Analysis on arithmetic quotients: SL(2)

Analysis on arithmetic quotients: SL(2)

Introduction to analysis on arithmetic quotients of SL(2)

Bill Casselman

University of British Columbia [email protected]

This course will be interested in the description of various spaces of functions on Γ\G, in analytic terms, 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 this business is to derive applications to number theory, and there are many such applications that depend ultimately on 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. You can already discern my bias if you compare my account with most others in the recent literature, which are largely concerned not with arithemetic 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 for me, since all the hard analysis on adelic quotients reduces ultimately to hard analysis on arithmetic quotients. Restricting to arithmetic quotients should make the basic points of analysis clearer. In compensation, the analysis can be fairly interesting.

If Γ\G is compact, the analysis is fairly straightforward (but the applications to number theory still not trivial). The unitary representation of G on L ² (Γ\G) is a direct sum L m _π π of irreducible unitary representations, and the multiplicity m _π of each irre- ducible π is finite. This is not hard to prove, and as far as analysis goes that’s a large percentage of the whole story. The rest of the story is about calculating the multiplicities m π , or at least saying something about them. This is what the Selberg trace formula is all about, but in this case most of what it tells you can also be derived from standard techniques on arbitrary compact manifolds. In any event, I’ll say nothing about the trace formula and its variants, either.

The most interesting case is when Γ\G is not compact. In this case it does at least have finite volume, and a fundamental domain for Γ\G can be described rather explicitly, at least in so far as its failure to be compact goes.

The simplest case is Γ = SL ₂ (Z), and indeed analysis on the quotient Γ\H, exhibits

most of the problems one has to deal with, and without distracting complications. So

from now on in this section and the next, I assume Γ to be SL ₂ (Z). The basic problem

now is to understand how to decompose various spaces, particularly L ² (Γ\H), in terms of eigenfunctions on ∆. If Γ\H is compact, L ² (Γ\H) would be a direct Hilbert space sum of finite-dimensional eigenspaces. This is no longer true if Γ = SL ₂ (Z), but the extent to which it is not true is well understood, and that’s what will be explained roughly here. I shall also generally look only at Γ\H, rather than Γ\G. This will eliminate some complicated algebra involved in classifying and describing irreducible representations of SL ₂ (R).

Contents

1. Geometry

2. Vector spaces important in analysis 3. Eisenstein series and the constant term 4. Cusp forms

5. Eisenstein series 6. References 1. Geometry

Throughout let

G = SL ₂ (R) K = SO ₂

H = {z | ^IM (z) > 0}

H Y = {z = x + iy | y ≥ Y }

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 ⁿ (p)   Γ = a proper subgroup of G

Γ P = Γ ∩ P

P _q etc. = stabilizer of the cusp q etc.

Thus

δ:

 t 0 0 1/t



7−→ t ² . I’ll explain in a moment what a proper discrete subgroup is.

• The group SL ₂ (C) acts on 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)

= ^IM (z)

|cz + d| ² ,

so SL ₂ (R) takes H to itself. The isotropy group of i is K, so H may be identified with G/K.

• The metric

ds ² = dx ² + dy ² y ² is G-invariant. The corresponding invariant area is

dx dy y ² . The corresponding invariant Laplacian is

∆ = ∆ _H = y ²

 ∂ ²

∂x ² + ∂ ²

∂y ²

 .

• M ¨obius transformations take circles to circles.

Lemma 1.1. The image of the region y ≥ Y with respect to

[geometriclemma]

g =

 a b c d



is the circle tangent to the real line at a/c, top at height 1/c ² Y . Proof. Exercise.

• On G define the norm

kgk = sup δ(a), δ ⁻¹ (a) if

g = k 1 ak 2 . Since kgk = sup _kvk=1 kg(v)k

kghk ≤ kgk khk .

• We have the integral formula Z

G

f (g) dg = Z

K×A

×K

f (k ₁ ak ₂ )

 δ(a) − δ ⁻¹ (a) 2



dk ₁ da dk ₂ .

The function

1/kgk ^1+ε is integrable on G.

