An Application of Microlocal
Analysis to the Representation
Theory of Compact Lie Groups
Dissertation
An Application of Microlocal Analysis
to the Representation Theory of
Compact Lie Groups
vorgelegt von
Sameh Keliny
Universit¨at Paderborn
Fakult¨at f¨ur Elektrotechnik, Informatik und Mathematik
March 19, 2009
3
Danksagung
Mit Dir kann ich Hindernisse ¨uberwinden.
Mit Dir springe ich ¨uber Mauern.
(Psalm 18, 30)
Der Glaube an Gott hat mir den R¨ucken gest¨arkt, Mut und Ausdauer verliehen.
Erstens m¨ochte ich mich herzlich bei Prof. Dr. S¨onke Hansen f¨ur seine Untersch¨utzung
im wissenschaftlichen Bereich sowie f¨ur seine T¨ur, die f¨ur mathematische Diskussio-
nen und Ratschl¨age immer offen stand, bedanken. Besonders zu sch¨atzen wusste ich
die Freiheit, die er mir sowohl in meiner Forschung als auch in der Lehre gegeben
hat. Zweitens gilt mein Dank Prof. Dr. Joachim Hilgert f¨ur die ¨
Uberlassung dieses
interessanten Dissertationsthemas, seine Hilfsbereitschaft und wichtigen Anregungen.
Weiterhin danke ich meinen Arbeitskollegen sowohl im wissenschaftlichen als auch im
nicht-wissenschaftlichen Bereich. Last but not least m¨ochte ich mich bei meinen Eltern,
die keine Kosten und M¨uhen gescheut haben, um mir die besten Voraussetzungen f¨ur
das Leben zu schaffen, bedanken. Weiterhin bedanke ich mich bei meinen Freunden,
insbesondere bei meiner Frau Heimke f¨ur ihre grenzenlose Geduld in Krisenzeiten.
Sameh Keliny
Ich danke den Gutachtern
Herrn Prof. Dr. S¨onke Hansen,
Institut f¨ur Mathematik, Universit¨at Paderborn
Herrn Prof. Dr. Joachim Hilgert,
Institut f¨ur Mathematik, Universit¨at Paderborn
Contents
Zusammenfassung 5
Abstract 6
Introduction 7
1 Preliminaries 9
1.1 Roots and Root Spaces ........................... 9
1.2 Analysis on Compact Groups ....................... 13
1.3 Tangent and Cotangent Bundles of a Lie Group ............. 15
1.4 Distributions ................................ 18
1.5 Microlocal Analysis ............................. 20
2 Wave Front Set and Fourier Coefficients 29
2.1 Wave Front Set of Truncated Distributions ................ 29
2.2 Wave Front Set of Convolution ...................... 38
2.3 Characterization of The Wave Front Set of Distributions ........ 39
3 Restriction of Characters 41
3.1 K-Characters ................................ 42
3.2 Restriction of Characters to a Closed Subgroup ............. 44
References 49
Index and Index of Notation 51
Zusammenfassung
Wir untersuchen Charaktere von reduziblen unit¨aren Darstellungen einer kompakten
zusammenh¨angenden halbeinfachen Liegruppe. Wir geben eine geometrische Bedin-
gung, unter welcher diese Charaktere auf abgeschlossenen Untergruppen eingeschr¨ankt
werden k¨onnen. Das wurde schon 1998 von T. Kobayashi untersucht, um ein Kriterium
f¨ur die diskrete Zerlegbarkeit von Einschr¨ankungen reduzibler unt¨arer Darstellungen
von reduktiven Liegruppen auf reduktive abgeschlossene Untergruppen zu erhalten. Der
Schl¨ussel in seinem Beweis besteht darin, Charaktere von reduziblen unit¨aren Darstel-
lungen einer kompakten zusammenh¨angenden Liegruppe und ihre Einschr¨ankungen auf
abgeschlossene Untergruppen mit Hilfe von Methoden der mikrolokalen Analysis f¨ur Hy-
perfunktionen zu betrachten. In dieser Arbeit benutzen wir mikrolokale Analysis f¨ur
Distributionen. Das Neue ist die Benutzung der Stetigkeit zwischen angepassten Distri-
butionenr¨aumen und der Einschr¨ankung auf abgeschlossene Untermannigfaltigkeiten.
Diese Stetigkeit ist im Falle der Hyperfunktionen nicht vorhanden.
Abstract
We consider characters of a reducible unitary representation of a compact connected
semisimple Lie group. We provide a geometric condition under which these characters
can be restricted to closed subgroups. This was already considered by T. Kobayashi in
1998 to establish a criterion for the discrete decomposability of restrictions of unitary
representations of reductive Lie groups to reductive closed subgroups. The crucial point
in his proof is to consider characters of a reducible unitary representation of a compact
connected Lie group and their restriction to closed subgroups using microlocal analysis
methods in the hyperfunctions setting. In this thesis we use microlocal analysis of dis-
tribution theory instead. The novelty consists in using the continuity between adapted
spaces of distributions and the restriction to closed submanifolds. This continuity is
not readily available in the hyperfunctions setting.
Introduction
The main motivation of this thesis is to help understanding how an irreducible repre-
sentation of a real reductive linear Lie group decomposes when restricted to a subgroup.
Following Kobayashi [Ko98] the main problem is studied by considering characters of
unitary representation of a connected compact semisimple Lie group and the decom-
position of its restriction to closed subgroups. Based on the the work of Kashiwara
and Vergne [KV79] Kobayashi used microlocal analysis methods in the hyperfunctions
setting. In this thesis we use microlocal analysis of distribution theory instead.
Chapter 1provides notations, basic definitions, various theorems, and examples to
be used.
In Chapter 2we consider a connected, compact, and semisimple Lie group Kwith a
maximal abelian torus T.kdenotes the semisimple Lie algebra of K,tits subalgebra
corresponding to T, and D0(K) the distributions defined on K. We identify the equiv-
alence classes of irreducible representation b
Kwith their highest weights, b
K=L∩C(
C⊂it∗,i=√−1, denotes the closure of the (dual) Weyl chamber and Lthe weight
lattice in it∗). Then due to Peter-Weyl Theorem, we get that for each u∈D0(K) we
have the Fourier expansion
u=X
λ∈L∩C
ϕλ,
If the Fourier coefficients vanish outside a closed cone Ω ⊂C\0, ϕλ= 0 if λ /∈Ω,
then the wave front set of usatisfies
WF(u)⊂K×Ad∗K(−iΩ) ⊂T∗K.
Furthermore, the Fourier series of uconverges to uin D0
Γ(K), where Γ := K×
Ad∗K(−iΩ) is a closed cone in T∗K. Here D0
Γ(K) denotes the space of distributions
having their wave front set in Γ. The convergence in D0
Γ(K) is an important tool in
defining and analyzing the restriction of a distribution to a closed submanifold. We will
also prove a more precise relationship between the wave front set of distribution and
the asymptotic behavior of the L2-norm of the Fourier coefficients (was first introduced
by [KV79]).
In the first part of chapter 3we introduce the K-character,
ΘK
τ=X
λ∈L∩C
mK(πλ:τ)χπλ,
of a unitary representation τof Kwith multiplicities mK(πλ:τ) which are polynomially
bounded in |λ|. Here πλdenotes an irreducible representation of Kcorresponding to
the highest weight λand χπλthe character of πλ. ΘK
τis a well-defined distribution on
K.
Furthermore, the set of λ’s with mK(πλ:τ)6= 0 is called the support of τ. The
support is approximated by the asymptotic K-support of τwhich is a closed conic
subset ASK(τ)⊂C. We show that the wave front set of this distribution satisfies
WF(ΘK
τ)⊂K×Ad∗K(−iASK(τ)) ⊂T∗K\0≃K×k∗.
8Contents
The rest of chapter 3is devoted to the restriction of ΘK
τto a closed subgroup. Here
we consider a closed subgroup Hof Kand their Lie algebras hand k, respectively. The
space of conormals is denoted by h⊥⊂k∗. If we assume that
ASK(τ)∩iAd∗K(h⊥) = ∅,
then the restriction ΘK
τ|His a well-defined distribution on H. Moreover, the restricted
distribution ΘK
τ|Hcoincides with the distribution ΘH
τ|H, i.e.,
ΘH
τ|H= ΘK
τ|H.
This dissertation gives a simplified proof, in the framework of distributions, of a
Theorem of Kobayashi [Ko98] on the restrictions of a K-character to closed subgroup.
The novelty consists in using the continuity between adapted spaces of distributions
and restriction to closed submanifolds. This continuity is not readily available in the
hyperfunctions setting which is used by Kobayashi [Ko98].
1 Preliminaries
1.1 Roots and Root Spaces
We will give in this section a brief overview of some facts about root (weight) spaces.
Let Kbe a compact, connected, and semisimple Lie group with a maximal torus T.
Let kand tbe the semisimple Lie algebra of Kand T. The algebra kC=k⊗RChas
the root-space decomposition
kC=tC⊕M
α∈Λ
kα(1.1)
where αis a linear form on tC, Λ = Λ(kC,tC) is called the set of roots and
kα:= {X∈kC|[Y, X] = hα, Y iX, ∀Y∈tC}
is the corresponding root space with respect to tC, where h·,·i denotes the duality
bracket t∗
C×tC→C, see [He01] p. 165.
An element X∈tis said to be regular in t, if hα, Xi 6= 0 for all roots α∈Λ,
treg := t\Sα∈Λker α. Let cbe a connected component of treg, it follows that for any
α∈Λ, i−1hα, Xieither >0 or <0 on c. Hence,
Λ=Λ+∪Λ−Λ+∩Λ−=∅,(1.2)
if we take
Λ+(c) = Λ+:= {α∈Λ|i−1α > 0 on c},(1.3)
Λ−={−α|α∈Λ+}. Conversely, if Λ+is a subset of Λ satisfying (1.2), then the set
c(Λ+) := {X∈t|i−1hα, Xi>0,∀α∈Λ+}(1.4)
is an open, convex polyhedral cone in t, contained in treg, and equal to a connected
component of treg, if and only if, the convex cone in it∗generated by Λ+is proper, i.e.,
if Λ+satisfies the following condition:
X
α∈Λ+
cαα= 0, cα≥0,∀α∈Λ+⇒cα= 0,∀α∈Λ+.(1.5)
Remark 1.1. A subset Λ+of Λ satisfying (1.2) and (1.5), is called a choice of positive
roots. The connected components of treg are called the Weyl chambers in t(a equivalent
definition of the Weyl chambers will be presented later on). Moreover, the relation (1.3)
and (1.4) defines a bijection between the choice of a positive roots Λ+and the Weyl
chambers c(see [DK00], p.144-147).
