Fourier expansion along geodesics

0.21 MB·English

by  Anton Deitmar

6 Fourier expansion along geodesics
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l Anton Deitmar
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] Abstract: A growth estimate on the Fourier coefficients along geodesics for
G
eigenfunctions of the Laplacian is given on compact hyperbolic manifolds.
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Along the way, a new summation formula is proved.
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v Contents
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6
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1 Generalised period integrals 3
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0 2 Intertwining functionals 5
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a 3 Triple products 10
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v 4 Proof of Theorem 2.4 12
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X
4.1 The summation formula in the case d = 2 . . . . . . . . . . . . 14
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a
4.2 The summation formula in the case d ≥ 3 . . . . . . . . . . . 17
1
FOURIER EXPANSION 2
Introduction
Let Y be a compact hyperbolic manifold and let c be a closed geodesic in Y.
We are interested in the Fourier coefficients
1
c (f) = f(c(l t))e−2πiktdt, k ∈ Z,
k c
Z0
of a smooth function f ∈ C∞(Y). Here l is the length of c. Under the
c
assumption that f be an eigenfunction of the Laplace operator on Y with
eigenvalue µ, one can relate c to an intertwining integral Iµ(f), which de-
k k
pends on µ and f, but not on c. Namely, there is an automorphic coefficient
a ∈ C such that
k
c = a Iµ(f).
k k k
The sequences (c ) and (Iµ(f)) are rapidly decreasing, so there is no a priori
k k
knowledge on the growth of (a ). Surprisingly, it is possible to prove the
k
following bounds, which constitute the main result of the paper.
If dimY = 2, then, as T → ∞,
|a |2 = O(T1/2).
k
k∈Z
X
|k|≤T
If dimY > 2, then
|a |2 < ∞.
k
k∈Z
X
The proof uses two expicit summation formulae (Theorems 4.1 and 4.3),
which are presented as results of own interest. The proof of these two relies
on the uniqueness of intertwining functionals (section 2) and the uniqueness
of trilinear products [2]. The idea that uniqueness of trilinear products to-
gether with explicit formulae can be used to derive growth estimates is due
to Bernstein and Reznikov [1], the idea that this can be used for Fourier
expansions along geodesics was formulated in [10], the arguments in section
3 are inspired by similar ideas in that paper.
We will explain the construction of the factors a in a bit more detail. Let
k
X be the universal covering of Y and Γ its fundamental group. Then Γ
acts on X by isometries and Y is the quotient Γ\X. So Γ injects into the
∼
isometry group G of X, which acts transitively on X, so X = G/K for a
FOURIER EXPANSION 3
subgroup K, which turns out to be compact. Let (π,V ) be an irreducible
π
unitary representation of G and let η : V → L2(Γ\G) be an isometric linear
π
G-map. Let P : L2(Γ\G) → L2(Γ\G)K = L2(Γ\G/K) = L2(Y) denote the
K
orthogonal projection onto the subspace of K-invariants. Demanding that
f ∈ C∞(Y) be an eigenfunction of the Laplacian amounts to the same as
demanding f to lie in the image of P ◦η for some π and η. The functional
K
Iγ = c ◦P ◦η onV thenhasanintertwining property withrespect toasplit
k k K π
torus A inside G. By a uniqueness result, proven in section 2, this implies
that Iγ is a multiple of a canonical intertwiner Iπ on V , which we named Iµ
k k π k
above. So we get the existence of a factor a ∈ C with Iγ = a Iπ = a Iµ as
k k k k k k
above.
1 Generalised period integrals
Let d be an integer ≥ 2 and let Y be an orientable compact hyperbolic
manifold of dimension d. For a closed geodesic c in Y let l(c) denote its
l(c)
length. The period integral I (f) = f(c(t))dt is the zeroth coefficient
c 0
of the Fourier-expansion of the function t 7→ f(c(t)). Therefore the higher
R
coefficients can be viewed as “generalised period integrals”. We compute this
expansion by temporarily turning to the more general setting of functions on
the sphere-bundle SY over Y. In the SY, the geodesic c lifts to a closed
orbit, again denoted c, of the geodesic flow φ . For a given point x ∈ c one
t 0
is interested in the Fourier-expansion of the function t 7→ f(φ x ), where now
t 0
f is in C∞(SY).
