automorphic forms in gl n) i: decomposition of the space of cusp

Automorphic Forms in GL(n) I:

Decomposition of the Space of Cusp Forms

and Some Finiteness Results

M.S. Raghunathan∗

School of Mathematics, Tata Institute of Fundamental Research,

Mumbai, India

Lectures given at the

School on Automorphic Forms on GL(n)

Trieste, 31 July - 18 August 2000

LNS0821001

∗[email protected]

Contents

0 Introduction 5

1 Fundamental Domains 6

2 Automorphic Forms 13

3 Cusp Forms 21

4 Proof of Theorem 2.13 33

References 37

Automorphic Forms in GL(n) 5

0 Introduction

These are notes of lectures given by the author at an Instructional School on

“Automorphic Forms in GL(n)” held at the Abdus Salam International

Centre for Theoretical Physics, Italy, in August 2000. The author would

like to thank the ICTP for their hospitality and their patience over the long

delay in making these notes available.

In this first chapter we introduce some concepts in the theory of Auto-

morphic Forms and prove some basic results. Although our central focus

is on the general linear group GL(n), in this chapter we formulate the re-

sults in the more general context of reductive algebraic groups over number

fields. This is because the techniques for handling the general case are no

different from those needed to handle GL(n). We have also indicated what

the notions used mean in the special case of GL(n) (or SL(n)), for readers

unfamiliar or uncomfortable with the general theory of algebraic groups. All

the results about algebraic groups needed here are to be found in the paper

[B-T] of Borel and Tits.

In §1 we describe a good fundamental domain for G(k) in G(A) where

k is a number field, A is its Adele ring and G is a reductive algebraic group

over k. This description follows from a theorem of A. Borel [B] that describes

a fundamental domain for an arithmetic subgroup Γ in∏

v∈∞ G(kv): here

∞ is the complete set of inequivalent archimedean valuations of k and for

v ∈ ∞, kv is the completion of k at u. Conversely Borel’s theorem can be

deduced from this description (of a fundamental domain for G(k) in G(A)).

A quick and elegant proof of this (due to R. Godement and A. Weil) is to

be found in [G].

In §2 the definition of automorphic forms is given and the central finite

dimensionality results are formulated. These concepts and results are due

to R. Langlands [L] (see also [H-C]).

In §3 we introduce the space of cusp-forms and prove that the repre-

sentation of G(A) in the space of L2-cusp-forms is completely reducible,

a result due to I. Gelfand and I. Piatetishi-Shapiro. This is needed in §4

where the finite dimensionality theorems formulated in §2 are proved. We

need in §3 some well known results from Analysis; these can be found in

R. Narasimhan’s book [N].

6 M.S. Raghunathan

1 Fundamental Domains

1.1. Notations Throughout these notes we will adopt the following no-

tations. We denote by k a number field and by V a complete set of mu-

tually inequivalent valuations of k. Let ∞ (resp. Vf ) denote the subset of

archimedean (resp. non-archimedean) valuations in V. For v ∈ V, kv de-

notes the completion of k with respect to v. We denote by Ov for v ∈ Vf the

ring of integers in kv and by pv the maximal ideal in Ov. The residue field

Ov/pv is denoted Fv and we set qv =| Fv |; also pv = characteristic of Fv .

We denote by | |v the absolute value on kv determined by v. We assume

the absolute value | |v, v ∈ V chosen so that the following holds: Let Qv

denote the closure of Q in kv and Nv : kv → Qv be the norm map. If v is

archimedean we set | x |v=| Nv(x) | where | | is the usual absolute value on

Qv ≃ R. If v ∈ Vf and Nv(x) = prv · y with r ∈ Z and y a unit in the ring

of integers in Qv(≃ Zpv), | x |v= p−rv . With this definition we have the well

known

1.2. Product formula If x ∈ k∗, | x |v= 1 for all but finitely may v ∈ V

and∏

v∈S | x |v= 1 where S = v ∈ V || x |v 6= 1.

1.3. We denote by A the ring of adeles: A = x = xvv∈V ∈∏

v∈V kv |

xv ∈ Ov for all v 6∈ S, S a finite subset of Vf: A is a ring under coordinate-

wise addition and multiplication. Let ∧ denote the subset∏

v∈∞ kv×∏

v∈VfOv;

then ∧ is subring of A. Under the product topology ∧ is a locally compact

ring topological ring. A has a natural structure of a locally compact ring

such that the inclusion of ∧ (with its product topology) in A is an isomor-

phism of topological rings on to an open subring. We denote by Af the

subset x ∈ A | xv = 0 for v ∈ ∞. The group of Ideles I of k is the set

of invertible elements in A. It is easy to see that I = x ∈ A | xv 6= 0

for v ∈ V and | xv |v= 1 for all but a finite number of v ∈ Vf. We also

set If = x ∈ I | xv = 1 for v ∈ ∞. The group I is given a topology as

follows: Consider the group∏

v∈∞ k∗v ×∏

v∈VfO∗

v where for v ∈ Vf ,O∗v is the

compact group of units in k∗v ; this group has a natural structure of a locally

compact group in the product topology. I is given the unique structure of a

topological group for which the natural inclusion of∏

v∈∞ k∗v∏

v∈VfO∗×

v in

I is an isomorphism (of topological groups) onto an open subgroup. If x ∈ I

and S = v ∈ Vf || xv |v 6= 1, we set | x |=∏

v∈V | xv |v=∏

v∈S | xv |v; then

x 7→| x | is a continuous homomorphism of I in R∗ and the kernel of this


homomorphism is denoted I1. Elements of I1 will be called “Ideles of norm

1”. We state without proof the following well known

1.4. Theorem The inclusion of k (resp. k∗) in A (resp. I1) imbeds k

(resp. k∗ ) as a closed discrete subgroup of the (additive) group A (resp.

(multiplicative) group I1) such that A/k (resp. I1/k∗) is compact.

1.5. Let G denote a connected reductive linear algebraic group over k.

(Our main interest is in the case when G = GL(n) the general linear group

and we will draw attention to what our terms and definitions mean in this

special case). Let S be a maximal k-split torus in G (when G = GL(n),S

can be taken to be the group of diagonal matrices in G). Let X∗(S) denote

the group of characters (all characters of S are defined over k) of S and

Φ ⊂ X∗(S) the subset of k-roots of G with respect to S: recall that a k-root

is a non-trivial character for the adjoint action of S on Lie algebra of G (when

G = GL(n),Φ consists of the characters d = diagonal (d1, · · · , dn) 7→ di/dj

for some pair (i, j), 1 ≤ i, j ≤ n, i 6= j). We fix an ordering on X∗(S) and

denote by Φ+ (resp. ∆) the set of positive (resp. simple) roots in Φ (when

G = GL(n), one usually chooses the ordering given as follows: any character

χ on S (= diagonal matrices) is of the form

χ(d) =∏

1≤i≤n

dni(χ)i

where d = diagonal (d1, · · · , dn) and ni(χ) are integers; we set χ > 0 if

nr(χ) > 0 where r(χ) = min i | 1 ≤ i ≤ n, ni(χ) 6= 0. The set of simple

roots for this order are the roots d 7→ di/di+1, 1 ≤ i < n). We fix once and

for all a realisation of G as an algebraic subgroup (over k) of GL(N) for

some N (when G = GL(n), we can take N = n). For v ∈ V then G(kv) is a

closed subset of GL(N, kv) hence a locally closed subset of M(N, kv) ≃ kN2

v

and thus acquires a locally compact topology. For v ∈ Vf we denote by

Mv the compact open subgroup G(kv) ∩GL(N,Ov)(Mv = GL(n,Ov) when

G = GL(n)) of G(kv). G(A) (resp. G(Af ) the A (resp. Af ) points of G

has a natural identification with g = gvv∈V(resp.Vf ) | gv ∈ G(kv) and

the set v ∈ Vf | gv 6∈Mv is finite. We have a natural inclusion

∏

v∈Vf

Mv → G(Af )

8 M.S. Raghunathan

and G(Af ) is made into a locally compact topological group by stipulating

that the above inclusion is an isomorphism of topological groups of the group

on the left onto an open subgroup (the topological group G(Af ) obtained in

this fashion is independent of the realisation of G as a k-algebraic subgroup

of GL(N) for some N). We set G∞ =∏

v∈∞ G(kv). We then have a natural

identification of G(A) with G∞ × G(Af ). We give to G∞ the product

topology on∏

v∈∞ G(kv) and to G(A) the product topology on G∞×G(Af ).

Let pf (resp. p∞) be the projection of G(A) on G(Af ) (resp. G∞). For

g ∈ G(A), we set gf = pf (g) and g∞ = p∞(g). Let df (resp. d∞) = pf d

(resp. p∞d). We have a natural inclusion d : G(k) → G(A) of the k-points

of G in G(A) as a closed discrete subgroup of G(A).

In the sequel we identify G(k) with d(G(k)). We will now describe

certain k-algebraic (resp. closed) subgroups of G and (resp. G(A)) that will

be needed in the sequel. Let Ad denote the adjoint representation of G on its

Lie algebra (over k) L(G). Under the adjoint action of S, L(G) decomposes

into a direct sum of eigenspaces, the eigencharacters being the roots and the

trivial character. For α ∈ Φ, let L(G)(α) be the eigenspace corresponding

to α and let L(U+) =∐

α∈Φ+ L(G)(α). Then there is a unique k-subgroup

U of G normalised by S and with L(U) as the Lie subalgebra of L(G)

corresponding to U. Also U is normalised by the Z(S) the centraliser of S

and P = Z(S) ·U is a parabolic subgroup defined over k : P is a minimal k-

parabolic subgroup of G. We denote by oP the intersection of the kernels of

all squares of characters on P defined over k; then P = S ·oP and oP = B ·U

where B is a reductive algebraic k-subgroup of oP which is anisotropic over

k (i.e. B does not admit a non-trivial k-split toral subgroup). As S is k-split

we may treat it as obtained from a Q-split torus by the base change Q → k-

thus in the sequel we will treat S as a Q split torus. We then have for each

v ∈ ∞ an inclusion S(Qv) → S(kv) when Qv is the closure of Qv in kv; and

since all the v ∈ ∞ induce on Q the unique archimedean topology with R

as the completion, we obtain a diagonal inclusion S(R) →∏

v∈∞ S(kv). Let∏v∈∞ S(kv) → S(A) be the inclusion s = svv∈∞ 7→ svv∈V where for

v ∈ V\∞, sv = 1. We denote by A the image of the identity component of

S(R) (a subgroup∏

v∈∞ S(kv)) in G(A). The group A is a closed subgroup of

G(A) isomorphic to a product of ℓ = dimS copies of the multiplicative group

R+ = x ∈ R | x > 0. For a constant c > 0 let Ac = x ∈ A | α(x) ≤ c for

α ∈ ∆: α is a character (defined over k) on S and as is easily seen on A

takes positive real values. With these notations we have


1.6. Theorem (R. Godement and A. Weil) There is a compact subset Ω

of oP(A) such that Ω ·o P(k) = P(A). There is a maximal compact (open)

subgroup K of G(A) and a constant c0 > 0 such that for all c ≥ c0 and

compact subsets Ω′ of oP(A) with Ω′ ⊃ Ω, one has K.Ac.Ω′.G(k) = G(A).

