Annales de l’institut Fourier (AIF)

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L’INSTITUT FOURIER
Michael BJÖRKLUND & Alexander FISH
Continuous Measures on Homogenous Spaces
Tome 59, no 6 (2009), p. 2169-2174.
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Ann. Inst. Fourier, Grenoble
59, 6 (2009) 2169-2174
CONTINUOUS MEASURES
ON HOMOGENOUS SPACES
by Michael BJÖRKLUND & Alexander FISH
Abstract. — In this paper we generalize Wiener’s characterization of contin-
uous measures to compact homogenous manifolds. In particular, we give necessary
and sufficient conditions on probability measures on compact semisimple Lie groups
and nilmanifolds to be continuous. The methods use only simple properties of heat
kernels.
Résumé. — Dans ce travail, on étend la caractérisation des mesures continues,
due à Wiener, à des variétés compactes et homogènes. Pour des groupes de Lie com-
pacts et semisimples, et pour des nilvariétés, on trouve des conditions nécessaires
et suffisantes pour qu’une mesure de probabilité soit continue. Les démonstrations
s’appuient sur des propriétés élémentaires des noyaux de la chaleur.
1. Introduction
N. Wiener gave a very simple, necessary and sufficient, condition for the
support of a probability measure µ on the unit circle T to contain points
with positive mass. In fact, he proves the following formula ([7])
lim
N→∞
1
2N + 1
∑
|k|6N
|µ̂(k)|2 =
∑
x∈T
|µ({x})|2.
This result has found many applications in various areas in mathematics;
most notably in ergodic theory and optimal control theory.
In this note we extend Wiener’s criterion to compact homogenous Rie-
mannian manifolds, i.e., compact Riemannian manifolds which admit a
transitive action by isometries. Our two main examples are symmetric
spaces of compact type (where other extensions of Wiener’s lemma are
Keywords: Probability measures on groups, heat kernels.
Math. classification: 60Bxx, 60B15, 30Cxx, 30C40.
2170 Michael BJÖRKLUND & Alexander FISH
known, due to Anoussis and Bisbas [2]) and nilmanifolds. The Fourier co-
efficient of the measure is replaced with the integral of an eigenfunction of
the Laplacian against the given measure. We will make use of the short–
time behaviour of heat kernels on compact manifolds to deduce our results.
2. A Generalized Wiener Lemma
2.1. Approximations of Diagonal Measures
In this section we will briefly discuss the simple idea which lies behind this
paper. Let (X,µ) be a probability space, and suppose there is a family of
complex–valued measurable functions on the direct product X ×X, which
is dominated in absolute value by some integrable function with respect
to the product measure µ × µ, and converges pointwise to the indicator
function of the diagonal of X ×X. By dominated convergence,
lim
t→0+
∫
X×X
Ft(x, y) dµ(x)dµ(y) =
∑
x∈X
|µ({x})|2.
Suppose (X, g) is a compact Riemannian manifold. Let ∆ be the Laplacian
associated to the Riemann metric g. It is a well-known fact [4] that ∆ is
a positive self-adjoint operator on L2(X), with respect to the Riemannian
volume measure. We let {λk}k>1 denote the eigenvalues of ∆, and {ϕk}k>1
the corresponding eigenfunctions, which are orthonormal in L2(X). Note
that the sequence of eigenvalues is non-negative and unbounded. The heat
kernel Kt is the unique solution to the equation
∂tKt(x, y) + ∆xKt(x, y) = 0,
with the initial condition limt→0+ Kt(x, y) = δy(x). The heat kernel can be
expressed as (see e.g., chapter 9 in [3])
Kt(x, y) =
∑
k>1
e−λktϕk(x)ϕk(y).
If (X, g) is a compact homogenous Riemannian manifold, i.e., of the form
G/K, where G is a Lie group, and K a closed subgroup of G, the Lapla-
cian is equivariant under the action of G on L2(X), and thus, Kt(x, x) is
independent of x ∈ X. If we assume that the volume has been normalized
to be 1, we conclude that
Kt(x, x) =
∑
k>1
e−λkt,
ANNALES DE L’INSTITUT FOURIER
CONTINUOUS MEASURES ON HOMOGENOUS SPACES 2171
for all x ∈ X, since the eigenfunctions are orthonormal in L2(X). We now
let
Ft(x, y) =
Kt(x, y)∑
k>1 e
−λkt
.
