, e) = ∏_{v∈S} W_{v}(e_{v}) ∏_{i=1}^{m} L_{S}(1 + is, π, r̃_{i})^{−1},

where for v ∈ S

W_{v}(e_{v}) = λ_{ψ_{v}}(sα̃, π_{v})(f_{v})

with f_{v} ∈ V (sα̃, π_{v}). Here we assume that the extension Φ of φ to G = G(𝔸) corresponds to f = ⊗_{v}f_{v} ∈ I(sα̃, π). The first important ingredient in the proof is the formula

λ_{ψ_{v}}(sα̃, π_{v})(f_{v}^{0}) = ∏_{i=1}^{m} L(1 + is, π_{v}, r̃_{i})^{−1}

for every v ∉ S due to Casselman and Shalika [CS]. Here f_{v}^{0} ∈ I(sα̃, π_{v}) is as in Theorem 9.1, i.e. f_{v}^{0}(k) = 1 for all k ∈ G(O_{v}).

Next one needs to apply (cf. [Sh7, Sh8]) some deep technical results at the archimedean places due to Casselman [C1] and Wallach [Wa] to show that W_{v}’s can be chosen so that W_{v}(e_{v}) ≠ 0 if v = ∞. The crude functional equation is now a consequence of that of Eisenstein series E(sα̃, Φ, g), namely

E(sα̃, Φ, g) = E(−sα̃, M(s)Φ, g),

and the definition of local coefficients, using

M(sα̃, π) = ⊗_{v}A(sα̃, π_{v}, w_{0}).

In the next paragraph we shall state the main global results of the method.

9.8. Main global results. F is now a global field and G, P, M, N, U, T, B, ψ_{M}, ψ, w̃_{0}, w_{0}, … are as before, and P is assumed maximal. We fix a ψ–generic cuspidal representation π = ⊗_{v}π_{v} of M = M(𝔸) and we choose representations r_{i} as before. The set S is always a finite set of places of F such that π_{v} and ψ_{v} are both unramified for v ∉ S.

Theorem 9.4 [Sh8]. Fix t ∈ ℝ. Then

∏_{i=1}^{m} L_{S}(1 + it\sqrt{−1}, π, r_{i}) ≠ 0.

Sketch of the Proof. We have

(9.4.1)  E_{ψ}(sα̃, Φ, e) = ∏_{v∈S} W_{v}(e_{v}) ∏_{i=1}^{m} L_{S}(1 + is, π, r̃_{i})^{−1}.

By the general theory of Eisenstein series, E(sα̃, −, −) is holomorphic if s is pure imaginary, i.e. s = \sqrt{−1}t, t ∈ ℝ. But E_{ψ} is obtained by integrating E over the compact set U(F)\U which remains holomorphic on the imaginary axis. Using the archimedean results [C1, Wa, Sh7, Sh8] discussed in the proof of Theorem 9.3, each W_{v}(e_{v}) can be made non–zero for a given s ∈ ℂ, v = ∞. Their non–vanishing for v < ∞ is easily verified. The theorem now follows from (9.4.1).

We refer to Theorem 7.4 and [Sh4, Sh8] for important special cases of Theorem 9.4 and the application of Theorem 7.4 to the important classification theorem of Jacquet and Shalika [JS2] (Theorem 8.1 here).

Theorem 9.5 a) (cf. [L2, Sh1]). Each L_{S}(s, π, r_{i}) extends to a meromorphic function of s on ℂ.

b) [Sh1] Assume m = 1 or m = 2 and dim r_{2} = 1. Then L_{S}(s, π, r_{1}) extends to a meromorphic function of s on ℂ with only a finite number of poles, if S is large enough to include all the conjugates of places over which π_{v} and ψ_{v} ramify.

We refer to [Sh1] for the proofs and the induction involved, as well as a list of examples where b) is valid.

Theorem 9.6. At each place v ∉ S and for each i, the local L–function

L(s, π_{v}, r_{i}) = det(I − r_{i}(A_{v})q_{v}^{−s})^{−1}

is holomorphic for Re(s) ≥ 1. Here A_{v} ∈ ^{L}M is a semisimple element in the conjugacy class of ^{L}M attached to π_{v}.

This is proved in [Sh1] using an induction on m whose full version was later proved in [Sh5], together with;

Lemma 9.7. For Re(s) ≥ 1, L_{S}(s, π, r̃_{i}) and the quotient

L_{S}(s, π, r̃_{i})/L_{S}(1 + s, π, r̃_{i})

both have only a finite number of poles and zeros, 1 ≤ i ≤ m.

