AUTOMORPHIC L-FUNCTIONS SALIM ALI ALTUGphlee/CourseNotes...SALIM ALI ALTUG NOTES TAKEN BY PAK-HIN LEE AND REVISED BY SALIM ALI ALTUG Abstract. Here are the notes I took for Salim Ali

AUTOMORPHIC L-FUNCTIONS

SALIM ALI ALTUĞ

NOTES TAKEN BY PAK-HIN LEE AND REVISED BY SALIM ALI ALTUĞ

Abstract. Here are the notes I took for Salim Ali Altuğ’s course on automorphic L-functions offered at Columbia University in Spring 2015 (MATH G6675: Topics in NumberTheory). Hopefully these notes will be completed by Spring 2016. I recommend that youvisit my website from time to time for the most updated version.

Due to my own lack of understanding of the materials, I have inevitably introducedboth mathematical and typographical errors in these notes. Please send corrections andcomments to [email protected] and [email protected].

Contents

1. Lecture 1 (January 20, 2015) 31.1. Introduction 31.2. Modular forms 32. Lecture 2 (January 22, 2015) 72.1. Last time 72.2. Spectral theory 7

The following has not been revised by Ali yet. Use with caution!3. Lecture 3 (January 29, 2015) 123.1. Last time 123.2. Real-analytic Eisenstein series 133.3. A (classical) computation 143.4. Why is the L2 theory helpful? 154. Lecture 4 (February 3, 2015) 164.1. Last time 164.2. Analytic continuation 174.3. Application: Prime number theorem 184.4. Application: Gauss class number problem 184.5. Application: Volumes of fundamental domains 195. Lecture 5 (February 5, 2015) 215.1. Convergence of averaging 215.2. Reminder (last time) 215.3. Modular forms as Maass forms 215.4. From H to group 225.5. To adelic group 235.6. Eisenstein series (adelically) 24

Last updated: November 19, 2015.1

http://www.math.columbia.edu/~altug/http://math.columbia.edu/~altug/Topics_in_NT/webpage/Topics_in_NT.htmlhttp://www.math.columbia.edu/~phlee/mailto:[email protected]:[email protected]

6. Lecture 6 (February 10, 2015) 266.1. Last time 266.2. Eisenstein series (adelic) 266.3. Constant term 277. Lecture 7 (February 12, 2015) 317.1. Algebraic group theory 317.2. Roots, weights, etc. 348. Lecture 8 (February 17, 2015) 378.1. Root systems 378.2. Parabolic (sub)groups 388.3. Decompositions of parabolics 409. Lecture 9 (February 19, 2015) 429.1. Last time 429.2. Character spaces 469.3. The Hp function 4610. Lecture 10 (February 24, 2015) 4910.1. Last time 4910.2. More on parabolics 4910.3. Parabolic induction 5010.4. Eisenstein series 5211. Lecture 11 (February 26, 2015) 5311.1. Last time 5311.2. A concrete example 5311.3. Intertwining 5411.4. Decomposition of L2(G(F )\G(A)) 5411.5. Drawings 5512. Lecture 12 (March 3, 2015) 5612.1. Integration 5612.2. Decomposition of Haar measures 5712.3. Applications 5812.4. Convergence of Eisenstein series 5912.5. A little bit of reduction theory 6012.6. Back to Eisenstein series 61

2

1. Lecture 1 (January 20, 2015)

1.1. Introduction. One central theme in number theory is automorphic L-functions. Inthe 1960’s it was unclear what a general L-function is supposed to be; there were certainlythe Riemann zeta function ζ(s), L(s, χ) due to Hecke et al., Artin L-functions, and alsosuggestions from Tamagawa. My goal is to show how automorphic L-functions arose since1967. The whole theory was born from the Eisenstein series, particularly their constantterms, and was really a coincidence, since Langlands was studying these constant terms justto kill time (more on this later).

I will go over Langlands’ book Euler Products1, which was the manuscript written on hislectures at Yale in 1967. In order to talk about this, we need to define a lot of things. Onewarning is that this book is not easy to read, as is anything that Langlands wrote. I willkeep updating the list of references on my website.

This theory was later known as the Langlands–Shahidi method, and was used to provefunctoriality of Sym3 and Sym4 of cuspidal automorphic representations of GL(2). Onedownside of this method is that it is limited; this will be made precise later.

I will try to keep things self-contained. The theory of Eisenstein series involves:

• reductive groups (roots, weights, parabolics, etc.),• representation theory of local groups (unramified principal series, Satake isomor-

phism, Langlands’ interpretation as L-groups),• reduction theory (Borel–Harish-Chandra),• computation.

This topic can be pretty dry if we just want to go over these things, so I will give moreexamples. There is also one more blackbox, which is the analytic continuation of Eisensteinseries. Essentially this is the heart of the matter.

If we can cover all of this, I am open to suggestions.

1.2. Modular forms. A reference for this is Serre’s A Course in Arithmetic.Historically, modular forms arose from elliptic integrals in the 1800’s. These are at the

moment irrelevant to L-functions. Elliptic integrals gave rise to elliptic curves and ellipticfunctions, and these gave rise to modular forms. I am not going to write down ellipticintegrals, but they are essentially integrals over elliptic curves. Let me start with ellipticfunctions, which came from trying to invert elliptic integrals.

An elliptic function is a meromorphic function f : C→ C such that there exist w1, w2 ∈ C,w1/w2 6∈ R with f(w1 + z) = f(w2 + z) = f(z). There are two names attached to thesefunctions: Weierstrass (1815–1897) and Jacobi (1804–1851).

• Weierstrass attached to any given any lattice Λ the function

℘Λ(z) =∑

w∈Λ\{0}

(1

(z − w)2− 1w2

)+

1

z2

which is meromorphic and periodic.• Jacobi came before Weierstrass, and literally took the elliptic integral

u =

∫ φ0

dt√1− k2 sin2 t

1Available at http://publications.ias.edu/sites/default/files/ep-ps.pdf.3

http://publications.ias.edu/sites/default/files/ep-ps.pdf

where 0 < k2 < 1. This is an elliptic integral of second kind. Jacobi’s first thetafunction (there are a total of twelve) is defined by

sn(u) = sin(φ).

It is a fact that this is an elliptic function, with

sn(u+ 2mK + 2niK ′) = sn(u)

where

K =

∫ π2

0

dt√1− k2 sin2 t

and K ′ =

∫ π2

0

dt√1− (1− k2) sin2 t

are the complete elliptic integrals.

I introduced Jacobi’s function only for fun, but Weierstrass’ function will be useful. Oneimportant property of ℘Λ(z) is

(℘′Λ(z))2 = 4℘3Λ(z)− 60G2(Λ)℘Λ(z)− 140G3(Λ).

This is actually bad notation since in a second we will be using the z ∈ H for the variable ofGk (the translation is by writing Λ = ω1Z + ω2Z such that ω1/ω2 ∈ H, then Λ correspondsto z = ω1/ω2 and vice versa), but we will change this in a second. Note that (℘Λ(z), ℘

′Λ(z))

gives an explicit point on the elliptic curve

y2 = 4x3 − 60G2x− 140G3.

Definition 1.1 (Eisenstein series). For k ∈ Z, the holomorphic Eisenstein series of weight2k is defined as

Gk(z) =∑

(m,n)∈Z2\{(0,0)}

1

(mz + n)2k.

This converges for k > 1. Given a lattice Λ, we get a point z ∈ H. By Gk(Λ) we mean Gkevaluated at z.

Here is a fun fact: if we set

u =

∫ ∞y

ds√4s3 − 60G2s− 140G3

,

then y = ℘(u).One property of Gk is that

Gk(z + 1) = Gk(z),

so we can write a Fourier series expansion

Gk(z) =∑n

anqn

where q = e2πiz. The whole class is essentially about calculating a0 for general Eisensteinseries. For holomorphic Eisenstein series, the constant term a0 is actually a constant. Whenwe introduce non-holomorphic Eisenstein series, the constant term will depend on y: thegeneral Fourier expansion looks like ∑

n

anfn(y)e2πinx.

4

Here is a word on the broader picture. In case you have seen this before, the whole theorydigresses into two branches: arithmetic and analytic. These holomorphic Eisenstein seriesare arithmetic objects. But they do not appear in L2(Γ\G) and have no place in the spectraltheory, which originated in the 1950’s.

Let us carry out an ad hoc calculation of the Fourier expansion. Recall that

π cotπz =1

z+∞∑m=1

(1

z +m+

1

z −m

).

Since

cotx =cos(x)

sin(x)= i

eix + e−ix

eix − e−ix,

we get

cot πz = i

(1 +

2

q − 1

)= i

(1− 2

∞∑n=0

qn

).

On the other hand, note that

d2k−1

dz2k−1π cotπz =

(−1)2k−1(2k − 1)!z2k

+ (−1)2k−1(2k − 1)!∑( 1

(z +m)2k+

1

(z −m)2k

)= (−1)2k−1(2k − 1)!

∑m∈Z

1

(z +m)2k.

By the q-expansion, we have

(−2πi)2k

(2k − 1)!

∞∑n=0

n2k−1qn =∑m

1

(m+ z)2k.

This is so ad hoc it appears to come out of nowhere.Let us go back to Gk.

Gk(z) =∑m,n

′ 1

(mz + n)2k

= 2ζ(2k) +∑m6=0,n

1

(mz + n)2k

= 2ζ(2k) + 2∞∑m=1

∑n

1

(mz + n)2k

= 2ζ(2k) + 2∞∑m=1

(−2πi)2k

(2k − 1)!

∞∑a=0

a2k−1qam

= 2ζ(2k) +2(2πi)2k

(2k − 1)!

∞∑α=1

σ2k−1(α)qα

where σt(α) =∑

d|α αt is the classical divisor sum function.

Here are some reasons for writing this down. Firstly, the special values of the zeta functionappear in the constant terms. For non-holomorphic Eisenstein series, we do get a zeta

5

function as the constant term, rather than just special values. Secondly,

σ2k−1(α)qα = σ2k−1(α)e

πiαxe−παy,

where σ2k−1(α) is the divisor function, eπiαx is a function on the unit circle, and e−παy is one

of the solutions to d2fdy2

= f (up to a constant). Classically over the Euclidean plane, d2fdy2

= f

has two solutions: ey and e−y. We see e−y in the Eisenstein series. This is a common theme:the Fourier coefficients of general Eisenstein series will satisfy certain differential equations.Only some of those solutions will appear.

This is the state of matters at the beginning of the 20th century. Things were case-by-caseat this point.

Definition 1.2 (Modular function). A modular function of weight 2k for SL2(Z) is a functionf : H→ C such that

• f((

a bc d

)z

)= (cz + d)2kf(z) for all

(a bc d

)∈ SL2(Z).

• f is meromorphic.

Definition 1.3 (Modular form). A modular form is a modular function that is holomorphiceverywhere. (By holomorphic at ∞, we mean that the q-expansion should not have anynegative powers.)

Definition 1.4 (Cusp form). A cups form is a modular form with zeroth coefficient inq-expansion 0.

Example 1.5.

• Gk is a modular form.• ∆(z) = (60G2(z))3 − 27(140G3(z))2 is a modular form of weight 12.• j(z) = 1728(60G2(z))

3

∆(z)is a modular function of weight 0, but not a modular form. It

has a simple pole with residue 1 at ∞.