• A parabolic domain in H is a transform of some H _Y , hence either some other H _Y or a circle tangential to R. A point p on P ¹ (R) is a cusp for Γ if Γ ∩ N _q is isomorphic to Z, in which case we can conjugate it to N _Z . If ∞ is a cusp for Γ, then there exists Y > 0 such that

Γ _P \H _Y ֒→ Γ\H , and c is bounded.

If Γ = SL ₂ (Z) this is more precise:

Lemma 1.2. The map

[P-reps]

 a b c d



7−→ (c, d)

is a bijection between Γ _P \Γ and pairs (c, d) of relatively prime non-negative integers with c > 0.

Proof. Exercise.

The image is called a parabolic domain of Γ\H. A proper discrete subgroup of G is one for which Γ\H is the union of a finite number of parabolic domains and a compact set.

It has finite area.

• On a Siegel set corresponding to a cusp, kgk and inf _γ kγgk are comparable.

Lie algebra, L _X , R _X , Casimir and Laplacian.

2. Vector spaces important in analysis

There are several spaces of interest when one is doing analysis on Γ\G:

S(Γ\G)

A(Γ\G)

A _mg (Γ\G)

A _umg (Γ\G)

A(Γ\G)

L ² (Γ\G)

L ^2,∞ (Γ\G)

C(Γ\G)

C(Γ\G) b

• G˚arding’s space—for f in C _c (G) define R _f ϕ =

Z

G

f (g)R _g ϕ dg and if f is in C _c ^∞ this is smooth also:

π(X)R f = R L

f .

Propsoition 2.1. If Φ is a tempered distribution on Γ\G) and f is in C _c ^∞ (G) then R _f Φ

[gaarding]

lies in A _umg .

• The right regular representation of G on L ² is defined by R _g F (x) = F (xg) .

It is a unitary representation, which means primarily that the L ² norm is G-invariant.

3. Eisenstein series and the constant term

I now look at two related notions, Eisenstein series and the constant term.

If f is a function on Γ _P N \G, we can define formally E _f (g) = X

Γ

\Γ

f (γg) .

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 _f (z) = X

Γ

\Γ

f ( ^IM 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.

Lemma 3.1. For ϕ a continuous function on Γ _P N \H with support in the half-plane

[ephi]

y ≥ y ₀ , the series

E _ϕ (z) = X

Γ

\Γ

ϕ(γz)

converges absolutely to a continuous function on Γ\H. If ϕ has compact support then so does E _ϕ . If ϕ is smooth then so is E _ϕ .

Proof.

Now to conclude the proof of Lemma 3.1. The second Lemma tells us that there exist

♣ ^[ephi]

at most a finite number of γ modulo Γ ∩ P with c less than a given bound. This implies

that if U is a relatively compact open subset of H the sum E _ϕ is uniformly finite over all of U . Thus the first assertion. The second follows similarly.

Lemma 3.2. For ϕ in C _c ^∞ Γ _P N \H

and F continuous on Γ\H

[evarphi]

hF ₀ , ϕi _Γ

_\H = hF, E _ϕ i _Γ\H .

This is a special case of a more general result that we’ll use many times. Suppose for the moment that ϕ(y) is any function on (0, ∞). One can define its Eisenstein series , at least formally:

E _ϕ (z) = X

Γ

\Γ

ϕ y(γz)

It might very well not converge, but if it does:

Lemma 3.3. Assume ϕ to be a continuous function on (0, ∞). If E _ϕ converges absolutely

[evarphi2]

and uniformly on compact subsets of H, then for F continuous on Γ\H hF 0 , ϕi _Γ

_\H = hF, E ϕ i _Γ\H .

Proof. Formally we have hF, E _ϕ i _Γ\H =

Z

Γ\H

F (z)E _ϕ (z) dx dy y ²

= Z

Γ\H

F (z)  X

Γ

\Γ

ϕ y(γz)
 dx dy y ²

= Z

Γ∩P \H

F (z)ϕ y(γz)
dx dy y ²

= Z _y

0

ϕ y(γz)
Z 1 0

F (x + iy) dx

 dy y ²

= Z _y

0

ϕ y(γz)

F ₀ (y) dy y ²

= hF ₀ , ϕi _Γ

_\H . I leave it as an exercise to justify this.