Let X be the universal covering of Y and let G be the group of orientation
preserving isometries of X if d ≥ 3, and for d = 2 let G be the group
of all isometries of X. Then G is isomorphic to the connected component
SO(d,1)0 of the special orthogonal group SO(d,1) if d ≥ 3, and for d = 2
one has g ∼= PGL (R). The group G acts transitively on X, which thereby
2
∼ ∼
can be identified with G/K, where K = SO(d) if d ≥ 3, and K = O(2) if
d = 2, is the maximal compact subgroup of G. The Riemannian metric on
X determines an invariant symmetric bilinear form b on the real Lie algebra
gR of G. Let kR ⊂ gR be the Lie algebra of K. Then b is negative definite
on kR and positive definite on its orthogonal complement pR. Let aR be a
one-dimensional subspace of pR, and let A = exp(aR) be the corresponding
FOURIER EXPANSION 4
subgroup of G. Then A is closed and non-compact and its centraliser equals
AM, where M is the centraliser of A in K. Let a be the complexification
of aR and a∗ the dual space of a. Then a∗ can be identified with the set of
continuous homomorphisms from A to C×. For λ ∈ a∗ we write a 7→ aλ =
eλ(loga) forthecorresponding homomorphism. Itisknown thatG/M → G/K
can be identified with the sphere-bundle of X = G/K in a way that the
geodesic flow is given by
φ (gM) = gexp(tH )M,
t 1
where H1 is a fixed element of aR with b(H1,H1) = 1.
The fundamental group Γ of Y acts on X by deck-transformations and can
thus beviewed asa uniformlatticeinG. It follows thatY = Γ\X = Γ\G/K.
The sphere bundle SY equals Γ\G/M.
A closed geodesic c in Y gives rise to a conjugacy class [γ] in Γ of elements
which “close” c. Any such γ ∈ Γ is hyperbolic in the sense that it is conjugate
in G to an element of the form a m ∈ AM with a 6= 1. The characters of
γ γ γ
thecompact abeliangroupA/ha iaregivenbythoseλ ∈ a∗ withaλ = 1, i.e.,
γ γ
λ(loga ) ∈ 2πiZ. Let λ be the unique element of a∗ with λ (loga ) = 2πi.
γ γ R γ γ
\
Then A/ha i = Zλ .
γ γ
Fix k ∈ Z Fix an element x ∈ G with γ = x a m x−1 and let
γ γ γ γ γ
Iγ : C∞(Γ\G/M) → C
k
1
ϕ 7→ ϕ(x a)a−kλγ da,
γ
l(γ)
ZA/haγi
where the Haar measure da is determined by the metric and l(γ) equals
|loga | = vol(A/ha i. Note that Iγ depends on the choice of x . Geomet-
γ γ k γ
rically, this corresponds to choosing a base-point on the closed orbit c. This
dependence is not severe, as x is determined up to multiplication from the
γ
right by elements of AM , where M is the centraliser in M of m . If we
mγ mγ γ
replace x by x a m , then Iγ is replaced by akλγIγ. So in particular, the
γ γ 0 0 k 0 k
map |Iγ| is uniquely determined by k and γ. Further, if γ is replaced by a
k
Γ-conjugate, say γ′ = σγσ−1 then one can choose x to be equal to σx and
γ′ γ
with this choice one gets Iγ′ = Iγ.
k k
The space C∞(Γ\G/M) can be viewed as the space C∞(Γ\G)M of M-
invariants in C∞(Γ\G). The volume element on X determines a Haar mea-
FOURIER EXPANSION 5
sure dg on G. Let R denote the unitary G-representation on L2(Γ\G) given
by right translations. As Γ is co-compact,
L2(Γ\G) ∼= N (π)π,
Γ
Mπ∈Gˆ
where the sum runsover the unitary dualGˆ ofG andthe multiplicities N (π)
Γ
are finite. Here and later we understand the direct sum to be a completed
direct sum in the appropriate topology. Then
C∞(Γ\G) = L2(Γ\G)∞ = N (π)π∞,
Γ
Mπ∈Gˆ
where π∞ is the representation on the Fr´echet space of smooth vectors. Fur-
ther,
C∞(Γ\G/M) = C∞(Γ\G)M = N (π)(π∞)M .