Moving on, there were Siegel modular forms, Jacobi theta functions, Hecke’s generalizationof Riemann’s work for higher number fields, Hilbert’s (and Blumenthal’s) generalization ofthe notion of modular forms over upper half spaces, and many other names over 1900–1940.And then came Maass (1949), a student of Hecke who worked on non-holomorphic modularforms. He studied the L2 theory, which was the introduction of spectral theory to the studyof modular forms. Of course, there was the Petersson inner product before Maass, but thatwas limited to cusp forms only.

The Petersson inner product is defined as

〈f, g〉SL2(Z) =∫

SL2(Z)\Hf(z)g(z)yk

dx dy

y2

for f and g of weight k. In order for this to be well-defined, at least one of f or g has to becuspidal. I did not go through the spectral theory for holomorphic forms, but keep in mindthat the Eisenstein series and cusp forms span all the holomorphic modular forms. Whenwe multiply f and g, all the cross terms except for the zeroth term are integrable, since qgrows as e−y.

6

The spectral theory involves a Hilbert space and a measure. Next time it will be clearwhy we do this. Let us work with the specific case of L2(SL2(Z)\H, µ), where

dµ =dx dy

y2

is the volume element of the classical hyperbolic metric. (For now we are defining thesespaces for weight 0. We note that one can extend these to all K-types for a maximalcompact subgroup K, i.e. for all weights k.)

In the case of Eisenstein series, we could write down all the Fourier coefficients. But forcusp forms, I did not write any down because they are difficult. The important point is thatthe cuspidal part of L2 is the same as the cuspidal part of automorphic forms.

Next time I will talk more about the classical setting, introducing the Laplacian ∆, spectraltheory, the non-holomorphic Eisenstein series and their constant terms. Then we will passto the adelic setting and groups.

2. Lecture 2 (January 22, 2015)

2.1. Last time. Last time was an introduction to modular forms. We went back to the1800’s and talked about holomorphic modular forms, which are roughly speaking holomor-phic functions on the upper half plane that are modular with respect to the modular group.In some sense these are just “periodic” functions. We defined the Eisenstein series

Gk(z) :=∑

(c,d)6=0

1

(cz + d)2k

and computed their Fourier expansions

Gk(z) = 2ζ(2k) +2(2πi)2k

(2k − 1)!

∞∑n=1

σ2k−1(n)qn

where q = e2πiz = e2πixe−2πy.Then Maass came and introduced the L2 theory; Maass forms came up in 1949. This

marked the introduction of spectral theory to the subject, initiated by Maass and Selberg.Spectral theory means we have a Hilbert space with a measure and we try to diagonalizecertain operators.

2.2. Spectral theory. Let Γ = SL2(Z), and consider the Hilbert space L2(Γ\H, dµ) wheredµ = dx dy

y2is the invariant measure on H and the inner product is given by

〈f, g〉 =∫

Γ\Hf · g dµ.

We will keep doing things classically for another lecture before moving to the adelic setting.We will see that the formula for dµ essentially comes from the Iwasawa decomposition.

Let ∆ be the Laplacian, which in these coordinates is normalized as

∆ = ∆0 = −y2(∂2

∂x2+

∂2

∂y2

).

Remark. This is the theory for weight 0 functions (i.e., K-invariant on the right).7

Let us analyze ∆ on H a bit. Eventually we will see that holomorphic modular forms (moreprecisely yk times a holomorphic form of weight 2k) and Maass forms are eigenfunctions for∆ (and its extensions for all weights k), and it is important to first study its eigenvalues.This gives motivation for the Selberg’s eigenvalue conjecture and the Ramanujan conjecture.

Spectral theory is nice if we have a self-adjoint operator. For infinite-dimensional spaces,we need to be a bit careful because differentiation is in general unbounded. Basically, theidea of self-adjointness is not straightforward, and we need to look at a dense subspace.Some properties of ∆ are:

(1) symmetric (i.e., 〈∆f, g〉 = 〈f,∆g〉),(2) non-negative (i.e., 〈∆f, f〉 ≥ 0).

Proof. These are very easy, and essentially just integration by parts. Let f, g ∈ C∞0 (H).Then

〈∆f, g〉 =∫H

∆f · g dx dyy2

=

∫H∇f · ∇g dx dy

where ∇f = (fx, fy). �

This is a straightforward application of integration by parts, but immediately shows thefollowing consequences. Remember we are trying to understand the eigenvalues, λ, of ∆. Atthe beginning, there is no reason λ has any restrictions, but because of (1) and (2) above wehave:

• Eigenvalues are real.• Eigenvalues are non-negative.

In fact more is true.

Proposition 2.1. If λ is an eigenvalue of ∆ on H, then λ ≥ 14.

This 14

will appear everywhere and is an interesting object.

Proof. Without loss of generality assume F is real-valued and ∆F = λF ; otherwise we canjust separate the real and imaginary parts of F (this is to avoid complex conjugations). Thenintegration by parts gives

〈∆F, F 〉 = −∫ (

y2(∂2

∂x2+

∂2

∂y2

)F

)· F dx dy

y2

=

∫(F 2x + F

2y ) dx dy ≥

∫F 2y dx dy (1)

and ∫F 2

dy

y2= 2

∫F · Fy

dy

y.

The second relation implies∫F 2

dx dy

y2= 2

∫∫F · Fy

dx dy

y≤(

4

∫F 2y dx dy

) 12

||F ||

by the Cauchy–Schwarz inequality, so

||F ||2 ≤ 4∫F 2y dx dy. (2)

8

By (1) and (2), we conclude

〈∆F, F 〉 ≥ 14〈F, F 〉. �

This is the story for H. What usually happens is that for a big space without any discretegroup action, it is easy to determine eigenvalues: here we have the half line [1

4,∞), which is a

continuous spectrum. Once we take the quotient Γ\H, we will get discrete eigenvalues, whichcorrespond to Maass forms. Γ\H is like the global (real) picture. We can also localize andconsider PGL2(Qp)/PGL2(Zp), which is like a tree, and we can try to analyze the spectrumof ∆p. This is a pure spectrum and everything is continuous, but once we take quotientswe will get something discrete. Selberg’s eigenvalue conjecture states that the same boundλ ≥ 1

4holds even after we pass to the quotient Γ\H.

Now let us move to the eigenfunctions ∆f = λf over H. Recall that

∆ = −y2(∂2

∂x2+

∂2

∂y2

).

Step 1: Look at eigenfunctions which only depend on y. So we are looking for functions

F (x, y) = φ(y)

such that

∆F = −y2 ∂2

∂y2φ = λφ.

There are some obvious eigenfunctions: φ(y) = ys and y1−s (linearly independent), wheres(1− s) = λ and s 6= 1

2. If s = 1

2, then we can consider the solutions {ys, ys log(y)}.

Let us parametrize λ = s(1 − s). Then λ ∈ R and λ ≥ 14. These functions are the

building blocks of real-analytic Eisenstein series, and will also come up when we study theintertwining operators.

Step 2: Assume that the eigenfunction is periodic in x. This is because if we want to geteigenfunctions on Γ\H, then they should be invariant under the action

f

((1 10 1

)z

)= f(z).

Then we can write F (z) = e(x)φ(2πy), where e(x) = e2πix, and

∆F = −y2(∂2

∂x2+

∂2

∂y2

)e(x)φ(2πy)

= −y2(∂2

∂x2e(x)

)φ(2πy)− y2e(x)φ′′(2πy)

= y24π2e(x)(φ(2πy)− φ′′(2πy)).In order for ∆F = λF , we want

φ′′(2πy) + φ(2πy)

(λ

(2πy)2− 1)

= 0.

This is Bessel’s differential equation of second type. The upshot is, people have studiedthese. We have the following facts:

(1) There are two linearly independent solutions: (i)√

2π−1yKs− 12(y), (ii)

√2πyIs− 1

2(y).

9

(2) Their behavior is: (i) ∼ e−y and (ii) ∼ ey as y →∞. (To convince oneself informally,consider the equation for y →∞. Then φ′′ = λφ has solutions ey and e−y.)

Remember that we are trying to work on L2(Γ\H, dµ). For harmonic analysis on this,only Ks− 1

2is relevant. From the above discussion we get the following proposition.

Proposition 2.2. Any F ∈ L2(Γ\H) that is an eigenfunction of ∆ with eigenvalue s(1− s)has a Fourier–Whittaker expansion∑

n∈Z

an√yKs− 1

2(2π|n|y)e(xn)

where an ∈ C.

Rough proof. Let F (z) = F (x+ iy) be such a function. Since

(1 10 1

)∈ Γ we know that the

function is periodic in the x-variable and has a Fourier expansion of the form

F (z) =∑n∈Z

a(n, y)e(xn)

Since F (z) is an eigenfunction each a(n, y) satisfies the Bessel differential equation, therefore

the solutions are expressible as a linear combination of√

2π−1yKs− 12(y) and

√2πyIs− 1

2(y).

Finally since F is in L2 only K-Bessel function appears. �

Let us go back to Γ\H and Maass forms. The whole point of the above discussion is thatthe eigenfunctions of ∆ on H give us building blocks of these functions, and their eigenvaluessatisfy certain properties.

We want functions f : H→ C which are• invariant under SL2(Z),• eigenfunctions of ∆,• of moderate growth (i.e., f should grow at most polynomially at ∞).

Invariance under SL2(Z) implies invariance under x 7→ x+1, so F has a Fourier expansion.There are various ways of cooking up such eigenfunctions, but the best way is following yournose: averaging. Start with φ, an eigenfunction on H that is invariant under x 7→ x+ 1, andmake it invariant under the rest of the group by averaging.

Example 2.3. Let ±Γ∞ =〈±(

1 11

)〉=

{±(

1 n0 1

) ∣∣∣∣ n ∈ Z} be the stabilizer of y.Define φ̃s(z) = (Im z)

s = ys, and

Ẽ(z; s) =∑

±Γ∞\SL2(Z)

φ̃s(γz).

This is the non-normalized non-holomorphic Eisenstein series.

Note that ∆Ẽ(z; s) = s(1− s)Ẽ(z; s). We can rewrite∑±Γ∞\SL2(Z)

φ̃s(γz) =∑

(c,d)∈Z2gcd(c,d)=1

c≥0

ys

|cz + d|2s.

10

Once we pass to the group-theoretic language, these things will be canonically defined, butfor now let us do an ad hoc calculation. We have

Im(γz) =Im(z)

|cz + d|2

and the bijectionΓ∞\ SL2(Z)←→ {(c, d) | gcd(c, d) = 1, c ≥ 0}.

This is because if

(a bc d

)∼(a′ b′

c d

), then ad−bc = a′d−b′c = 1 and (a−a′)d+(b′−b)c = 0.

These calculations are easy to do for GL(2).I will do the following normalization

φs(z) = (Im z)s+ 1

2 ,

so that

∆E(z; s) =

(1

4− s2

)E(z; s).

Note that the above equality is true only for those s so that E(z, s) makes sense, and wethen get an eigenfunction with eigenvalue 1

4− s2.

Note that for any eigenfunction that is actually in L2, by the previous calculation, theeigenvalue has to satisfy 1

4− s2 ∈ [0,∞), and therefore s ∈ iR ∪ [0, 1

2].

Selberg’s eigenvalue conjecture says that for a Maass cusp form, the eigenvalue cannotbe in [0, 1

2). These eigenvalues parametrize representations of GL(2): iR corresponds to the

principal series and [0, 12) corresponds to the complementary series. Thus Selberg says the

representations coming from Maass forms have to be in the unitary principal series.Let me finish by stating some properties of these eigenfunctions, and we will prove them

next time.