1.7. Corollary Kf .Ω′f .dfG(k) = G(Af ) where Ω′

f = pf (Ωf ).

1.8. Let Lf be a compact open subgroup of G(Af ) = g ∈ G(A) | gv = 1

for all v ∈ ∞. Let ΓLf= γ ∈ G(k) | pf (γ) ∈ Lf. Now G∞ × Lf

has a natural identification with an open subgroup of G(A) and ΓLfis

in a natural fashion a closed discrete subgroup of G∞ × Lf . Since Lf is

compact, the projection d∞(ΓLf) of ΓLf

on G∞ is a discrete subgroup of G∞.

The maximal compact subgroup K above decomposes as a direct product

K∞ ×Kf following the product decomposition G(A) = G∞ × G(Af ) with

K∞ (resp. Kf ) a maximal compact subgroup of G∞ (resp. G(Af )). We

may also assume that Ω is of the form Ω∞ × Ωf , where Ω∞ (resp. Ωf ) is

a compact subset of oP∞(resp. oPAf )). It follows then that if g ∈ G∞,

there exists γ ∈ G(k) such that gγ∞ ∈ K∞Ac · Ω∞ and γf ∈ Kf · Ωf . Now

Lf being an open compact subgroup and Kf · Ωf being a compact set of

G(Af ), we can find finitely many elements θ1, · · · , θr ∈ G(Af ) such that

KfΩf ⊂ ∪1≤i≤rθiLf . In particular we have γf = θiℓi for some ℓi ∈ Lf

and 1 ≤ i ≤ r. Now if γf , γ′f are such that γf = θiℓi and γ′f = θiℓ

′i with

ℓi, ℓ′i ∈ Lf , then γ′f = γf .ℓ

−1i ℓ′i = γfζ with ζ ∈ Lf . We thus see that we may

assume that the θi = ξif with ξi in G(k). Let Ξ = ξ1, · · · , ξr ⊂ G(k).

Then one finds that

K∞Ac · Ω∞ · Ξ∞ · d∞(ΓLf) = G∞.

We summarize the above discussion in

1.9. Theorem (A. Borel) Let L be a compact open subgroup of G(Af )

and Γ = d∞(ΓL), where ΓL is the subgroup d−1f (L) ⊂ G(k). Then Γ a

discrete subgroup of G∞. There exists a finite subset Ξ in G(k) a constant

c0 > 0 and a compact set Ω∞ ⊂ P∞ such that

K∞ ·Ac · Ω′∞ · Ξ∞ · Γ = G∞

where K∞ is a maximal compact subgroup of G∞. Further for c′ > c if

Sc′ = K∞ · Ac′ · Ω′∞, the set

γ ∈ Γ | Sc′ξ ∩ Sc′ξ′ 6= φ

10 M.S. Raghunathan

is finite for ξ, ξ′ ∈ Ξ∞.

(The last assertion is an additional piece of information that cannot be

deduced from Theorem 1.6.)

1.10. Suppose now that the centre of G does not contain a split torus

(the centre of GL(n) is a 1-dimensional split torus so the considerations of

this paragraph are not applicable to GL(n); they are however applicable to

the group SL(n)). In this case by using the Iwasawa decomposition in G∞

it can be shown that the Haar measure of the set Sc′ is finite. Since a left

translation invariant Haar measure on G∞ is also right translation invariant,

one sees that a Haar measure on G∞ defines a left translation (under G∞)

invariant finite measure on G∞/Γ. [We will elaborate on this in the case

G = SL(n). In this case G∞ =∏

v∈∞

SL(n, kv). We can identify SL(n, kv)

with SL(n,R) or SL(n,C) according to kv ≃ R or C. The compact group

K∞ is a product∏

v∈∞

Kv where for v ∈ ∞,Kv = SO(n) or SU(n) according

to kv ≃ R or C. The group A can be identified with the group of diagonal

matrices in SL(n,R) with positive real diagonal entries - the inclusion of Q

in k induces for each v ∈ ∞, an inclusion SL(n,R) in SL(n, kv) and hence

a diagonal inclusion of SL(n,R) in G∞. For c′ > 0, Ac′ = d = diagonal

(d1, . . . , dn) | di > 0 for 1 ≤ i ≤ n, d1 · d2 . . . dn = 1 and di/di+1 ≤ c for

1 ≤ i < n. The group P consists of upper triangular matrices in SL(n)

and let U be the subgroup of upper triangular unipotent matrices in P. The

group P∞ =∏

v∈∞

oP(kv) and oP(kv) = B(kv). U(kv) where B(kv) = d =

diagonal (d1 · · · dn) | di ∈ k∗v , | di |v= 1 for 1 ≤ i ≤ n. One sees then that

B∞ =∏

v∈∞

B(kv) is compact and contained in K∞ and as it centralises A, we

see that taking Ω′ to be left translation invariant under B∞,Sc = K∞·Ac′ ·Ω′1

with Ω′1 a suitable compact subset of U∞. The natural map

K∞ ×A× U∞ → G∞

is an analytic isomorphism of manifolds. The Haar measure on G∞ pulled

back to K∞×A×U∞ takes the form ρ2(a)·dk ·da·du where dk (resp. da, du)

is a Haar measure on K∞ (resp. A,U) and ρ2 is the homomorphism of A

in R+ determined as follows: Inner conjugation by a ∈ A carries the Haar

measure du on U+∞ into ρ2(a) · du (U∞ is normalised by A). It is also easy

to see that∫A′

cρ2(a)da <∞. Since K∞ and Ω′

1 are compact, it is immediate


that the Haar measure of Sc′ is finite.] We continue with the assumption

that G contains no nontrivial k-split central torus (so that G∞/Γ has finite

Haar measure). We assert now that in this case the measure induced on

G(A)/G(k) by the Haar measure on G(A) is finite as well. To see this we

denote by G1 the closure of G(k) (imbedded diagonally in G(Af ). Now

G1 = G(k) if G∞ is compact; in this case one knows (Godement Criterion)

that G(A)/G(k) is compact and the finiteness of the Haar measure follows.

Thus we assume that G∞ is not compact (this is the case when G = SL(n)

with n ≥ 2). Then one knows that G(Af )/G1 is compact and that G1 is

normal in G(Af ) (when G = SL(n) one has in fact G1 = G(Af )-this is

seen easily using the fact that SL(n, kv) is generated by upper and lower

triangular unipotent matrices and the fact (by the Chinese remainder the-

orem) that k is dense in Af . π : G(A) → G∞ · G1\G(A) ≃ G1\G(Af )

be the natural map. From the definition of G1 it is clear that π factors

through G(A)/G(k). Since G1\G(Af ) is compact we conclude that in or-

der to show that G(A)/G(k) has finite Haar measure it suffices to show

that G∞ ·G1/G(k) has finite Haar measure. Now if L is any compact open

subgroup of G(Af ),G∞ · G1 ⊂ G∞ · L · G(k) and the quotient L · G1/G1

is compact. Thus it suffices to show that G∞ · L · G(k)/G(k) has finite

Haar measure for a compact open subgroup L ⊂ G(Af ). Clearly one has a

natural identification of G∞ · L ·G(k)/G(k) with G∞ · L/(G∞ ·L∩G(k));

since G∞/Γ (Γ = projection of ΓL = G∞ · L ∩ G(k) on G∞) has finite

Haar measure and L is compact G∞ ·L ·G(k)/G(k) has finite Haar measure

proving our contention. We have thus

1.11. Theorem Assume that G has no non trivial k split torus in its

centre. Then G(A)/G(k) has finite Haar measure. If G contains no non

trivial k-split torus G(A)/G(k) is compact.

1.12. Suppose now that C ⊂ G is the maximal central k split torus. Then

G = G/C contains no non trivial central split torus. In the case of main

interest to us viz when G = GL(n),C(≃ GL(1)) is of dimension 1 and is the

entire centre and G is the group PGL(n). By Theorem 1.10, G(A)/G(k)

has finite Haar measure. Now it is known that if π : G → G is the natural

map, π(G(A)) is a closed normal subgroup of G(A) and G(A)/π(G(A)) is

compact. Also the kernel of π : G(A) → G(A) is evidently C(A). It follows

that G(A)/π(G(k)) is compact so that G(A)/C(A) · G(k) is compact as

well. Once again we elaborate on this in the special case G = GL(n).

12 M.S. Raghunathan

Here for any field k′ ⊃ k the natural map GL(n)(k′) → PGL(n)(k′) is

surjective - this is an immediate consequence of Hilbert Theorem 90. Using

the realisation of PGL(n) as an algebraic k subgroup of GL(n2) got from

the adjoint representation, one shows that GL(n,Ov) → PGL(n)(Ov) is

surjective for all v ∈ Vf . It is then immediate that the map GL(n,A) →

PGL(n)(A) is surjective. Observe next that C(A)(≃ GL(1,A)) ≃ I in a

natural fashion and in the sequel we will treat continuous characters χ : I →

C∗ as characters on C(A) under this isomorphism. Let Fχ(G(A),G(k))

denote the vector space of all functions f : G(A) → C such that we have

f(gc) = χ(c)−1f(g) for c ∈ C(A) andf(gx) = f(g) for x ∈ G(k)