Γ
Mπ∈Gˆ
Note that, as M centralises A, the representation R| can be pushed down
A
to C∞(Γ\G/M). The linear functional Iγ satisfies
k
Iγ(R(a)ϕ) = akλγIγ(ϕ)
k k
for every a ∈ A. This means that Iγ is an intertwining functional.
k
2 Intertwining functionals
Let P = MAN be a parabolic with split component A. Let mR,nR be the
real Lie algebras of M and N and let m,n be their complexifications. The
modular shift ρ ∈ a∗ is defined by the equation a2ρ = det(a|n), where n is the
complexified Lie algebra of N on which A acts by the adjoint representation.
We establish an ordering on the one dimensional real vector space aR by
tρ > 0 ⇔ t > 0.
Let (σ,V ) be a finite dimensional irreducible representation of M and let λ
σ
be an element of the dual space a∗. Let π be the corresponding principal
σ,λ
series representation, which we normalise to live on functions f : G → V
σ
satisfying f(manx) = aλ+ρσ(m)f(x). Let V be the space of π and let
σ,λ σ,λ
FOURIER EXPANSION 6
V∞ be the space of smooth vectors in it. These can be viewed as smooth
σ,λ
sections of the vector bundle E over P\G given by the P-representation
σ,λ
(man) 7→ aλ+ρσ(m). Note that restriction of functions to K identifies V∞
σ,λ
with the space of all smooth sections of the homogeneous vector bundle E
σ
on M\K given by the representation σ. These can be interpreted as smooth
functions f : K → V with f(mk) = σ(m)f(k) for m ∈ M, k ∈ K.
σ
Let k ∈ Z. A continuous linear functional l : V∞ M → C is called a
σ,λ
k-intertwiner, if
(cid:0) (cid:1)
l(π(a)v) = akλγl(v)
holds for every a ∈ A and v ∈ V∞ M. Let V∞(k) be the space of all
σ,λ σ,λ
k-intertwiners.
(cid:0) (cid:1)
Let w be a representative in K of the non-trivial element of the Weyl group
0
W(G,A) and let n ∈ N be a fixed element different from 1.
0
Proposition 2.1 Assume throughout that Re(λ) > −ρ. Then we have
dimV∞(k) = dimV .
σ,λ σ
More precisely, let l ∈ V∗. Then the integral
σ
Iσ,λ(f) = l(f(w n a))a−kλγ da
k,l 0 0
ZA
converges for every f ∈ V∞ M. The map l 7→ Iσ,λ is a linear bijection
σ,λ k,l
(cid:0) V(cid:1) ∗ ∼= // V∞(k).
σ σ,λ
Dependence on the choice of n : If we replace n by another non-trivial
0 0
element n′, then there exists a m ∈ AM such that n′ = a m n (a m )−1.
0 0 0 0 0 0 0 0 0
Then Iσ,λ gets replaced with a−λ−ρ−kλγIσ,λ, where l′ = l◦σ(w m w−1).
k,l 0 k,l′ 0 0 0
Proof: The base space P\G of the bundle E consists of three orbits under
σ,λ
the group AM, namely the open orbit [w n ], and the two closed orbits [1],
0 0
[w ] which are indeed points. We consider the closed orbits first. Fix l ∈ V∗
0 σ
and let T denote the distribution
0
T (f) = l(f(1)).
0
FOURIER EXPANSION 7
Then T ◦ R(a) = aλ+ρT , so T is a k-intertwiner for akλγ = aλ+ρ. Let
0 0 0
N¯ = θ(N) and let n¯R be its Lie algebra. Then the tangent space of P\G at
the unit is isomorphic to n¯R. So for X ∈ n¯R we set
T (f) = l(Xf(1)).