(1) E(z; s) converges for Re(s) > 12.

(2) E(γz; s) = E(z; s) for all γ ∈ SL2(Z).(3) ∆E(z; s) = (1

4− s2)E(z; s).

(4) E(z; s) has analytic continuation and a functional equation relating s↔ −s.(5) E(z; s) has the Fourier expansion

E(z; s) =∑n6=0

an√yKs(2π|n|y)e(nx) + a0(y),

where

an =4|n|sσ−2s(|n|)ζ(2s+ 1)

and

a0 = ys+ 1

2 +ξ(2s)

ξ(2s+ 1)· y

12−s

with ξ(s) the completed Riemann zeta function.

The −2s in the divisor function dictates the eigenvalue. The ξ(2s)ξ(2s+1)

in a0 is the completed

ζ-function. Once we know the analytic continuation of ζ (classically obtained using integralrepresentation, Poisson summation and partial summation formula), we get the analyticcontinuation of E(z; s).

11

Spectral theory reverses this argument: we would like to start from the analytic contin-uation of E(z; s), and deduce from this the analytic continuation of ζ! This works moregenerally for GL(n) and Rankin products of L-functions, which appear as constant terms ofEisenstein series. This is the upshot for the whole class. We try to do this for all groups.

Next time we will calculate a0, and obtain analytic continuation and functional equationfrom a very soft spectral argument.

3. Lecture 3 (January 29, 2015)

3.1. Last time. Let us start with some clarifications from last time. We introduced the L2theory: we have the Laplacian ∆ acting on L2(Γ\H). Let me summarize what I meant tosay.

Consider ∆ on the universal covering H. It has two eigenfunctions: for each s, we haveys and y1−s. All the other eigenfunctions are generated by these. One can show that it issufficient to take s = 1

2+ it to get the whole spectrum.

But we are interested in Γ\H, so what about eigenfunctions that are invariant under

T =

(1 ∗0 1

), i.e., under x 7→ 1 + x? They have to be of the form

F (x) =∑n∈Z

an√yKs− 1

2(2π|n|y)e(nx)

where Ks− 12

is the modified Bessel function of second type. This is just basic harmonic

analysis on H. We have shown that eigenfunctions have to be of this form, but we haven’tconstructed any yet.

Consider the Dirichlet problem on any domain D in H: we want to find a function F suchthat ∆F = λF and F = 0 on the boundary of D. By the formal argument on H from lasttime (integral trick), we have λ ∈ R+. In fact we have the stronger bound λ ≥ 14 .

Question: why does this not work for Γ\H? It is only a formal argument after all. Inparticular, why would eigenvalues on Γ\H not be at least 1

4? In fact, it does work a bit.

Assume for simplicity Γ = SL2(Z) (the argument will work for any discrete subgroup).Let us fix a fundamental domain, which is in particular a subset of H. We can try to solvethe Dirichlet problem for a domain D inside the fundamental domain. For ∆F = λF , weget λ ≥ 1

4. But this is not a function on Γ\H. We can try to make this a function on Γ\H

by averaging ∑γ∈stabilizer\SL2(Z)

F (γz).

Whenever this guy converges, the eigenvalue will be at least 14, but there could be other

eigenfunctions not of this form! For example, Eisenstein series that contribute to L2 willhave eigenvalues greater than 1

4.

Selberg’s conjecture states that λ ≥ 14

for Γ arithmetic. In representation theory languagethis is the Ramanujan conjecture over the real place. The bound λ ≥ 0 still holds, but weknow λ ≥ 1

4only for SL2(Z) and congruence subgroups of small level (up to 7).

12

3.2. Real-analytic Eisenstein series. Following the same game as above, we start withthe eigenfunction ys+

12 and take the sum

E(z, s) =∑

γ∈Stab(y)\ SL2(Z)

Im(γz)s+12 =

∑±Γ∞\Γ

Im(γz)s+12 ,

where Γ∞ =

{(1 n0 1

) ∣∣∣∣ n ∈ Z} is the unipotent radical. We take s+ 12 because things willbe more symmetric this way. Note that(

a bc d

)z =

az + b

cz + d⇒ Im(γz)s+

12 =

Im(z)s+12

|cz + d|2s+1.

We have the following facts:

(1) E(z, s) converges for Re(s) > 12. (This is easy.)

(2) E(γz, s) = E(z, s). (This is clear by construction.)(3) ∆E(z, s) = (1

4− s2)E(z, s).

(4) E(z, s) has an analytic continuation to C (in the s-variable) with a simple pole ats = 1

2with residue 3

π.

(5) E(z, s) has the Fourier expansion

E(z, s) =∑n6=0

an(s)√yKs(2π|n|y)e(nx) + a0(y, s),

where

an(s) =4|n|sσ−2s(|n|)ζ(2s+ 1)

and

a0(y, s) = ys+ 1

2 +ξ(2s)

ξ(2s+ 1)y

12−s.

Here ξ(s) = π−s2 Γ( s

2)ζ(s) is the completed zeta function.

The important thing is there is a ratio of L-functions in a0 and an L-function in an. Atthe end of the day, we want to do the following. In this very special case one can calculatethe Fourier expansion. Then one gets the analytic continuation and functional equation ofE(z, s) by the known properties of ζ. BUT we don’t want to do this by writing the Fourierexpansion. Instead we would like to deduce properties about these L-functions from theanalytic properties of E(z, s). For example, if we know (4) and (5) by some other means,

then we know ξ(2s)ξ(2s+1)

has analytic continuation. For more general groups, we get a product

of quotients of various L-functions in the constant term, but we can still extract informationabout the individual L-functions by an induction argument.

Although the first Fourier coefficient seems to contain strictly more information than theconstant term, it is harder to compute and may not even make sense for general groups. TheShalika–Casselman formula was not available yet. Here is a bit of historical perspective. Weare at about 1965. Shalika was quite senior but Casselman was not yet. Then Shahidi camein around 1975.

13

3.3. A (classical) computation. The constant term is given by

a0(y, s) =

∫ 10

E(x+ iy, s) dx.

Before we start computing this, let us remark that there is a one-to-one correspondence

±Γ∞\ SL2(Z)←→ {(c, d) | gcd(c, d) = 1, c ≥ 0},

Exercise. Prove this bijection.

So, ∫ 10

E(z, s) dx =

∫ 10

∑(c,d)=1c≥0

ys+12

|cz + d|2s+1dx

= ys+12 +

∞∑c=1

∑d∈Z\{0}(c,d)=1

∫ 10

ys+12

|cz + d|2s+1dx+

∫ 10

ys+12

|x+ iy|2s+1dx.

The sum from c = 2 to ∞ contributes

ys+12

∞∑c=2

c−1∑α=1

(α,c)=1

∑k∈Z

∫ 10

dx

|cx+ α + kc+ ciy|2s+1

=ys+12

∞∑c=2

1

c2s+1

∑α

∑k∈Z

∫ 10

dx

|x+ αc

+ k + iy|2s+1

=ys+12

∞∑c=2

1

c2s+1

∑α

∑k∈Z

∫ k+1k

dx

|x+ αc

+ iy|2s+1

=ys+12

∞∑c=2

1

c2s+1

∑α

∫ ∞−∞

dx

|x+ αc

+ iy|2s+1

=ys+12

∞∑c=2

1

c2s+1

∑α

∫ ∞−∞

dx

|x+ iy|2s+1

=ys+12

∞∑c=2

ϕ(c)

c2s+1

∫ ∞−∞

dx

|x+ iy|2s+1.

We made a big fuss about this calculation, because this is the unfolding that will appearnaturally later. The integral

ys+12

∫R

dx

|x+ iy|2s+1

is a specialization of the beta function. More precisely, it is equal to

= y12−s∫

dx

(x2 + 1)s+12

= y12−sΓ(

12)Γ(s)

Γ(s+ 12).

14

Combining everything (including the c = 1 term), we get

a0(y, s) = ys+ 1

2 + y12−s

∞∑c=1

ϕ(c)

c2s+1Γ(1

2)Γ(s)

Γ(s+ 12).

Note by multiplicativity

∞∑c=1

ϕ(c)

c2s+1=∏p

(∞∑k=0

ϕ(pk)

pk(2s+1)

)=∏p

1− 1p2s+1

1− 1p2s

=ζ(2s)

ζ(2s+ 1)

and so

a0(y, s) = y12

+s +ξ(2s)

ξ(2s+ 1)y

12−s.

3.4. Why is the L2 theory helpful? Remember we are trying to give a historical contexttoo. Before 1965, there was no spectral theory and no Hilbert space in the theory of automor-phic forms. Everything was complex-analytic and people were just using the Petersson innerproduct. I am not sure if people proved the analytic continuation and functional equation ofEisenstein series, but I can now give a very soft analytic continuation result. We know that:

• E(z, s) converges for Re(s) > 12.

• E(z, s) ∼ ys+ 12 + (∗)y 12−s (where (∗) is constant in the y-variable). All the otherterms have exponential decay so are negligible.

Claim. E(z, s) is the unique eigenfunction of ∆ with eigenvalue 14− s2, Re(s) > 1

2, and

asymptotic growth ∼ ys+ 12 +O(y−�) for some � > 0.

This is not hard to prove. Analytic continuation is often proved in two ways: by studyingspecial integral transforms and their kernels (which is how the theta and zeta functions ariseclassically), or by using some uniqueness statements. For example, Jacquet–Langlands uses(in the local case) the uniqueness of Kirillov models, unlike Tate’s thesis which uses integrals.

Proof. Suppose there exists an F (z) satisfying this. Consider the difference E(z, s)− F (z).(1) ∆(E(z, s)− F (z)) = (1

4− s2)(E(z, s)− F (z)).

(2) E(z, s)− F (z) ∈ L2(Γ\H).This means E(z, s)−F (z) is an eigenfunction of ∆ on L2(Γ\H), so the eigenvalue has to be≥ 0 because ∆ is non-negative and symmetric on L2(Γ\H). This is where the L2 theory isimportant.

As the final step, the eigenvalue is 14− s2. The assumption is that Re(s) > 1

2, so either

s ∈ R with s > 12, or s has an imaginary part. The former implies that 1

4− s2 < 0 and the

latter implies that 14− s2 is not real. In either case, 1

4− s2 cannot be an eigenvalue of ∆ in

L2(Γ\H).But E(z, s)− F (z) actually has eigenvalue 1

4− s2, so we conclude F (z) = E(z, s). �

The implication is that for Re(s) > 12, we have

E(z, s) = E(z,−s)a(s)for some function a(s) independent of z. Why?

• s 7→ −s does not affect the eigenvalue.15

• SinceE(z, s) ∼ ys+

12 +

ξ(2s)

ξ(2s+ 1)y

12−s,

we know

E(z,−s) ∼ y12−s +

ξ(−2s)ξ(−2s+ 1)

y12

+s

and soξ(−2s+ 1)ξ(−2s)

E(z,−s) = y12

+s + (∗)y12−s.

Therefore

E(z, s) =ξ(−2s+ 1)ξ(−2s)

E(z,−s).

Remark. First of all we needed the fact that E(z,−s) was defined with the same eigenvaluefor this argument to work (which the Fourier expansion gives), so in a sense we are cheating,but this does give the correct functional equation for Re(s) > 1

2. However in a sense this is

useless for the L2 theory. E(z, s) converges on Re(s) > 12, but does not belong to L2 and does

not contribute to L2. By the functional equation we have obtained the same for Re(s) < −12.