Sketch of the proof. Assume that for Re(s) ≥ 1 and each i, 2 ≤ i ≤ m, the corresponding quotient has a finite number of poles and zeros. Then by finiteness of poles of M(sα̃, π) for Re(s) ≥ 0, the relation

M(sα̃, π) = ⊗_{v}A(sα̃, π_{v}, w_{0}),

and Theorem 9.1, the quotient for i = 1 has a finite number of poles for Re(s) ≥ 1. Now using this line of absolute convergence for each L_{S}(s, π, r̃_{i}), one concludes that each L_{S}(s, π, r̃_{i}), 1 ≤ i ≤ m, has a finite number of poles for Re(s) ≥ 1.

On the other hand by (9.4.1)

∏_{i=1}^{m} L_{S}(1 + is, π, r̃_{i})

has a finite number of zeros for Re(s) ≥ 0. Applying the finiteness of poles for each L_{S}(s, π, r̃_{i}) for Re(s) ≥ 1 and 2 ≤ i ≤ m, we conclude that L_{S}(s, π, r̃_{1}) has also a finite number of zeros for Re(s) ≥ 1. Consequently the quotient for i = 1 has a finite number of poles and zeros for Re(s) ≥ 1, completing the induction.

Sketch of the proof of Theorem 9.6. We enlarge S to include v and use

M(sα̃, π)(⊗_{u}f_{u}) = ⊗_{u∈S}A(sα̃, π_{u}, w_{0})f_{u} ⊗ ∏_{i=1}^{m} L_{S}(is, π, r̃_{i})/L_{S}(1 + is, π, r̃_{i})f̃_{S}^{0}

for f = ⊗_{u}f_{u} for which f_{u} = f_{u}^{0} for u ∉ S and

f̃_{S}^{0} = ⊗_{u∉S}f̃_{u}^{0}

with f_{u}^{0} and f̃_{u}^{0} as in Theorem 9.1.

Now suppose A(s, π_{v}, w_{0}) has a pole for some s with Re(s) ≥ 1. For each u ∈ S, u ≠ v, choose f_{u} such that

A(sα̃, π_{u}, w_{0})f_{u}(w_{0}) ≠ 0.

The operator A(s, π_{v}, w_{0}) having values which are rational functions of q_{v}^{−s} will then have infinitely many poles parallel to the imaginary axis. By Lemma 9.7 so must be M(sα̃, π), a contradiction. Consequently A(s, π_{v}, w_{0}) is holomorphic for Re(s) ≥ 1. This then implies

∏_{i=1}^{m} L(is, π_{v}, r̃_{i})/L(1 + is, π_{v}, r̃_{i})

is holomorphic for Re(s) ≥ 1.

Assume each local quotient is holomorphic for Re(s) ≥ 1 when 2 ≤ i ≤ m. Then using Re(s) large enough so that local L–functions are holomorphic there, one concludes that each L(s, π_{v}, r̃_{i}), 2 ≤ i ≤ m, is holomorphic for Re(s) ≥ 1. Consequently each local quotient for 2 ≤ i ≤ m is also non–zero for Re(s) ≥ 1, implying the holomorphy of the quotient for i = 1. We again use induction and the holomorphy of L(s, π_{v}, r̃_{1}) for Re(s) large, to conclude it for Re(s) ≥ 1. The induction is now complete and the Theorem 9.6 follows.

One important corollary to this which gives the best estimate presently proved for Hecke eigenvalues of cusp forms over an arbitrary number field can be formulated as:

Corollary 1 [Sh1, Sh2]. Let π = ⊗_{v}π_{v} be an irreducible cuspidal representation of GL_{2}(𝔸), where 𝔸 is the ring of adeles of a number field F. For each v where π_{v} is unramified, let the corresponding conjugacy class in GL_{2}(ℂ) be given by

t_{v} = ( \begin{matrix}α_{v} & 0 \\ 0 & β_{v} \end{matrix} ) .

Then

q_{v}^{−1/5} < |α_{v}| and |β_{v}| < q_{v}^{1/5}.

Proof in the adjoint case. Let us give a proof when π has trivial central character. The general case requires a little more knowledge of exceptional groups. We refer to [Sh2] for detail of the general case. The basic idea is the same.