Combining Lemmas, we see that it is consistent to define F ₀ for any distribution F on Γ\H, and in particular for F in L ² , as a distribution on Γ _P \H by the formula

hF ₀ , ϕi _Γ

_\H = hF, E _ϕ i _Γ\H .

4. Cusp forms

Any function F on Γ\H is a function on H invariant under integral translations z 7→

z + n. If F is a smooth function, it may therefore be expanded into a Fourier series F (x + iy) = X

n

F n (y)e ^2πinx ,

where the coefficients F n (y) are smooth functions of y. The ‘constant term’ F 0 (y) is of perticular interest. If F lies in L ² , one can write formally

F ₀ (y) = Z 1

0

F (x + iy)e ^−2πinx dx

and it is not hard to see that it is locally L ² on (0, ∞) with respect to the metric dy/y.

But there is something better to be done.

The subspace L ² _cusp of L ² is defined to be the subspace of F in L ² with F ₀ = 0. According to the definition, this is also the orthogonal complement of the image of the E _ϕ .

Since ∆ commutes with the constant term, it defines a self-adjoint operator on L ^2,∞ _cusp . Proposition 4.1. The space L ² _cusp decomposes into a discrete sum of eigenspaces of ∆,

[cusp-discrete]

each of finite dimension, with eigenvalues passing off to infinity.

I’ll prove this later, in the course of proving something slightly more general. Roughly, the point is that suitable functions in L ² _cusp vanish rapidly at ∞. In other words, such functions behave as if the quotient were compact—the space of cusp forms looks just like L ² of a compact quotient.

Discuss compact quotient case first, some facts about L ² . 5. Eisenstein series

What about the orthogonal complement of the cusp forms?

The orthogonal complement of the cusp forms consists of the closure in L ² of the func- tions E _ϕ , where ϕ(y) is a function of compact support on (0, ∞), but more canonically a function on Γ _P N \H that is invariant under all horizontal translations. One slight peculiarity to keep in mind is that tHe measure on this space is dy/y ² . It is natural to suppose that the space L ² (0, ∞), dy/y ²

has something to do with the orthogonal complement of the cusp forms in L ² .

Since Z _∞

0

 F (y)   ² dy

y ² = Z _∞

0

F (y) y ^1/2

2 dy

y

the map F (y) 7→ F (y)y ^−1/2 identifies L ² (0, ∞), dy/y ²

with that of functions on (0, ∞) assigned the multiplicatively invariant measure dy/y, so we could (but won’t) look at this more classical space instead.

For ϕ(y) of compact support on (0, ∞) I define its Fourier transform to be

ϕ(s) = b Z ∞

0

ϕ(y)y ^(−s+1)/2 dy y ² .

This is a holomorphic function on all of C. The reason for the choice of (−s + 1)/2 for exponents is that it will simplify notation later on, and ought to become clearer shortly.

The classical theory for the multiplicative group shows that its inverse is given by ϕ(y) = 1

2πi Z

RE

(s)=σ ϕ(s)y b ^(s+1)/2 ds

for any real number σ. This map induces an isomorphism of L ² (0, ∞), dy/y ²
with the space of L ² (iR), with the inverse map Φ 7→ F being

F (y) = lim

C→∞

1 2πi

Z _iC

−iC

Φ(s)y ^(s+1)/2 ds .

Formally, at least, we can now set E ϕ = 1

2πi Z

RE

(s)=σ ϕ(s)E b s ds where

E _s (z) = X

Γ∩P \Γ

y ^(s+1)/2) γ(z)

, y(z) = ^IM (z)
.

This makes sense formally since y(z) is invariant under Γ∩P . But a number of problems arise, since the series defining E _s converges only if ^RE (s) is large enough. This is not a trivial point, since E _s in fact has a pole at s = 1, which is related to the fact that we somehow have to get the constant functions in L ² from our analysis.