Note that in order that Fχ(G(A),G(k)) be non-trivial we need χ to be

trivial on C(k). If χ is unitary i.e., maps C(A) into S1 = z ∈ C∗∣∣| z |= 1,

then the function g 7→| f(g) | on G(A) is invariant under C(A) · G(k)

and hence defines a function f on G(A)/C(A) ·G(k). We will say that f ∈

Fχ(G(A),G(k)) is square integrable if f is square integrable on G(A)/C(A)·

G(k). Let G′ = [G,G] be the commutator subgroup. Then the connected

component S′ of the identity in S ∩ G′ is a maximal split torus in G′. The

natural map C × S′ → S is not in general an isomorphism (it is not in the

case G = GL(n) where G′ = SL(n)). However the group A in G(A) breaks

up into a direct product (C(A) ∩ A) × (S′(A) ∩ A) i.e., the natural map

(C(A) ∩ A) × (S′(A) ∩ A) → A is an isomorphism. Let A0 = C(A) ∩ A

and A′ = S′(A) ∩ A so that A ≃ A0 ×A′. We now define a subgroup A10 in

A0 : A10 = a ∈ A0 ⊂ C(A) |

∏

v∈∞

| χ(av) |v= 1 for every character χ of G

defined over k. Since χ(av) ∈ R+ for all a ∈ A0 and any character χ on

C, and the subgroup of characters on C which are restrictions of characters

on G has finite index in the group of all characters on C, we see that A10 =

a ∈ A0 |∏

v∈∞

| χ(av) |v= 1 for all characters χ on C. The sequence

1 → A10 → A0 → A0/A

10 → 1

splits and has in fact a natural splitting. To see this fix an isomorphism of C

with a product GL(1)r of r copies of GL(1). Note that such an isomorphism

yields compatible decompositions of A0 and A10 as products of r copies of

the same groups. This means that we need to give the ‘natural splitting’

in the case when C = GL(1). Here C(A) = I and A0 is the product of


| ∞ | copies of R+ imbedded in∏

v∈∞

k∗v through the inclusions R ⊂ kv ≃ R

or C(v ∈ ∞). The group A10 is then naturally isomorphic to the subgroup

x ∈∏

v∈∞

k∗v | xv ∈ R+,∏

v∈∞

xv = 1 and a natural supplement to A10 in A is

the group x ∈∏

v∈∞

k∗v | xv = x for all v ∈ ∞ with x ∈ R+. Thus we have

obtained a splitting using an isomorphism of C with (GL(1))r . It is easy to

see that the splitting is independent of the choice of this isomorphism. Now

let G∗ = g ∈ G(A) | | χ(g) |= 1 for every character χ of G defined over

k. Then G∗ ⊃ A10 and if ′A0 is the natural supplement to A1

0 in A0, then′A0 ∩ G∗ = 1 and the natural map

′A0 × G∗ → G(A)

is an isomorphism (of locally compact groups). The product formula (1.2)

shows that G(k) ⊂ G∗. Using the fact that I1/k∗ is compact, it is now easy

to deduce the following from Theorem 1.10.

1.13. Theorem G∗/G(k) has finite Haar measure.

1.14. It is easy to see that any compact subgroup of G(A) is contained in

G∗. Consequently there is a c > 0 and a compact subset Ω ⊂ oP(A)∩G∗ =

P∗ such that for all Ω ⊃ Ω′,Ω ⊂ oP(A) ∩ G∗, and c′ ≥ c we have

G∗ = K ·A10 ·A

′c′ · Ω

′ · G(k)

where A′c′ = a ∈ A1 | α(a) ≤ c′ for all α ∈ . Let C(A)1 = g ∈ C(A) | |

χ(g) |= 1 for all characters χ on C. Then C(k) ⊂ C(A)1 (product formula)

and C(A)1/C(k) is compact. Hence we can find a compact subset C of

C(A)1 such that C ·C(k) = C(A)1. Clearly C ·Ω′ = Ω0 is a compact subset

of P∗ and one has

G∗ = K ·A′c · Ω0 · G(k).

2 Automorphic Forms

2.1. We continue with the notations introduced in §1. We fix a unitary

character χ : C(A) → S1 on C(A) and assume that χ is trivial on C(k). We

introduce now some function spaces. Recall that we defined Fχ(G(A),G(k))

14 M.S. Raghunathan

as the vector space of all functions f : G(A) → C satisfying the following

condition

f(xgγ) = χ(x)−1f(x)

for all g ∈ G(A), x ∈ C(A) and γ ∈ G(k). In the sequel we often write Fχ

for Fχ(G(A),G(k)) (where there is no ambiguity about the group G we are

talking about). We set

Cχ = Cχ(G(A),G(k)) = f ∈ Fχ | f continuous

Cχc = Cχ

c (G(A),G(k)) = f ∈ Cχ | f has compact support modulo C(A)G(k).

A function f : G(A) → C in Fχ is square integrable iff f is Borel measurable

and ‖ f ‖22=

∫G(A)/C(A)G(k) | f(g) |2 dµ(g) < ∞− | f(xgγ) |=| f(g) | for g ∈

G(A), x ∈ C(A) and γ ∈ G(k) so that | f(g) | can be treated as a function

on G(A)/C(A)G(k); µ is the Haar measure on this homogeneous space of

the locally compact group G(A)/C(A). We denote by Lχ2 = Lχ

2 (G(A),G(k))

the set of all square integrable functions in Fχ modulo the equivalence of

equality almost everywhere with respect to µ. It is a Hilbert space under

the inner product

(f, f ′) 7→

∫

G(A)/C(A)G(k)f(g)f ′(g)dµ(g).

We have (as usual) an inclusion Cχc → Lχ

2 . A function f : Ω → C,Ω an open

set in G(A) is said to be C∞ if the following conditions hold:

(i) for any g0 ∈ Ω, there is a neighbourhood Ω′ of g0 in Ω and a compact

open subgroup L ⊂ G(Af ) such that LΩ′ ⊂ Ω and f(xg) = f(g) for

all g ∈ Ω′ and x ∈ L.

(ii) For any g0 ∈ Ω, there is neighbourhood U of the identity in G∞ such

that Ug0 ⊂ Ω and the map x 7→ f(xg0) is C∞ on U .

Following the decomposition G∞ × G(Af ) ∼= G(A) as a direct product we

may regard any complex-valued function f on G(A) as a function of two

variables (one in G∞ and the other in G(Af )). If f is C∞, then it is C∞ in

the first variable and locally constant in the second variable; also the partial

derivatives of f with respect to the first variable are themselves C∞ on G(A).

We denote by Cχ∞ = Cχ∞(G(A),G(k)) (resp. Cχ∞c = Cχ∞

c (G(A),G(k)) the

vector space of C∞ functions (resp. C∞ functions with support compact

modulo C(A) ·G(k)). We have evidently inclusions

Cχ∞c → Cχ

c → Lχ2 .


2.2. In the sequel E will denote any one of the function spaces Fχ, Cχ,

Cχc , Cχ∞, Cχ∞

c , Lχ2 introduced above and for an open compact subgroup L

of G(Af ), EL will denote the subspace of functions in E which are invariant

under left translations by L. We observe that a function f in E is determined

by its restriction to G∗ (since ′A0G∗ = G(A) and ′A0 ⊂ C(A)). On the

other hand

G∗ = K · A10 ·A

′c · Ω

′ ·C ·G(k)

with C ⊂ C(A) (cf. 1.14). Further K = K∞ · Kf with K∞ ⊂ G∞ and

Kf ⊂ G(Af ) and one may assume that Ω′ = Ω∞ · Ωf with Ω∞ (resp. Ωf )

a compact subset of P∞ (resp. P(Af )). Now Kf ·Ωf is a compact subset

of G(Af ). It follows that if Lf ⊂ G(Af ) is a compact open subgroup of

G(Af ), there exists a finite set H′ ⊂ G(Af ) such that Kf · Ωf ⊂ Lf · H′.

It follows from Corollary 1.7 that we have Lf · H′ · df (G(k)) = G(Af ). We

introduce an equivalence relation on H′ as follows: for θ, θ′ ∈ H′, θ ∼ θ′ iff

there exists ρ ∈ G(k) with θρfθ′−1 ∈ Lf (it is easily checked that this is

an equivalence relation). Let H ⊂ H′ be a subset containing exactly one

element in each equivalence class. We then assert that any f ∈ ELf is

determined by its restriction to G∞LfH (and in view of Lf -invariance) by

its restriction to G∞ · H. Evidently to prove this it suffices to show that

G∞ · Lf · H · C(A) ·G(k) = G(A). We have seen that

G(A) = ′A0 ·G∗ = ′A0 ·K∞ · A1

0 · A′c · Ω∞ ·Kf · Ωf ·C(A)G(k)

⊂ G∞ · Lf · H′ · C(A) ·G(k).

Thus it suffices to show that if θ ∼ θ′,

G∞ · Lf · θ ·G(k) ⊃ G∞ · Lf · θ′.

Since θ ∼ θ′, there exists ρ ∈ G(k) with θρfθ′−1 = ℓ ∈ Lf so that

θ′ = ℓ−1θρf . It follows that G∞Lfθ′ρ−1 = G∞Lfθρfρ

−1 = G∞ρ−1∞ Lfθ =

G∞Lfθ. It follows that Lf · H · df (G(k)) = G(Af ). For θ ∈ H, let rθ(f) :

G∞ → C be the function rθ(f)(g) = f(gθ) = f(θg). Then rθ(f)(gγ∞) =

rθ(f)(g) for all γ ∈ G(k) such that γf ∈ θ−1Lfθ. In fact one has

rθ(f)(gγ∞) = f(gγ∞θ) = f(gγ∞θγ−1) = f(gθγ−1

f θ−1θ) = f(gθ).

Let Γθ = d∞(G(k) ∩ d−1f (θ−1Lfθ)) i.e.,

Γθ = d∞γ | γf ∈ θ−1Lfθ.

16 M.S. Raghunathan

Let Cχ(G∞/Γθ) (resp. Cχc (G∞/Γθ), resp. Cχ∞(G∞/Γθ), resp. Cχ∞

c (G∞/Γθ),

resp. Lχ2 (G∞/Γθ)) be the space of all continuous functions (resp. contin-

uous functions with compact support modulo C∞ (resp. C∞ functions,

resp. C∞ functions with compact support modulo C∞, resp. square inte-

grable functions) on G∞/Γθ satisfying the condition f(gx) = χ(x)−1f(g)

for x ∈ C∞. Then if E = Cχ, (resp. Cχc , resp. Cχ∞, resp. Cχ∞

c , resp. Lχ2 ),

and f ∈ ELf , rθ(f) ∈ Cχ(G∞/Γθ) (resp. Cχc (G∞/Γθ), resp. Cχ∞(G∞/Γθ),

resp. Cχ∞c (G∞/Γθ), resp. Lχ

2 (G/Γθ). We thus obtain maps

∐

θ∈H

rθ : CχLf →∐

θ∈H Cχ(G∞/Γθ)

∐

θ∈H

rθ : CχLfc →

∐θ∈H Cχ

c (G∞/Γθ)

∐

θ∈H

rθ : Cχ∞Lf →∐

θ∈H Cχ∞(G∞/Γθ)

∐

θ∈H

rθ : Cχ∞Lfc →

∐θ∈H Cχ∞

c (G∞/Γθ)

∐

θ∈H

rθ : LχLf

2 →∐

θ∈H Lχ2 (G∞/Γθ).

2.3. Proposition The maps above are isomorphisms of topological vector

spaces.

We record here for future use the following fact proved above.

2.4. Lemma Lf · H · df (G(k)) = G(Af ).

2.5. Proof of 2.3 The topologies on these spaces are the standard ones.

On CχL and Cχ(G∞/Γθ) (resp. Cχ∞L and Cχ∞(G∞/Γθ)) it is the topology

of uniform convergence (resp. together with all derivatives) on compact

sets. On CχLc and Cχ∞

c (resp. Cχc (G∞/Γθ) and Cχ∞

c (G∞/Γθ)) it is the

inductive limit of the topology of uniform convergence (resp. together with

all derivatives) on closed sets of G(A) which are compact modulo C(A)·G(k)

and closed subsets of G∞ which are compact modulo C∞ · Γθ respectively.