Let G be a split group of type F_{4}. Its Dynkin diagram is

∘_{α_{1}}——∘_{α_{2}}——∘_{α_{3}}——∘_{α_{4}}

with α_{1} and α_{2} long and α_{3} and α_{4} short. Let Π be the Gelbart–Jacquet lift of π. Since Ramanujan–Petersson is valid for monomial cusp forms, we may assume Π is a cuspidal representation of P GL_{3}(𝔸). Let M = M_{θ}, where θ = {α_{1}, α_{2}, α_{4}}. There is an exact sequence

0 ⟶ A ⟶ M ⟶ P GL_{3} × P GL_{2} ⟶ 0,

leading to a surjection

M ⟶ P GL_{3}(𝔸) × P GL_{2}(𝔸) ⟶ 0

by the lemma in page 36 of [L2]. The cuspidal representation Π × π of P GL_{3}(𝔸) × P GL_{2}(𝔸) then defines a cuspidal representation ρ of M. Let

t_{v} = ( \begin{matrix}α_{v} & 0 \\ 0 & α_{v}^{−1} \end{matrix} ) ∈ SL_{2}(ℂ) = ^{L}P GL_{2}

represent the conjugacy class attached to π_{v}. Then among the factors dividing L(s, ρ, r_{1})^{−1} is

(1 − α_{v}^{5}q_{v}^{−s})(1 − α_{v}^{−5}q_{v}^{−s}).

By Theorem 9.6 this must be non–zero for Re(s) ≥ 1. This implies

|α_{v}^{5}|q_{v}^{−1} < 1

as well as

|α_{v}^{−5}|q_{v}^{−1} < 1,

proving the Corollary.

The corollary can also be proved by letting G be a group of type E_{6}

∘ α_{2}
|

∘_{α_{1}}———∘_{α_{3}}———∘_{α_{4}}———∘_{α_{5}}———∘_{α_{6}}

and M = M_{θ} with θ = {α_{1}, α_{3}, α_{2}, α_{5}, α_{6}}, and the cuspidal representation Π × π × Π of P GL_{3}(𝔸) × P GL_{2}(𝔸) × P GL_{3}(𝔸).

Corollary 2 [Sh1]. Every L–function L_{S}(s, π, r_{i}) is absolutely convergent for Re(s) > 2.

The fact that π is globally generic is crucial. In fact one expects that such forms satisfy Ramanujan–Petersson’s conjecture to the effect that each local component π_{v} of π is tempered, i.e. their matrix coefficients are in L^{2+ε}(M_{v}) for each ε > 0.

Local factors and functional equations. Let F be local and G a split group over F. We fix B, T, U, P, M, N, ..., as before. We always assume N ⊂ U and T ⊂ M. Moreover assume P is maximal and let α be the unique simple root in N. Define α̃ = ⟨ρ, α⟩^{−1}ρ as before. Let ψ be a generic character of U = U(F) extending ψ_{M}, one of U_{M} = (U ∩ M)(F). We assume ψ is defined by a character of F, denoted the same way. Let r_{1}, , …, r_{m} be representations of ^{L}M on ^{L}𝔫 as before. Let s ∈ ℂ.

Theorem 9.8. Let σ be an irreducible admissible ψ_{M}–generic representation of M. Then for each i, 1 ≤ i ≤ m, there exists a root number ε(s, σ, r_{i}, ψ) which is a monomial in q^{−s} if F is non–archimedean, and an L–function L(s, σ, r_{i}), whose inverse is a polynomial in q^{−s} with value 1 at zero, again if F is non–archimedean, with the following properties.

1) If F is archimedean or σ has an Iwahori fixed vector (any component of an unramified principal series), let φ′ : W′_{F} ⟶ ^{L}M be the homomorphism of the Deligne–Weil group W′_{F} parametrizing σ. Then

ε(s, σ, r_{i}, ψ) = ε(s, r_{i} · φ′, ψ)

and

L(s, σ, r_{i}) = L(s, r_{i} · φ′),

where the factors on the right are the corresponding Artin root numbers and L– functions.

2) For each i, 1 ≤ i ≤ m,

C_{ψ}(sα̃, σ, w_{0}) = ∏_{i=1}^{m} ε(is, σ, r̃_{i}, \overline{ψ})L(1 − is, σ, r_{i})/L(is, σ, r̃_{i}).

3) Inductive property (cf. part 3, Theorem 3.5 and equations (7.10) and (7.11) of [Sh5]).