If we can analytically continue the Eisenstein series to the critical line Re(s) = 0, these arethe ones that contribute to L2.

“Contribution” means the following. What is the dual of (R,+), i.e., Hom(R,C×)? Forany a ∈ C×, we have x 7→ e(ax), which gives an isomorphism C× ∼= Hom(R,C×). In orderto look at L2(R), Fourier theory says

f(x) =

∫Rf̂(z)e(−zx) dz.

We see that it is enough to take all the characters that are unitary. Only the ones that areon the unitary axis are “contributing”. Note also that none of the harmonics e(−zx) are inL2, but they form L2.

Next time we will compute the volume of the fundamental domain, prove the prime numbertheorem, and prove that there are finitely many imaginary quadratic fields with class number1. These are kind of random, but as you can guess they share a common theme: they allfollow from the computations for E(z, s).

4. Lecture 4 (February 3, 2015)

4.1. Last time. We defined the Eisenstein series

E(z; s) =∑

±Γ∞\ SL2(Z)

Im(γz)s+12

where Γ∞ =

{(1 n0 1

) ∣∣∣∣ n ∈ Z} is the unipotent radical. We talked about how the L2theory helped: using a very soft argument, we proved the

Proposition 4.1. E(z; s) is the unique eigenfunction of ∆ satisfying:

• ∆E(z; s) = (14− s2)E(z; s),

• growth ys+ 12 +O(y−�) for some � > 0, and16

• Re(s) > 12.

Remark. Re(s) > 12

is necessary for the convergence of E(z; s).

As a corollary, we get the functional equation.

Corollary 4.2. E(z, s) = A(s)E(z,−s) where A(s) is meromorphic and Re(s) > 12.

Proof. We have

E(z, s) = ys+12 +

ξ(2s)

ξ(2s+ 1)y

12−s +O(e−

y2 )

where ξ(s) = π−s2 Γ( s

2)ζ(s) is the completed zeta function. We know

∆E(z, s) =

(1

4− s2

)E(z, s)

and so

∆E(z,−s) =(

1

4− s2

)E(z,−s).

We also have

E(z,−s) ∼ y−s+12 +

ξ(−2s)ξ(−2s+ 1)

y12

+s

and hence

E(z,−s)ξ(−2s+ 1)ξ(−2s)

∼ ys+12 +O(y−�).

By the proposition,

E(z, s) =ξ(−2s+ 1)ξ(−2s)

E(z,−s). �

This result does not really give the analytic continuation of E(z, s). It simply flips theregions Re(s) > 1

2and Re(s) < −1

2. This only gives a philosophical argument for the

functional equation. The whole point is to go the other way round:

(1) Analyze Eisenstein series (this is hard!).(2) Calculate constant terms.

(2.5) Observe that they are L-functions.(3) Deduce the properties of these L-functions from (1).

4.2. Analytic continuation. Today we will go back to the original theory. Let us forgetabout all the stuff above.

Theorem 4.3 (Analytic continuation of Eisenstein series). E(z, s), originally defined forRe(s) > 1

2, satisfies the following properties:

• E(z, s) has analytic continuation to the whole C (in the s-variable).• E(z, s) = A(s)E(z,−s), where A(s) is meromorphic.• ∆E(z, s) = (1

4− s2)E(z, s).

• E(z, s) has a simple pole at s = 12

with residue 3π

.

17

4.3. Application: Prime number theorem. We will prove the following form of theprime number theorem.

Theorem 4.4 (Prime number theorem). ζ(s) 6= 0 on the line Re(s) = 1.

Proof. Recall

E(z, s) = ys+12 +

ξ(2s)

ξ(2s+ 1)y

12−s +

4

ξ(2s+ 1)

∑n6=0

|n|sσ−2s(|n|)Ks(2π|n|y)e(nx).

Consider Re(s) = 0. By the theorem, E(z, s) has no poles there. Looking at the right handside, we see that ξ(2s+ 1) 6= 0 for Re(s) = 0. �

This is a fairly straightforward argument, but it gives strictly less information than theoriginal proof of the prime number theorem by de la Vallée-Poussin, which gives a zero-freeregion of the form

σ > 1− clog(|t|+ 2)

where s = σ + it and c > 0 is a constant.One can analyze the spectral decomposition further to get a zero-free region. This involves

looking at the inner products of truncated Eisenstein series. A reference is Sarnak’s articleon his webpage2.

Note that once we have the Fourier expansion of E(z, s), the formal argument above willactually prove the functional equation.

4.4. Application: Gauss class number problem.

Theorem 4.5 (Deuring, Landau, Gronwall). There are finitely many imaginary quadraticfields with class number 1.

Side remark. Landau and Gronwall knew that the claim follows from the Riemann hypothesisfor L(s,

(−D

)) for all fundamental discriminants D < 0.

Proof. Suppose the Riemann hypothesis is false, so there exists s0 such that ζ(s0 +12) = 0

with Re(s0) 6= 0. Let z0 ∈ H be a CM point, i.e., az20 + bz0 + c = 0 for some a, b, c ∈ Z withb2 − 4ac = D. Then we have two facts:

• For Γ = SL2(Z), Γz0 corresponds to the ideal class of(a, b+

√D

2

)in Q(

√D).

• We can write

E(z0, s) =∑u,v∈Z

′ Im(z0)s+ 1

2

|uz + v|2s+1

=

(√|D|2

)s+ 12 ∑ ′ 1

|av2 − buv + cv2|s+ 12

=

(√|D|2

)s+ 12

ζz0

(s+

1

2

)2Available at http://web.math.princeton.edu/sarnak/ShalikaBday2002.pdf.

18

http://web.math.princeton.edu/sarnak/ShalikaBday2002.pdf

where ζz0 is the zeta function for the ideal class corresponding to z0. Recall that forany Galois extension K/Q,

ζK/Q(s) =∑I

1

N(I)s=

∑c∈Cl(K)

∑α∈K

1

(αc)s

and we define

ζc(s) =∑α∈K

1

(αc)s.

Summing the above over z0 ∈ ΛD,

∑z∈ΛD

E(z, s) =

(|D| 12

2

)s+ 12

ζQ(√D)

(s+

1

2

).

Note that

ζQ(√D)

(s+

1

2

)= ζ

(s+

1

2

)L

(s+

1

2,

(−D

)),

so evaluating everything at the hypothetical zero gives∑z∈ΛD

E(z, s0) = 0.

In particular, if the class number h(D) = 1, then this gives E(zD, s0) = 0, where zD =1+√D

2.

By the Fourier expansion of E(z, s),

E

(1 +√D

2, s0

)=

(√|D|2

)s0+ 12+ (∗)

(√|D|2

) 12−s0

+O

(e−√|D|2

).

This cannot be all 0 for large |D| if s0 is not on the line Re(s) = 0. Therefore, this can onlyhappen finitely many times, or the Riemann hypothesis is true in which case we can use theabove argument. �

Of course, we were cheating in this proof because we only assumed the falsity of theRiemann hypothesis for ζ. We can indeed push the argument a bit further for the Riemannhypothesis for quadratic L-functions.

4.5. Application: Volumes of fundamental domains. Our last application will be quitegeneral: the computation of the volumes of the fundamental domains. This was one ofLanglands’ first applications.

We want to calculate the volume of Γ\H, which is π3. The idea is to calculate∫

Γ\HE(z, s) dz

by shifting the contour and picking up the pole at s = 12. But there are two problems:

(1) In general E(z, s) /∈ L1(Γ\H) when it converges.(2) This integral is 0 when Re(s) = 0.

19

We are going to modify this idea and somehow make it work.This is how we are going around. Take f ∈ C∞0 (R>0), and consider the series

θf (z) =∑±Γ∞\Γ

f(Im(γz)).

Recall that the Mellin transform is given by

f̃(s) =

∫ ∞0

f(y)ysdy

y.

The Mellin inversion is

f(y) =1

2πi

∫ ∞−∞

f̃(s)y−s ds.

The idea is to rewrite f as its own Mellin transform, which is a very general method inanalytic number theory. The sum becomes the Eisenstein series

θf (z) =1

2πi

∫Re(s)=s0

f̃(s)E

(z,−s− 1

2

)ds

where s0 > 1.One can compute ∫

Γ\Hθf (z) dz

in two ways: by shifting the contour, and by unfolding. Shifting the contour gives∫Γ\H

θf (z) dz = f̃(−1) ·3

πvol(Γ\H) + 1

2πi

∫Re(s)=− 1

2

f̃(s)

∫Γ\H

E

(z,−s− 1

2

)dµ(z) ds,

whereas unfolding gives∫Γ\H

θf (z) dµ(z) =

∫±Γ∞\H

f(Im(γz)) dµ(z) =

∫ ∞0

f(y)dy

y2= f̃(−1).

So we have the following identity

f̃(−1) = f̃(−1) 3π

vol(Γ\H) + 12πi

∫s∈iR

f̃

(−s− 1

2

)∫Γ\H

E(z, s) dµ(z) ds.

For Re(s0) = 0, the integral ∫Γ\H

E(z, s0) dµ(z)

is 0, so we get

vol(Γ\H) = π3.

Among the things we did this is the most general. Langlands calculated this for splitChevalley groups over Q. At the end of the day, once we have an Eisenstein series, this willwork for more general groups.

Next time we will leave the classical language and move to groups and the adelic language.20

5. Lecture 5 (February 5, 2015)

5.1. Convergence of averaging. Let me comment on something way earlier. We obtainedeigenfunctions ϕ by considering the Dirichlet problem on any domain D in the upper halfplane H. Assume that we can extend ϕ across the boundary of D. Then since Γ actsdiscontinuously, for a fixed g, γg only hits D finitely many times. So

∑ϕ(γg) converges.

5.2. Reminder (last time). Last time we were calculating the volume of the fundamentaldomain. We got the following equality

f̃(−1)(

3

πvol−1

)=

1

2πi

∫Re(t)=0

I(t)f̃

(−t+ 1

2

)dt,

where

I(t) =

∫Γ\H

E(z, t)dx dy

y2.

We had a lemma last time.

Lemma 5.1. Suppose I(t) is defined. Then it is identically zero.

A heuristic reason is that the Eisenstein series is orthogonal to the constant function.

Proof. Consider1

2πi

∫Re(t)=0

I(t)f̃

(−t+ 1

2

)dt = cf̃(−1)

for all f ∈ C∞c (R≥0). We will show that if this is true, then I(t) ≡ 0.We can shift the eigenvalue by taking F (y) = y

12f(y). Then F̃ (s) = f̃(s + 1

2). So we can

consider F in place of f :

1

2πi

∫Re(s)=0

F̃ (−s)I(s) ds = cF̃(−3

2

).

Now let G(y) = yF ′ + 32F . Then G̃(s) = (s + 3

2)F̃ (s). Since the above is true for all F̃ ,

we have the same relation

1

2πi

∫G̃(−s)I(s) ds = cG̃

(−3

2

)≡ 0,

so1

2πi

∫F̃ (−s)

(−s+ 3

2

)I(s) ds ≡ 0

for all F . This implies I(s) = 0. �

Therefore vol = 3π. This is a roundabout argument.

5.3. Modular forms as Maass forms. Today we will end our discussion on H and passto G. We talked about modular forms and real-analytic Eisenstein series.