Since we are interested in the spectral analysis of ∆, it is good news that

∆y ^(s+1)/2 = (s ² − 1)

4 y ^(s+1)/2 . Since ∆ commutes with Γ, it seems at least plausible that

∆E s = s ² − 1

4 E s .

In fact, the functions E _s can be defined as a meromorphic function of s, with good behaviour in the region ^RE (s) ≥ 0. They satisfy a functional equation relating E _s and E _−s which incorporates the Riemann function ξ(s), and their behaviour in the region

RE (s) < 0 depends on the location of its zeroes.

Everyone has his own favourite construction of E _s throughout C. Mine takes place, for reasons that will become apparent, in three steps via (a) convergence, (b) spectral resolution, (c) integral operators. I’ll not give the argument in this introduction, but I shall make some remarks that might make things clearer.

Remark 1. If ^RE (s) is large, then y ^(s+1)/2 decreases rapidly as y → 0. In view of Lemma 3.1, it shouldn’t be surprising that E _s then converges. In fact, it converges for

♣ [ephi]

RE (s) > 1. The proof is very close to that of Lemma 3.1.

♣ ^[ephi]

Remark 2. Suppose F to be a function on Γ ∩ P \H such that ∆F = λF . Since ∆ commutes with N , we have also ∆F 0 = λF ₀ . In other words, we get the ordinary differential equation

y ² ∂ ² F ₀ (y)

∂y ² = λF ₀ (y) .

This is a differential equation of Euler type, and to solve it we look for solutions of the form y ^t . This gives us

t(t − 1)y ^t − λ
y ^t = 0

so t is a root of the indicial equation t(t − 1) − λ = 0. If in particular t = (s + 1)/2, the roots are (±s + 1)/2. Therefore the constant term of the Eisenstein series E _s is a linear combination

b(s)y ^(s+1)/2 + c(s)y ^(−s+1)/2 In fact a relatively simple calculation gives b(s) = 1 and

c(s) = ξ(s) ξ(s + 1) ,

where ξ is Riemann’s function. The function E s turns out to be, up to scalar multiplica- tion, the only eigenfunction of ∆ with eigenvalue (s ² − 1)/4 that is orthogonal to cusp forms. But E _−s has the same eigenvalue, and is also orthogonal to cusp forms. We therefore have an equation

E _s = c(s)E _−s ,

which is the functional equation of Eisenstein series. It implies that c(s)c(−s) = 1. It is compatible with, but does not imply, the Riemann’s functional equation.

Remark 3. Later on we shall see an argument involving contour movements that

derives the constant functions as the residues of Eisenstein series at s = 1, and then

show that the subspace L ² _Eis orthogonal to cusp forms is the sum of constants and the space of L ² integrals

C→0 lim 1 2πi

Z _iC

−iC

F (s)E s ds

where F (s) is in L ² (iR) and F (s) = c(−s)F (−s). The subtle fact about Eisenstein series is that it is not at all easy to define E _s for s in iR, which is precisely the most interesting range.

Remark 4. At any rate, in the end we see that L ² (Γ\H) as a module over ∆ is the direct sum of the space of cusp forms, itself a direct sum of finite-dimensional eigenspaces; the constants; and a space isomorphic to the subspace of F (s) in L ² (iR) with F (s) = c(−s)F (−s). On this last ∆ acts as multiplication by (s ² − 1)/4.

Proofs of all these things will be sketched eventually, if not proved in detail. The argument will not be very direct, and it will prove along the way a more refined result, an analogue of the classical Paley-Wiener theorem for C _c ^∞ (R). The results to be encountered are not new, but recent results of Keith Ouellette allow a great simplification.

The space L ² (Γ\H) decomposes into an orthogonal sum of two components, discrete and continuous:

L ² (Γ\H) = L ² _disc (Γ\H) ⊕ L ² _cont (Γ\H) .

The discrete part is a sum of finite-dimensional eigenspaces of ∆, and the continuous part is related to a continuous family of eigenfunctions that fall just short of being in L ² (Γ\H).

6. References

1. R. P. Langlands, ‘Eisenstein series’, talk at the 1965 Boulder conference.

2. K. Ouellette, web site at Mount Holyoke

3. W. A. Casselman, contribution to the Shalika birthday volume