4) Functional equations. Let K be a number field and G̃ a split group over K. Let P̃ = M̃Ñ, Ñ ⊂ Ũ, be a maximal parabolic subgroup of G̃. Fix a generic character ψ̃ = ⊗_{v}ψ_{v} of Ũ = Ũ(𝔸_{K}), trivial on Ũ(K), defined by one of K\𝔸_{K}, still denoted the same way. Let ψ̃_{M} = ψ̃|M̃ ∩ Ũ. Let π = ⊗_{v}π_{v} be a ψ̃_{M}–generic cuspidal representation of M̃ = M̃(𝔸_{K}). Finally let r_{1}, …, r_{m} be representations of ^{L}M̃ or ^{L}𝔫̃ as before. For each i, 1 ≤ i ≤ m, set

L(s, π, r_{i}) = ∏_{v} L(s, π_{v}, r_{i})

and

ε(s, π, r_{i}) = ∏_{v} ε(s, π_{v}, r_{i}, ψ_{v}),

where the first product is a finite product and the second one converges absolutely for Re(s) large (say bigger than 2). Then:

L(s, π, r_{i}) = ε(s, π, r_{i})L(1 − s, π, r̃_{i})

for each i, 1 ≤ i ≤ m.

Moreover, let

(9.8.1)  γ(s, σ, r_{i}, ψ) = ε(s, σ, r_{i}, φ)L(1 − s, σ, r̃_{i})/L(s, σ, r_{i}).

Then properties 1, 3, and 4, determine complex functions γ(s, σ, r_{i}, ψ), 1 ≤ i ≤ m, uniquely.

Several comments are in order:

1. When F is archimedean one may take W_{F} instead of W′_{F} since W_{F} suffices to parametrize all the irreducible admissible representations of real groups [L4]. We refer to [T] and pages 286–288 of [Sh5] for W′_{F}.

2. The γ–functions γ(s, σ, r_{i}, ψ) are defined first using local coefficients inductively. When σ is tempered and F is non–archimedean, the L–function L(s, σ, r_{i}) is defined as the inverse of the normalized numerator of γ(s, σ, r_{i}, ψ), a polynomial in q^{−s} whose constant term is 1. The root number is then defined by means of (9.8.1). To define root numbers and L–functions for any irreducible admissible generic σ, one uses analytic continuation of the ones defined by tempered σ into the positive Weyl chamber and Langlands classification, exactly as in the archimedean one. It is then clear that uniqueness of γ(s, σ, r_{i}, φ) leads to the same thing for root number and L–functions.

3. Finally, there is one important property of these L–functions that can be formulated as following conjecture, since it is not yet proved in general.

Conjecture 9.9. Assume σ is tempered. Then each L(s, σ, r_{i}) is holomorphic for Re(s) > 0.

This is conjecture 7.1 of [Sh5] which is true if F = ℝ or ℂ. Its validity is proved in following cases when F is p–adic:

1) m = 1, σ is generic tempered;

2) m = 2 and dim r_{2} = 1, σ is generic tempered;

3) σ is generic supercuspidal, i.e. its matrix coefficients are compact modulo the center of the group, and finally

4) G is of classical type, but σ is any generic tempered representation of M. This includes all the classical groups.

Cases 1), 2), and 3) are proved in [Sh5], while 4) which is more involved was just recently settled [CSh]. In the next section we will study certain applications of Conjecture 9.9.

§10. Some local applications

Throughout this section F is local (of characteristic zero), i.e. either F = ℝ or ℂ or a finite extension of ℚ_{p} for some prime p. G will be a split reductive group over F and B, T, U, P, M, N, ψ, ψ_{M}, ... are all as before.

10.1 Normalization of intertwining operators. Let P = P_{θ}, θ ⊂ Δ, w̃ ∈ W. For simplicity of formulation assume w̃(θ) = θ. Let N_{w} = U ∩ w̃N^{−}w^{−1}, where N^{−} = N_{−θ}. Let ^{L}𝔫_{w} be the Lie algebra of ^{L}N_{w}. The L–group ^{L}M of M acts on L𝔫_{w} by adjoint action. Let r_{w} be the representation of ^{L}M on ^{L}𝔫_{w}. Use r̃_{w} to denote its contragredient.

Let σ be an irreducible unitary ψ–generic representation of M. Fix ν ∈ 𝔞_{ℂ}^{∗} and define A(ν, σ, w), the intertwining operator attached to ν, σ, and w as in §9.5, where w is a representative for w̃. We recall for completeness that

A(ν, σ, w)f(g) = ∫_{N_{w}} f(w^{−1}ng)dn (f ∈ I(ν, σ)).