Remark. All modular forms (holomorphic of weight k) are Maass forms.

So far we have talked about Maass forms that are invariant, but we need to define themmore generally.

Definition 5.2. A Maass form of weight k (for trivial Nebentypus) is f : H→ C such that21

• f(γz) =(cz + d

|cz + d|

)kf(z) for all γ =

(a bc d

)∈ Γ0(N),

• ∆kf(z) = s(1 − s)f(z), where ∆k = −y2(∂2

∂x2+

∂2

∂y2

)+ iky

∂

∂xis the weight k

Laplacian, and• f is of moderate growth.

With this, modular forms become Maass forms.

Proposition 5.3. If f is a weight k holomorphic form, then F (z) = yk2 f(z) is a Maass

form of weight k and eigenvalue k2(1− k

2).

Proof. Exercise. �

This unifies the picture of modular forms and Maass forms.

5.4. From H to group. We defined these Eisenstein series on the upper half plane, whichis a manifold. Recall that the upper half plane is a symmetric space

H = SL2(R)/ SO(2).

Instead of looking at H, we want to look at functions on SL2(R) and more generally onGL2(R).

First we will go from a modular form f on H to a function on

GL+2 (R) = {g ∈ GL2(R) | det(g) > 0}.Let

j(g, z) =cz + d√det(g)

where g =

(a bc d

). If g ∈ SL2(Z), then det(g) = 1. This function satisfies the following

properties:

• j(gh, z) = j(g, hz)j(h, z) (cocycle property). (Using this, we can define half-integralweight modular forms over any groups, for example over function fields. If we dothe same calculation over anything that is not R, the formula for j(g, z) does notwork but the cocycle condition is what generalizes. This turns out to be the Hilbertsymbol.)

• j(gkθ, i) = eiθj(g, i) where kθ =(

cos θ sin θ− sin θ cos θ

)∈ SO(2).

If f is a modular form of weight k, we construct

φf (g) = j(g, i)−kf(gi).

This lifts to the group G and is invariant modulo Γ, i.e., φf is a function on Γ\G.More generally, if f is a Maass form of weight k, we define the cocycle

jMaass(g, z) =

(cz + d

|cz + d|

)1√

det(g)

then φf (g) = jMaass(g, i)−kf(gi) is a function on the group.

22

5.5. To adelic group. Now we want to pass from a function φ on GL2(Z)\G(R) to Φ onG(A).

Recall strong approximation. Suppose we have a group G/Q with S a finite set of places.The pair (G,S) satisfies strong approximation if∏

v∈S

G(Qv)×G(Q)

is dense in G(A). In practice we usually take S to be the archimedean places. This essentiallymeans we can do Chinese remainder theorem for the group G.

It is a fact that (SL(2), {∞}) has strong approximation. More generally, we have the

Theorem 5.4 (Kneser). If G is simply connected and∏

v∈S G(Qv) is not compact, then(G,S) satisfies strong approximation.

But we will not need such generality.Let G = SL2. Then G(A) = G(Q)G(R)Kf , where we take Kf =

∏v finite G(Zv). Note

G(Zv) is a maximal compact for each v. Thus G(R)� G(Q)\G(A)/Kf is onto.

Claim. G(Z)\G(R)↔ G(Q)\G(A)/Kf is a one-one correspondence.

Proof. Suppose g∞, g′∞ ∈ G(R) map to the same point in the double coset. Then

g′∞ = γg∞kf

where γ ∈ G(Q) and kf ∈ Kf . We can write γ = γ∞ · γf for the diagonal embeddingG(Q) ↪→ G(A), i.e., γ∞ = (γ, 1, 1, · · · ) and γf = (1, γ, γ, · · · ). We have

g′∞ = γg∞kf = γ∞γfg∞kf = γ∞g∞γfkf .

Since g′∞ and γ∞g∞ are 1 at all finite places, we must have γf = k−1f , so

γf ∈∏v finite

SL2(Zv) ∩ SL2(Q) = SL2(R)

and γ ∈ SL2(Z). �

We have already gone from f : Γ\H → C, to φf : Γ\G(R) → C. Finally, we can go fromφf to a function on G(Q)\G(A)/Kf by defining

F (gQg∞kf ) = φf (g∞).

The claim above shows that this is well-defined.Remember the point of this course is to compute constant terms. What happens to them

under this identification? We have∫ 10

f(x+ iy) dx =

∫N(Z)\N(R)

φf (ng) dn

where N =

{(1 ∗0 1

)∈ G(R)

}is the standard unipotent radical, and dn = dx. The coor-

dinates are

z = x+ iy ←→ gz =

(√y x√

y

0 1√y

)23

and ng =

(√y x+n√

y

0 1√y

).

5.6. Eisenstein series (adelically). Recall that

E(z, s) =∑c≥0

gcd(c,d)=1

ys+12

|cz + d|2s+1=

∑±Γ∞\ SL2(Z)

Im(γz)s+12 .

We lift this toESL2(g, s) := E(gi, s).

We do not need to deal with the j-factors because E(z, s) is already invariant. ESL2 is definedon SL2(Z)\ SL2(R).

Remark. Diagonal matrices γ =

(a

a

)act trivially, so we can lift ESL2 to functions on

GL2 that are invariant under the center as well.

To define the adelic Eisenstein series, let us review the Iwasawa decomposition. In general,G(k) = N(k)A(k)K(k), where N is the unipotent radical, A is the diagonal matrices and Kis a maximal compact. K looks significantly different for archimedean and non-archimedeanplaces.

For g ∈ GL2(R),

g =

(a bc d

)=

(det(g)√c2+d2

ac+bd√c2+d2

0√c2 + d2

)(d√

c2+d2−c√c2+d2

c√c2+d2

d√c2+d2

)where the two matrices are in NA and K respectively. We will need this for calculationslater.

Exercise. Prove this.

Then we define

EGL2(g, s) = E(gi, s) =∑

γ∈P (Z)\GL2(Z)

| Im(γgi)|s+12

where P = NA =

{(∗ ∗0 ∗

)∈ G

}is the parabolic. This can be expressed in terms of the

function ϕs,∞ : GL2(R)→ C defined by

g = gPgK 7→∣∣∣∣αgγg∣∣∣∣s+ 12

where gP =

(αg βg0 γg

).

Claim.E(i, s) =

∑γ∈P (Z)\G(Z)

ϕs,∞(γ).

Exercise. Prove this.24

In general,

EGL2(g, s) =∑

γ∈P (Z)\G(Z)

ϕs,∞(γg).

Now we are ready to work adelically. We will need to define a function at each place. Letp be a prime and define ϕs,p : G(Qp)→ C by sending

g 7→∣∣∣∣αgβg∣∣∣∣s+ 12p

where g = ngagkg with ag =

(αg

βg

). Finally, ϕs : G(A)→ C is defined as

ϕs(g) = ϕs,∞(g∞)∏p

ϕs,p(gp).

Definition 5.5.

E(g, ϕs) =∑

γ∈P (Q)\G(Q)

ϕs(γg).

We need to convince ourselves that P (Q)\G(Q) ↔ P (Z)\G(Z), which is believable byclearing denominators. We will do this next time.

Let me end by describing the constant terms. Given any f : G(A)→ C, its constant termalong the parabolic P is

cP (f, g) =

∫N(Q)\N(A)

f(ng) dn.

Proposition 5.6.

cP (E(−, ϕs), g) = ϕs(g) +ξ(2s)

ξ(2s+ 1)ϕ−s(g).

We will do this next time. This is analogous to the classical constant term

ys+12 +

ξ(2s)

ξ(2s+ 1)y

12−s.

More generally, for any adelic characters µ1, µ2 : Q×\A× → C, we can define

ϕµ1,µ2,s

(α

β

)= µ1(α)µ2(β)

∣∣∣∣αβ∣∣∣∣s+ 12

and extend to all of G(A). Then the Eisenstein series E(g, ϕµ1,µ2,s) has constant term

cP (E(−, ϕµ1,µ2,s), g) = ϕµ1,µ2,s(g) +L∗(2s, µ1µ

−12 )

L∗(2s+ 1, µ1µ−12 )

ϕµ2,µ1,−s(g)

where L∗ is the completed L-function.25

6. Lecture 6 (February 10, 2015)

6.1. Last time. Last time we finally left the realm of classical forms. We started withf : H → C, a Maass form or modular form on SL2(Z), and associated to this φf :SL2(Z)\GL2(R) → C by multiplying with the cocycle and f evaluated at i. Using strongapproximation, we cooked up a function φ : GL2(Q)\GL2(A)→ C.

We will almost never turn back to the upper half plane picture, unless we need someintuition. Instead we will focus on functions on the adelic group.

6.2. Eisenstein series (adelic). Let me start by recalling the Iwasawa decomposition

G = NAK.

(These decompositions are known before they were so named, at least for special groups.)We will talk about general reductive groups later, but for now, we have G = GL(2), N ={(

1 ∗1

)}⊆ G, A =

{(∗∗

)}⊆ G and K is a maximal compact. Over any ring, N and

A stay the same form but K will change.Over R, we have

G(R) = N(R)A(R)K(R)where we take K(R) = SO(2), not O(2) because we can change the determinant by A. Wewrite

g =

(a bc d

)= pgkg

where pg ∈ NA = P and kg ∈ K. More explicitly,

g =

(det(g)√c2+d2

ac+bd√c2+d2

0√c2 + d2

)(d√

c2+d2−c√c2+d2

c√c2+d2

d√c2+d2

).

We will use this in a second.

Exercise. Prove this.

There is an analogous decomposition over any p-adic field. Now we are looking at

G(Qp) = N(Qp)A(Qp)K(Qp)

where K(Qp) = GL2(Zp). Multiplying by an integral matrix on the right corresponds tocolumn operations, so the valuations of c and d play a role. Indeed,

g =

(a bc d

)= pgkg =

(det gd

b

0 d

)(1 0cd

1

)ifc

d∈ Zp,

(det gc

a

0 c

)(0 −11 d

c

)ifd

c∈ Zp.

We will not need this, because this decomposition is not unique and we will start talkingabout functions that are invariant under the maximal compact.

Exercise. Prove this.26

We will start with a function on A, extend to P = NA trivially (note that A normalizesN), and induce to all of G.

We will adopt the following convention: p will denote a finite prime of Q, and v will denoteany place of Q (including ∞3).

Definition 6.1. Define ϕs : G(A)→ C by

ϕs =∏

ϕs,v

where each ϕs,v : G(Qv)→ C is given by

ϕs,v(gv) = ϕs,v(ngvagvkgv) = ϕs,v(agv) =

∣∣∣∣αgvβgv∣∣∣∣s+ 12v

where agv =

(αgv 00 βgv

).

Definition 6.2.

E(g, ϕs) =∑

γ∈P (Q)\G(Q)

ϕs(γg).

This is ad hoc notation, which will be improved when we start talking about generalEisenstein series.

Exercise. If we take g = (g∞,z, 1, · · · , 1, · · · ) where g∞,z =

(√y x√

y1√y

), then E(z, s) =

E(g∞,z, ϕs,∞).

6.3. Constant term. The general calculation will follow the same line, modulo a few com-plications. If we can break the group into double cosets modulo P , then we are in goodshape. Recall the Bruhat decomposition

GL2 = B tB(

11

)N

where B =

{(∗ ∗0 ∗

)}⊂ GL2.

Exercise. Prove this.