The following definitions will be needed later on. For each x∈Kthe map Ad x:
y7→ xyx−1is the conjugation by xin the group K. Because Ad x(e) = e, the tangent
mapping of Ad xat eis a linear mapping
Ad x:= Te(Ad x) : k−→ k
10 1.1 Roots and Root Spaces
called the adjoint mapping of x. That is, the mapping
Ad : K−→ GL(k)
is a homomorphism of groups and called the adjoint representation of Kin k=TeK.
Accordingly, the linear map ad := Te(Ad) is given by
ad : k−→ L(k,k),
ad X(Y) := [X, Y ], X, Y ∈kwhere [·,·] is called the Lie bracket of Xand Y(see
[DK00], p. 3). The bilinear form on a Lie algebra is given by
B(X, Y ) := tr(ad X◦ad Y), X, Y ∈k,(1.6)
Bis called the Killing form of k. This bilinear form is symmetric, because in general
tr(A◦B) = tr(B◦A), for linear endomorphisms Aand B. Since the real trace of a
linear mapping is equal to the complex trace of its complex linear extension, the Killing
form extends to a complex bilinear form on kCby B(X, Y ) = trC(ad X◦ad Y), where
X, Y ∈kCand ad X◦ad Yis considered as an element of LC(kC,kC) (see [DK00], p. 148).
Moreover, for each Xin the Lie algebra kof a compact Lie group, ad X∈LC(kC,kC) is
diagonalizable, with only purely imaginary eigenvalues (see [DK00], Lemma 3.5.1).
Remark 1.2. B|tC×tCis nondegenerate, consequently to each root αcorresponds Xα
in tCwith hα, Xi=B(X, Xα) for all X∈tC(see [Kn02], Proposition 2.17). On tC×tC
the Killing form is given by
B(X, X0) = X
α∈Λhα, Xihα, X0i(1.7)
see [Kn02], Corollary 2.24. Moreover, we transfer the restriction to tCof the Killing
form to a bilinear form on the dual t∗
Cby the definition
hα, λi=B(Xα, Xλ) = hα, Xλi=hλ, Xαi(1.8)
for α, λ ∈t∗
C(see [Kn02], p. 144). Combining (1.7) and (1.8), we obtain
hα, λi=B(Xα, Xλ) = X
β∈Λhβ, Xαihβ, Xλi=X
β∈Λhβ, αihβ, λi.(1.9)
The restriction of h·,·i to t∗×t∗is a positive definite inner product (see [Kn02], p.
147-149).
Definition 1.3. Asimple Lie algebra is a non-abelian Lie algebra whose ideals are
0 or itself. A semisimple Lie algebra is a direct sum of simple Lie algebra. A Lie
group called semisimple if its Lie algebra is semisimple. A reductive Lie algebra is
a sum of abelian and semisimple lie algebra. A Lie group called reductive if its Lie
algebra is reductive.
11 1.1 Roots and Root Spaces
Definition 1.4. An abstract root system, in a finite-dimensional real inner product
space Vwith inner product h·,·i which induces a norm |·|, is a finite set Λ of non-zero
elements of Vsuch that;
(i) Λ spans V,
(ii) the orthogonal transformation sα(β) = β−2hα, βi
|α|2α, for α∈Λ, map Λ to itself,
(iii) 2hβ, αi
|α|2is an integer whenever αand βare in Λ.
An abstract root system is said to be reduced system if α∈Λ implies 2α /∈Λ. If
αis a root and 1/2αis not a root, we say that αis reduced element (see [Kn02],
p. 152). We say that a member λof Vis dominant if hλ, αi ≥ 0 for all α∈Λ+(see
[Kn02], p. 168).
The next theorem will give the relation between roots and abstract root systems.
Theorem 1.5. The root system of a complex semisimple Lie algebra kCwith respect to
a Cartan subalgebra tCforms a reduced abstract root system in t∗with respect to the
inner product h·,·i defined in 1.9 (see [Kn02], Proposition 2.42).
In the following we will present some definitions and results which are needed to state
Theorem of The Highest Weight.
The presence of the groups Kand Tgives us additional information about the root-
space decomposition (1.1). In fact, Ad(T) acts by orthogonal transformations on k
relative to our given inner product (Killing form). If we extend this inner product on kto
a Hermitian inner product on kC, then Ad(T) acts on kby a commuting family of unitary
transformations. Such a family must have a eigenspace decomposition, that is (1.1).
The action of Ad(T) on the 1-dimensional space kαis a 1-dimensional representation
of T, necessarily of the form
Ad(t)X=χα(t)Xfor t∈T, (1.10)
where χα:T→S1is a continuous homomorphism of Tinto the group of complex
numbers of modulus 1. We call χαamultiplicative character (see [Kn02], p. 254).
Proposition 1.6. If λ∈t∗, then the following are equivalent:
(i) Whenever H∈tsatisfies exp(H) = 1, then hλ, Hiis in 2πiZ.
(ii) There is a multiplicative character χλof Twith χλ(exp(H)) = ehλ,Hifor all H∈t
(see [Kn02], Proposition 4.58)
Remark 1.7. A linear form satisfying (i) and (ii) is called analytically integral.
12 1.1 Roots and Root Spaces
Proposition 1.8. If λ∈t∗is analytically integral, then λsatisfies the following con-
dition
λ(α)∨:= hλ, αi
|α|2is in Zfor each α∈Λ (1.11)
(see [Kn02], Proposition 4.59)
Remark 1.9. A linear form satisfying condition (1.11) is called algebraically inte-
gral.
Proposition 1.10. Let πbe an irreducible finite-dimensional representation of Kthen
we have:
1. If λis the highest weight of π, then λ(α)∨≥0for all α∈Λ+.
2. For a weight λof π, the following are equivalent:
(i) λis a highest weight of π.
(ii) If α∈Λ+, then α+λis not a weight of π.
(iii) For any weight µof π, we have µ=λ−Pα∈Λ+nααfor some nα∈N0
(see [DK00], Proposition 4.9.4)
Definition 1.11. One introduces a partial ordering ≤by writting µ≤λif and only
if µ=λ−Pα∈Λ+nααfor some nα∈N0. The customary definition is to call a weight
λa highest weight of an irreducible representation π, if it is a maximal element of the
set of weights of π, with respect to the partial ordering ≤; this is just condition (iii) in
proposition (1.10) (2) (see [DK00], p. 260).
Definition 1.12. The restriction of the Killing form to tCand to tis defined in (1.8)
and (1.9).
C:= {λ∈it∗| hλ, αi>0,∀α∈Λ+}(1.12)
is called (dual) Weyl chamber. This definition is compatible with the definition of
the Weyl chamber we gave in remark 1.1, since we have a bijection between the choice
of a positive roots Λ+and the Weyl chamber c(Λ+) (see (1.4) and remark 1.1).
Definition 1.13. Suppose that Π = {α1,...αl}is any set of lindependent reduced
elements αi(see definition 1.4), such that every expression of a member α∈Λ as
α=Piniαihas all non-zero niof the same sign. We call Π a system of simple
roots (see [Kn02], p. 164).
The continuous representation σ:K→GL(U) and τ:K→GL(V), where Vand
Uare finite-dimensional vector spaces, respectively are said to be equivalent if there is
a toplogical linear isomorphism Lfrom Uonto V, such that L◦σ(k) = τ(k)◦Lfor all
k∈K. The set of equivalence classes of irreducible representations of Kis called the
dual b
Kof K(see [DK00], p. 210).
Let πbe an irreducible representation of kon a finite-dimensional complex vector
space Vand Vλ:= {v∈V|π(H)v=λ(H)v, ∀H∈t}be a weight space of V
(eigenspace with respect to π). Due to proposition 1.10 and definition 1.11 the largest
weight in the partial ordering is called the highest weight of π.
13 1.2 Analysis on Compact Groups
Theorem 1.14 (Theorem of The Highest Weight).Up to equivalence the irreducible
finite-dimensional representations πof kare in one-one correspondence with the dom-
inant algebraically integral linear functionals on tC, the correspondence being that λ
is the highest weight of πλ. The highest weight λof πλhas the following additional
properties:
1. λdepends only on the simple system Πand not on the ordering used to define Π.
2. The weight space Vλfor λis 1-dimensional.
3. Each root vector Eαfor an arbitrary α∈Λ+annihilates the members of Vλ, and
the elements of Vλare the only vectors with this property.
4. Every weight of πλis of the form λ−Piniαiwith ni∈N0and the αiin Π.
(see [Kn02], Theorem 5.5)
Definition 1.15. Let LT:= X∈t|eX= 1. Then
L:= {α∈it∗| hα, Xi ∈ 2πiZ,∀X∈LT}
is called the weight lattice in it∗(see [DK00]. p. 271).
Remark 1.16. Due to the Theorem of Highest Weight 1.14 we can identify L∩Cwith
b
Kwhere Cthe closure of the Weyl chamber and b
Kthe set of equivalence classes of
irreducible representations of K.
1.2 Analysis on Compact Groups
In this section we will present the Peter-Weyl Theorem.
Let Kbe a compact connected Lie group and πa continuous representation of Kon
the Hilbert space H. Then πis said to be unitary representation if each π(x), for
x∈K, is a unitary transformation in H, i.e.,
(π(x)v, π(x)w) = (v, w),∀v, w ∈H.
Furthermore, since Kis a compact group, then there exists a Hermitian inner product
H(i.e., symmetric sesquilinear form) for which the representation πis unitary (see
[DK00], Corollary 4.2.2.). Let b
Kdenote the equivalent classes of unitary irreducible
representation of K.
