Analysis on arithmetic quotients Introduction

Analysis on arithmetic quotients Introduction

Bill Casselman

University of British Columbia [email protected]

This course will be interested in the description, in analytic terms, of various spaces of functions on Γ \G where G is the group of real points on a reductive group defined over Q, and Γ is an arithmetic subgroup.

What do I mean by ‘analytic terms’? Well, the ultimate goal in the theory of automorphic forms is to derive applications to number theory, particularly those relating to a wide range of L-functions. There are many applications of this kind that depend ultimately on sophisticated analysis, and indeed can often be formulated only given a number of results in analysis. I shall occasionally mention these, but for the most part I shall not say much about them, even when the topic at hand is relevant. You can already discern my bias if you compare my account with most others in the recent literature, which are largely concerned not with arithmetic quotients Γ \G but with adelic quotients G(Q)\G(A), where G(Q) is the Q-rational group implicitly determining Γ, and A is the ring of rational adeles.

This is no loss, at least as far as my purposes are concerned, since all the hard analysis on adelic quotients reduces ultimately to hard analysis on arithmetic quotients. Further, it seesm to me that working directly on adele groups often obscures the essential nature of arguments. Restricting to arithmetic quotients should make the basic problems of analysis, as well as their solutions, clearer.

The course will come in two parts. In the first, I’ll explain what happens for SL

, where almost all important phenomena already appear, but in much simplified fashion. Here I’ll give complete proofs.

In the second part, I’ll discuss groups of higher rank, but with necessarily less complete coverage.

Necessarily, because the approach I propose is largely still conjectural.

In the body of this introduction, I’ll sketch without details—and in particular without proofs—what happens for SL

, remarking from time to time on what happens more generally.

One topic I’ll not deal with, at least not on the first pass, is the Arthur-Selberg trace formula.

Contents

1. Geometry

2. Compact quotients 3. SL(2,Z)

4. The Fourier expansion

5. The topological vector spaces of interest 6. Cusp forms

7. A model continuous spectrum 8. The construction of Eisenstein series 9. Truncation

10. Wave packets

11. Fourier-Eisenstein transform 12. The volume formula

13. Automorphic forms and de Rham cohomology

14. References

1. Geometry Let

G = SL

(R) K = SO

H = {z |

(z) > 0}

Γ = an arithmetic subgroup

• The group SL

(C) acts on P

(C) = C ∪ {∞} by M¨obius transformations. If

g =

 a b c d



then 

a b c d



: z 7−→ az + b cz + d . If g is real, then

IM

g(z)

=

(z)

|cz + d|

,

so G takes H to itself. The isotropy group of i is K, so H may be identified with G/K.

• The metric

ds

= dx

+ dy

y

is G-invariant. The corresponding invariant area is

dx dy y

.

The corresponding invariant Laplacian, acting on functions on H, is

∆ = ∆

= y

 ∂

∂x

+ ∂

∂y

 .

• Functions on Γ\H may be identified with functions on Γ\G invariant on the right by K.

Since

dg(z)

dz = dz

(cz + d)

 g =

 a b c d



the group G acts on holomorphic forms f (z) dz

according to the formula (in Shimura’s notation) f (z) 7−→ f | [g

]

where

f | [g]

(z) = f g(z)

j(g, z)

j(g, z) = cz + d
.

Since j(gh, z) = j(g, h(z))j(h, z) these forms lift covariantly to functions F : G → C such that

F (gk) = ε(k)

F (g) .

where

ε: k =

 c −s

s c



7−→ ε(k) = j(k, i) = c + is . Explicitly, the correspondance sets

F (g) = f | [g]

(i) .

• Elements of the universal enveloping algebra U(g) act on functions on G/K, f 7→ R

f . The centre Z(g) of the enveloping algebra is a polynomial algebra in the Casimir element C. The right action of C on functions on G fixed by K is the same as the left action of ∆ on functions on H.

A basis of sl

is

α =

 1 0 0 −1



κ =

 0 −1

1 0



σ =

 0 1

1 0

 . Since

C = α

4 + σ

2 − κ

2 , the element Ω = C

+ κ

is an elliptic operator in U (g).