From the above discussion, it should be clear that Ft satisfies the properties
above.
If µ is a probability measure on X, we define
µk =
∫
X
ϕk(x) dµ(x), k > 1.
We can now formulate our first result:
Lemma 2.1. — Let X be a compact Riemannian homogenous space. If
µ is a probability measure on X, then
lim
t→0+
∑
k>1 |µk|2e−λkt∑
k>1 e
−λkt
=
∑
x∈X
|µ({x})|2.
In particular, if X is the standard d–dimensional torus, the lemma trans-
lates to
lim
t→0+
∑
k∈Zd |µ̂(k)|2e−4π
2||k||2t∑
k∈Zd e
−4π2||k||2t =
∑
x∈Td
|µ({x})|2,
where µ̂ is the Fourier transform of µ and ||k|| is the l2–norm of k ∈ Zd.
We will generalize this formula in two directions: To symmetric spaces of
compact type and to compact Heisenberg manifolds.
2.2. Continuous Measures on Compact Lie Groups
Let G be a compact, simply connected and semisimple Lie group. It is a
well-known fact [8] that the set of irreducible unitary representations of G
is parameterized by dominant weights. We let D denote the set of dominant
weights of the Lie algebra of G, and if λ is in D, we let πλ denote the as-
sociated representation and dλ the dimension of this representation. Recall
that all irreducible representations are finite–dimensional. The matrix co-
efficients of πλ are common eigenvectors of the Casimir element restricted
to the representation, and they all have the same eigenvalue, which we will
denote by cλ. The character χλ is defined to be the normalized trace of πλ,
and thus χλ(e) = 1, where e is the identity in G. The heat kernel on G can
be expressed as [6]
Kt(x, y) =
∑
λ∈D
χλ(x)χλ(y) e−tcλ .
TOME 59 (2009), FASCICULE 6
2172 Michael BJÖRKLUND & Alexander FISH
If µ is a probability measure on G, we define
µλ =
∫
X
χλ(x) dµ(x).
By lemma 2.1, we now have
Theorem 2.2. — Let G be a compact, simply connected and semisim-
ple Lie group, and suppose µ is a probability measure on G. Then,
lim
t→0+
∑
λ∈D |µλ|2e−tcλ∑
λ∈D e
−tcλ
=
∑
g∈G
|µ({g})|2.
This formula should be compared to Anoussis and Bisbas result in [2].
They prove, if G is a compact semi–simple Lie group and {An}n>1 is a
sequence of subsets of dominant weights, satisfying some technical assump-
tions, then
lim
n→∞
1
|An|
∑
λ∈An
d−1λ ||µ̂(πλ)||
2
2 =
∑
x∈G
|µ({x})|2,
where || · || denotes the Hilbert–Schmidt norm, and
µ̂(πλ) =
∫
G
πλ(x−1) dµ(x)
is the operator–valued Fourier transform of µ.
Example 2.3. — We now consider the group G = SU(2). In this case,
the dominant weights are positive half–integers [8], and cλ = 12λ(1 + λ).
The character associated to the weight λ (restricted to the maximal torus
T1) is given by:
χλ(t) =
sin ((2λ+ 1)πt)
(2λ+ 1) sin (πt)
, t ∈ R,
Thus, theorem 2.2 takes the form,
lim
t→0+
∑
2λ∈Z+ |µλ|
2e−
1
2λ(1+λ)t∑
2λ∈Z+ e
− 12λ(1+λ)t
=
∑
x∈G
|µ({x})|2.
The extension of the above result to symmetric spaces of compact type
is straight–forward.
ANNALES DE L’INSTITUT FOURIER
CONTINUOUS MEASURES ON HOMOGENOUS SPACES 2173
2.3. Continuous Measures on Nilmanifolds
The real Heisenberg group Hn is defined to be the group
Hn =


1 x1 . . . xn z
. . . y1
. . . 0
...