We set

A(σ, w) = A(0, σ, w)

if the right hand side is well defined.

Using inductive property 3) of Theorem 9.8, one can define a root number ε(s, σ, r_{w}, ψ) and an L–function L(s, σ, r_{w}) for r_{w} as in Page 312 of [Sh5]. The following normalization of intertwining operators was first conjectured by Langlands in [L5]. Let

A(σ, w) = ε(0, σ, r̃_{w}, ψ)L(1, σ, r̃_{w})L(0, σ, r̃_{w})^{−1}A(σ, w),

where the right hand side is determined as a limit.

The normalized operator A(σ, w) is supposed to satisfy a list of properties set forth in Theorem 2.1 of [A3], a larger list than what was originally demanded by Langlands in [L5]. When F = ℝ on ℂ, they are all verified by Arthur in [A3]. Theorem 7.9 of [Sh5] verifies them when F is p–adic and σ is generic, except for condition R_{7} of Theorem 2.1 of [A3]. Condition R_{7}, which was not originally demanded in [L5], follows immediately from Conjecture 9.9. For the sake of completeness, let us state Theorem 7.9 of [Sh5] here as:

Theorem 10.1 (Langlands’ Conjecture). The normalized operator A(σ, w) satisfies:

a) A(σ, w_{1}w_{2}) = A(w_{2}(σ), w_{1})A(σ, w_{2}) and

b) A(σ, w)^{∗} = A(w(σ), w^{−1}), where A(σ, w)^{∗} is the adjoint of A(σ, w), i.e. A(σ, w) is unitary.

In general, i.e. when w̃(θ) is not necessarily equal to θ, the representation r_{w} is the adjoint action of ^{L}M on the Lie algebra of the L–group of \overline{N} − ; _{w} = (w^{−1}N_{w}w)^{−}, the unipotent group opposed to w^{−1}N_{w}w. Observe that if w̃(θ) = θ, then \overline{N}− ; _{w} =

N_{w}.

We refer to Section 9 of [Sh5] for how one expects to extend the normalization to representations which are not necessarily generic.

10.2. Irreducibility of standard modules for generic representations. Let 𝔞 be the real Lie algebra of the split component of M. Take ν ∈ 𝔞_{ℂ}^{∗}. For each simple root α in N, realized as a root of T, let H_{α} ∈ 𝔱 be the corresponding coroot. Consider ν as as element of 𝔱_{ℂ}^{∗}. We will say ν is in positive Weyl chamber of A if

Re⟨ν, H_{α}⟩ > 0

for every simple root α in N.

Let σ be a tempered representation of M and consider I(ν, σ). If ν is in the positive Weyl chamber, then I(ν, σ) is called a standard module. It has a unique quotient J(ν, σ), called the Langlands quotient of I(ν, σ) (cf. [L4, BoWa, Si2]). Every irreducible admissible representation of G is of the form J(ν, σ) for some ν and σ. The choice of ν and σ are unique up to conjugation.

When F = ℝ or ℂ, it was proved by Vogan [V], that if J(ν, σ) is generic, then I(ν, σ) = J(ν, σ), i.e. the standard module of a generic representation is irreducible. In this section we will exploit this to deduce results on reducibility of induced representations as well as study the non–archimedean case. The following is the subject matter of a joint work with Casselman [CSh].

Proposition 10.2. Let I(ν_{0}, σ) be the standard module for a ψ–generic representation. Assume

lim_{ν⟶ν_{0}} C_{ψ}(w_{0}(ν), w_{0}(σ))A(w_{0}(ν), w_{0}(σ), w_{0}^{−1})

is well defined on I(w_{0}(ν_{0}), w_{0}(σ)). Then I(ν_{0}, σ) is irreducible.

Proof. Choose f ∈ I(w(ν_{0}), w_{0}(σ)) such that

λ_{ψ}(w_{0}(ν_{0}), w_{0}(σ))(f) ≠ 0.

Since

C_{ψ}(w_{0}(ν), w_{0}(σ))A(w_{0}(ν), w_{0}(σ), w_{0}^{−1})

is well defined at ν = ν_{0}, the image of f, and consequently the image of I(w_{0}(ν_{0})w_{0}(σ)), under it will be generic. By Rodier’s theorem, it must equal J(ν_{0}, σ). Thus I(ν_{0}, σ) = J(ν_{0}, σ). Converse is clear.