• Any function in L

(Γ\G) may be considered a distribution. The elliptic operator R

with domain C

(Γ\G) is esentially self-adjoint. The domain of its closure is the same as the F in L

(Γ\G) such that R

F , considered as a distribution, is again in L

(Γ\G). The domain of any power Ω

is the same as the intersection of the domains of all X in U

(g).

• On G define the ^norm

kgk = sup δ(a), δ

(a) if

g = k

ak

. Since kgk = sup

kg(v)k

kghk ≤ kgk khk .

2. Compact quotients

Suppose for the moment that Γ \G is compact. For example, Γ could be the group of units of norm 1 in a quaternion algebra over Q, or more generally in a quaternion algebra B over a totally real number field, such that B ⊗

R ∼ = M

(R) for exactly one real embedding of F into R.

There are two related results of interest: (1) The unitary representation of G on L

(Γ\G) decomposes into an orthogonal direct sum of irreducible representations of G, each with finite multiplicity. (2) On the subspace of L

(Γ\G) of functions transforming with respect to the right action of K by ε

, the operator Ω decomposes into an orthogonal direct sum of eigenspaces, with a discrete set of eigenvalues passing off to ∞.

There is more that can be said, for example about the asymptotic distribution of eigenvalues, but it will

not be said here.

3. SL

( ^Z )

More interesting is the case in which Γ \G is not compact. As I have already said, I’ll look here at only one example, but one for which most interesting phenomena already appear. Let

Γ = SL

(Z)

P = group of upper triangular matrices A = group of diagonal matrices

|A| = connected component of A

N = group of upper triangular unipotent matrices δ = the modulus character of P : p 7→ 

detAd

(p)   H

= {z = x + iy | y ≥ Y }

Γ

= Γ ∩ P Thus

δ:

 t 0 0 1/t



7−→ t

.

The region

D = {z ∈ H | |z| ≥ 1, |

(z)| ≤ 1/2}

is a fundamental domain for Γ.

i

1 /2 + i √ 3 /2

This region has finite area, less than Z

√3/2

dx dy y

. In this region

kgk ≍ inf

γ∈Γ

kγgk ≍ y .

• For Y ≥ 1 the canonical map

Γ

\H

→ Γ\H

is an embedding. Its image is called a parabolic domain of Γ \H. In other words, the quotient Γ\H is the

union of a parabolic domain, with simple geometry, and a compact part which is more complicated.

4. The Fourier expansion

If F is any continuous function on Γ \H then it is invariant under horizontal translatiosn z 7→ z + n. We may define its Fourier series

X

Z

F

(z)e

where

F

(z) = Z

0

F (x + z)e

dx .

Since H modulo horizontal translations may be identified with iR

, this may be identified with a function F

(y) on (0, ∞). If F is smooth enough, this series will converge.

For us, the most important term will be the constant term F

, although the other terms are not without interest. For example, if F (z) is a classical automorphic form one can express it as a converging analytic series

F (z) = c

+ c

e

+ c

e

+ c

e

+ · · · = X

c

e

e

, with F

(y) = c

e

, and the coefficients c

are of great interest.

If F is a continous function on Γ \G one may define its ‘constant term’

F

(g) = Z

Γ∩N\N

F (ng) dn

which will be a function on N Γ

\G. The Iwasawa decomposition G = N |A| K tells us that this is in bijection with |A| × ({±I}\K).

In fact, the constant term can be defined for any distribution on Γ\G. If f is a function on Γ

N \G, we can define formally the ‘Eisenstein series’

E

(g) = X

ΓP\Γ

f (γg) .

If this converges, it will be invariant Γ. In particular, if f is fixed by K it may be identified with a function f (y) on N \H ∼ = {iy | y > 0}, and then

E

(z) = X

ΓP\Γ

f

(γz)
.

There are many criteria for the convergence of this series. The basic feature in all of them is that f (nak) = f (ak) must grow small as δ(a) → 0 (or, in the special case, that f(y) grow small as y does).

For now we’ll just need this one:

Lemma 4.1. For ϕ a continuous function on Γ

N \H with compact support on Γ\G, the series

[ephi]

E

(z) = X

ΓP\Γ

ϕ(γz)

converges absolutely to a continuous function of compact support on Γ\H. It is a right G- and g- covariant map from C

(Γ

N \G) to C

(Γ\G).