The L2 spaces are of course given the Hilbert space structure. That the

maps are continuous injections is easy to see. We need to show that the

maps are surjective - the open mapping theorem would then guarantee that

the maps are isomorphisms. Let fθθ∈H be a collection of functions on

G∞/Γθ, θ ∈ H belonging to one of the above spaces. Define a function f

on G∞LH by setting f(gℓθ) = fθ(g). We extend the function f to all of


G(A) by setting f(g) = f(gγ) where γ ∈ G(k) is an element such that

gγ ∈ G∞LH. We need only check that f is well defined i.e., if gγ′ ∈ G∞LH

also for some γ′ ∈ G(k), f(gγ) = f(gγ′). Let gγ = hℓθ and gγ′ = h′ℓ′θ′.

Then setting γ−1γ′ = ζ, we have hℓθζ = h′ℓ′θ′. It follows that ℓθζf = ℓ′θ′

leading to θζfθ′−1 = ℓ−1ℓ′ ∈ L i.e., θ ∼ θ′ and hence θ = θ′; and when

θ = θ′, ζf ∈ θ−1Lθ so that ζ ∈ Γθ, and fθ is Γθ-invariant. This proves that f

is well defined. That f has the required properties of continuity, smoothness

etc. if the fθθ∈H have them is immediate from the definitions.

2.6. Let U denote the universal enveloping algebra of the real Lie algebra

L(G) of G. Then U operates on the space of C∞ functions on G(A) leaving

stable the subspaces Cχ∞(G(A),G(k)) and Cχ∞c (G(A),G(k)) as well as the

subspace of L-invariants in these spaces for any compact open subgroup L of

G(Af ). A function f ∈ Cχ is K-finite if the C-linear span of Lkf | k ∈ K is

finite dimensional. The group K acts continuously on this finite dimensional

vector space V and hence the image of the profinite group Kf in GL(V ) is

finite. Thus if ϕ isK-finite, ϕ is invariant under an open compact subgroup of

G(Af ); and the C-span of Lkϕ | k ∈ K∞ is finite dimensional. Conversely

if ϕ satisfies these two conditions ϕ is K-finite. Let Z be the centre of U .

A C∞ function ϕ in Cχ∞ is Z-finite iff the C-linear span of zf | z ∈ Z is

finite dimensional. Functions relevant to harmonic analysis on G(A)/G(k)

are those that do not grow too rapidly at infinity. To describe the kind

of growth at infinity that we need to impose we need some preliminary

definitions. We assume, as we may, that the imbedding of G in GL(N) maps

G into SL(N) (when G = GL(n) we can take N = 2n and the imbedding

to be

g 7→

(g 00 tg−1

)g ∈ GL(n)).

This is done so that one takes care that the entries of g as well as those

of g−1 are handled simultaneously. We define for g ∈ G(A) the height of

g denoted ‖ g ‖ in the sequel as∏

v∈V

(max1≤i,j≤N | gvij |v) where gvij , 1 ≤

i, j ≤ N are the entries of the v-adic component gv ∈ G(kv) of g. Observe

that since gv ∈ GL(N,Ov) for all v ∈ V\S for a finite subset S containing

∞,max1≤i,j≤N | gvij |= 1 for all but a finite number of v and thus the

product above over all v ∈ V reduces to a finite product. It is easy to see

that g 7→‖ g ‖ is a continuous function of G(A) in R+. In the sequel we

will say that a right G(k)−invariant function f : G(A) 7→ C has moderate

18 M.S. Raghunathan

growth if there exist constants c, r > 0 such that for all g ∈ G(A) one has

| f(g) |≤ c ‖ g ‖r

with this notion of moderate growth we have

2.7. Definition An automorphic form (with central unitary character χ)

on G(A) is a C∞ function f : G(A) → C satisfying the following conditions

(i) Uf is of moderate growth for all U ∈ U .

(ii) f(gx) = χ(x)−1f(g) for all g ∈ C(A) and all x ∈ C(A).

(iii) f(gγ) = f(g) for all g ∈ G(A) and γ ∈ G(k).

(iv) f is K-finite

(v) f is Z finite.

The automorphic forms with central character form a vector space which we

denote Aχ

2.8. Remarks (i) If a nonzero automorphic form is to exist, with χ as

central character, evidently one must have χ(ρ) = 1 for all ρ ∈ C(k) in view

of the condition (iii). Thus we will consider only characters χ on C(A) that

are trivial on C(k). Also note that since χ is unitary, the growth condition

does not lead to any contradiction.

(ii) We have assumed that the group G is reductive. This means that the

Lie algebra L(G∞) of G∞ is a direct product of an abelian Lie algebra h and

a semisimple Lie algebra s. The Lie algebra s has a Cartan - decomposition

s = k⊕ p with the Killing form <,> of s restricted to k (resp. p) negative

(resp. positive) definite. Let Xi, 1 ≤ i ≤ r, Yj , 1 ≤ j ≤ s and Zk, 1 ≤ k ≤ t

be bases of h,k and p respectively so chosen that < Yj , Yj′ >= −δjj′ and

< Zk, Zk′ >= δkk′ . Then the element C =∑

1≤i≤r

X2i +

∑

1≤p≤t

Z2k −

∑

1≤j≤s

Y 2j is

a central element of the enveloping algebra U . Let C ′ =∑

1≤j≤s

Y 2j . Then

C+2C ′ is an element of U and the corresponding differential operator on

G∞ is an elliptic operator. Let B be the subalgebra of End Cχ(G(A),G(k))


generated by the centre Z of U and Lk | k ∈ K (Lk is the left translation

by k). Then the C-span of T (f) | T ∈ B is finite dimensional for the

automorphic form f . Now the span of the Lkf, k ∈ K contains Uf for all

U in the subalgebra of U generated by k. Thus C and C ′ belong to B and

hence so does . It follows that nf | n ∈ N spans a finite dimensional

vector space over C. We conclude that there is a monic polynomial P in one

variable such that P ()f = 0. Now P () is an elliptic operator on G∞

with analytic coefficients. By the regularity theorem for elliptic operators

with analytic coefficients we conclude that f is analytic (in the variable in

G∞).

(iii) The third comment is that if U ∈ U and f is an automorphic form

then so is Uf . It is clear that it suffices to check this when U ∈ L(G). Now

the map L(G∞)⊗ Cχ∞ given by (U, f) 7→ Uf is compatible with the action

of G(A) on the two sides. G(A) acts on L(G) by the adjoint representations

of the factor G∞ and on Cχ∞ by left translation; we take the tensor product

representation on the left-hand side. Suppose now V is a finite dimensional

subspace of Cχ∞ which is K as well as Z stable. Then the image of L(G)⊗V

in Cχ∞ under the above map is finite dimensional Z-stable as well as K-

stable. As this space contains Uf , for U ∈ L(G∞), we see that Uf is K-

finite and Z-finite. That U ′(Uf) = (U ′U)(f) has moderate growth is clear

from the definitions.

(iv) The vector space of all automorphic forms Aχ with central character χ

is stable under left translations by elements of G(Af ). This is seen as follows:

for g ∈ G(Af ), gKg−1 ∩K has finite index in K and Lgf is (g−1Kg ∩K)-

finite and hence K-finite as well. Since the action of G(Af ) on the left

and the action of U on the space of C∞ functions commute, we see that

(Lgf) is Z-finite for g ∈ G(Af ). The growth condition is immediate from

the fact that for g0 ∈ G(Af ) one has a constant C > 0 such that for all

g ∈ G(A), ‖ g0g ‖≤ C ‖ g ‖.

2.9. Lemma (Harish-Chandra: Acta Math. 116 p.118). If f is a K∞-

finite Z-finite C∞ function on G∞, there is an α ∈ C∞c (G∞) (= C∞ func-

tions with compact support in G∞) such that α ∗ f = f .

This lemma shows that the growth condition (i) in (2.6) can be replaced

by the weaker condition: (i′)·f is of moderate growth. In fact fix α ∈ C∞c (G)

such that α ∗ f = f . Then one has for U ∈ U , Uf = U(α ∗ f) = Uα ∗ f ; and

20 M.S. Raghunathan

it is easy to see that for any β ∈ C∞c , β ∗ f has moderate growth. In the

sequel we will make use of Lemma 2.8 in other contexts too.

2.10. Let χ be a unitary character on C(A)/C(k). Let σ : K → GL(W )

be a finite dimensional representation of the compact subgroup K of G(A)

and λ := Z → EndCW′ a finite dimensional representation of Z. Let

Aχ(G(A),G(k), σ, λ) denote the space of complex values C∞ functions f

on G(A) satisfying the following conditions (cf. 2.8).

(i) f has moderate growth.

(ii) f(gγ) = f(g) for g ∈ G(A), γ ∈ G(k).

(iii) f(gh) = χ(h−1)f(g) for g ∈ G(A), h ∈ C(A).

(iv) The K-span of f (in Fχ) is a quotient of W as a K-module.

(v) The Z-span of f (in Fχ) is a quotient of W ′ as a Z-module.

With this notation our central result is

2.11. Theorem dim Aχ(G(A),G(k), σ, λ) <∞.

2.12. The representation σ of K when restricted to the totally discon-

nected subgroup Kf has a kernel Lf of finite index. Then it is easily seen

that Aχ(G(A),G(k), σ, λ) is contained in the space of C∞ functions f on

G(A)/G(k) such that f(gx) = χ(x)−1f(g) for g ∈ G(A), x ∈ C(A), f(kg) =

f(g) for k ∈ Lf , the K∞-span of f is a quotient of σ∞ as a K∞-module and

the Z-span of f is a quotient of λ. We can now appeal to Proposition 2.4 to

conclude that Theorem 2.11 is equivalent to:

2.13. Theorem Let Γ ⊂ G(k) be an arithmetic subgroup. Let Aχ(G∞,

Γ, σ∞, λ) be the space of C∞ functions f on G such that

(i) f(gγ) = f(g) for all g ∈ G∞ and γ ∈ Γ.

(ii) f(gc) = χ(c)f(g) for all g ∈ G∞ and c ∈ C∞.

(iii) The K∞-span of f is a quotient of σ∞.

(iv) The Z-span of f is a quotient of λ.


(v) there is a constant c > 0 and integer r > 0 such that | f(g) |≤ c ‖ g ‖r

for g ∈ G∞.

Then Aχ(G∞,Γ, σ∞, λ) is finite dimensional.