X
Then T ◦R(a) = aλ+ρ+α0, where α is the positive root, i.e., α = 2 ρ. For
X 0 0 d−1
X1,...Xk ∈ n¯R set
T (f) = l(X ...X f(1)).
X1...Xk 1 k
Then TX1...Xk ◦R(a) = aλ+ρ+d2−k1ρ. Since these span the space of all distribu-
tions supported at 1 we see that we only get a k-intertwiner supported on 1 if
akλγ = aλ+ρ+d2−k1ρ for some k ≥ 0. The latter condition implies Re(λ) ≤ −ρ,
which is outside the range of the proposition.
We turn to the other closed orbit [w0]. In this case, let X1,...,Xk ∈ nR and
define
S (f) = l(X ...X f(w )).
X1...Xk 1 k 0
Then SX1...Xk ◦ R(a) = a−λ−ρ−d2−k1ρ and likewise, all intertwiners supported
on [w ] lie at Re(λ) ≤ −ρ. Since we assume Re(λ) > −ρ, this means that
0
every k-intertwiner which vanishes on the open orbit is zero. As an M-space,
∼
the open orbit [w n ] is isomorphic to M/M ×A, where M = SO(d−2) ⊂
0 0 0 0
∼
SO(d−1) = M. So the M-invariant sections of E | may be viewed as
σ,λ [w0n0]
∼
the sections of a vector bundle F on [w n ]/M = A. By Lemma 2.2 of [2]
0 0
every equivariant distribution on F is given by a smooth section which must
be A-invariant, hence is uniquely determined by its value at one point. So
the space of such distributions is in bijection with the fibre V∗. Since it is not
σ
clear a priori that any intertwiner defined on the open orbit indeed extends
to the whole space P\G, this argument only shows dimV∞(k) ≤ dimV .
σ,λ σ
For the other direction we need to show the convergence of the integral Iσ,λ.
k,l
So let f ∈ V∞ and compute formally
σ,λ
Iσ,λ(f) = a(w n a)λ+ρl(f(k(w n a)))a−kλγ da,
k,l 0 0 0 0
ZA
where a and k are the projections G → K and G → A of the ANK-Iwasawa
decomposition of G. Now l(f(k(w n a)))akλγ is bounded, so we only need to
0 0
show the convergence of
a(w n a)Re(λ)+ρda = a−Re(λ)−ρa(w na)Re(λ)+ρda,
0 0 0 0
ZA ZA
FOURIER EXPANSION 8
where na = a−1n a. By the explicit expressions derived in [2] this equals
0 0
d−1(1+Re(λ¯))
1 2
dt,
et +e−t
R
Z (cid:18) (cid:19)
¯ ¯
where λ = λρ, with Re(λ) > −1. The convergence is clear.
Finally we have to show the injectivity of l 7→ Iσ,λ. For this it suffices to
k,l
consider sections f which are M-invariant and vanish in a neighbourhood
of {1,w }. Since no two points of the form w n a are M-conjugate, we can
0 0 0
prescribe the value of f(w n a) arbitrarily as long as it is smooth in a and
0 0
(cid:3)
compactly supported. This implies injectivity.
A sequence (cj)j∈N of complex numbers is said to be of moderate growth, if
there exist N ∈ N such that
|c | = O(jN),
j
as j → ∞. The sequence is called rapidly decreasing, if for every N ∈ N one
has
|c | = O(j−N)
j
as j → ∞. The product of two moderately growing sequences is moderately
growing and the product of a moderately growing sequence and a rapidly
decreasing sequence is rapidly decreasing.
Proposition 2.2 Let Re(λ) > −ρ and σ be given. Then for any l ∈ V∗ and
σ
f ∈ (V∞)M the sequence
σ,λ
Iσ,λ
k,l
k∈Z
(cid:16) (cid:17)
is rapidly decreasing.