The point of all this is that the constant term and Eisenstein series are closely related:

Lemma 4.2. Assume ϕ to be a continuous function on Γ

N \G. If E

converges absolutely and uniformly

[evarphi2]

on compact subsets of H, then for F continuous on Γ\H

hF

, ϕi

= hF, E

i

.

As a result, we see that it is consistent to define Φ

for any distribution Φ on Γ \H, and in particular for Φ in L

, as a distribution on Γ

\H by the formula

hΦ

, ϕi

= hΦ, E

i

.

5. The topological vector spaces of interest

There are several topological vector spaces of interest when one is doing analysis on Γ \G. In one group are these:

C

(Γ\G) C

(Γ\G) S(Γ\G) A(Γ\G) A

(Γ\G) A

(Γ\G) A(Γ\G) and in another are these:

L

(Γ\G) L

(Γ\G) C(Γ\G) C(Γ\G) b

The difference between the two groups is that the first are largely involved with distributions, the second just with square-integrable functions. Expressed in other terms, the first are concerned with arbitrary continuous representations of G, the second only with unitary ones.

The topologies themselves are not likely to appear in an important role, except for the unitary represen- tations on Hilbert spaces.

• C

(Γ\G)

As a topological vector space, this is soemwhat complicated. If U is a relatively compact open subset of Γ\G one has on C

(U ) the semi-norms

kfk

= sup

g∈U,X∈Uⁿ(g)

 R

f (g)   .

The space C

itself is the direct limit of these spaces. What is important is that a linear functional on the direct limit is continuous if and only if it is continuous on each C

(U ).

• C

(Γ\G)

This is the space of distributions on Γ \G, the continuous linear functions on C

(Γ\G).

• S(Γ\G)

This I call the Schwartz space of Γ \G, that of all smooth functions f such that all R

f vanish more rapidly at infinity than any 1/y

. The topology is defined by semi-norms

kfk

= sup

g∈Γ\G,X∈Uⁿ(g)

 kgk

R

f (g)   .

• A(Γ\G)

This is the continuous dual of S, the space of moderate distributions .

• A

(Γ\G)

Functions F such that every R

F is of moderate growth.

• A

(Γ\G)

Functions F of uniform moderate growth —there exists one N such that all R

F are O(kgk

). For a fixed N the space of F with F = O( kgk

) is a Banach space, and this is the subspace of its smooth vectors.

• A(Γ\G)

The space of automorphic forms on Γ \G—those smooth functions of moderate growth contained in a finite-dimensional K-stable subspace and annihilated by some non-zero polynomial P (C). Classical holomorphic automorphic forms are automorphic forms, precisely since they are required to have Laurent expansions

f (z) = X

n

c

e

with c

= 0 for n < 0.

• L

(Γ\G)

Square-integrable functions—a Hilbert space.

• L

(Γ\G)

Functions F in L

such that all R

F , considered as distributions, are also in L

. It is contained in A

(Γ\G).

• C(Γ\G)

This I call the Harish-Chandra space of Γ \G, that of all smooth functions f such that all R

f vanish more rapidly at infinity on D than any y

/ log

y. These functions are just barely square-integrable.

The topology on this space is defined by semi-norms kfk

= sup

g∈Γ\G,X∈Uⁿ(g)

log

kgk· 

R

f (g)   .

• b C(Γ\G)

The continuous linear dual of C.

6. Cusp forms

There is one fundamental fact about the constant term—the difference F − F

generally decreases as y → ∞. There are various technical versions of this, but one of the useful is:

Proposition 6.1. If F lies in A

then every R

F − R

F

is rapidly vanishing at ∞.

[vanishingf0]

This is certainly the case for classical automorphic forms F , for which F − F

= O(e

).

A distribution of moderate growth is called cuspidal if its constant term vanishes, or equivalently if and only if it annihilates all E

.

Let L

be the subspace of cuspidal functions in L

(Γ\G). By definition, we can write L

(Γ\G) = L

⊕ L

where L

is the closure in L

of the subspace spanned by the E

. This also gives us L

(Γ\G) = L

⊕ L

.