If πis a unitary representation of Kon some Hilbert space H, then the functions
φu, v(x) = (π(x)u, v), u, v ∈H
are called matrix elements of π. If uand vare elements of an orthonormal basis {ej}
for H, then φu, v(x) is one of the entries of the matrix for π(x) with respect to that
basis, namely
πij(x) = φej, ei(x)=(π(x)ej, ei).(1.13)
We denote the linear span of the matrix elements of πby Eπ.Eπis a subspace of C(K)
and hence also of Lp(K) for all p(see [Fo95a], p. 129).
14 1.2 Analysis on Compact Groups
Remark 1.17. We denote the character tr(π(x)) of πby χπ(x). The convolution is
defined as follows
f∗χπ(x) := ZK
f(xy−1)χπ(y)dy =ZK
f(y)χπ(y−1x)dy
We note that the space L1(K) is an algebra with respect to convolution.
Proposition 1.18. Eπdepends only on the unitary equivalence class of π. It is invari-
ant under left and right translation and is a two-sided ideal in L1(K). If dim H=n <
∞then dim Eπ≤n2(see [Fo95a], Proposition 5.6).
We note that Ewhich is given by
E= the linear span of [
π∈
b
K
Eπ,
is an algebra (see [Fo95a], Proposition 5.10).
Any unitary representation πof Kon Hdetermines another representation πon the
dual space H0of H, namely π(x) = π(x−1)twhere the tdenotes the transpose. Here we
identify a Hilbert space with its dual. Thus, if we choose an orthonormal basis for H,
so that π(x) is represented by a matrix M(x), then the matrix for π(x) is the inverse
transpose of M(x), and since πis unitary this is nothing but the complex conjugate of
M(x) and πis called the contragredient of π. We set dπ= dim H(dπ= dim H<∞
if πis irreducible see proposition 1.18).
Theorem 1.19 (Peter-Weyl Theorem).Let Kbe a compact group. Then Eis uni-
formly dense in C(K),L2(K) = Lπ∈
b
KEπ(direct Hilbert sum). Let πij be defined as
in (1.13), then
{pdππij |i, j = 1, . . . , dπ, π ∈b
K}
is an orthonormal basis for L2(K). Each π∈b
Koccurs in the right and left regular
representations of Kwith multiplicity dπ. More precisely, for i= 1, . . . , dπthe subspace
of Eπ(respectively Eπ) spanned by the i-th row (respectively the i-th column) of the
matrix (πij)(respectively (πij)) is invariant under the right (respectively. left) regu-
lar representation, and the latter representation is equivalent to πthere (see [Fo95a],
Theorem 5.12).
Due to the Peter-Weyl theorem, if f∈L2(K) we have
f=X
π∈
b
K
dπ
X
i, j=1
cπ
ijπij, cπ
ij =dπZK
f(x)πij(x)dx. (1.14)
Note that this decomposition of L2is dependent on the choice of an orthonormal
basis for H. However, it is possible to reformulate the equation (1.14) to avoid this
dependency.
15 1.3 Tangent and Cotangent Bundles of a Lie Group
Definition 1.20. If f∈L1(K) and π∈b
K, we define the operator b
f(π) on Hby:
b
f(π) = ZK
f(x)π(x)∗dx =ZK
f(x)π(x−1)dx, (1.15)
where dx denotes the Haar measure. This map f7−→ b
fis called the group Fourier
transformation of fat π(see [Fo95a], p.134). If we choose an orthonormal basis for
Hso that π(x) is represented by the matrix (πij(x)), then b
f(π) is given by the matrix
b
f(π)ij =ZK
f(x)πji(x)dx =1
dπ
cπ
ji.
But then X
i, j
cπ
ijπij(x) = dπX
i, j b
f(π)jiπij(x) = dπtr hb
f(π)π(x)i
so that (1.14) becomes a Fourier inversion formula,
f(x) = X
π∈
b
K
dπtr hb
f(π)π(x)i(1.16)
We get that
tr hb
f(π)π(x)i=ZK
f(y) tr π(y−1)π(x)dy
=ZK
f(y) tr π(y−1x)dy
=f∗χπ(x),
so equation (1.16) becomes
f=X
π∈
b
K
dπf∗χπ=X
λ∈L∩C
dπλf∗χπ.(1.17)
In particular, dπf∗χπis the orthogonal projection of fonto Eπ.
1.3 Tangent and Cotangent Bundles of a Lie Group
Let Kbe a real Lie group with Lie algebra k. Let TK and T∗Kdenote the tangent and
contangent bundle of K.kis defined as the tangent space TeKat the identity element
e∈K. The Lie bracket is given by the Lie bracket of the left invariant vector fields
v, w ∈C∞(K, TK) as follows: [X, Y ] = [v, w] (e) if X=v(e), Y=w(e). Elements
X∈kare also viewed as generators of one-parameter group, X=d
dt |t=0 etX. Let
Lx:K→K,y7→ xy, denote left translation. Then we identify the tangent bundle
TK with K×kusing the bundle isomorphism
K×k∼
−→ TK
(x, X)7−→ (dLx)(X) = d
dt|t=0 x etX .
16 1.3 Tangent and Cotangent Bundles of a Lie Group
This gives a global trivialization of the tangent bundle. Similarly, we identify the
cotangent bundle T∗Kwith K×k∗as follows: Let dLx:TeK→TxK,dL∗
x:T∗
xK→
T∗
eK, and (dL∗
x)−1:T∗
eK→T∗
xK. The last map defines the identification
K×k∗∼
−→ T∗K
(x, ξ)7−→ (dL∗
x)−1(ξ).
Definition 1.21. The map Ad∗:K→GL(k∗) is called the co-adjoint representa-
tion of Kwhich is defined by: hAd∗k(λ), Xi:= hλ, Ad k−1(X)ifor all λ∈k∗,k∈K,
and X∈kwhere h·,·i denotes the duality bracket k∗×k→C
Proposition 1.22. Let φ:K×K→K,(x, y)7→ (yxy−1). Using the identification
TxK=kvia X7→ d
dt|t=0 x etX . Then the derivative of φis given as follows:
dφ(x, y) : k×k−→ k
(X, Y )7−→ dφ(x, y)(X, Y ) = Ad yX−Y+ Ad x−1(Y).
The adjoint map dφ(x, y)∗:k∗−→ k∗×k∗is given by
dφ(x, y)∗:ξ7−→ (Ad∗y−1(ξ),Ad∗xy−1(ξ)−Ad∗y−1(ξ))
Proof. We start first by fixing y, then we have
dφ(x) : k−→ k
d
dt|t=0 x etX 7−→ d
dt|t=0
y x etX y−1,
and d
dt|t=0y x etX y−1=d
dt|t=0
y x y−1etAd y(X).
Then we get
dφ(x) : X7→ Ad y(X).
Similarly, now fix x. Then we have
dφ(y) : k−→ k
d
dt|t=0 y etY 7−→ d
dt|t=0 y etY x e−tY y−1.
We compute
d
dt|t=0 y etY x e−tY y−1=d
dt|t=0 y xetAd x−1(Y)e−tY y−1
=d
dt|t=0 y x etAd x−1(Y)y−1e−tAd y(Y)
=d
dt|t=0 y x y−1etAd yx−1(Y)e−tAd y(Y)
=d
dt|t=0 y x y−1etAd yx−1(Y)−tAd y(Y)+O(t2)
=d
dt|t=0 y x y−1etAd y(Ad x−1(Y)−Y)+O(t2).
17 1.3 Tangent and Cotangent Bundles of a Lie Group
Then
dφ(y) : Y7→ Ad y(Ad x−1(Y)−Y).
Therefore,
dφ(x, y) : k×k−→ k
(X, Y )7→ dφ(x)(X) + dφ(y)(Y)
= Ad y(X) + Ad y(Ad x−1(Y)−Y)
= Ad y(X−Y+ Ad x−1(Y))
For any ξ∈k∗we get
hξ, dφ(x, y)(X, Y )i=hξ, Ad y(X+ Ad x−1(Y)−Y)i
=hAd∗y−1(ξ), X + Ad x−1(Y)−Yi.
Then the adjoint map is given by
dφ∗:k∗−→ k∗×k∗
ξ7−→ (Ad∗y−1(ξ),Ad∗xy−1(ξ)−Ad∗y−1(ξ))
We will need later on two more derivatives which we will compute in the following
examples.
Example 1.23. Let ω:K×K→K, (x, y)7→ x. The derivative of this function is
denoted by dω. First we fix y, then we get
dω :k×k−→ k
(X, Y )7−→ X.
Then the adjoint map of is given by
dω∗:k∗−→ k∗×k∗
ξ7−→ (ξ, 0)
Example 1.24. Let Φ : K×K→K, (x, y)7→ xy. The derivative of this function is
denoted by dΦ. We start first by fixing y, then we have
dΦ(x) : k−→ k
d
dt|t=0 x etX 7−→ d
dt|t=0 x etX y
where d
dt|t=0 x etX y=d
dt|t=0 x y etAd y−1(X).
18 1.4 Distributions
Then
dΦ(x) : X7→ Ad y−1(X).
Similarly, we will fix x, then we have
dΦ(y) : k−→ k
d
dt|t=0 y etY 7−→ d
dt|t=0 x y etY
Then
dΦ(y) : Y7→ Y.
Therefore,
dΦ(x, y) : k×k−→ k
(X, Y )7→ dΦ(x)(X) + dΦ(y)(Y)
= Ad y−1(X) + Y
Then the adjoint map of is given by
dΦ∗:k∗−→ k∗×k∗
ξ7−→ (Ad∗y(ξ), ξ)
1.4 Distributions
Definition 1.25. Let Xbe an open set in Rn. A distribution uin Xis a linear form
on C∞
0(X) such that for every compact set B⊂Xthere exist a constant Cand integer
ksuch that:
|hu|ϕi| ≤ CX
|α|≤k
sup
B|∂αϕ|,∀ϕ∈C∞
0(B).(1.18)
The set of all distribution in Xis denoted by D0(X) (see [Ho83], Definition 2.1.1).
Distributions can be restricted to open subsets. Let u∈D0(X), then the support of
u, denoted by supp u, is the set of points in Xhaving no open neighborhood to which
the restriction of uis 0.