Proof: By definition it is clear that the sequence a = Iσ,λ(f) is bounded,
k k,l
as λ is purely imaginary. Next, using integration by parts one sees that for
γ
N ∈ N,
kNa = l(f(w n exp(tH))kNe−ktλγ(H)dt
k 0 0
R
Z
N
= λ (H) l(HNf(w n exp(tH))e−ktλγ(H)dt
γ 0 0
R
Z
= λ (H)NIσ,λ(HNf).
γ k,l
FOURIER EXPANSION 9
(cid:3)
This, again, is bounded and we get the claim.
If σ is the trivial representation, then V = C and we can choose l to be the
σ
identity. In that case we write Iλ for Iσ,λ.
k k,l
An irreducible unitary representation (π,V ) is called a class one or spherical
π
representation, if the space of K-invariant vectors VK is non-zero. In that
π
case one has dimVK = 1 (see [3]). Let GˆK denote the set of all π ∈ Gˆ which
π
are spherical.
Proposition 2.3 Let π ∈ GˆK. If π 6= 1, then
dimV∞(k) = 1.
π
If π = 1, then V∞(k) is zero unless k = 0 in which case it is one dimensional.
π
Foreveryπ ∈ GˆK H{1} there is a canonical generatorIπ given by the integral
k
of Proposition 2.1.
Proof: Since GˆK H{1} consists of the principal series π1,λ for λ ∈ iaR
and the complementary series π for 0 < λ < ρ the claim follows from
1,λ
(cid:3)
Proposition 2.1.
By an automorphic representation (π,V ,η) we mean an irreducible unitary
π
representation (π,V ) of G together with an isometric G-equivariant linear
π
map η: V → L2(Γ\G). Then η maps the space V∞ of smooth vectors into
π π
C∞(Γ\G). The automorphic representation (π,V ,η) is called a spherical
π
automorphic representation, if π is spherical. In that case, by Proposition
2.3 it follows that for γ as in section 1 and k ∈ Z there exists a unique
complex number aη,γ such that
k
Iγ(η(v)) = aη,γIπ(v)
k k k
for every v ∈ V∞. The factor aη,γ carries the “automorphic” information.
π k
Since both sequences, Iγ(η(v)) and Iπ(v) are rapidly decreasing, there is no
k k
a priori knowledge on the growth of (aη,λ) . All the more striking is the
k k
following theorem, which constitutes the main result of this paper.
FOURIER EXPANSION 10
Theorem 2.4 If d = 2, then, as T → ∞,
|aη,γ|2 = O(T1/2).
k
k∈Z
X
|k|≤T
If d ≥ 3, then we have
|aη,γ|2 < ∞.
k
k∈Z
X
The proof extends over the next two sections.
3 Triple products
Let (π,V ,η) be a spherical automorphic representation. Since π is unitary,
π
there is an anti-linear isomorphism to the dual c : V → V∗. Let π˘ denote
π π
the dual representation on V∗ = V . Let ¯· be the complex conjugation on
π π˘
L2(Γ\G) and let η˘ be the composition of the maps
V c−1 // V η // L2(Γ\G) ¯· // L2(Γ\G).
π˘ π
Then η˘ is a G-equivariant linear isometry of V into L2(Γ\G), so (π˘,V ,η˘) is
π˘ π˘
an automorphic representation as well.
Let∆ : Γ\G → Γ\G×Γ\Gbethediagonalmap. Let∆∗ : C∞(Γ\G×Γ\G) →
C∞(Γ\G) be the corresponding pullback map and let E = V∞⊗ˆV∞, where
π π˘
⊗ˆ denotes the projective completion of the algebraic tensor product. Let
η denote the natural embedding E → C∞(Γ\G)⊗ˆC∞(Γ\G) ∼= C∞(Γ\G×
E
Γ\G). For γ as in the first section we get an induced functional on E,
l = Iγ ◦∆∗ ◦η .
∆(γ) 0 E
In other words, for w ∈ E we have
1
l (w) = η (w)(x a,x a)da.
∆(γ) E γ γ
l(γ)
ZA/haγi