3 Cusp Forms

3.1. We need a number of facts from analysis which we collect together in

3.2 - 3.11 below for convenient later use. We refer to R. Narasimhan [N] for

proofs of results not given here. We begin with the following

3.2. Proposition Let X be a locally compact space and µ a Borel prob-

ability measure on X. Let Cb(X) denote the space of bounded continuous

functions on X and for ϕ ∈ Cb(X), let ‖ ϕ ‖∞= lim Supϕ(x) | x ∈ X.

Suppose T : L2(X,µ) → L2(X,µ) is a linear map such that T (ϕ) ∈ Cb(X)

for all ϕ ∈ L2(X,= µ) and there is a constant C > 0 such that ‖ T (ϕ) ‖∞≤

C ‖ ϕ ‖2. Then T is an operator of the Hilbert Schmidt type (and hence

compact).

Proof. For x ∈ X, the linear form ϕ 7→ T (ϕ)(x) on L2(= L2(X,µ)) is

bounded. It follows that there exists kx ∈ L2 such that ‖ kx ‖2≤ C and

T (ϕ)(x) =< kx, ϕ > for all ϕ ∈ L2. Let k(x, y) = kx(y) for x, y ∈ X : k(x, y)

is defined for all most all y for each x. Clearly for each fixed x, kx(y) is

measurable so that k(x, y) is measurable on X ×X. Now

∫X×X | k(x, y) |2 dµ(x) · dµ(y) =

∫X dµ(x)

∫Y | kx(y) |2 dµ(y)

≤∫X dµ(x)C2 = C (since µ(X) = 1)

Thus k(x, y) ∈ L2(X ×X,µ× µ) and

Tϕ(x) =

∫

Xkx(y)ϕ(y)dµ(y) =

∫

Xk(x, y)ϕ(y)dµ(y).

Thus T is a Hilbert-Schmidt operator.

3.3. The next result we want to state is the Sobolev inequality. We need

to introduce some preliminary notation to state the result. Let M be be a

smooth manifold. Let Ω, be a relatively compact subset of M ; the compact

closure of Ω is denoted Ω. Let Ui, | 1 ≤ i ≤ m be an open covering of

22 M.S. Raghunathan

the closure Ω of Ω by coordinate open sets; let Vi, 1 ≤ i ≤ m, be open

subsets of Ui such that the closure V i of Vi is compact and contained in Ui

and further ∪1≤i≤rVi ⊃ Ω. We fix a C∞ volume form on M and denote

the corresponding measure by µ. If x1, · · · , xn are the coordinates in Ui, we

denote by Dβ the operator ∂1

∂x1n· · · ∂1

α

∂xnn

for a multi index β = (β1, · · · , βn) of

non-negative integers. We introduce on the space of C∞ functions on Ω (i.e.

C∞ functions defined a neighbourhood of Ω) the following norms ‖ ‖r,p

where r is an integer ≥ 0 and p > 1. For a C∞ function f on Ω,

‖ f ‖r,p= Sup ‖ Dβf ‖pV i| | β |≤ r, 1 ≤ i ≤ m.

Here for h a C∞ function on Ω,

‖ h ‖pV i=

∫

V i

| h |p dµ.

The norm defined above depends on the covering and the shrinking chosen,

the coordinates chosen in the covering open sets, and the volume form on

M . However the equivalence class of the norm ‖ ‖r,p depends only on r

and p and not on the choices made above. With these definitions we can

now state Sobolev’s inequality.

3.4. Theorem There is a constant C,C(p,Ω) > 0 such that for all f,C∞

on Ω, ‖ f ‖∞≤ C ‖ f ‖[n/p]∗,p where [n/p]∗ is the minimal integer ≥ n/p (n =

dimM).

3.5. Remark In the special case when M = Rn and we take the coor-

dinates to be standard coordinates, the volume form as the standard one,

and Ω is a disc of radius ρ centred at a point x0, we have for any f,C∞

on Ω, ‖ f ‖∞≤ C ‖ f ‖[n/p]∗,p with C = C(ρ) independent of the point x0.

This is because the measure as well as the vector fields ∂/∂xi are translation

invariant.

3.6. Our next result is a very special case of a general theorem about elliptic

operators on a compact manifold. We consider a connected real nilpotent

Lie group N with a discrete subgroup Φ such that N/Φ is compact. Let

L(N) denote the Lie algebra of N and X1, · · · ,Xn be a basis of L(N) over

R. We consider the Xi to be right translation - invariant vector fields on

N , hence they define vector fields on N/Φ which we continue to denote Xi.


Then ∆ = −∑X2

i is a non-negative self adjoint elliptic operator on the

space of C∞- functions on N/Φ with respect to the inner product <,>:

(f, g) 7→∫N/Φ fgdn where dn is the Haar measure on N/Φ. One has in fact

for X ∈ L(N),

< Xf, g >=< f,−Xg >

so that

− < X2f, f >=< Xf,Xf >≥ 0

and hence < ∆f, f >≥ 0. In fact one has

< ∆f, f >=∑

< Xif,Xif >

so that ∆f = 0 if and only if Xif = 0 for all i; and since the Xi, 1 ≤ i ≤ r

give a basis for the tangent space at every point this means that f is a

constant. Now the general theory of self adjoint elliptic operators applied

here shows that L2(N/Φ), dn) decomposes as an orthogonal direct sum

C ⊕∐

1≤n<∞

H(λn)

where C is identified with the space of constant functions on N/Φ, λn, 1 ≤

n < ∞, a monotone sequence of positive real numbers tending to ∞ and

H(λn) = f a C∞ function on N/Φ | ∆f = λnf are finite dimensional

subspaces of L2. It is clear from this that if f ∈ C∞(N/Φ, C) and∫N/Φ fdn =

0 then f ∈∐

1≤n<∞ H(λn). It is also immediate from this that we have the

following fact which has a crucial role in the sequel:

3.7. Lemma If ϕ ∈ C∞(N/Φ, C) is such that∫N/Φ ϕdn = 0, then we have

for every integer r > 0,

‖ ∆rϕ ‖2≥ λr1 ‖ ϕ ‖0,2

where ‖ ‖2=‖ ‖0,2 is the L2 norm with respect to the Haar measure.

3.8. We now recall Friedrich’s inequality for elliptic operators as applied

to our special case. For a multi-index β = (β1, β2, · · · βn) of non-negative

integers we set Xβ = Xβ1

1 · · ·Xβnn - a differential operator on N/Φ. As usual

| β | is the sum∑

1≤i≤n βi. With this notation Friedrick’s inequality asserts

the following

24 M.S. Raghunathan

3.9. Theorem There are constant C,C ′ > 0 such that for any α with

| α |≤ 2r and any ϕ ∈ C∞(N/Φ),

‖ Xαϕ ‖2≤ C ‖ ∆rϕ ‖2 +C ′ ‖ ϕ ‖2 .

Note The norm Sup ‖ Xβϕ ‖2| | β |≤ r is equivalent to the norm

‖ ‖r,2 introduced earlier; this is easily seen from the fact that the Xi can

be expressed as a linear combination of the ∂/∂xj of the local coordinates

with coefficients that are C∞.

3.10. Corollary If ϕ ∈ C∞(N/Φ) is such that∫N/Φ ϕdn = 0, then for

| α |≤ 2r, one has

‖ Xβϕ ‖2≤ C ‖ ∆rϕ ‖2

for a constant C > 0 (independent of ϕ).

3.11. Corollary Let (N,Φ) be as in 3.6. Then there is a constant C > 0

such that for ϕ ∈ C∞(N/Φ) with∫N/Φ ϕ(n)dn = 0, we have | ϕ(x) |p≤

C∫N/Φ | ∆rϕ(x) |p dx for r ≥ [n/p]∗. This is immediate from Theorem 3.4

and Corollary 3.10. From now on we go back to the notations of §1 and §2.

3.12. A function ϕ ∈ Cχ (cf. 2.1) is cuspidal if the following holds: let P′

be a proper parabolic subgroup of G defined over k one U′ its unipotent

radical. Then for almost all g ∈ G(A) the function u 7→ ϕ(gu), u ∈ U′(A) is

integrable for the Haar measure on U′(A)/U′(k)) and∫U′(A)/U′(k) ϕ(gu)du =

0 for almost all g ∈ G(A).

3.13. Remarks

(i) ‘Almost all g’ above means for all g ∈ G(A) outside a set of Haar

measure zero.

(ii) It is known that every parabolic subgroup defined over k is conjugate

to a parabolic subgroup of G defined over k containing a fixed minimal

parabolic subgroup P of G defined over k. Using the G(k) invariance

of elements of Fχ, it is easily seen that for ϕ to be cuspidal, it suf-

fices that∫U′(A)/U′(k) ϕ(gu) vanish for almost all g ∈ G(A) for U′ the

unipotent radical of any of the (finitely many) k-parabolic subgroups

P′ containing P.


(iii) We observe next that for ϕ to be cuspidal, it sufficient that the integral

vanish for maximal parabolic subgroups defined over k and containing

P. In fact if P′ is a k-parabolic subgroup and P′′

is a maximal k-

parabolic subgroup containing P′, then the unipotent radical U′ of P′

contains the unipotent radical U′′

of P′′

as a normal subgroup. We

have thus a fibration U′(A)/U′(k) → U′(A)/U′′(A)U

′′(k) whose fibres

are U′′(A)/U

′′(k); our contention now follows from Fubinis theorem

which is valid for this fibration.

3.14. Notation If E denotes one of the spaces in E = Fχ,Fχ, Cχc , Cχ∞

and Lχ2 we denote by oE the subspace

ϕ ∈ E | ϕ cuspidal.

3.15. If E is one of the spaces in E , then oE is invariant under the left

action of G(A) on G(A)/G(k). This is clear from the definition of cuspidal

functions. Also if E ∈ E and E is not Fχ, for ϕ ∈ E and α ∈ F∞c (G(A)), α∗

ϕ ∈ E as is easily seen from the definition of the convolution operation

((α ∗ ϕ)(g) =∫G(A) α(h)ϕ(h−1g)dh (dh Haar measure on G(A))). It is

further easy to see that ϕ 7→ α∗ϕ is a continuous linear operator on E which

leaves the subspace oE stable (note that oE is closed in E (for E 6= Fχ)).

By the definition of C∞c (G(A)), there is a compact open subgroup K(α)

of G(Af ) such that α is invariant under left translations by K(α). From

the definition of α ∗ ϕ, it is immediate that it is K(α)-invariant on the

left. Let A∗ : E → E(E ∈ E , E 6= Fχ) be the averaging over K(α) :

(A∗ϕ)(g) =∫K(α) ϕ(kg)dk (dk Haar measure on K(α), g ∈ G(A)). Then A∗

is a continuous projection of E on EK(α) and of oE on oEK(α). We note that

for all ϕ ∈ E,α ∗ A∗(ϕ) = A∗(α ∗ ϕ)α ∗ ϕ. In particular α ∗ ϕ = A(α ∗ Aϕ)

on E as well as oE. We now state the central result of this chapter.