These are straightforward for GL2. In general, this is essentially Gaussian elimination andwe will have

G =∐w∈W

BwN

where W is the Weyl group, or even more generally

G =∐w∈WP

PwNP

for any group with a Tits system.

3John Conway likes to call this −1.27

One consequence of the Bruhat decomposition is that

B\GL2 ↔(

11

)t wN

where w =

(1

1

). One needs to check that these are not equivalent under B.

Here B is a Borel, but can be replaced by any parabolic subgroup.Given a function on any group, we can define its constant term along any parabolic. We

can think of parabolics as ways of going to infinity. For GL2, there is only one way. For GL3,there are three ways. We can think of a manifold with pinches, where the pinches can havevarious dimensions as the dimension of the manifold goes higher.

Recall that

cP,E(−,ϕs)(g) =

∫NB(Q)\NB(A)

E(ng, ϕs) dn.

Classically, the integral is computed overN(Z)\N(R). From now on the argument is straight-forward:

cP,E(−,ϕs)(g) =

∫NB(Q)\NB(A)

∑B(Q)\G(Q)

ϕs(γng)

dn=

∫NB(Q)\NB(A)

∑( 1 1 )twNB(Q)

ϕs(γng)

dnby the Bruhat decomposition, where w =

(1

1

). Dropping the dependence of NB on B,

we write

cB(g) =

∫N(Q)\N(A)

ϕs(ng) dn+

∫N(Q)\N(A)

∑γ∈wN(Q)

ϕs(γng) dn.

We will see that these two terms correspond to ys+12 and y

12−s in the classical constant term

respectively.The first integral is easy. Since ϕs(ng) = ϕs(g) for all n ∈ N , we have∫

N(Q)\N(A)ϕs(ng) dn = ϕs(g)

∫N(Q)\N(A)

dn.

We take the measure normalization∫N(Q)\N(A) dn = 1. This is really the classical normaliza-

tion as done by Tate: N(A) = A and N(Q) = Q, so Q\A ' Z\R · Ẑ with measures dx onZ\R and dµ on Qp respectively, normalized such that µ(Zp) = 1.

The second integral can be written as∫N(Q)\N(A)

∑n0∈N(Q)

ϕs(wn0ng) dn =

∫N(A)

ϕs(wng) dn =∏v

∫N(Qv)

ϕs,v(wnvgv) dnv.

28

Again this corresponds to the classical computation∑n

∫ 10

dx

|x+ n+ iy|c=

∫R

dx

|x+ iy|c

by unfolding. Note we chose our function ϕs as a product function; not all adelic functionsare product functions. Let me call

cB,v(g) =

∫N(Qv)

ϕs,v(wngv) dn.

We are left with local computations, which are not very hard.

(1) At v =∞,

cB,∞(g) =

∫N(R)

ϕs,∞(wng∞) dn

=

∫Rϕs,∞

((1

1

)(1 n0 1

)g∞

)dn.

Here is a simple calculation. Say g∞ = ngagkg.

Claim. (−1

1

)(1 N0 1

)(A

B

)=

(AB√

A2+N2B2−B2N√A2+N2B2

0√A2 +N2B2

)modulo SO(2) on the right.

Exercise. Prove this (using the Iwasawa decomposition).

This implies that

wng∞ =

αgβg√α2g+(n+ng)2β2g ∗0

√α2g + (n+ ng)

2β2g

modulo SO(2), where ag =

(αg

βg

), so

ϕs,∞(wng∞) =

∣∣∣∣ αgβgα2g + (ng + n)2β2g∣∣∣∣s+ 12∞

and the constant term is

cB,∞(g) =

∫R

|αgβg|s+12

(α2g + n2β2g)

s+ 12

dn.

Under n 7→ αgβgn, we get

cB,∞(g) = |αgβg|s+12

∣∣∣∣αgβg∣∣∣∣ ∫

R

dn

(α2g + n2α2g)

s+ 12

=|βs−

12

g αs+ 3

2g |

|α2s+1g |

∫R

dn

(1 + n2)s+12

29

=

∣∣∣∣αgβg∣∣∣∣ 12−s ∫

R

dn

(1 + n2)s+12

= ϕ−s,∞(g∞) ·Γ(1

2)Γ(s)

Γ(s+ 12)

which comes from a specialization of the beta function

Γ(a)Γ(b)

Γ(a+ b)= 2

∫ ∞0

v2a−1

(v2 + 1)a+bdv.

We have seen that the trivial Bruhat cell gives ϕs, and the non-trivial one will giveϕ−s with an intertwining operation.

(2) At v = p,

cB,p(g) =

∫N(Qp)

ϕs,p(wngp) dn.

We want to get the Iwasawa decomposition of wngp.

Claim.

(−1

1

)(1 N

1

)(A

B

)=

(AN−B

0 NB

)if

A

NB∈ Zp,

(B 0

0 A

)ifNB

A∈ Zp,

modulo Kp on the right.

This implies, modulo Kp, that

wngp =

(αg

n+ng−βg

0 (n+ ng)βg

)if

αg(n+ ng)βg

∈ Zp,

(βg 0

0 αg

)if

(n+ ng)βgαg

∈ Zp,

so

ϕs,p(wngp) =

∣∣∣∣ αgβg(n+ ng)2∣∣∣∣s+ 12p

ifαg

(n+ ng)β∈ Zp,∣∣∣∣βgαg

∣∣∣∣s+ 12p

if(n+ ng)β

αg∈ Zp.

This implies∫Qpϕs,p

(w

(1 n0 1

)gp

)dn

=

∫vp(n)≤vp

(αpβp

) ϕs,p((αg

nnβg

))dn+

∫vp(n)>vp

(αpβp

) ϕs,p((

βgαg

))dn.

30

Recall that dn is such that∫Zp dn = 1. The rest is an easy exercise, and we get∣∣∣∣αgβg

∣∣∣∣ 12−sp

{∞∑k=0

pk

p2sk+k

(1− 1

p

)+

1

p

}=

∣∣∣∣αpβp∣∣∣∣ 12−sp

1− 1p2s+1

1− 1p2s

.

This is a nice calculation essentially following Tate’s thesis. The only thing to keepin mind is that

∫Z×pdn = 1− 1

p. So the constant term at p is

cB,p(gp) = ϕ−s,p(gp) ·ζp(2s)

ζp(2s+ 1).

Therefore, the constant term of E(g, ϕs) is

cB,E(−,ϕs)(g) = ϕs(g) +ξ(2s)

ξ(2s+ 1)ϕ−s(g).

For GL(n), the Weyl group is Sn. If we try to do these integrals, for each Weyl elementwe will bring the 1’s to the appropriate row and column using the Iwasawa decomposition.

This boring calculation above is illustrative. It brings out the intertwining operator be-tween s and −s clearly.

At the beginning of the lecture we defined ϕs : G(A) → C. Let me now define a slightlymore general ϕs. Let µ1, µ2 : Q×\A× → C× be two characters, with µ1 =

⊗µ1,v and

µ2 =⊗

µ2,v. Define

ϕ(µ1,µ2,s)(g) =∏v

ϕ(µ1,v ,µ2,v ,s)(gv)

where the functions ϕ(µ1,v ,µ2,v ,s) are again defined using the Iwasawa decompostion gv =nvavkv.

For v where µ1 and µ2 are unramified, i.e., µ1|Z×v = µ2|Z×v = 1 (so they depend on theuniformizer only), we have

ϕ(µ1,v ,µ2,v ,s)(gv) = µ1,v(αgv)µ2,v(βgv)

∣∣∣∣αgvβgv∣∣∣∣s+ 12v

where av =

(αgv

βgv

).

Exercise. Calculate∫N(Qp)

ϕ(µ1,p,µ2,p,s)(wngp) dn = ϕ(µ2,p,µ1,p,−s)(gp) ·Lp(2s, µ1µ

−12 )

Lp(2s+ 1, µ1µ−12 )

.

In general, for the ramified places, we can consider an analogous decomposition where themaximal compact K is replaced by something smaller.

Next time I will start talking about the general theory of reductive groups.

7. Lecture 7 (February 12, 2015)

7.1. Algebraic group theory. Today I will go over things like roots, weights and thestructure theory of groups. We will give examples as we go on. This is necessary becausethe Eisenstein series is defined by some data on the group. If you have not seen this before,

31

the basic objects are characters and cocharacters. Let us fix a field F , which we assume isa finite extension of Q.

There is a theory over F and F , which can be connected using cohomological descent.We will study the theory over F only, i.e., focus on split groups, because that is the caseconsidered in Langlands’ book Euler Products. Eventually we may do something with quasi-split groups.

Definition 7.1. An algebraic group defined over F is an F -variety with multiplication andinversion F -morphisms.

Algebraic groups naturally split into two categories, namely abelian varieties and linearalgebraic groups, but we will not talk about abelian varieties.

Definition 7.2. A linear algebraic group over F is a Zariski closed subgroup of GL(N).

From now on, whenever we say “algebraic group” it will mean “linear algebraic group”.

Example 7.3.

(1) G = GL(N).(2) G = SL(N).

Once we leave the realm of these two, we encounter the problem of representingthese groups. We will need to fix a symplectic form, which we might change overtime.

(3) G = Sp(2n) = {g ∈ GL(2n) | tgJg = J}, where

J =

−1

. ..

−11

. ..

1

.

(4) Let K/F be a separable extension, and H be an algebraic group over K. Then thereexists an algebraic group G = ResK/F (H) over F such that G(F ) = H(K). Thereis an actual construction of this using the coordinate ring, but I will tell you a moreconcrete example.

For example, let K/F be a quadratic extension, so K = F (√d) where d ∈

F×\(F×)2. Take H = Gm. Then

ResK/F (H)(F ) =

{(a bdb a

) ∣∣∣∣ a2 − b2d 6= 0, a, b ∈ F} .All tori over quadratic étale algebras arise this way. Note ResC/R(Gm) = S is theDeligne torus, which is related to Hodge theory.

(5) Unitary group. For this example, let F/F+ be a quadratic extension with conjugationc, and V be an n-dimensional vector space over F with a Hermitian pairing h :V × V → C, i.e., h(α, ϕβ) = c(ϕ)h(α, β). Then

U(V ) = {g ∈ GL(V ) | h(gα, gβ) = h(α, β) for all α, β ∈ V }.Remark. Given any one of these defined over Z, one can consider its points over any ring.

32

Example 7.4. GLN(R) = {g ∈ Matn(R) | det(g) ∈ R×}.

Now we will talk about the radical of a group, because we want to avoid the solvable part.

Definition 7.5 (Radical). Let G be a connected algebraic group. The radical R(G) of G isa maximal connected solvable normal subgroup of G.

Definition 7.6 (Unipotent radical). The unipotent radical Ru(G) is a maximal connectedunipotent normal subgroup of G.

Example 7.7.

• For G = GL(N), R(G) = Z0(G) and Ru(G) = 1.• For G = SL(N), R(G) = 1 and Ru(G) = 1.• For B the Borel subgroup of GL(N) consisting of the upper triangular matrices,R(B) = B and Ru(B) is the set of upper triangular matrices with 1’s on the diagonal.

Note Ru(G) is always a subgroup of R(G). We defined these in order to give the

Definition 7.8. An algebraic groupG is semisimple if R(G) = 1, and reductive if Ru(G) = 1.

Thus GLn is reductive but not semisimple, SLn and PGLn are semisimple, and B is neitherreductive nor semisimple.