Definition 1.26. If u∈D0(X), then the singular support of u, denoted by sing supp u,
is the set of points in Xhaving no open neighborhood to which the restriction of uis
a smooth function.
If u∈D0(X) has a compact support, then hu|ϕican be defined for all ϕ∈C∞(X).
Let ψ∈C∞
0(X) and ψ= 1 in a neighborhood of supp u, so we define hu|ϕi:= hu|ψϕi.
This definition does not depend on the choice of ψ. It follows from (1.18) and the
product rule that
|hu|ϕi| ≤ X
|α|≤k
sup
B|∂αϕ|,∀ϕ∈C∞(X),
19 1.4 Distributions
where B= supp ϕ. Conversely, suppose that we have a linear form von C∞(X) such
that for some constant C, integer k, and some compact set L⊂X
|hv|ϕi| ≤ X
|α|≤k
sup
L|∂αϕ|,∀ϕ∈C∞(X).
Then the restriction of vto C∞
0(X) is a distribution with support contained in L. We
denote the space of distribution with compact support E0(X) (see [Ho83], p. 44).
Remark 1.27. The set E0(X) can be identified with the set of distributions in E0(Rn)
with supports contained in X(see [Ho83], Theorem 2.3.1 ).
Theorem 1.28. The Fourier transformation of a distribution u∈E0(Rn)is the func-
tion
bu(ξ) = hux|e−ihξ|xii.(1.19)
The right-hand side is also defined for every complex vector ξ∈Cnand is entire analytic
function of ξ, called the Fourier-Laplace transformation of u(see [Ho83], Theorem
7.1.14).
Remark 1.29. If Bis a compact set in Rn,u∈E0(Rn) is a linear form on C∞(Rn)
with supp u⊂B, such that, if Ω is a neighborhood of B,
|hu|ϕi| ≤ X
|α|≤k
sup
Ω|Dαϕ|,∀ϕ∈C∞(Rn).
where Dα:= (1/i)∂α. One can extend hu|ϕiby continuity to all ϕ∈C∞(Ω).
The derivatives of an analytic function can be estimated in a compact set by the
maximum of its absolute value in a neighborhood. Therefore the following definition
makes sense.
Definition 1.30. Let B⊂Cnbe a compact set, then A0(B) , the set of analytic
functionals carried by B, is the space of linear forms uon the space Aof entire
analytic functions in Cn, such that for every neighborhood Ω of B
|hu|ϕi| ≤ CΩsup
Ω|ϕ|,∀ϕ∈A.
Definition 1.31. Let Xbe an open and bounded set in Rn. The spaces of hyper-
functions which is denoted by B(X) can be defined as follows:
B(X) := A0(X)/A0(∂X)
(see [Ho83], Definition 9.2.1).
The following theorem is a special case of Theorem 1.56 below.
20 1.5 Microlocal Analysis
Theorem 1.32. Let Uj⊂Rnj,j= 1,2, be an open sets, and f:U1→U2aC∞map
such that f0(x)is surjective for all x∈U1. Then there is a unique continuous linear
map f∗:D0(U2)→D0(U1)such that f∗u=u◦fwhen u∈C(U2). One calls f∗uthe
pull-back of uby f(see [Ho83], Theorem 6.1.2).
Definition 1.33 (Distributions on a Manifold).Let Mbe a smooth manifold.
Assume that to every coordinate system κ:Uκ⊂M→Vκ⊂Rnin Mwe have a
distribution uκ∈D0(Vκ) such that
ueκ= (κ◦eκ−1)∗uκin eκ(Uκ∩Ueκ).(1.20)
Here (·)∗denotes the pull-back of the map κ◦eκ−1. We call the system uκa distribution
uin M. The set of all distributions in Mis denoted by D0(M).
Theorem 1.34. Let Fbe an atlas for M. If for every κ∈Fwe have a distribution
uκ∈D0(Vκ), and (1.20) is valid when κand eκbelongs to F, it follows that there exists
one and only one distribution u∈D0(M)such that u◦κ−1=uκfor every κ∈F(see
[Ho83], Theorem 6.3.4).
In our case the smooth manifold is a compact Lie group K. We can define the space
of distributions as the dual space of C∞
0(K). Then we can identify functions with
distributions via the Haar measure dk:
L2(K)→D0(K), f 7→ (ϕ7→ ZK
f(k)ϕ(k)dk)
Theorem 1.35. Let Kbe a a connected compact Lie group and ube a function on K
which has the Fourier expansion u=Pλ∈L∩Cϕλdue to Peter-Weyl Theorem (compare
(1.17)), then
1. u=Pλ∈L∩Cϕλis a C∞-function on Kif and only if for any m∈N, there exists
a positive number Lmsuch that
kϕλkL2≤Lm(1 + |λ|)−m.
2. u=Pλ∈L∩Cϕλis a distribution on Kif and only if there are positive numbers
mand Lsuch that
kϕλkL2≤L(1 + |λ|)m.
(see [Se65])
1.5 Microlocal Analysis
We will give the definitions of the wave front set of a distribution, the pull-back, and
the push-forward. Let Γ ⊆Rn\{0}conic set (i.e., if ξ∈Γ, t > 0⇒tξ ∈Γ,∀t∈R)
21 1.5 Microlocal Analysis
Definition 1.36. Let ube a distribution in an open subset Xof Rn. The wave front
set of uis the subset WF(u)⊆X×(Rn\{0}) defined as follows: (x0, ξ0)/∈WF(u) iff
there exists a conic neighborhood Γ of ξ0and ϕ∈C∞
0(X) with ϕ(x0)6= 0 such that:
|cϕu(ξ)| ≤ CN(1 + |ξ|)−N, N = 1,2, . . . , ξ ∈Γ,(1.21)
(see [Ho83], p. 252). Notice there is an equivalent statement which often used
cϕu(ξ) = O(|ξ|)−N,as Γ 3ξ−→ ∞ (1.22)
for all N. Another equivalent formulation is: There is a neighborhood Uof xsuch that
(1.22) holds for every ϕ∈C∞
0(U). Now 1.22 is equivalent to
hu|e−ithξ|·iϕi=O(t−N) for t−→ ∞ (1.23)
uniformly in |ξ|= 1 where ξ∈Γ, for all Nwhere R3t≥1. Here we tested
the distribution uwith oscillatory test function e−ithξ|·iϕ(x) and then investigated the
asymptotic behavior letting the frequency variable tgo to ∞(see [Du96], p.15).
Remark 1.37. For any linear differential operator with C∞-coefficients Pwe have
WF(Pu)⊂WF(u)
(see [Ho83], p. 256).
Definition 1.38. Let Xbe an open set in Rnand Γ be a closed cone in X×Rn\{0},
then we define the following:
D0
Γ(X) := nu∈D0(X)|WF(u)⊂Γo
Lemma 1.39. A distribution u∈D0(X)is in D0
Γ(X)if and only if for every φ∈
C∞
0(X)and every closed cone in V⊂Rnwith
Γ∩(supp(φ)×V) = ∅(1.24)
we have;
sup
ξ∈V|ξ|N|c
φu(ξ)|<∞N∈N
Proof. See [Ho83], Lemma 8.2.1.
Definition 1.40. For a sequence uj∈D0
Γ(X) and u∈D0
Γ(X) we say that uj→uin
D0
Γ(X) if
uj→uin D0(X)(weakly) (1.25)
sup
ξ∈V|ξ|N|c
φu(ξ)−d
φuj(ξ)| → 0, j → ∞,(1.26)
for N∈Nif φ∈C∞
0(X) and Vis a closed cone in Rnsuch that (1.24) is valid. Since
(1.25) implies that d
φuj→c
φu uniformly on every compact set and Nis arbitrary in
(1.26), we can replace (1.26) by
sup
j
sup
ξ∈V|ξ|N|d
φuj(ξ)|<∞N∈N(1.27)
22 1.5 Microlocal Analysis
Theorem 1.41. For every u∈D0
Γ(X)there is a sequence uj∈C∞
0(X)such that
uj→uin D0
Γ(X).
Proof. See [Ho83], Theorem 8.2.3.
We need more definitions. Let Pbe a differential operator of order mwith C∞-
coefficients defined on a manifold X. In local coordinates we have
P=P(x, D) = X
|α|≤m
aα(x)Dα.
The principal symbol Pmis invariantly defined on T∗X\0
Pm(x, ξ) = X
|α|=m
aα(x)ξα.
The characteristic variety (set) Char Pis defined by
Char P:= {(x, ξ)∈T∗X\0|Pm(x, ξ) = 0}.(1.28)
(See [Ho83], p. 271.)
Theorem 1.42. If Pis a differential operator of order mwith C∞-coefficients on a
manifold X, then
WF(u)⊂Char P∪WF(Pu), u ∈D0(X),
(see [Ho83], Theorem 8.3.1).
If Pis elliptic, that is, Pm(x, ξ)6= 0 in T∗X\0, then
WF(u) = WF(Pu), u ∈D0(X).
(see [Ho83], Corollary 8.3.2)
Proposition 1.43. Let Xbe a manifold and Ya submanifold with co-normal bundle
N∗(Y) := (y, ξ)|y∈Y, ξ ∈T∗
ι(y)X, hξ, TyYi= 0,
where ι:Y ,→Xdenotes the inclusion. For every distribution uin Xwith
WF(u)∩N∗(Y) = ∅
the restriction u|Yto Yis well defined distribution on Y, the pull-back by the inclusion.
Proof. see [Ho83], Corollary 8.2.7.
Remark 1.44. Let Xbe a smooth manifold. We will give an equivalent definition of
D0
Γ(X) using pseudo-differential operator instead of Fourier transformation. First we
will give the definition of a pseudo-differential operator.
23 1.5 Microlocal Analysis
Definition 1.45. Let r∈R.a(x, η)∈C∞(Rm×Rn) is a symbol of order ≤riff
|∂β
x∂α
ηa(x, η)| ≤ Cα, β(1 + |η|)r−|α|(∀α, β ∈NN
0).(1.29)
The lowest upper bounds of the constants in (1.29) are seminorms on the symbol space
Srturning it into a Fr´echet space. S∞=∪rSrand S−∞ =∩rSr(see[Ho85], Definition
18.1.1).