But L

is containd in A

, so

L

(Γ\G) ⊆ S(Γ\G) . As a consequence we can now also write

S = S

⊕ S

where S is the closure in S of the span of the E

.

The spaces L

and S

look just like the corresponding spaces for a compact quotient. I’ll say no more about them here, although in the course itself I’ll prove these statements. The complements L

and S

are more interesting, from an analyst’s point of view. From a number theorist’s point of view the cusp forms are certainly more interesting, but it turns out that in order to understand them one has also to deal with the complement.

7. A model continuous spectrum

The constant functions lie in L

since they are certainly orthogonal to cusp forms. The complement of the constants in L

is a continuous integral of unitary representations. Before I explain exactly what happens, I’ll look in this section at a simple example.

For the moment, let G = R

, the multiplicative group of positive real numbers, and Γ = {1}. Let D be the multiplicatively invarant derivative xd/dx. Here too we can define spaces that will be of interest:

• C(0, ∞)

This is the space of all f in C

(0, ∞) such that all D

f decrease more rapidly than any 1/| log

x| near 0 and ∞.

The map x 7→ log x identifies the group (0, ∞) with R, and this space is isomorphic to the additive Schwartz space S(R). The Fourier transform

f 7−→ b f (s) = Z

0

f (x)x

dx x is an isomorphism of this with S(iR). The inverse map is

f (x) = 1 2πi

Z

RE(s)=0

F (s)x

ds .

• b C(0, ∞)

This the space dual to C(0, ∞), that of tempered distributions.

• S(0, ∞)

This is the space of all C

(0, ∞) such that all D

f decrease more rapidly than any x

at ∞ and more rapidly than any x

at 0. It looks more like C

(0, ∞) than C(0, ∞). The Fourier transform b f is holomorphic, characterized by the condition that for all N and σ > 0 there exists C

such that

 F (s) 

 ≤ C

(1 + |s|)

in the region

(s) ≤ σ.

• A(0, ∞)

This is the space of all functions on (0, ∞) annihilated by some polynomial P (D). It has as basis the functions x

log

x.

• L

(0, ∞)

One can verify that for f in C Z

0

 f(x) 

dx = 1 2πi

Z

RE(s)=0

  b f (s)  

ds

Therefore isomorphism of C(0, ∞) with C(iR) extnds to one of L

(0, ∞) with L

(iR).

8. The construction of Eisenstein series

Again let Γ = SL

(Z). I want to describe roughly what the continuous spectrum of Γ\G looks like. But to simplify things slightly, I’ll look just at Γ \H.

The starting point is that if F is an eigenfunction of ∆ on H, then F

is one of ∆ on Γ

N \H. So we then have

y

∂

F

∂y

= D

F − DF = λF ,

so that F

is an automorphic form of the kind we have seen in the previous section. The solutions are of the form y

, with λ = λ

= s(s − 1). It seems natural to try constructing eigenfunctions of ∆ by means of Eisenstein series. Since ∆ commutes with Γ, the Eisenstein series

E

(z) = X

ΓP\Γ

IM

(γz)

,

if it converges, ought also to be an eigenfunction of ∆ with eigenvalue s(s − 1). It doesn’t always

converge, but it does extend meromorphically to values of s throughout C. Everyone has his own

favourite way to carry out this meromorphic extension—mine takes place in three steps described by

this diagram:

real part of s = 1/ 2

s = 1

CONVERGENCE.

The function y

=

(z)

is an eigenfunction for ∆ on H, with eigenvalue s(s − 1).

invariant under N .

Proposition 8.1. The series converges absolutely and uniformly on compact subsets of H to an eigen-

[es-convergence]

function of ∆ that satisfies the condition that

E

− y

= O(y

) for y ≫ 0, with σ =

(s).

The constant term of E

satisfies an ordinary differential equation that tells us it is equal to A(s)y

+ B(s)y

for some holomorphic functions A, B. It turns out that A(s) = 1, and that in fact E

∼ y

+ ξ(2s − 1)

ξ(2s) y

where ξ(s) = π

Γ(s/2)ζ(s).

RESOLVANT.