Corollary 1. Suppose F is p–adic. Assume σ is supercuspidal. Then Vogan’s theorem is valid, i.e. J(ν, σ) is generic if and only if I(ν, σ) = J(ν, σ).

Proof. This follows immediately from the validity of Conjecture 9.9 for σ supercuspidal which implies the holomorphy of C_{ψ}(w_{0}(ν), w_{0}(σ)) for all ν in the positive Weyl chamber, as well as the fact that A(w_{0}(ν), w_{0}(σ), w_{0}^{−1}) has no poles as long as ν is regular.

More can be proved.

Theorem 10.2. Let F be any local field (of characteristic zero). Fix ν in the positive Weyl chamber and take σ to be tempered. Assume

C_{ψ}(w_{0}(ν), w_{0}(σ))A(w_{0}(ν), w_{0}(σ), w_{0}^{−1})

is well defined on all of I(w_{0}(ν), w_{0}(σ)). Then

a) Suppose J(ν, σ) is ψ–generic. Then I(ν, σ) = J(ν, σ), i.e. I(ν, σ) is irreducible.

b) Assume F is non–archimedean. Suppose J(ν, σ) is not ψ–generic. Let W be the direct sum of the irreducible subspaces of I(ν, σ), counted with their multiplicities. Let

σ_{ν} = σ ⊗ q^{⟨ν,H_{M} ( )⟩}.

Denote by δ = δ_{P} the modulus character of P = MN. Assume σ_{ν}δ^{1/2} does not appear as a subquotient anywhere in the Jacquet module (V/W)_{N} = V_{N}/W_{N} of V/W, where V = V (ν, σ). Then the ψ–generic subquotient of V (ν, σ) is a subrepresentation.

We should only remind the reader of what Jacquet modules are. Let G be a (split) reductive group over an archimedean local field F. Choose a parabolic subgroup P = MN of G. Let (π, V) be a smooth representation of G, i.e. a representation for which every v ∈ V is fixed by an open compact subgroup of G. Let

V (N) = {π(n)v − v|n ∈ N, v ∈ V }.

Since M normalizes N, it acts on V (N) as well as on V/V (N) = V_{N}, giving a representation π_{N} of M on V_{N}. One can then prove [C2]:

1. Let U ⟶ V ⟶ W be an exact sequence of smooth representations of G. Then U_{N} ⟶ V_{N} ⟶ W_{N} is exact.

2. (Frobenius reciprocity) Let (π, V) be a smooth representation of G and let σ be a smooth representation of M on a space U. Then the P –morphism

Λ : (I(σ), V (σ)) ⟶ (σδ_{P}^{1/2}, U)

defined by

f ↦ f(e)

induces an isomorphism:

Hom_{G}(V, V (σ)) ≅ Hom_{M}(V_{N}, U),

where U is given the M–structure σδ_{P}^{1/2}.

Then by 1) and 2) it is clear that σδ_{P}^{1/2} appears as a quotient of (V_{0})_{N} exactly as many times as the number of subrepresentations of V_{0} with multiplicities. Beside those, (V_{0})_{N} might also have σδ_{P}^{1/2} as subquotients which are not quotients. The condition in Theorem 10.3.b then means that there will be no more appearance of σδ_{P}^{1/2} in V_{N}, except possibly in (ker λ_{ψ}(ν, σ))_{N}.

The following result can also be proved.

Proposition 10.4. Let P = MN be a maximal parabolic subgroup of G, a split reductive group over a local field F (of characteristic zero). Fix s ∈ ℂ with Re(s) > 0. Let σ be an irreducible generic tempered representation of M. Assume that whenever J(sα̃, σ) is generic, I(sα̃, σ) = J(sα̃, σ). Then I(sα̃, σ) is irreducible if and only if

∏_{i=1}^{m} L(1 − is, σ, r_{i})^{−1} ≠ 0,

if conjecture 9.9 is valid, when F is non–archimedean.

Corollary. Suppose F = ℝ or ℂ. Let

φ : W_{F} ⟶ ^{L}M

be the parametrization of σ. For each i, 1 ≤ i ≤ m, let L(s, r_{i} · φ) be the corresponding Artin L–function. Then I(sα̃, σ) is irreducible if and only if

∏_{i=1}^{m} L(1 − is, r_{i} · φ)^{−1} ≠ 0.

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