Let Sdenotes the Schwartz class and S0the space of tempered distributions.
Theorem 1.46. Let a∈Smand u∈S. Then
Au(x) = a(x, D)u(x) = (2π)−nZRn
eihx|ξia(x, ξ)bu(ξ)dξ (1.30)
defines a function a(x, D)u∈S, and the bilinear map (a, u)7→ a(x, D)uis continuous.
The commutator with Djand the multiplication by xjare
[a(x, D)u, Dj] = i(∂xja(x, D)) (1.31)
[a(x, D), xj] = −i(∂ξja(x, D)).(1.32)
One calls A=a(x, D)apseudo-differential operator of order mand denoted by
A∈Ψm(see [Ho85], Theorem 18.1.6).
Remark 1.47. Due to the definition of buin (1.30), it follows that the Schwartz
kernel Kof Ais given by
K(x, y) = (2π)−nZRn
eihx−y|ξia(x, ξ)dξ, (1.33)
which is a partial Fourier transformation of a(see [Ho85], p. 69).
Theorem 1.48. Let a∈Sm. We denote by K∈S0(Rn×Rn)the Schwartz Kernel of A
defined by (1.33). Then K∈Cj(Rn×Rn)if m+j+n < 0, and K∈C∞(Rn×Rn\4)
for any mif 4is the diagonal {(x, x), x ∈Rn}. More precisely,
WF(K)⊂ {(x, x, η, −η)|x, η ∈Rn}(1.34)
which is the the co-normal bundle of 4. We have, for every u∈S0,
WF(Au)⊂WF(u),(1.35)
sing supp Au ⊂sing supp u. (1.36)
If a∈S−∞, then Au ⊂C∞(see [Ho85], Theorem 18.1.16).
We can define pseudo-differential operators on a manifold as follows
24 1.5 Microlocal Analysis
Definition 1.49. Let Xsmooth manifold. A pseudo-differential operator of order mon
Xis a continuous linear map A:C∞
0(X)→C∞(X) such that for every local coordinate
patch Uκ⊂Xwith coordinates Uκ3x→κ(x)∈Vκ⊂Rnand all φ, ψ ∈C∞
0(Vκ) the
map
S0(Rn)3u−→ φ(κ−1)∗Aκ∗(ψu) (1.37)
is in Ψm(X). We can extend A, by continuity, to a map E0(X)→D0(X). (see [Ho85],
Definition 18.1.20)
Definition 1.50. The (pseudo-differential) operator Ain Xis said to be properly
supported if both projections from the support of the kernel in X×Xto Xare proper
maps, that is, for every compact subset B⊂Xthere is a compact set B0⊂Xsuch
that
supp u⊂B⇒supp Au ⊂B0.(1.38)
(see [Ho85], Definition 18.1.21)
Definition 1.51. If a∈Sm(T∗X) is a principal symbol of A∈Ψm(X) then Ais said
to be non-characteristic at (x0, ξ0)∈T∗X\0 if ab−1∈S−1in a conic neighborhood of
(x0, ξ0) for some b∈S−m(T∗X). The set of characteristic points is denoted by Char A.
The operator is said to be elliptic at a non-characteristic point (see [Ho85], Definition
18.1.25). This definition is independent of the choice of a. The proof of Theorem 18.1.9
in [Ho85] shows that in local coordinates an equivalent condition is that; Ais elliptic
at (x0, ξ0) if and only if there exists a neighborhood Uof x0, a conic neighborhood Vof
ξ0, and a constant C > 0 such that |a(y, η)|≥|η|n/C for y∈U,η∈V,n∈R,|η|> C.
If Ahas a homogeneous principal symbol a, the condition is equivalent to a(x0, ξ0)6= 0,
then the last definition of Char Acoincides with (1.28) for differential operator.
Since WF(K), Kdenotes the kernel of the pseudo-differential operator A, is contained
in the diagonal of T∗X\0×T∗X\0 it is natural to identify it with a conic subset of
T∗X\0. We shall write
WF(A) = {γ∈T∗X\0|(γ, γ)∈WF(K)}(1.39)
Theorem 1.52. If u∈D0(X)we have for every m∈R
WF(u) = \Char A(1.40)
where the intersection is taken over all properly supported A∈Ψm(X)such that Au ∈
C∞(X)(see [Ho85], Theorem 18.1.27).
Definition 1.53. Let Xbe a smooth manifold and Γ be a closed cone in T∗X\0. The
convergence of a sequence, uj→uin D0
Γ(X) is equivalent to
uj→uin D0(X)(weakly) (1.41)
and that there exists for every (x, ξ)∈(T∗X\0) \Γ, a pseudo-differential operator
A∈Ψm(X) such that (x, ξ)/∈Char A, WF(A)∩Γ = ∅, and Auj→Au ∈C∞(X). (see
[Ho85], p. 89 remark following Theorem 18.1.28)
25 1.5 Microlocal Analysis
Remark 1.54. To prove that uj→uin D0
Γ(X) it suffices to show that for every closed
cone Γ1⊂T∗X\0 such that Γ ⊂◦
Γ1the following holds
uj→uin D0
Γ1(X).(1.42)
This is clear from the last definition since for a given (x, ξ)∈(T∗X\0) \Γ we can
find Γ1such that (x, ξ)/∈Γ1, a pseudo-differential operator A∈Ψm(X) such that
(x, ξ)/∈Char A, WF(A)∩Γ1=∅, and Auj→Au ∈C∞(X).
Proposition 1.55. Let Kbe a Lie group and Γbe a closed cone in T∗K\0. Let
(uj)be a sequence in D0
Γ(K)with uj→uin D0(K). Assume that for every (x0, ξ0)∈
(T∗K\0) \Γ, there exist an open neighborhood U⊂Kof x0, a real-valued function
ϕ∈C∞(U×k∗), linear in the second variable, η0∈k∗where ξ0=ϕ0
x(x0, η0), and an
open conic neighborhood W0⊂k∗\0of η0where det ϕ00
xη(x, η)6= 0 for (x, η)∈(U×W0),
such that
sup
j
sup
η∈W0|η|N|huj|e−iϕ(·,η)ψi| <∞ ∀N∈N,(1.43)
and all ψ∈C∞
0(U). Then ujconverges to uin D0
Γ(K).
Proof. Let (x, ξ)∈(T∗K\0) \Γ. Choose U, W0, and ϕas in the hypothesis, with Ua
coordinate neighborhood, and ϕ00
xη 6= 0 on U×W0. Let ψ∈C∞
0(U) and a∈S0(k∗) is
elliptic at η0with supp(a)bW0. This implies that (x0, ξ0)/∈Char(A). Consider the
operator A:C∞(K)→D0(K) with Schwartz Kernel compactly supported in U×U.
The operator Ais, in terms of local coordinates, given by
Au(y0) = ZW0ZU
eiϕ(y0,η)−iϕ(y,η)ψ(y0)a(η)ψ(y)u(y)dydη
=ZW0
eiϕ(y0,η)ψ(y0)a(η)hv|e−iϕ(·,η)ψidη, (1.44)
where y0, y ∈U. Observe that the singular support of the Schwartz Kernel of Ais
contained in the diagonal (compare theorem 1.48) and Kcan be covered with open sets
U⊂K. Let u∈D0(K). Using theorem 1.52 we obtain that WF(u) = TChar A, where
the intersection is taken over all properly supported A∈Ψ0(K) such that Au ∈C∞(K).
The distributions space D0
Γ(K) is equipped with a local convex topology (see definition
1.24). Moreover, for every u∈D0
Γ(K) there exists a sequence uj∈C∞
0(X) such that
uj→u∈D0
Γ(K) (see theorem 1.41). From definition 1.53 we obtain that a sequence
uj→u∈D0
Γ(K) is equivalent to uj→u∈D0(K) (weakly) and the existence of,
for every (x, ξ)∈(T∗K\0) \Γ, a pseudo-differential operator A∈Ψ0(K) such that
(x, ξ)/∈Char A, and Auj→Au ∈C∞(K).
We want to show that (Auj) is bounded sequence in C∞(K). Since A∈Ψ0(K), then
we can estimate the integral in (1.44) using (1.43),
|Auj(y0)| ≤ ZW0|eiϕ(y0,η)ψ(y0)a(η)|dη, ≤cZW0|η|−Ndη ≤C(1.45)
26 1.5 Microlocal Analysis
where the constants c, C > 0 are independent of j. This implies that (Auj) and
(DαAuj) are bounded sequences in L∞(K) for some α∈NN
0. This implies that (Auj)
is bounded sequence in C∞(K). Using Ascoli’s theorem and the continuity of Aon
D0(K) we conclude that Auj→Au ∈C∞(K). Since we can choose aand ψsuch that
WF(A)∩Γ = ∅, then ujconverges to uin D0
Γ(K).
Let Xand Ybe an open subsets of Rmand Rnrespectively, and f:X→Y
be a smooth map. The map f∗:C∞(Y)→C∞(X), u7→ u◦f, have a unique
continuous linear f∗:D0(Y)→D0(X) if f0is surjective (see Theorem 1.32) or under
some conditions which will be presented in theorem 1.56. First we recall the definition
of the pull-back of a distribution.
The pull-back of distribution is define as follows: Let Xand Ybe an open subsets
of Rmand Rnrespectively, and f:X→Ybe a smooth map. For u∈C∞
0(Y) and
ϕ∈C∞
0(X) we have
hf∗u|ϕi= (2π)−nZRmZRn
eihη,f(x)ibu(η)ϕ(x)dηdx (1.46)
We will rewrite this equality to generalize the pull-back operator to distributions which
have their wavefront sets in suitable position. The assumption on the wave front set
arises from the geometry. Let Kbe compact subset of Xwhich denotes the support of
ϕand Vbe closed cone in Rn\{0}. Moreover, we assume
x∈K, η ∈V⇒(f0(x))tη6= 0.
Choose Φ ∈S0such that supp(Φ) ⊂Vand |(f0(x))tη| ≥ c|η|when x∈Kand
Φ(η)6= 0. We note that
∂
∂xj
eihη|f(x)i=ihη|∂f(x)
∂xjieihη|f(x)i.