For s in the striped region

RE

(s) > 1/2, s / ∈ (1/2, 1]

the value of λ

= s(s − 1) does not lie in (−∞, 0], which includes the spectrum of the closure of ∆. The resolvant (∆ − λ

I)

is therefore a bounded function on L

in this region. We use this, roughly to find a function E

which is asymptotic to y

in this region in the sense that E

− y

is square-integrable for y ≫ 0. It is therefore a continuation of E

.

THE CRITICAL LINE.

For the final step, there are two competing techniques, one (from [Colin de Verdi`ere:1972] usng the Friedrichs extension, the other (from [Jacquet:1981]) using integral operators.

Neither one seems definitive, but I know no better.

In the end, one obtains not only meromorphic extension, but a functional equation E

= c(s)E

.

The function c(s) satisfies c(s)c(1 − s) = 1, and hence  c(s) 

 = 1 on the critical line

(s) = 1/2.

Explain the functional equation for general E

, using intertwining operators for principal series, talk

about Γ(s) ^factors.

9. Truncation

In constructing Eisenstein series we have seen many variations on the theme of cutting off a function in the neighbourhood of infinity. For Y ≥ 1 define the ^truncation of Φ informally by the condition that over D, with z the projection of g onto H:

Λ

Φ(g) =

 Φ(g) − Φ

(g) if

(z) ≥ Y

Φ(g) otherwise.

A little more formally, define

C

Φ(g) = E

Φ

(g)

(for any fixed g the series defining this has only a finite number of terms) and Λ

Φ = Φ − C

Φ .

If Φ lies in A

then Λ

Φ is rapidly decreasing at infinity, and in particular square-integrable. The decomposition

Φ = Λ

Φ + C

Φ

is orthogonal whenever that makes sense. The Maass-Selberg relation expresses hΛ

E

, Λ

E

i

in an explicit formula. (This is used to prove that E

had no poles on the critical line. There is no circular reasoning, I promise.)

10. Wave packets

For f in C

(1/2 + iR) the function

E

= 1 2πi

Z

RE(s)=1/2

f (s)E

ds

lies in C, and hence may be paired with E

. This is proved by expressing E

as E

= Λ

E

+ C

E

Since it lies in L

, it is in A

.

11. Fourier-Eisenstein transform

For f in S(Γ\H) its Fourier-Eisenstein transform is the meromorphic function f (s) = hE b

, f i

.

There are some obvious properties this function F (s) satisfies:

(a) It is meromorphic in C, holomorphic in

(s) ≤ 0 except for a simple pole at s = −1;

(b) it satisfies the functional equation F (s) = c( −s)F (−s);

and some more subtle ones:

(c) it is L

on

(s) = −1/2, along with all s

F (s);

(d) in a region σ

< σ =

(s)h1/2,

(s)iε it is bounded up to constant by all 1

σ

(s)

There exists a map back from such F (s) to S, inducing an isomorphism with S

. Part of the inverse map is an integral

1 2πi

Z

RE(s)=1/2

F (s)E

ds , and another part is the residue of F at s = −1.

Consequence: description of all automorphic forms.

12. The volume formula

The volume of the fundamental region D is not easy to compute. The Eienstein series E

has a simple pole at s = 1, with residues 1/ξ(2). This turns out to tell us that the volume is ξ(2).

13. Automorphic forms and de Rham cohomology

The Paley-Wiener theorem for S implies Borel’s conjecture, that the cohomology of Γ\H is the coho- mology of the complex of differential forms that are also automorphic forms. This is a kind of Hodge theorem for this non-compact space.

14. References

1. J. Bernstein, Meromorphic continuation of Eisenstein series, notes based on lectures, undated but probably sometime around 1985. A more elaborate and more general version of Selberg’s integral operator proof. ^2. A. Borel, Automorphic forms on SL

(R), Cambridge University Press, 1997.

3. W. A. Casselman, ‘Automorphic forms and a Hodge theory for congruence subgroups of SL(Z)’, pp.

103–140 in Lie group representations II , Lecture Notes in Mathematics ¹⁰⁴¹ , 1984.