0
. . . yn
1

| x, y ∈ Rn, z ∈ R

,
and we let Γn denote the restriction of Hn to integer points. It is a well-
known fact that the quotient Xn = Hn/Γn is a compact Riemannian mani-
fold (the Riemann structure is induced from Euclidean structure on the Lie
algebra of Hn). The spectrum of the Laplacian on Xn was calculated by
Deninger and Singhof in [5]. We will briefly recall their result. The spectrum
is decomposed into two parts: the first part is parameterized by k, h ∈ Zn
with eigenfunctions
fk,h(x, y, z) = e2πi(〈k,x〉+〈h,y〉)
with eigenvalues
λk,h = −4π2(||k||2 + ||h||2).
The second part of the spectrum consists of eigenfunctions of the form,
gq,m,h(x, y, z)
= e2πi(mz+〈q,y〉)
n∏
j=1
[∑
k∈Z
Fhj
(√
2π|m|
(
xj +
qj
m
+ k
))
e2πikmyj
]
where m ∈ Z− {0} , q ∈ Zn and h = (h1, . . . , hn) ∈ Nn0 , with eigenvalues
λq,m,h = −2π|m|(2h1 + . . .+ 2hn + n+ 2π|m|).
Here, {Fν}ν>0 denotes the Hermite functions,
Fν(t) = (−1)νet
2/2 d
ν
dtν
e−t
2
, ν > 0.
As before, if µ is a probability measure on Xn, we define
µk,h =
∫
Xn
fk,hdµ
and
µq,m,h =
∫
Xn
gk,m,h dµ.
We can now formulate a generalization of Wiener’s lemma on the nilmani-
folds Xn.
TOME 59 (2009), FASCICULE 6
2174 Michael BJÖRKLUND & Alexander FISH
Theorem 2.4. — Suppose µ is a probability measure on Xn, then
lim
t→0+
∑
k,h |µk,h|2e−tλk,h +
∑
q,m,h |µq,m,h|2e−tλq,m,h∑
k,h e
−tλk,h +
∑
q,m,h e
−tλq,m,h
=
∑
x∈Xn
|µ({x})|2.
Remark 2.5. — Lemma 1 can of course also be applied to compact
locally symmetric spaces X, where the covering space is a symmetric space
of non–compact type. It is easy to see that the heat kernel on X is the
periodization of the heat kernel on the covering space. Explicit formulae
for heat kernels on symmetric spaces of non–compact type can be found
in [1].
BIBLIOGRAPHY
[1] J.-P. Anker & P. Ostellari, “The heat kernel on noncompact symmetric spaces.
Lie groups and symmetric spaces”, Amer. Math. Soc. Transl. Ser. 2 210 (2003),
p. 27-46, Providence, RI.
[2] M. Anoussis & A. Bisbas, “Continuous measures on compact Lie groups”, Ann.
Inst. Fourier (Grenoble) 50 (2000), no. 4, p. 1277-1296.
[3] M. Berger, A panoramic view of Riemannian geometry, Springer-Verlag, Berlin,
2003, xxiv+824 pp. ISBN: 3-540-65317-1.
[4] I. Chavel, Riemannian Geometry: A Modern Introduction, Cambridge University
Press, New York, 1994.
[5] C. Deninger & W. Singhof, “The e-invariant and the spectrum for compact nil-
manifolds covered by Heisenberg groups”, Invent. Math. 78 (1984), p. 101-112.
[6] H. Fegan, “The heat equation on a compact Lie group”, Trans. Amer. Math. Soc.
246 (1978), p. 339-357.
[7] C. C. Graham & O. C. McGehee, Essays in Commutative Harmonic Analysis,
Springer-Verlag, New York, 1979.
[8] M. Taylor, “Noncommutative Harmonic Analysis”, in Math. Surveys and Monogr,
no. 22, American Math. Society, Providence, R.I., 1986.
Manuscrit reçu le 25 août 2008,
accepté le 16 janvier 2009.
Michael BJÖRKLUND
Royal Institute of Technology (KTH)
Department of Mathematics
Lindstedtsvägen 25
S-100 44 Stockholm (Suède)
mickebj@math.kth.se
Alexander FISH
The Ohio State University
Department of Mathematics
231 W. 18-th Ave., Columbus
OH 43210 (USA)
afish@math.ohio-state.edu
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