For any such groups, we have the Levi decomposition, which will be used all the time inthe parabolic case. Let G be connected over F . Then there exists a reductive M such that

G = M ·Ru(G).This is a semi-direct product, where M normalizes Ru(G).

If G is reductive, then G = R(G) ·G′ where G′ = [G,G] is the derived group and we havethat R(G) ∩ [G,G] is finite. For example, for GL(N), this is just the roots of unity.

Example 7.9 (Levi decomposition). For the Borel, B = M · Ru(B) where M is the set ofdiagonal matrices and Ru(B) is the set of upper triangular matrices with 1’s on the diagonal.

Example 7.10. For G = P2,3,1 ⊆ GL(6) consisting of block upper triangular matrices withblock sizes (2, 3, 1), M is the block diagonal matrices and Ru(G) is the subgroup of matriceswith identity matrices on the block diagonal, i.e.,

∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

∗

∗

=∗ ∗∗ ∗

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

∗

·

11

∗ ∗

11

1∗

1

.Now comes the important piece: the torus. The structure theory goes through this.

Definition 7.11 (Algebraic torus). An algebraic group T defined over F is called a torus ifT ' Gkm over F .

Example 7.12.

• Diagonal matrices in GL(N).• ResK/F (Gm).

33

Definition 7.13 (Split torus). A torus is called split over F if T ' Gkm over F .

I will say a couple of their properties in a second.

Definition 7.14 (Borel). A Borel subgroup is a maximal connected solvable subgroup of G.

This may not exist over the base field, but always exists over C.

Definition 7.15 (Quasi-split group). A group is called quasi-split over F if the Borel isdefined over F .

Everything I wrote down is quasi-split so far. The theory of automorphic forms goes intotwo parts: quasi-split groups, and the inner forms of the group. We will not talk about theinner forms. In order to define the Eisenstein series, we need the Borel.

Note that tori and Borel subgroups are not canonically defined, but they are all conjugateto each other. More precisely, we have the following properties:

(1) All Borels are F -conjugate. (More generally, all minimal parabolics are F -conjugate.)(2) All maximal split tori are F -conjugate.

Definition 7.16 (Rank). The F -rank of a group is the dimension of a (hence all) maximalsplit torus over F .

Once we have all these definitions, we will start talking about weights and roots attachedto them. We linearize the theory by looking at the Lie algebra, which can be decomposedinto subspaces depending on the characters of the maximal torus. Since they are nice vectorspaces, we can calculate things with them instead of looking at the group itself.

7.2. Roots, weights, etc.

Definition 7.17 (F -character lattice). X∗F (G) = HomF (G,Gm).

Example 7.18. Take G = Gm. Then X∗(G) ' Z is given by t ∈ Gm(F ) 7→ tk.

Example 7.19. Take K = C and F = R, and consider G = ResK/F (Gm). Then

G(R) ={(

a −bb a

) ∣∣∣∣ a, b ∈ R, a2 + b2 6= 0} ' R+ · SO(2) ' C×and G(C) ' G2m. We have X∗R(G) ' Z and X∗C(G) ' Z2.

Definition 7.20 (F -anisotropic). A torus is called F -anisotropic if X∗F (T ) = {1}.

Example 7.21. SO(2,R) is anisotropic, because all (non-trivial) characters are defined overC.

Every torus has an anisotropic (compact) part and a split part.

Definition 7.22 (Cocharacter lattice). X∗(T ) = HomF (Gm, T ).34

Example 7.23. X∗(Gm) ' Z, and more generally X∗(GLn) ' Zn given by

α 7→

1. . .

1α

1. . .

1

.

Weyl group will lead to the classification of roots and weights.

Definition 7.24 (Weyl group). Let T be a maximal F -split torus of G. Then

WF (G) := NG(T )/ZG(T ).

Every s ∈ NG(T ) acts on T via s 7→ (ws : t 7→ sts−1).

Example 7.25.

• For G = GL(N), WF (G) = Sn.• For G = Sp(2n), WF (G) = (Z/2Z)n o Sn.• For G = SO(2n+ 1), WF (G) = (Z/2Z)n o Sn.

For the rest of the class, I will give a classification of Chevalley groups. Langlands’ bookis on Chevalley groups, which are defined over Z. They are split, and essentially one of thesimplest groups one can consider, but also fairly general.

From now on, we fix the following notations:

• G will be a connected reductive group over F .• Lie(G) is the Lie algebra ofG. One can define it using derivations, or more canonically

as

Lie(G) = ker(G(F [t]/t2)→ G(F )

).

• T is a maximal split torus of G. Then T acts on Lie(G) by X 7→ tXt−1.We get

Lie(G) = gT0 ⊕∑α∈Φ

gTα

where

gTα = {X | tXt−1 = α(t)X}for α ∈ X∗(T ).

Definition 7.26 (Roots). The elements of Φ are called F -roots.

Example 7.27. Take G = GL(n) and T the torus consisting of all diagonal matrices. Let

αj : T → C× be

a1 . . .an

7→ aj. Then the roots are given by eij = αi − αj for all1 ≤ i 6= j ≤ n, and geij is the matrix with 1 at the (i, j)-entry and 0’s elsewhere.

35

Example 7.28. Take G = Sp(4) and T =

t1

t2t−12

t−11

. Define

α1

t1

t2t−12

t−11

= t1 and α2t1

t2t−12

t−11

= t2.Then the roots are

Φ = {±(α1 ± α2),±2α1,±2α2}.The Lie algebra of G is

g =

{(A BC D

) ∣∣∣∣ A = (u vw x), D =

(−x −v−w −u

), B =

(α ββ′ α

), C =

(γ θθ′ γ

)},

and we have

gα1−α2 =

0 v0 0

0 −v0 0

.Note that X∗F (T ) has a Weyl action. Take χ ∈ X∗F (T ). Then χ ◦ ws = χ(s • s−1). Weyl

group moves the Weyl chambers around, so if we study one chamber we will know somethingabout the whole space. Choosing a Weyl chamber means fixing a Borel subgroup.

There is a natural pairing between X∗ and X∗. Recall that HomF (Gm,Gm) ∼= Z byα 7→ αk. Define 〈·, ·〉 : X∗ ×X∗ → Z by 〈α, µ〉 = k, where α ◦ µ : Gm → Gm so there existsk such that α ◦ µ(t) = tk, i.e.,

α ◦ µ(t) = t〈α,µ〉.Definition 7.29 (Coroots). For a ∈ Φ, let a∨ ∈ X∗(T ) be such that 〈α, α∨〉 = 2 (essentially).

Example 7.30. For G = GL(n), eij = αi − αj sends

t1 . . .tn

7→ titj

. Then the dual

e∨ij : Gm → T sends t 7→

. . .

t. . .

t−1

. . .

(where t and t−1 appear on rows i and j

respectively).

Example 7.31. For Sp(4), we have

(α1 − α2)∨ : t 7→

tt−1

tt−1

36

and

(2α1)∨ : t 7→

1

tt−1

1

.Definition 7.32 (Root system). LetG be semisimple with maximal torus T . Then (X∗(T ),ΦT ,WT )is called a root system.

Definition 7.33 (Simple roots). A root α ∈ Φ is called simple if α 6= β + γ for β, γ ∈ Φ+.Denote the set of simple roots by ∆.

We are not going to define the positive roots Φ+, but a down-to-earth way of thinkingabout them is that they give the Borel. For example, for GL(n) they are eij for i ≥ j.

Theorem 7.34 (Chevalley).

(1) Given any abstract root system (X,Φ,W ), there exists a connected semisimple G suchthat X = X∗(G) and Φ = ΦG.

(2) If (X1,Φ) and (X2,Φ) are such that X1 ↪→ X2 fixing Φid→ Φ, then there exists an

isogeny G1φ→ G2.

Theorem 7.35 (Classification). There are four infinite families and five exceptional irre-ducible root systems (X,Φ,W ):

• An = SL(n+ 1), Bn = SO(2n+ 1), Cn = Sp(2n), Dn = SO(2n);• E6, E7, E8, F4, G2.

We know all these groups. We will look at their parabolic subgroups and define theEisenstein series.

8. Lecture 8 (February 17, 2015)

8.1. Root systems. Last time we had an overview of the theory of algebraic groups. Tosummarize, we classified the Chevalley groups over any fields.

We did not have time to define an abstract root system Φ:

• Φ is a finite set, and V = SpanR(Φ).• For every α ∈ Φ, there exists a reflection sα such that:

– sα(Φ) ⊆ Φ.– sα fixes a codimension 1 subspace.

• If α, β ∈ Φ, then sα(β) − β ∈ αZ. (So we can define the coroot α∨ by sα(β) =β − 〈β, α∨〉α.)

We say that

• Φ is reduced if α ∈ Φ and n · α ∈ Φ (n ∈ Z) imply n = ±1.• Φ is irreducible if Φ 6= Φ1 ⊥ Φ2 with Φi 6= ∅.

Theorem 8.1. There is a one-to-one correspondence

{reduced irreducible root systems} ↔ {An, Bn, Cn, Dn, E6, E7, E8, F4, G2},where

• An = SL(n+ 1), n ≥ 1,37

• Bn = SO(2n+ 1), n ≥ 1,• Cn = Sp(2n), n ≥ 3,• Dn = SO(2n), n ≥ 4.

Note that all of these can be realized by semisimple split groups (in fact by Chevalleygroups). A very similar classification exists for reductive groups. Let G be a reductive groupand T be a maximal torus.

Definition 8.2 (Root datum). A root datum is Ψ = (X, Y,Φ,Φ∨) where X and Y are freeabelian groups of the same rank with a pairing 〈·, ·〉 : X × Y → Z (so that X ∼= Hom(Y,Z)and Y ∼= Hom(X,Z)), Φ and Φ∨ are finite subsets with a bijection Φ↔ Φ∨, and there is anaction of Weyl group.

This is an abstract root datum. For reductive groups, we will have

(X, Y,Φ,Φ∨) = Ψ(G, T ) = (X∗, X∗,Φ(G, T ),Φ(G, T )∨).

A root datum is reduced if 2α /∈ Φ whenever α ∈ Φ.Theorem 8.3 (Classification). Given (X, Y,Φ,Φ∨), there exists a unique (up to isomor-phism) connected reductive group G over Q and maximal torus T such that Ψ = Ψ(G, T ).

Langlands only considered split Chevalley groups in his book. In 1967–68, the theory ofreductive groups was not very well understood. To talk about Eisenstein series, one needs todo parabolic induction and talk about points over Zp. These groups are all defined over Z, sowe can immediately go on to the calculation of constant terms without any linear algebraicgroup theory.

8.2. Parabolic (sub)groups. Informally, a parabolic subgroup is a subgroup that measuresinfinity. It is a measure of the obstruction to G being anisotropic (informally, compact). IfP is a parabolic subgroup, then G/P will be compact (for example, look at the Iwasawadecomposition).G will always be a connected reductive group.

Definition 8.4 (Borel subgroup). A maximal connected solvable closed subgroup B of G iscalled a Borel subgroup.

Definition 8.5 (Parabolic subgroup). A parabolic subgroup P is a closed subgroup contain-ing a Borel. (More intrinsically, a parabolic subgroup is one for which G/P is projective.)

The standard example of a Borel is the upper triangular matrices, and a parabolic is anysubgroup containing it.

The main properties of parabolic subgroups P are:

• P is connected.• NG(P ) = P .• If P is conjugate to P ′ with P, P ′ ⊃ B, then P = P ′.• Bruhat decomposition: G =

∐w∈W BwB.