Therefore we can find a differential operator L=Pjaj(x, η)∂/∂xjwith coefficients
aj(x, η)∈S−1such that
Φ(η)Leihη|f(x)i= Φ(η)eihη|f(x)i.
Now let u∈E0(Y), WF(u)⊆Y×V. Using partial integrations, we rewrite equation
(1.46) as follows:
hf∗u|ϕi= (2π)−nZRmZRn
eihη,f(x)iΦ(η)bu(η)(Lt)Nϕ(x)dηdx
+(2π)−nZRmZRn
eihη,f(x)i(1 −Φ(η))bu(η)ϕ(x)dηdx
When N∈Nis sufficiently large the integrals exist and we take this equation as the
definition of the pull-back of distribution u(see [Du96], p. 19). More precisely, the
following is true.
27 1.5 Microlocal Analysis
Theorem 1.56. Let Xand Ybe an open subsets of Rmand Rn, respectively, and let
f:X→Ybe a smooth map. Denote the set of the normals of the map by
Nf={(f(x), η)∈Y×Rn|(f0(x))tη= 0}.
where (f0(x))tis the transpose of f0(x). Then the pull-back f∗ucan be defined in one
and only way for all u∈D0(Y)with
Nf∩WF(u) = ∅(1.47)
so that f∗u=u◦fwhen u∈C∞and for any closed conic subset Γ⊂Y×Rn\{0}
with Γ∩Nf=∅we have a continuous map f∗:D0
Γ(Y)→D0
f∗Γ(X), where
f∗Γ = {x, (f0(x))tη|(f(x), η)∈Γ}.
In particular we have for every u∈D0(Y)satisfying (1.47)
WF(f∗u)⊂f∗WF(u).
(see [Ho83], Theorem 8.2.4)
Remark 1.57. It is useful to introduce the following sets
Cf:= n(x, y;ξ, η)|y=f(x),(f0(x))tη+ξ= 0o(1.48)
and
C0
f:= {(x, y;ξ, η)|(x, y;−ξ, η)∈Cf}(1.49)
Then
Nf=C0
f◦{(x, ξ)|ξ= 0}(1.50)
and
f∗Γ = C0
f◦Γ.(1.51)
Remark 1.58. If Xis a smooth manifold and u∈D0(X) we can now define WF(u)⊂
T∗X\{0}so that the restriction to a coordinate patch Xκis equal to κ∗WF(u◦κ−1).
In fact when fis a diffeomorphism between open set in Rnit follows from last theorem
1.56 that WF(f∗u) is the pull-back of WF(u) considered as a subset of the cotangent
bundle. Hence the definition of the pull-back is independent of the choice of local
coordinates. Moreover, WF(u) is a closed subset of T∗X\ {0}which is conic in the
sense that the intersection with the vector space T∗
xXis a cone for every x∈X(see
[Ho83], p. 265).
The following remark is a supplement to proposition 1.43 using theorem 1.56.
28 1.5 Microlocal Analysis
Remark 1.59. Let Xbe a manifold and Ya submanifold with the inclusion ι:Y ,→X.
From proposition 1.43 we obtain that, for every distribution udefined on Xwith
WF(u)∩N∗(Y) = ∅, the restriction u|Yis a well-defined distribution on Yand u|Yis
the pull-back of uby the inclusion ι. From theorem 1.56 and remark 1.57 we obtain
the following
Cι:= n(y, x;η, ξ)|x=ι(y),(ι0(y))tξ+η= 0o,(1.52)
here (ι0(y))t:T∗
ι(y)X→T∗
yYand
C0
ι:= {(y, x;η, ξ)|(y, x;−η, ξ)∈Cι}(1.53)
Then
Nι=C0
ι◦{(y, η)|η= 0}=N∗(Y),(1.54)
and
C0
ι◦WF(u) = {(y, η)∈T∗Y| ∃(x, ξ)∈WF(u), x =ι(y),(ι0(y))tξ=η}.(1.55)
We can define the tensor product u⊗v∈D0(X×X) of two distributions u, v ∈
D0(X). The corresponding bilinear map is separately sequentially continuous. In case
u, v ∈C(Rn) we have u⊗v(x, y) := u(x)v(y).
Theorem 1.60. Let X, Y be smooth manifolds and u∈D0(X),v∈D0(Y), then
WF(u⊗v)⊂(WF(u)×WF(v)) ∪((supp u×{0})×WF(v))
∪(WF(u)×(supp v×{0})) (1.56)
Proof. See [Ho83], Theorem 8.2.9.
Proposition 1.61. Let Xand Ybe a smooth manifolds. Let Φ : X→Ybe a proper
map. The dual map Φ∗= (Φ∗)t:D0(X)→D0(Y)is defined and continuous. It is called
the push-forward by Φ. Moreover, for a given closed cone Γ⊂T∗X\0the restriction
Φ∗:D0
Γ(X)→D0
Φ∗Γ(Y)is sequentially continuous. Here we get that the set Φ∗Γis
contained in
{(y, η)∈T∗Y\{0} | y= Φ(x)∧(x, (dxΦ(x))tη)∈Γ, x ∈X}
∪ {(y, η)∈T∗Y\{0} | y= Φ(x)∧(dxΦ(x))tη= 0}(1.57)
where dΦ(x) : T∗
xX→T∗
Φ(x)Y.
Proof. See [Du96], Proposition 1.3.4. and [FJ98], Proposition 11.3.3.
2 Wave Front Set and Fourier Coefficients
In this chapter we are going to calculate the wave front set of a distribution defined on
connected, compact, and semisimple Lie groups.
Let Kbe a connected, semisimple, and compact Lie group and πan irreducible
representation of Kon L2(K). We fix a Cartan subalgebra tCof kCand a positive
system of roots Λ+= Λ+(kC,tC). We denote by C⊂it∗the closure of the Weyl chamber
(see definition 1.12) and L⊂it∗the weight lattice (see definition 1.15). Recall that b
K
denotes the set of equivalence classes of irreducible representations of K. We identify
b
Kwith λ∈L∩Cusing the theorem of Highest Weights 1.14. Due to the Peter-Weyl
Theorem 1.19 we have
L2(K) = M
λ∈L∩C
Eλ.(2.1)
For u∈L2(K) we have the Fourier series
u=X
λ∈L∩C
ϕλ
where ϕλis an element of Eλ(compare theorem 1.19 Eλ=Eπλ). Here ϕλ=dπλu∗χπλ,
dπλis the dimension of the representation π∈b
K,χπλis the character of π. If u∈D0(K)
then we have the same Fourier series because; χπλ∈C∞
0(K), u∗χπλ∈Eλ⊂C∞(K),
and the convolution of a distribution with a C∞
0(K)-function is defined by u∗χπλ(x) =
Rku(xy−1)χπλ(y)dy, where the duality bracket written as an integral. Hence, the Fourier
series uconverges to uin D0(K).
In the following Ωwill always denote a closed cone contained in C\0⊂it∗.
2.1 Wave Front Set of Truncated Distributions
In this section we aim to calculate the wave front set of a truncated distribution uΩ=
Pλ∈L∩Ωϕλ.
Remark 2.1. Let X∈kand e
Xbe the cooresponding left invariant vector field (see
section 1.3). Let (x, ξ)∈K×k∗≃T∗K. We want to show that the principal symbol of
e
X, denoted by σ(e
X), at the point (x, ξ), equals ihξ, Xiwhere h·,·i denotes the duality
brackets k∗×k→C. Choose ψ∈C∞(K) with dψ(e) = ξ, where eis the identity
element of K. Set ϕ:= (Lx−1)∗ψwhere (Lx−1)∗denotes the pull-back of ψby the left
translation Lx−1for x∈K. Then dψ(x)=(x, ξ) under the identification T∗
xK≃k∗.
Then the principal symbol of e
X, in terms of local coordinates, can be define as follows,
30 2.1 Wave Front Set of Truncated Distributions
for 0 < t ∈Rand x∈K,
σ(e
X)(dψ(x)) = lim
t→∞
1
te−itϕ(x)e
Xeitϕ(x)(x)
= lim
t→∞
1
te−it(Lx−1)∗ψ(x)e
Xeit(Lx−1)∗ψ(x)(x)
= lim
t→∞
1
t(Lx−1)∗e−itψ(x)(Lx−1)∗e
Xeitψ(x)(x)
= lim
t→∞
1
t(Lx−1)∗e−itψ(x)e
Xeitψ(x)(x)
= lim
t→∞
1
te−itψ(x)e
Xeitψ(x)(e)
= lim
t→∞
1
te−itψ(e)eitψ(e)it e
Xψ(x)(e)
=ihdψ(e),e
X(e)i
=ihξ, Xi.
This imply that σ(e
X)(x, ξ) = ihξ, Xi.
Proposition 2.2. Let uΩ=Pλ∈L∩Ωϕλbe a distribution on Kand all ϕλare highest
weight vectors with respect to the left action. Then uΩconverges in D0
Γ(K)where
Γ := K×(−i)Ω is a closed cone in T∗K. In particular WF(uΩ)⊂Γ.
Proof. We assume also that Ω is convex. Let each ϕλ∈Eλbe a highest weight vector
of the highest weight λwith respect to the left action on L2(K). We denote by h·,·i
denotes the duality bracket t∗
C×tC→C. Let n:= Lα∈Λ+kα, then we have
X·ϕλ= 0 for X∈n(2.2)
X·ϕλ=hλ, Xiϕλfor X∈t(2.3)
for the left action. Since X∈kis not only an element of the Lie algebra but acts as a
differential operator of first order with a smooth coefficients which we denote by a tilde
e
Xto distinguish from the Lie algebra element, then X·ϕλ=e
Xϕλ.
From (2.2) and because e
Xis a continuous operator we get
e
XX
λ∈L∩Ω
ϕλ=X
λ∈L∩Ωe
Xϕλ= 0
Furthermore, due to theorem 1.42 we get
WF(uΩ)⊂Char e
X, for X∈n
where Char e
Xdenotes the characteristic variety of the operator e
X(see (1.28)). More-
over, using remark 2.1, we get that
Char e
X=K×X⊥for X∈n.