4. W. A. Casselman, ‘Extended automorphic forms on the upper half plane’, Mathematische Annalen 296 (1993).

5. W. A. Casselman, ‘On the Plancherel measure for the continuous spectrum of the modular group’, in Automorphic forms, automorphic representations, and arithmetic , Proc. Symp. Pure Math. ⁶⁶ (1999).

6. W. A. Casselman, ‘Harmonic analysis of the Schwartz space of Γ \SL

(R’ 163–192 in Contributions to

automorphic forms, geometry, and number theory , Johns Hopkins Press, 2004.

7. Yves Colin de Verdi`ere, ‘Une nouvelle d´emonstration de prolongement m´eromorphe des s´eries d’Ei- senstein’, C. R. Acad. Sci. de Paris ²⁹³ (1981), 361–363.

8. I. Y. Efrat, The Selberg Trace formula for PSL

(R)

, Memoirs of the American Mathematical Society 359 , 1987. The simplest published form of Selberg’s proof of meromorphic continuation.

9. Harish-Chandra, Automorphic forms on semi-simple Lie groups , Lecture Notes in Mathematics ⁶² , Springer, 1968.

10. Roger Godement, ‘Decomposition of L

(G/Γ)’, 211-224 in Algebraic groups and discontinuous subgroups , Proceedings on Symposia in Pure Mathematics ^IX , 1966. This is the Boulder conference.

11. E. Hecke, ‘Eisensteinreihen h ¨ohere Stufe und ihre Anwendungen auf Funktionentheorie uns Arith- metic’, Abh. Math. Sem. Hamburg ⁵ (1927), 199–224. First mention of non-holomorphic automorphic forms.

12. D. A. Hejhal, The Selberg Trace Formula for PSL)

(R), Lecture Notes in Mathematics ¹⁰⁰¹ , Springer, 1983. Appendix F contains a proof of Selberg’s from which the proof given in Efrat’s book is derived.

13. H.Jacquet, ‘Note on the analytic continuation of Eisenstein series’, 407–412 in Proceedings of Sym- posia in Pure Mathematics ⁶¹ , AMS, 1997.

14. Tomio Kubota, Elementary theory of Eisenstein series , Kodansha, 1965.

15. R. P. Langlands, ‘Eisenstein series’, pp. 235–252 in IX of the Proceedings of Symposia in Pure Mathematics , AMS, 1966.

16. R. P. Langlands, ‘The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups’, in IX of the Proceedings of Symposia in Pure Mathematics , AMS, 1966.

17. R. P. Langlands, talk on notion of an automorphic form at Corvallis.

18. R. P. Langlands, On the functional equations satisfied by Eisenstein series Lecture Notes in Mathematics ⁵⁴⁴ , Springer, 1976.

19. H. Maass, ‘ ¨ Uber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichen’, Math. Ann. ¹²¹ (1949), 141–183. First mention of the Eisenstein series E

, whose meromorphic continuation is proven by using known properties of ζ(s) applied term-by-term to its Fourier expansion.

20. Colette Moeglin and Jean-Loup Waldspurger, Spectral decomposition and Eisenstein serioes , Cambridge University Press, 1995.

21. W. Roelcke, ‘ ¨ Uber die Wellengleichung bei Grenzkreisgruppen erster Art’, Sitzbericht Heidelberg Akad. Wiss. 1953–1955, 159–267. First attempt to use spectral analysis to prove continuation.

22. W. Roelcke, ‘Analytische Fortsetzung der Eisensteinreihen zu den parabolischen Spitzen von Gren- zkreisgruppen erster Art’, Math. Ann. ³² (1956), 121–129. Carries the program further, but inconclu- sively.

23. Keith Ouellette, ‘On the Fourier inversion formula for the full modular group’, preprint available on his home page at Holy Cross College. To appear in Representation theory.

24. A. Selberg, ‘Discontinuous groups and harmonic analysis’, pages 177–189 in Proc. Int. Cong. Math., Stockholm, 1962. An announcement of the first complete proof of the meromorphic continuation relying only on spectral methods. Also the first proof of the spectral completeness of cusp forms and the residues of the Eisenstein series.

25. A. Selberg, ‘Introduction to the G¨ottingen lecture notes’, in the Collected Works. A personal account

of the subject.