All Borels are conjugate in G(F ) whenever they are defined over F . In a sense this saysthat Borels are intrinsic to infinity. The same is true for all minimal parabolics (which mightnot be the Borel, if the Borel is not defined).

It would take a whole course to prove these results, so instead I will put up some references.Some standard ones are:

38

• Humphreys, Linear Algebraic Groups.• Springer, Linear Algebraic Groups.• Borel, Linear Algebraic Groups.

Example 8.6.

• Let G = GL(n). We can take B to be the set of all upper triangular matrices

B =

∗ ∗. . .

0 ∗

∈ GL(n) .

An example of a parabolic is

P =

{(GL(n1) ∗

0 GL(n2)

)∈ GL(n)

∣∣∣∣ n1 + n2 = n} .• Let G = SL(n), and B be the set of upper triangular matrices. Then

P =

{(GL(n1) ∗

0 GL(n2)

)∈ SL(n)

∣∣∣∣ n1 + n2 = n}is a parabolic. Even if we are studying semisimple groups, reductive groups show upnaturally.

Parabolic subgroups are subsets of simple roots. One should think of each conjugacy classof parabolics as a boundary component of the group. We are interested in functions onsymmetric spaces, such as G/K. In GL(2), there is only one way to go to infinity:(

aa−1

)(1 n0 1

).

In GL(3), there is more than one way. In order to decompose L2(Γ\G), we need to understandthe behavior at infinity, so we want to know the constant terms, which are integrals overparabolics.

Simple roots are a subset ∆ ⊆ Φ such that:• ∆ spans Φ.• For any β ∈ Φ, we have β =

∑α∈∆ cαα where cα are either all ≥ 0 or all ≤ 0.

Given simple roots ∆, the positive roots are Φ+ = {∑

α∈∆ cαα | cα ≥ 0}.Let us look at the prime example of GL(n).

Example 8.7. Let G = GL(n). Let αi : T → C× be the root

t1 . . .tn

7→ ti andeij : T → C× be

t1 . . .tn

7→ titj

. (Eventually we will linearize everything and look at

these in additive notations. It is sufficient to look at the tangent space (Lie algebra) becausethe fundamental group does not really matter to the behavior at infinity.) Then we can take

Φ = {eij | 1 ≤ i 6= j ≤ n},39

∆ = {ei,i+1 | 1 ≤ i ≤ n− 1},Φ+ = {eij | 1 ≤ i < j ≤ n}.

We have the following facts:

(1) The choice of a Borel is the same thing as fixing Φ+. Note that Φ+ involves orderingthe roots and looking at a positive cone. This is because the unipotent radical canbe written as

Ru(B) =∏α∈Φ+

Uα.

For GL(n), Ueij = Uij is the set of unipotent matrices with 1 at the (i, j)-th entry.Then

∏eij∈Φ+

Uij =

1 ∗. . .

1

.

(1’) Levi decomposition: B = T nRu(B), where the Levi component is a split torus T .(1”) Given a base ∆, the group generated by T and Uα (α ∈ Φ+) is a Borel.(2) The Borels containing a fixed T are in one-one correspondence with the set of bases

of Φ.

There are a lot of choices in the theory. We started with a reductive group, with the choiceof a torus. Then we choose a Borel. But once we have fixed these, we can start talking aboutthings which are standard to those choices.

Definition 8.8 (Standard parabolics). Fix a Borel (or a minimal parabolic) B. Any Pcontaining B is called standard.

The structure of L2(Γ\G) is very inductive: it will be composed of pieces, each of whichcorresponds to a parabolic subgroup. This was understood in the 1960’s by Gelfand and histroop. They did this for GL(2) and probably for SL(n). The Eisenstein series tells us thatfor each parabolic, there is a map into L2. These form the continuous spectrum, and theintertwining maps come from the Eisenstein series.

8.3. Decompositions of parabolics. Fix G, T,B,∆ as before.

8.3.1. Levi decomposition. Fix a parabolic P ⊇ B. Then

P = MPNP

where MP is the Levi component and NP = Ru(P ) is the unipotent radical. Moreover,

• MP normalizes NP .• NP is uniquely determined by P .• MP is unique up to conjugation by P .

The same thing works if B is replaced by any minimal parabolic.40

8.3.2. Langlands decomposition.

Definition 8.9 (Split component). A split component of G is a maximal split torus of theconnected component of the center of G.

This is an annoying definition. For GL(n), the split component is Z0(G). For SL(n), thesplit component is trivial. The reason for looking at this is the following.

By the Levi decomposition P = MPNP . This can be further broken down into

P = M1PAPNP

where M1PAP = MP and AP is the split component of MP .

Example 8.10. For GLn(R), if P is the parabolic of block upper triangular matrices withblock sizes (n1, · · · , nr), then

MP =

GLn1

GLn2. . .

GLnr

and NP =In1 ∗

In2. . .

Inr

are the block diagonal matrices and block unipotent matrices respectively. In this case, wehave

AP =

z1. . .

z1z2

. . .z2

. . .zr

. . .zr

and M1P =

g1

g2. . .

gr

where zi > 0 and gi ∈ GL(ni) with det(gi) = ±1.

8.3.3. Explicit construction of the Langlands decomposition. Fix G, T,B. For each P ⊇ B,there exists ∆P ⊆ ∆ such that

P = G(Σ∆P0 )T∆P0 U∆P0 = M1PAPNP

where ∆P0 = ∆\∆P . This notation is in accordance with Arthur’s article on the traceformula.

Example 8.11. Consider the Dynkin diagram of An−1 corresponding to GL(n). Each par-abolic is a block of GL(ni)’s, and we are removing the roots between GL(ni) and GL(ni+1).

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ · · · ◦ ◦ ◦ ◦

◦GL(n1)

◦ ◦ ◦ ◦GL(n2)

◦ ◦ ◦ · · · ◦GL(nr)

◦ ◦ ◦

In general, ∆P is the set of roots that we are removing from the Dynkin diagram. We set

Σ∆P0 = {nα | α ∈ ∆P0 , n ∈ Z} ∩ Φ

41

and

Σ+∆P0

= {nα | α ∈ ∆P0 , n ∈ Z} ∩ Φ+.

Then the Langlands decomposition can be constructed as:

• AP =

⋂α∈∆P0

ker(α)

0.• G(Σ∆P0 ) is the analytic subgroup over R corresponding to m1 in the Lie algebra

decomposition

m = a⊕m1where a is the Lie algebra of AP .

• U∆P0 =∏α∈ΦP

Uα, where ΦP = Φ+\Σ+

∆P0.

They satisfy the following properties:

(1) MP is the centralizer of AP .(2) G(Σ∆P0 )

0 = [MP ,MP ]0.

(3) AP ∩G(Σ∆P0 ) is finite.

Example 8.12. Let G = GL(n), T be the diagonal matrices, B be the upper triangularmatrices, Φ = {eij | 1 ≤ i 6= j ≤ n}, and ∆ = {ei,i+1 | 1 ≤ i ≤ n − 1}. If we take ∆P = ∆,then ∆P0 = ∅, i.e., we are taking out all the roots. Then

AP = T0,

M1P =

∗ . . .∗

with each ∗ ∈ {±1}, and

NP =∏α∈Φ+

Uα =

1 ∗. . .

1

.

9. Lecture 9 (February 19, 2015)

9.1. Last time. Last time we talked about parabolics, which correspond to subsets of thesimple roots.

Let G be a connected reductive group, T be a maximal torus and B a Borel (minimalparabolic), Φ be the roots and ∆ be the simple roots. There is a one-one correspondence

{standard parabolics} ↔ {subsets ∆P ⊆ ∆}.

The correspondence goes like this. For ∆P , we set ∆P0 = ∆\∆P and define

Σ+∆P0

= (Z-span of ∆P0 ) ∩ Φ+.42

Then

AP =

⋂α∈∆P0

ker(α)

0 , NP = ∏Φ+\Σ+

∆P0

Uα.

We have also defined M1P .For GL(N), we have the following picture:

Dynkin diagram ∆P P

∅ ∆ B◦ ◦ αn1−αn1+1 ◦ ◦ ···αnr−αnr+1 ◦ ◦ {αn1 − αn1+1, · · · , αnr − αnr+1} Pn1,··· ,nr

◦ ◦ αi−αi+1 ◦ ◦ {αi − αi+1} Pi,n−i◦ ◦ ◦ ◦ ◦ ◦ ∅ G

Example 9.1. Sp(4) = {g ∈ GL(4) | tgJ0g = J0} where J0 =(

0 −JJ 0

)and J =

(1

1

).

More explicitly, we can write its Lie algebra as

sp(4) =

{(A BC D

)∣∣∣∣A = (u vw x), B =

(b11 b12b21 b11

), C =

(c11 c12c21 c11

), D =

(−x −v−w −u

)}

and we choose

t =

u

x−x

−u

.

The roots are

Φ = {±(α1 ± α2),±2α1,±2α2},

and some of the root spaces (corresponding to the unipotent subgroups Uα) are:43

α gα

α1 − α2

0 v0 0

0 −v0 0

α2 − α1

0 0w 0

0 0−w 0

α1 + α2

b 00 b

2α1

0 b0 0

2α2

0 0b 0

Let B be the upper triangular matrices in G = Sp(4), corresponding to the simple roots

∆ = {α1 − α2, 2α2}. Then the standard parabolics are:

∆P0 ∆P P

∅ ∆ B

{2α2} {α1 − α2}

∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

∗

{α1 − α2} {2α2}

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗∗ ∗

(Siegel parabolic)∆ ∅ G = Sp(4)

When one learns Lie theory, the root spaces are usually one-dimensional, but this is notalways the case.

Example 9.2 (Non-quasisplit). Fix a base field F and consider the quadratic form

QF = x1xn + x2xn−1 + · · ·+ xqxn−q+1 +Q0(xq+1, · · · , xn−q),44

where Q0 is anisotropic. The matrix is

1

. ..

1Q0

1

. ..

1

.

A maximal F -split torus of SO(Q) is

T =

t1. . .

tq1

. . .

1t−1q

. . .

t−11

.

The Levi component is the centralizer of the maximal torus:

CG(T ) =

∗. . .

∗SO

∗. . .

∗

.

The minimal parabolic is

Pmin =

∗ ∗

. . .

�. . .

∗

.

Note that the Borel is not defined over F .

Remark. Root spaces are not always 1-dimensional. In this example, they are either 1-dimensional or (n− 2q)-dimensional.

45

9.2. Character spaces. Now I will start defining spaces which will be some linearizationof groups, like the Lie algebra. The notations will be heavy.

From now on (G,B) (and all related data) will be fixed. Let P be a standard parabolic,with Levi decomposition

P = MPNP = APM1PNP .

Recall the characters X∗(G) = Hom(G,Gm).

Definition 9.3.a∗P = X

∗(MP )⊗ R, a∗P,C = a∗P ⊗R C,aP = Hom(X

∗(MP )Q,R), aP,C = aP ⊗R C.

Example 9.4. Take P = G = GL(N). Then X∗(GL(N)) = 〈det〉, so a∗P ' R.Example 9.5. Take P = Pn1,n2,n3 . Then X

∗(MP ) = 〈det(n1), det(n2), det(n3)〉, so a∗P ' R3.Remark. X∗(AP ) ⊇ X∗(P ) as Z-modules, but are not necessarily equal. These are equal forthe Borel, but that is essentially the