31 2.1 Wave Front Set of Truncated Distributions
Hence,
WF(uΩ)⊂\
X∈n
K×X⊥=K×t∗.
Recall the identification of the cotangent bundle T∗K=K×k∗(see 1.3).
Now we consider equation (2.3). Let U0be a neighborhood of 0 ∈kand Vx
0be
a neighborhood of x∈Kdefined by Vx
0=Lx◦exp(U0), where Lxdenotes the left
translation on K. Let κ:Vx
0⊂K→U0⊂k,y7→ exp−1(x−1·y) be a chart (y∈Vx
0).
Then κis a diffeomorphism: Vx
0→U0(see [DK00], Theorem 1.6.3).
Let e
Xand ≈
Xbe differential operators of first order with real smooth coefficients such
that the following diagram commutes
D0(Vx
0)
e
X//
(κ−1)∗
D0(Vx
0)
(κ−1)∗
D0(U0)
≈
X//D0(U0)
where (κ−1)∗denotes the pull-back operator (see theorem 1.56). In local coordinates
≈
Xis given by Pjcj(y)∂/∂yj.
The pull-back of Haar measure is (κ−1)∗dy =Jκ(Y)dY ,Jκ(Y) denotes the Jacobian
of κ−1. The function Jκ(Y) is smooth and will be calculated explicitly in remark 2.5.
To characterize the wave front set we use definition 1.53 and proposition 1.55. Let
ϑ∈C∞(U0×t∗) be a real-valued function satisfying the conditions in proposition 1.55.
h·|·idenotes the duality brackets between D0(K)×C∞
0(K) and ≈
X
t
denotes the transpose
of ≈
X. We choose a suitable localization function ψ∈C∞
0(k). Then, for Y∈U0⊂k,
X∈t,ξ∈k∗, we have
hλ, Xi(κ−1)∗(ϕλ)|ψe−iϑ(·,ξ)=Zkhλ, Xiϕλ(κ−1(Y))ψ(Y)e−iϑ(Y,ξ)Jκ(Y)dY
(2.3)
=Zk
(e
Xϕλ)(κ−1(Y))ψ(Y)e−iϑ(Y,ξ)Jκ(Y)dY
=Zk
≈
X(ϕλ(κ−1(Y)))ψ(Y)e−iϑ(Y,ξ)Jκ(Y)dY
=−Zk
ϕλ(κ−1(Y)) ≈
X
tψ(Y)e−iϑ(Y,ξ)Jκ(Y)dY
=−Z
k
ϕλ(κ−1(Y)) ≈
X
t
(ψ(Y)Jκ(Y))e−iϑ(Y,ξ)dY
−Z
k
ϕλ(κ−1(Y))ψ(Y)Jκ(Y)≈
X
t
(e−iϑ(Y,ξ))dY
32 2.1 Wave Front Set of Truncated Distributions
Recall that ≈
X=Pjcj(Y)∂/∂yjin local coordinates. Then
≈
X
t
(e−i ϑ(Y,ξ)) =
n
X
j
∂
∂yjcj(Y)e−i ϑ(Y,ξ)
=
n
X
j
cj(Y)∂
∂yj
e−i ϑ(Y,ξ)+
n
X
j∂
∂yj
cj(Y)e−i tϑ(Y,ξ)
=≈
Xe−i ϑ(Y,ξ)+
n
X
j∂
∂yj
cj(Y)
| {z }
=:Z(Y)
e−i tϑ(Y,ξ)(2.4)
Claim 2.3. Let ≈
Xbe a vector field with a real smooth coefficients. Then due to the
method of characteristics we obtain
≈
X(e−i ϑ(Y,ξ)) = −ihξ, Xie−i ϑ(Y,ξ),(2.5)
where ϑ∈C∞(U0×t∗) is a real-valued function satisfying the conditions in proposition
1.55.
Proof of claim 2.3.It is sufficient to consider the equation
≈
X ϑ(Y, ξ) = hξ, Xi
n
X
j
cj(y)∂ϑ
∂yj
(Y, ξ) = hξ, Xi(2.6)
This is a linear partial differential equation of first order. Assume, without loss of
generality, cn(y)6= 0, then due to the method of characteristics (see [Fo95], p. 34-38)
we can find a solution ϑ. It satisfies along any characteristic curve, i.e., a solution of
dyj
dt =cj(Y)
yj(0) = (y0)j,
the differential equation,
hξ, Xi=dϑ
dt (Y, ξ) =
n
X
j
dyj
dt
∂ϑ
∂yj
(Y, ξ) =
n
X
j
cj(Y)∂ϑ
∂yj
(Y, ξ)
ϑ(Y, ξ) = hξ, Y ion yn= 0.
Observe that in local coordinates we can give a basis (y1,··· , yn) = Y∈U0⊂k, and
(ξ1,··· , ξn) = ξ∈k∗, respectively.
33 2.1 Wave Front Set of Truncated Distributions
Due to claim 2.3 and (2.4) we get that
hλ, Xi(κ−1)∗(ϕλ)|ψe−iϑ(·,ξ)=−Z
k
ϕλ(κ−1(Y)) ≈
X
t
(ψ(Y)Jκ(Y))e−iϑ(Y,ξ)dY
−Z
k
ϕλ(κ−1(Y))ψ(Y)Jκ(Y)≈
X
t
(e−iϑ(Y,ξ))dY
=−Z
k
ϕλ(κ−1(Y)) ≈
X
t
(ψ(Y)Jκ(Y))e−iϑ(Y,ξ)dY
+Z
k
ϕλ(κ−1(Y))ψ(Y)Jκ(Y)hiξ, Xie−i ϑ(Y,ξ)dY
−Z
k
ϕλ(κ−1(Y))ψ(Y)Jκ(Y)Z(Y)e−i ϑ(Y,ξ)dY
Then we have the following
Zkhλ−iξ, Xiϕλ(κ−1(Y))ψ(Y)e−i ϑ(Y,ξ)Jκ(Y)dY =
−Z
k
ϕλ(κ−1(Y)) ≈
X
t
(ψ(Y)Jκ(Y))e−iϑ(Y,ξ)dY
−Z
k
ϕλ(κ−1(Y))ψ(Y)Jκ(Y)Z(Y)e−i ϑ(Y,ξ)dY.
The term on the left hand side equals to hλ−iξ, Xi(κ−1)∗(ϕλ)|ψe−iϑ(·,ξ). The terms
on the right hand side, using Cauchy-Schwarz inequality, are bounded by a constant
C1times kϕλk. Iterating 2Ntimes we obtain
|hλ−iξ, Xi2N(κ−1)∗(ϕλ)|ψe−iϑ(·,ξ)| ≤ C1kϕλk,
where C1is independent of λ∈L∩Ω and ξ∈k∗. Since the L2-norms of the Fourier
coefficients are polynomially bounded we obtain, for some N0∈N
|(κ−1)∗(ϕλ)|ψe−iϑ(·,ξ)| ≤ C1|hλ−iξ, Xi|−2N|λ|N0.(2.7)
Claim 2.4. Let Ω be a non-empty convex cone and Va closed cone contained in ik∗
such that Ω ∩V=∅. Then there exists C2:= CΩ, V >0 and X∈tsuch that the
following estimate holds true:
min
|λ|+|ξ|=1
iξ∈V, λ∈Ω
|hλ−iξ, Xi| ≥ C2.(2.8)
In particular, we get
|hλ−iξ, Xi| ≥ C2(|λ|+|ξ|)∀λ∈Ω,∀iξ ∈V(2.9)
34 2.1 Wave Front Set of Truncated Distributions
Proof of claim 2.4.Due to the condition above we get from Hahn-Banach separation
theorem (see [We05], Theorem III.2.5): There exists an X∈t∀λ∈Ω,∀iξ ∈Vsuch
that hλ−iξ, Xi 6= 0. The function |hλ−iξ, Xi| is continuous and defined on a compact
set where |λ|+|ξ|= 1, then it will take its minimum on this set which proves assertion
(2.8). Moreover, due to the homogeneity of λ, ξ, X we get assertion (2.9).
Therefore, we obtain from claim 2.4 and estimate (2.7), for iξ ∈V
|(κ−1)∗(ϕλ)|ψe−iϑ(·,ξ)| ≤ C1|hλ−iξ, Xi|−2N|λ|N0≤C1C−2N
2|λ|−N+N0|ξ|−N.(2.10)
Hence, for Nsufficiently large we obtain
sup
λ∈Ω
sup
ξ∈V|ξ|N|(κ−1)∗(ϕλ)|ψe−iϑ(·,ξ)|<∞ ∀N∈N.(2.11)
Set Γ = K×(−i)Ω. Then due to definition 1.53 and proposition 1.55 we get that the
series uΩconverges to uΩin D0
Γ(K). Hence,
WF(uΩ)⊂K×(−i)Ω ⊂T∗K≃K×k∗
We made an additional assumption at the beginning of the proof that Ω is convex. If
this were not true; let (Ωj)1≤j≤Nbe a family of closed convex cones such that Ω ⊂ ∪Ωj.
Choose Aj⊂Ωj,Aj∩Ak=∅for j6=k, such that u=Pjuj,uj=Pλ∈Ajϕλ. Then
WF(uj)⊂K×(−iAj), hence WF(u)⊂K×∪(−iΩj).
Remark 2.5. The pull-back of the Haar measure is (κ−1)∗dy =Jκ(Y)dY . We will
determine the Jacobian of κ−1which we denote by Jκ(Y) = det((κ−1)0) we first have
to calculate the derivative of κ−1at Y∈k
(Lx◦exp)0(Y) = dLx(exp(Y)) ◦dexp(Y) = dLx(exp(Y)) ◦dLexp(Y)(e)◦1−e−ad Y
ad Y,
the second equality follows from Theorem 1.7 in [He01]. Since the Haar measure is left
invariant, it is sufficient to consider only the term 1−e−ad Y
ad Y. Then we get, for Y∈t
det 1−e−ad Y
ad Y= det ead Y/2−e−ad Y/2
ad Y/2=Y
α∈Λ
ehα,Y i/2−e−hα,Y i/2
hα, Y i=: Jκ(Y).