3.16. Theorem The operator ϕ 7→ α ∗ ϕ of oLχ2 into itself is a compact

operator.

3.17. We will reformulate the theorem for a space of functions on G∞. In

the light of the remarks made at the end of 3.15, it is clear that it suffices to

show that convolution with α is a compact operator on oLχK(α)2 . Consider

now the isomorphism r =∐

θ∈H rθ (defined in 2.2: we take for Lf of 2.2 the

group K(α)) of LχK(α)2 K(α) on

∐θ∈H L

χ2 (Gα/Γθ).

26 M.S. Raghunathan

3.18. Claim For θ ∈ H, let oLχ2 (G∞/Γθ) be the subspace of all those

functions ϕ in Lχ2 (G∞,Γθ) for which

∫U′

∞∩Γ ϕ(gu)du = 0 for almost all

g ∈ G∞ and for all U′, the unipotent radical of proper k parabolic subgroup

P′ of G. Then r maps oLχ2 isomorphically onto

∐θ∈H

oLχ2 (G∞/Γθ).

3.19. Proof It is known that the group df (U′(k)) is dense in U′(Af )

(this is known as the strong approximation property). Let Mθ = U′(Af ) ∩

θ−1K(α)θ; then Mθ is an open compact subgroup of U′(Af ). Consequently

U′(Af ) = Mθ ·df (U′(k)). It follows that U′(A) = U′∞ ·Mθ ·U

′(k) and hence

the natural map

(U′∞ ·Mθ)/((U

′∞ ·Mθ) ∩ U′(k)) ≃

−→ U′(A)/U′(k) (∗)

is a homeomorphism. For ϕ ∈o Lχ2 , we have

∫U(A)/U′(k) ϕ(θgu)du = 0 for

θ ∈ H and g ∈ G∞. In view of the identification (*) described above, we see

that ∫

U′∞·Mθ/U′

∞·Mθ∩U′(k)ϕ(θgu)du = 0.

Now if we set u = u∞ · uf with u∞ ∈ U′∞ and uf ∈ U′(Af ), we have

ϕ(θgu) = ϕ(θgu∞uf ) = ϕ(θufgu∞) = ϕ(θufθ−1 · θgu∞) = ϕ(θgu∞) (since

by definition of Mθ, uf ∈ Mθ implies that θufθ−1 ∈ K(α) and ϕ is left

K(α)-invariant). Thus integrating over Mθ first we find that

0 =

∫

(U′∞·Mθ)/((U′

∞ ·Mθ)∩U′(k))ϕ(θgu)du =

∫

U′∞/U′

∞∩Γθ

ϕ(θgu)du

proving that r maps oLχ2 into

∐oθ∈H L

χ2 (G∞ | Γθ). We need to show that

the map is onto this subspace. To see this let ϕ ∈ LχK(α)2 be such that rθ(ϕ)

is in oLχ2 (G∞/Γθ) for all θ-we know that such a ϕ exists and is unique. We

need to show that ϕ ∈o Lχ2 i.e. we have to show that

∫

U′(A)/U′(k)ϕ(gu)du = 0 (∗∗)

for almost all g ∈ G(A) for the unipotent radical U′ of any proper k-parabolic

subgroup P′ of G. Now g = g∞ · gf where g∞ ∈ G∞ and gf ∈ G(Af ). By

Lemma 2.4, gf = kf ·θ·df (γ) with kf ∈ K(α), θ ∈ H and γ ∈ G(k). Since ϕ is

left K(α)-invariant, ϕ(gu) = ϕ(g∞gfu) = ϕ(g∞kfθdf (γ)u) = ϕ(g∞θdf (γ)u)

(since ϕ is left K(α)-invariant) = ϕ(g∞θγ−1∞ γ∞γfu) = ϕ(g′∞θγu).


Set U′′

= γuγ−1. Then one has

∫U′(A)/U′(k) ϕ(g′∞θγu)du =

∫U′(A)/U′(k) ϕ(g′∞θ · γuγ

−1)du

=∫U

′′ (A)/U′′ (k) ϕ(θg′∞u′)du′ =

∫(U′′ ·M ′

θ)/(U′′

∞ ·M ′θ∩U

′′ (k)) ϕ(θg′∞u′)du′

where M ′θ = (U

′′(Af ) ∩ θ−1K(α)θ); and the integral in the last integral

is an M ′θ-invariant function on (U

′′

∞ ·M ′∞)(U

′′

∞ ·M ′∞ ∩ U

′′(k)). It follows

that integral equals∫U

′′∞/U′′

∞∩Γθϕ(θg′∞u)du and is thus zero since rθϕ ∈

oLχ2 (G∞/Γθ). Thus Claim 3.15 is established.

3.20. Lemma For α ∈ Cc(G(A)), let α∞ ∈ Cc(G∞) be the function

defined by α∞(g) =∫G(Af ) α(gh)dh. Then rθ(α ∗ ϕ) = α∞ ∗ ϕ for all ϕ ∈

LχK(α)2 . This is immediate from the definition of rθ and the K(α)-invariance

of α. The lemma shows that Theorem 3.13 is equivalent to the following:

3.21. Theorem Let Lf be a compact open subgroup of G(A) and set

Γ = d∞(ΓLf). Let χ be a unitary character on C(A)(C = centre of G)

which is trivial on C(k). Let α ∈ Cc(G∞). Then ϕ 7→ α ∗ ϕ is a compact

operator on oLχ2 (G∞/Γ).

3.22. We begin with the observation that there is no loss of generality in

assuming that the centre of G is finite. To see this observe first that if G′ is

the commutator subgroup of the identity connected component of G, then

C∞ · G′∞ = H has finite index in G∞. Let ξ1 . . . ξr be a complete set of

inequivalent representatives for G∞/H in G∞. For 1 ≤ i ≤ r and ϕ ∈ Lχ2 ,

let ϕi : C∞ · G∞ → C be the function defined by ϕi(g) = ϕ(ξi · g). Then

ϕi(gγ) = ϕi(g) for g ∈ C∞·G∞′ = H and γ ∈ H∩Γ. Also ϕi(gc) = χ(c)ϕi(g)

for c ∈ C∞. Now every unipotent element of G∞ is contained in G∞′. From

this we see immediately that if ϕ is a cuspidal function so is ϕi restricted

to G∞′. Further ϕi is completely determined by its restriction ϕ′

i to G∞′.

Evidently the map

oLχ2 (G∞/Γ) −→

∏

r copies

oLχ′

2 (G∞′/Γ′)

28 M.S. Raghunathan

where χ′ = χ |(G′∩C)∞ and Γ′ = G∞′ ∩ Γ is an isomorphism onto a closed

subset. Moreover for 1 ≤ i ≤ r

(α ∗ ϕ)i(g) =∫G∞

α(h)ϕ(h−1ξig)dh =∫G∞

α(ξ−1i h)ϕ(h−1g)dh∫

H(∑

1≤j≤r

α(ξ−1i ξjh)ϕ(h−1ξ−1

j g)dh =∫H

∑

1≤j≤r

αij(h)ϕj(h−1g )dh

=∑

1≤j≤r

αij ∗ ϕj

where αij ∈ C∞c (H) is the function αij(h) = α(ξ−1

i ξhh). The natural map

π : C∞ × G∞ → H has for kernel the finite group C∞ ∩ G∞ whose order

we denote by q. We regard functions on H as functions on C∞ × G∞ by

composing with π. With this convention we have for α ∈ C∞c (H) and ϕ ∈

Lχ2 (H/H ∩ Γ), (α ∗ ϕ)(g) =

∫H α(h)ϕ(h−1g)dh = 1

q

∫C∞×G∞

α(h)ϕ(h−1g)dh

and setting h = c.x with c ∈ C∞ and x ∈ G∞ we have for g ∈ G∞

(α ∗ ϕ)′(g) = 1q

∫G′

∞dx

∫C∞

α(xc)ϕ(c−1x−1g)dc

= 1q

∫G′

∞dx

∫C∞

α(xc)χ(c−1g)dc = α′ ∗ ϕ′(g)

where α′ in C∞c (G∞) is defined by α′(x) = 1

q

∫C∞

α(xc)χ(c)dc. This discus-

sion shows that we need to prove the theorem only in the case when G∞ =

G∞′ has a finite centre. Now the group Γ admits a torsion free subgroup

Γ′ of finite index and Γ′ evidently intersects the centre trivially. We have

a natural inclusion of oLχ2 (G∞

′(Γ) in oL2(G∞′/Γ′) (where oL2(G

′∞/Γ

′) =

ϕ ∈ L2(G∞′/Γ′) |

∫U/U∩Γ′ ϕ(gu)du = 0 for all unipotent radicals of proper

k-parabolic subgroups of G∞′) compatible with convolution by elements of

C∞c (G∞

′) on both the spaces. We have thus to prove the following

3.23. Theorem Let G be a connected semisimple algebraic k-group and

Γ an arithmetic subgroup of G∞. Let oL2(G∞/Γ) be the space of all Γ-

unvariant functions ϕ on G∞ which are square summable on G∞/Γ and

in addition are cuspidal i.e. satisfy the following condition: If U is the

unipotent radical of a proper k-parabolic subgroup P of G,∫U∞∩Γ

ϕ(gu)du =

0 for almost all g ∈ G∞. Then the operator ϕ 7→ α ∗ ϕ on oL2(G/Γ) is a

compact operator for all α ∈ C∞c (G∞).

Since G∞ is semisimple, G∞/Γ has finite Haar measure. We now see

from Proposition 3.2 that Theorem 3.20 is a consequence of the following:


3.24. Proposition Let G∞,Γ, α be as above. Then there is a constant

C = C(α) > 0 such that for ϕ ∈ L2(G∞/Γ), | α ∗ ϕ(x) |≤ C ‖ ϕ ‖2 for all

x ∈ G∞ (note that α ∗ ϕ is C∞ for ϕ ∈ L2(G∞/Γ)).

3.25. For c > 0 and a compact subset Ω ⊂o P∞ (notation introduced in

§1.5 - 1.9) we define S(c,Ω) to be the subset K∞ ·Ac ·Ω of G∞. According to

Theorem 1.9, there is a finite set Ξ ⊂ G(k), a constant c0 > and a compact

set Ω0 ⊂ oP∞ with the following properties:

(i) (ξΩ0ξ−1)(ξoP∞ξ

−1 ∩ Γ) = ξoP∞ξ−1.

(ii) If c > c0 and Ω ⊃ Ω0 is a compact subset of oP , then S(c,Ω)ΞΓ = G∞.

(iii) If c,Ω are as above the set γ ∈ Γ | S(c,Ω)Ξγ ∩ S(c,Ω)Ξ 6= φ is finite.

Condition (iii) implies in particular the following. There is a constant M =

M(c,Ω) > 0 such that for ξ ∈ Ξ and all ϕ ∈ L2(G∞/Γ), we have

∫

S(c,Ω)ξ| ϕ(g) |2 dg ≤M ‖ ϕ ‖2

2 (∗)

3.26. We have seen that oP is a semi-direct product B · U and we have

correspondingly a semi-direct product decomposition oP∞ = B∞ ·U∞. The

group B∞ is reductive and K∞ ∩ B∞ is a maximal compact subgroup of

B∞. We then have an Iwasawa decomposition of B∞ as (K∞ ∩ B∞) · F

where F is a connected solvable closed Lie subgroup of B∞. Moreover as

B∞ commutes with A, we see that for ξ ∈ Ξ the map

K∞ × F ×A×U∞ → G∞ (∗)

given by (k, f, a, u) 7→ k.f.a.u.ξ is an analytic diffeomorphism. Suppose now

that P′ is a proper maximal k-parabolic subgroup of G containing P. Then

P′ is the semidirect product Z(S′) · U′ where Z(S′) is the centraliser of a

suitable 1-dimensional sub-torus of S and U′ is the unipotent radical of P′.

We have correspondingly a semidirect product decomposition Z(S′)∞ ·U′∞ of

P′∞. Now U′

∞ is a normal subgroup of U∞ and (hence) U∞ is the semidirect

product (Z(S′)∞ ∩ U∞) · U′∞. Let Z(S′)∞ ∩ U∞ = U

′′

∞. Then we have a

further refinement of the product decomposition (*) above:

Φξ : K∞ × F × a× U′′

∞ ×U′∞ → G

30 M.S. Raghunathan

given by (k, f, a, u′′, u′) 7→ k, f, a, u

′′, u′, ξ. It is convenient to denote by

H1,H2,H3,H4 and H5 the groups K∞, F,A,U′′

∞ and U′′

∞ respectively. We

assume as we may say that the compact set Ω0 (resp. Ω) in 3.22 is chosen to

be of the form (K∞∩oP∞).Ω02.Ω04.Ω05 (resp. (K∞∩oP∞).Ω02.Ω04.Ω05 with

Ω0i (resp. Ωi) a compact subset of Hi, 1 = 2, 4, 5 with Ω0i contained in the

interior of Ωi. We also assume Ω05 so chosen that ξ−1Ω05.ξ.(ξ−1H5ξ ∩ Γ) =

ξ−1H5ξ. The Lie algebra L(G∞) of G∞ is identified with the Lie algebra

of right translation invariant vector fields on G∞. The Lie algebras L(Hi)

of the Hi, 1 ≤ i ≤ 5 are identified with subalgebras of L(G∞) and hence

their elements will also be regarded as right translation invariant vector

fields on G∞. On the other hand an element X of L(Hi) determines a

right translation invariant vector field on the group Hi. We denote this

vector field by X ′ (to distinguish it from the vector field X on G∞). For

X ∈ L(Hi), 1 ≤ i ≤ 5 we define a vector field X on G∞ as follows: X is the

image under the analytic diffeomorphism Φξ of the vector field on the product∏1≤i≤5Hi whose component in Hi is X while all the other components are

zero. If g ∈ G∞ is such that g = h1.h2.h3.h4.h5.ξ with hi ∈ H then we have

- as is easily checked - for Xi ∈ L(Hj),

X(g) = (Ad(∏

i<j

hi)(X))(g) (∗)

Here when j = 1,∏

i<j hi is the identity element while for j > 1, the product

is taken in the order of increasing i : h1h2 · · ·hj−1.

3.27. We now fix a basis B = Xi1≤i≤p∪Yi1≤i≤q ∪Zi1≤i≤r of L(G∞)

with the following properties:

(i) Xi and Yj are eigenvectors for Ad(S), the corresponding character

on S being denoted ηi and ζj respectively: Adt(Xi) = ηi(t)Xi and

Adt(Yi) = ζi(t)Yi.

(ii) Zi | 1 ≤ i ≤ r ∩ L(Hi) is a basis for L(Hi) for 1 ≤ i ≤ 3.

(iii) Xi | 1 ≤ i ≤ p (resp. Yi | 1 ≤ i ≤ q) is a basis for L(H5) (resp.

L(H4)).

From our choice of ordering on the character group of S, ηi, 1 ≤ i ≤ p and

ζi, 1 ≤ i ≤ q, are positive roots so that ηi =∏

α∈∆ αmiα , ζi =

∏α∈∆ α

niα

i with

miα, niα ≥ 0. For a multi-index β = (β1, · · · , βp) (resp. γ = (γ1, · · · , γ1),

resp. δ = (δ1, · · · δr), (β1, γj , δk non-negative integers) we set


Xβ = Xβ1

1 Xβ2

2 · · · Xβpp , X1β = X1β1X1β2

2 · · · X1βpp

Y γ = Y γ1

1 Y γ2

2 · · · Yγqq , Y 1γ = Y 1γ1

1 Y 1γ2

1 · · · Y1γqq ,

Zδ = Zδ1Zδ22 · · · Zδr

r , Z1δ = Z1δ1

1 · Z1δ22 · · · Z1δr

r .

With this notation we have

3.28. Lemma Let g = h1, h2, h3, h4, h5, ξ ∈ G∞ with hi ∈ Hi. Then for

multi-indices β, γ, δ as above we have, setting m =| β + | γ | + | δ |,

Z′δY

′γX′β = ηβ(h3)ζ

γ(h3)∑

|β′|+|γ′|+|δ′|≤m

Cβ′,γ′,δ′(h1, h2, h3, h4, h−13 )

with Cβ′γ′δ′ , C∞-functions on H1 ×H2 ×H4 (note that H3 normalises H4).

Here ηβ =∏

1≤i≤p ηβi

i and ζγ =∏

1≤i≤q = ζγi

i .

Proof. One argues by induction on | β | + | γ | + | δ |. When | β |

+ | γ | + | δ |= 1, this follows from (*) of 3.23 and the fact that the Xi

and Yj are eigenvectors with eigencharacter ηi and ζj respectively. Suppose

that | β | + | γ | + | δ |> 1; then we can find β′, γ′, δ′ and with | β′ | +

| γ′ | + | δ′ |=| β | + | γ | + | δ | −1 and Z′δY

′βX′β = T ′Z

′δY′γX

′β′with

T ∈ B. Using the induction hypothesis we have an expansion for Z′δY

′γ′X

′β′

in terms of the X,Y,Z of the desired kind. The result for Z′δY

′δX′β now

follows from the following observations: T ′ has the desired kind of expansion;

secondly for a character χ on H3 treated as a function on G∞, (T′, χ)(h3) = 0

unless T ∈ LH3 and if T ∈ LH3, (T′χ)(h3) = χ(T )

·χ (h3) where

·χ is the

tangent map L(H3) → R induced by χ. This proves the lemma.

3.29. We fix an element g0 = h01, h

02, h

03, h

05, ξ in G∞ with hi ∈ Hi, 1 ≤ i ≤

5. We take P′ to be the parabolic subgroup defined as follows: let α0 ∈ ∆

be such that α0(h0) ≤ α(h′3) for all α ∈ ∆; then P′ is the proper maximal

k-parabolic subgroup determined by α0. The Lie algebra L(H5) is the sum

of the eigenspace in L(G∞) for AdH3 corresponding to the eigencharacters

(i.e. k-roots) ψ of the form∏

α∈∆ αmα with mα integers ≥ 0 and mα0

> 0- in

other words in the notation of Lemma 3.25, ηi =∏

α∈∆ αmiα with miα0

> 0.

Let (c′0,Ω0) and (c,Ω) be as in 3.22. Then there is neighbourhood V of

the identity in A such that for g0ξ−1 = h01, h

02, h

03, h

04, h

05 ∈ S(c0,Ω0) and

x ∈ V, h01, h

02, h

03x, h

04, h

05 ∈ S(c,Ω). Fix such a neighbourhood V of 1 in

32 M.S. Raghunathan

A once and for all (for a fixed (c,Ω)). Let ∆ denote the elliptic operator∑1≤i≤pX

′2i on H5 and let ∆′ =

∑1≤i≤p X

′2i . Then we have the L2-Sobolev

inequality (Theorem 3.4) if 2p > r/2,

| ϕ(g0) |2≤ C

∫

Ω5

| ∆′pϕ(h0

1.h02.h3.h

04.h

05.ξ) |

2 dh5 . . . (I)

where ϕ ∈ C∞c (G∞). Treating the right-hand side as a function on H1×H2×

H3 ×H4 and applying the L1 Sobolev inequality combined with Schwarz’s

inequality we have

| ϕ(g0) |2≤ C ′

∫

K∞×Ω2×h03V ×Ω4×Ω5

∑

|δ|+|γ|≤q+4

| X′δY

′γ∆′pϕ |2

dh1, dh2, dh3, dh4, dh5 (II)

Now ∆′p =

∑

1≤i1,i2,···ip≤p

ai1 · · · ioX′i1X ′

i2· · · X ′

ip. It is now immediate from

Lemma 3.25 that we have

| ϕ(g0) |2≤ C′′

∫

E

∑

1≤i1,i2,···ip≤p|β|+|γ|+|δ|≤2p+q+r

| ηi1ηi2 · · · ηip(h3) |2| ZδY γXβϕ |2

dh1, dh2, dh3, dh4, dh5 (III)

where E = K∞×Ω2×h03V ×Ω4×Ω5. Now for h′3, h3V one has (ηi1ηi2 · · · ηip)

2

(h′3) ≤ λp · αp0(h

′3) for a suitable constant λp > 0. This is because we have

for any i, with 1 ≤ i ≤ r, ηi =∏

α∈∆ αmiα with miα ≥ 0,miα0

≥ 1 integers;

also α(h′3) ≤ c for α ∈ ∆ so that (ηi1ηi2 · · · ηip · · · ηip)(h′3) ≤ cNα0(h

′3)

p

where N = (∑

1≤r≤p

∑α∈∆mirα − p). Next observe that if p is sufficiently

large we have α(h3)p < C1δ

2(h3) where δ2 =∏

1≤i≤p ηi∏

1≤j≤q ζj. This

is seen as follows: since α0(h03) ≤ α(h0

3) for all α ∈ ∆, we see that there

is a constant b > 0 such that α0(h3) ≤ bα(h3) for all α ∈ ∆ and h3 ∈

h03V . Now δ2(h3) =

∏α∈∆ α(h3)

να with να integers ≥ 1 so that δ2(h3) ≥∏α∈∆(b−1α0(h3))

να ≥ C−11 α0(h3)

ν where ν =∑

α∈∆ να. Now if p ≥ ν, we

have α0(h3)ν ≤ α0(h3)

p · cν−p ≤ cν−pC1δ2(h3). We see from (III) above now

that we have

| ϕ(g0) |2≤ C

′′C1

∫

S(c,Ω)·ξ

∑

|β|+|γ|+|δ|≤2p+q+r

| (ZδY γXβ)ϕ |2 dg (IV)


Note that E ⊂ S(c,Ω) and dh1.dh2.dh3.dh4.dh5.δ2(h3) = dg. Proposition

3.21 now follows for the inequality IV above: We have for α ∈ A∞c (G∞) and

D in the enveloping algebra of L(G∞),D(α ∗ ϕ) = Dα ∗ ϕ so that by (IV)

(α ∗ ϕ)(g0) |2 ≤ C

′′C1

∫S(c,Ω)ξ

∑|β|+|γ|+|δ|≤2p+q+r | (ZδY γXβα) ∗ ϕ |2 dg

≤ C(α) ‖ ϕ ‖22

where C(α) = C′′C1M(α) withM(α) =

∑|β|+|γ|+|δ|≤2p+q+r ‖ Z

δY γXβα ‖01.

Note that we have for α ∈ C∞c (G∞) and ϕ ∈ L2(G∞/Γ), ‖ α ∗ϕ ‖2 ≤‖ α ‖2.

This proves the proposition and hence the Theorem.

4 Proof of Theorem 2.13

We begin by proving the following:

4.1. Theorem Let oAχ(G∞,Γ, σ∞, λ) = f ∈ Aχ(G∞,Γ, σ∞, λ) | f cuspidal .

Then oAχ(G∞,Γ, σ∞, λ) is finite dimensional.

4.2. We will first show that any ϕ ∈o Aχ(G∞,Γ, σ∞, λ) decreases to

zero rapidly at ∞ so that in particular ϕ ∈ L2(G∞/Γ) (and hence inoLχ

2 (G∞/Γ)). By Lemma 3.25 combined with (I) of 3.26, we have for p′ ≥ p

and g0 = h01.h

02.h

03.h

04.h

05.ξ,

| ϕ(go) |2≤

∫

Ω5

∑

1≤i.i1···ip′≤p′

| (ηi1ηi1ηi2 · · · ηip(h03) |

2

∑

|β|+|α|+|δ|≤2p′

| cβγδZδY γXβϕ(g0ξ−1)h5ξ) |

2 dh5

Let I be the set of p′-tuples i1, . . . , ip′ with 1 ≤ i1 ≤ i2 · · · ≤ ip′ ≤ p′ and for

I ∈ I, let ηIηi1ηi2 · · · ηip . Since ϕ and its derivatives have moderate growth

there is constant B > 0 and an integer r > 0 such that | (ZδY γXβϕ(g0) |2≤

B ‖ g0 ‖r for | α | + | β | + | γ |≤ 2p′. Thus, we have any p′ ≥ p

| ϕ(g0) |2 ≤ C′′ ∫

Ω5

∑

I∈I

(ηI(h03))

2∑

|δ|+|γ|+|β|≤2p′

| (ZδY γXβϕ)(g0ξ−1h5ξ) |2 dh5

≤ C′′′α0(h

03)

p′ ‖ g0 ‖r

34 M.S. Raghunathan

Now ‖ g0 ‖r≤ b′Supα∈∆ α(h03)

−s for some integer s for g0 ∈ S(c0,Ω0). It

follows that we have

| ϕ(g0) |2≤ b′′α0(h

03)

p′ · α0(h03)

−s = b′′α(h0

3)p′−s.

Since p′ is at our choice we see that | ϕ(g0) |2≤ is bounded in S(c0,Ω0).

Hence ϕ ∈o L2(G∞/Γ).

4.3. Lemma (Godement) Let X be a locally compact space and µ a

probability measure. Let V ⊂ L2(X,µ) be a closed subspace. Suppose that

every ϕ ∈ V is essentially bounded. Then dimH <∞.

Proof V is a closed subspace of L∞ as well since L∞-convergence implies

L2 convergence. It follows also that the identity map of V is a continuous

homomorphism of V with L∞ topology on V with the L2 topology. By the

open mapping theorem there is a constant c > 0 such that ‖ ϕ ‖∞< c ‖ ϕ ‖2

for all ϕ ∈ V . Suppose ϕ1, · · · , ϕn is an orthonormal set in V , and a =

(a1, · · · , an) ∈ Cn; then we have for almost all x ∈ X,

|n∑

i=1

aiϕi(x) |≤ C ‖n∑

i=1

‖2= C(∑

| aiϕi |2)

1

2

It follows that if D is a dense countable subset of Cn, there is a set Y of

measure zero in X such that we have for all x ∈ X\Y and all a ∈ D,

|

n∑

i=1

aiϕi(x) |≤ c(

n∑

i=1

| ai |2)

1

2

SinceD is dense in Cn, the inequality holds for all a ∈ Cn. Taking ai = fi(x),

we conclude thatn∑

i=1

| fi(x) |2≤ c2

Integrating over X, we have n ≤ c2. Thus dimV <∞.

4.4. Proof of 4.1. We have seen that we may assume that G has no

central split torus so that G∞/Γ has finite measure and further that χi

trivial. We claim that oA(G∞,Γ, σ∞, λ) is a closed subspace of oL2(G∞/Γ).

In fact if ϕn ∈ oA(G∞,Γ, σ∞, λ) converges to ϕ in oL2, then ϕn → ϕ in


the sense of distributions. All the ϕn are in the kernel of a fixed elliptic

operator with C∞ coefficients on G∞/Γ hence the distribution ϕ is also

in this kernel. But any distribution in the kernel of an elliptic operator

with C∞ coefficients is a C∞ function. Thus ϕ is a C∞ function in oL2.

Since it is the limit of the ϕn as a distribution, the K∞-span of ϕ is a

quotient of σ∞ and the Z-span a quotient of λ. Further as ϕ is k-finite and

Z finite there is an α ∈ Z∞c (G) such that α ∗ ϕ = ϕ and one has thus

‖ ϕ ‖∞≤‖ α ∗ϕ ‖∞≤ C ‖ α ‖2G∞‖ ϕ ‖G∞/Γ. It follows that ϕ has moderate

growth so that ϕ ∈ oA((G∞,Γ, σ∞, λ). Lemma 4.3 now clearly implies

Theorem 4.1.

4.5. Clearly Theorem 4.1 is equivalent to the assertion that o(A)χ(G(A),

G(k), σ, λ) is finite dimensional. To prove Theorem 2, we will argue by

induction on the k-rank of [G,G]. Let PiicI be the collection of all the

maximal parabolic subgroups of G containing P. Let Ui be the cusp radical

of Pi and Gi ⊃ S. Let supplement to Ui,Pi. Let ri : C∞(G(A)/G(k))

→ C∞(Gi(A)/Gi(k)) be the map defined as follows:

γi(ϕi)(x) =

∫

Ui(A)/Ui(k)ϕ(xu)du.

We assert that for a suitable χi and λi and σi∞, λi(ϕ) ∈ Aχ(Gi∞(A),Gi∞(k),

λi, σi∞) if ϕ ∈ Aχ(G(A),G(k), λ, σ∞). Assume that this is true. Ob-

serve that k-rank ([Gi,Gi]) = k rank G − 1. Thus by induction hy-

pothesis Aχi(Gi(A),Gi(k), σ, λ∞) is finite dimensional for all i ∈ I. On

the other hand if ϕ is in ∩i∈I ker λi, ϕ is a cuspidal automorphic form inoAχ(G(A),G(k), σ, λ∞); and this last space is finite dimensional by Theo-

rem 4.1. This completes the proof of Theorem 2 assuming the following:

4.6. Lemma Given χ, σ, λ there exists for every i ∈ I, χi, σi, λi, χi = χ

(note that C ⊂ Gi(σi)), representation of K∞ ∩ Gi and λi : Zi → EndW ′i ,

(Zi the centre of the enveloping algebra Ui of L(Gi)) such that

ri(Aχ(G(Af ), G(k), σ, λ)) ⊂ Aχi(Gi(Af ),Gi(k), σi, λi)

Proof Let U− be the unipotent algebraic k-subgroup of G normalised

by S and whose Lie algebra is spanned by L(G)(β) | β a root, β =∑

α∈∆

mαα,mαi< 0. Let P−

i be the normaliser of U−i . One then has

36 M.S. Raghunathan

Pi ∩ P−i = Gi and L(G) = L(U−

i ) ⊗ L(Gi) ⊗ L(Ui). Correspondingly

the enveloping algebra U of L(G) can be written as

U(L(U−i )) · Ui · U(L(Ui))

where Ui is the enveloping algebra of L(Gi). Now U(L(U−i )) = L(U−

i )(U(L(U−i ))⊕

C and similarly U(L(Ui)) = U(L(Ui))L(Ui) ⊕ C. Consequently we have

U = L(U−i )U(L(U−

i )) · Ui · U(L(Ui))L(Ui)⊕ L(U−

i )U(L(U−i )) · Ui

⊕ Ui · U(L(Ui))L(Ui) ⊕ Ui

Let hi denote the projection of U on Ui following this direct sum decom-

position. Then according to a theorem of Harish-Chandra hi restrict to

U maps Z injectively into Zi and Zi is finitely generated as a Z-module.

Also the component of z ∈ CU in the direct factors L(U−i )(U(L(U−

i )) · Ui

and UiU(L(Ui))L(Ui) are trivial. From this it is easy to see that ri(zϕ) =

hi(z)ri(ϕ) for ϕ ∈ (Aχ(G(A),G(k), σ, λ). It follows that if we set W ′i =

W ⊗Z Zi,W′i is a finite dimensional vector space and we have a homomor-

phism λi : Zi → End W ′i . Evidently Ui(ri(ϕ)) is a quotient of W ′

i as a

Zi-module. Also, if Ki = K ∩Gi(Af ), it is clear that the Ki (K-span of ϕ)

and thus is a quotient of the Ki-module W (obtained by restriction of the

K-module structure to K0). Lemma 4.5 is immediate from this.


References

[B] A. Borel, Introduction aux groupes arithmetiques, Hermann (1969),

Paris.

[B-T] A. Borel and J. Tits, Groupes reductifs, Publ. Math. de ℓ’IHES, 27

(1965), 55–150.

[G] R. Godement, Domaines fondamentaux de groupes arithmetiques emi-

naire Bourbaki, (1962/63), Fas 3, No.257, Paris.

[H-C] Harish-Chandra, Automorphic forms on semisimple Lie groups, LN

62, Springer-Verlag (1968).

[L] R.P. Langlands, On the functional equations satisfied by Eisenstein se-

ries, LN 544, Springer-Verlag (1976).

[N] R. Narashimhan, Analysis on real and complex manifolds, Advanced

Studies in Pure Mathematics, Vol. 1, North Holland, Amsterdam,

(1968).

automorphic forms in gl n) i: decomposition of the space of cusp

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