The first equality follows from Corollary 5.5 [BGV92] and the second equality follows
from [BGV92], p. 253. In general, for each Y∈kthere exists a k∈Ksuch that
Ad k(Y)∈t, then ad Y= Ad k−1◦ad(Ad k(Y)) ◦Ad k. Since the trace is invariant
under the adjoint action then the second equality is valid for all Y∈k. Observe that the
enumerator and the denominator of Jκ(Y) has the same simple zero where hα, Y i= 0.
Remark 2.6. Note that in the last proposition the elements ϕλare assumed to be
highest weight vectors in Eλ(see (1.13)). Thus very special elements.
35 2.1 Wave Front Set of Truncated Distributions
2.1.1 Dirac-Distributions
Let us consider the Dirac-distribution δwhich is supported at e(the identity element
of the group) and defined by
hδ|ψi=ψ(e),
where ψ∈C∞
0(K). We note that the characters {χπλ},χπλ= trace(πλ), form an
orthonormal basis for the space of Ad K-invariant L2-functions on K.χπλis the unique
Ad K-invariant function in Eλ. We obtain
δ=X
λ∈L∩C
dπλχπλ,
where dπλ= dim πλ. Note that δis the identity of the convolution, i.e., u∗δ=u(see
also [DK00], p. 237).
2.1.2 Truncated Dirac-Distributions
In proposition 2.2 we assumed that the element ϕλin the Fourier expansion Pλ∈L∩Ωϕλ
are highest weight vectors.
Let πλbe a continuous irreducible representation of Kon the Hilbert space Hπ
and uλbe a highest weight vector of πλ, provided with a K-invariant hermitian inner
product (·,·). We set
Ψλ(y) = (πλ(y−1)uλ, uλ)
where y∈K. Since the Haar measure is an invariant measure ([DK00], p. 184) we
obtain ZK
Ψλ(yxy−1)dy =ZK
Ψλ(ykx(yk)−1)dy =ZK
Ψλ(ykxk−1y−1)dy.
Then x7→ RKΨλ(yxy−1)dy is an Ad K-invariant function in Eλ, hence proportional to
χπλand we obtain ZK
Ψλ(yxy−1)dy =1
dπλ
χπλ(x),(2.12)
(see [KV79] for more details). We define
ΨΩ:= X
λ∈L∩Ω
d2
πλΨλ∈D0(K),
(see remark 2.7). We note that
1. The pull-back of Ψλby φ:K×K→K, (x, y)7→ yxy−1, is
(φ∗Ψλ)(x, y)=Ψλ(φ(x, y)) = Ψλ(yxy−1)
2. The push-forward of Ψλby ω:K×K→K, (x, y)7→ x, is
ω∗Ψλ(x) = ZK
Ψλ(x, y)dy,
36 2.1 Wave Front Set of Truncated Distributions
Then
ω∗φ∗Ψλ(x) = ZK
φ∗Ψλ(x, y)dy =ZK
Ψλ(φ(x, y)) dy =ZK
Ψλ(yxy−1)dy
and we get from (2.12)
d2
πλZK
Ψλ(yxy−1)dy =dπλχπλ(x).
The truncated Dirac-Distribution δΩis defined by the push-forward and pull-back
of the distribution ΨΩ. Since the push-forward and pull-back are linear continuous
operators we obtain
δΩ:= ω∗φ∗ΨΩ=X
λ∈L∩Ω
dπλχπλ.
Remark 2.7. dπλ=Qµ∈Λ+
hµ,λ+ρi
hµ,ρi, where ρis the half sum of the positive roots (it is
called Weyl dimension formula which holds for connected and compact group (see
[DK00], Theorem 4.9.2)). Then due to the characterization theorem 1.35 we get that
ΨΩand δΩare distribution on K.
Furthermore, due to Theorem 5.9 in [Fo95a] we have bijection between the matrix
element Ψλ∈Eλand any vector u∈Hπ, in particular the highest weight vector
uλ. Hence, Ψλsatisfies equation (2.2) and (2.3). From proposition 2.2 we obtain that
ΨΩ=Pλ∈L∩Ωd2
πλΨλconverges in D0
Γ(K) where Γ = K×(−i)Ω. In particular
WF(ΨΩ)⊂K×(−i)Ω
In the following we will study the effect of the pull-back and the push-forward of ΨΩ
respectively, on the wave front set.
Lemma 2.8. Let φbe defined as above and u∈D0
K×(−iΩ)(K). Then
WF(φ∗u)⊂φ∗WF(u)⊂T∗K×T∗K.
More precisely, φ∗WF(u)is contained in the set e
Γwhich is defined by
{(x, y;ξ, η)| ∃ζ: (yxy−1, ζ)∈WF(ΨΩ), ξ = Ad∗y−1(ζ), η = (Ad∗x−I) Ad∗y−1(ζ)}
Proof. Following theorem 1.56 and using proposition 1.22 we get
dφ(x, y) : k×k−→ k
(X, Y )7→ Z=dφ(x, y)(X, Y ) = Ad y(X−Y+ Ad x−1(Y))
Moreover, the graph of φis given by
G(φ) = {(x, y, z)|z=yxy−1=φ(x, y)}.
37 2.1 Wave Front Set of Truncated Distributions
and the tangent space of this graph at (x, y, z) is given by
Tx, y, zG(φ) = {(X, Y, Z)|Z=dφ(x, y)(X, Y ) = Ad y(X−Y+ Ad x−1(Y))}.
We will denote by ξ, η, ζ ∈k∗the dual variable corresponding to X, Y, Z ∈k. Let Nφ
be the the normals of the map φ(compare theorem 1.56 and remark 1.57)
Nφ={(z, ζ)|z=yxy−1,hζ, dφ(x, y)(X, Y )i= 0,∀X, Y ∈k},
The set Cφequals
Cφ={(x, y, z;ξ, η, ζ)| ∀(X, Y, Z)∈Tx, y, zG(φ),hζ, Zi+hξ, Xi+hη, Y i= 0},
Using the formula for dφ(x, y)(X, Y ) we obtain
Cφ=(x, y, z;ξ, η, ζ)|ξ=−Ad∗y−1(ζ), η =−Ad∗xy−1(ζ) + Ad∗y−1(ζ).
Hence,
C0
φ={(x, y, yxy−1;ξ, η, ζ)|ξ= Ad∗y−1(ζ), η = (Ad∗x−I) Ad∗y−1(ζ)}.
Then we get that WF(φ∗u)⊂φ∗WF(u) = C0
φ◦WF(u) equals
{(x, y;ξ, η)| ∃ζ: (yxy−1, ζ)∈WF(u), ξ = Ad∗y−1(ζ), η = (Ad∗x−I) Ad∗y−1(ζ)}.
and
WF(φ∗u)⊂φ∗WF(u)⊂T∗K×T∗K.
Proposition 2.9. Set e
Γas in lemma 2.8. Let ωbe defined as above u∈D0
e
Γ(K×K).
Then we get that
WF(ω∗u)⊂K×Ad∗K(−iΩ).(2.13)
Proof. Following proposition 1.61 and using the calculation in example 1.23 we get
dω∗:k∗−→ k∗×k∗
ξ7−→ (ξ, 0)
It follows from proposition 1.61
WF(ω∗u)⊂ {(x, ξ)| ∃y: (x, y, ξ, 0) ∈e
Γ}
Hence, by definition of e
Γ,
WF(ω∗u)⊂K×Ad∗K(−iΩ)
38 2.2 Wave Front Set of Convolution
Proposition 2.10. The wave front set of the truncated Dirac-Distributions is contained
in Γ := K×Ad∗K(−iΩ) and
δΩ=X
λ∈L∩Ω
dπλχπλconverges in D0
Γ(K).(2.14)
Proof. Set A:= ω∗◦φ∗and ΨΩ∈D0
(K×−iΩ)(K). Then
A:D0
(K×−iΩ)(K)−→ D0
(K×Ad∗K(−iΩ))(K)
is linear and continuous. Hence, (2.14) converges in D0
Γ(K).
2.2 Wave Front Set of Convolution
The goal of this section is to show that WF(u∗δΩ)⊂K×Ad∗K(−iΩ). Observe that
D0(K) = E0(K).
Fact 2.11. Let u, v ∈E0(Rn) be a distribution with compact support. Then WF(u∗v)
is contained in
{(x+y, ξ)∈Rn×Rn\{0} | (x, ξ)∈WF(u),(y, ξ)∈WF(v)}(2.15)
Proof. See [FJ98], p. 158 .
Remark 2.12. The convolution in fact 2.11 is defined for distributions on Rn. For
distribution defined on Kwe will first express the convolution via tensor product.
Next we will calculate the wave front set of the tensor product.
Let Φ : K×K−→ Kdenote the smooth map (x, y)7→ x·y. Then we write the
convolution of δΩ, u ∈E0(K), with the duality brackets written as integrals, as follows:
hu∗δΩ|ψi=ZKZK
u(xy−1)δΩ(y)ψ(x)dydx
x=zy
=ZKZK
u(z)δΩ(y)ψ(zy)dydz
=ZKZK
u(z)δΩ(y)Φ∗ψ(z, y)dydz
=hu⊗δΩ|Φ∗ψi
=hΦ∗(u⊗δΩ)|ψi(2.16)
where ψ∈C∞
0, Φ∗is the pull-back of ψby Φ, and Φ∗is the push-forward of u⊗δΩby
Φ. It follows that u∗δΩ= Φ∗(u⊗δΩ).
Proposition 2.13. Let δΩ, u ∈E0(K)and Φ : K×K→K,(x, y)7→ x·y. Then we
obtain
WF(Φ∗(u⊗δΩ)) ⊂K×Ad∗K(−iΩ)
39 2.3 Characterization of The Wave Front Set of Distributions
Proof. From proposition 1.60 we obtain that the wave front set of the tensor product
satisfies
WF(u⊗δΩ)⊂(WF(u)×WF(δΩ)) ∪((supp(u)×{0})×WF(δΩ))
∪(WF(u)×(supp(δΩ)×{0})) .
Following proposition 1.61 and using the calculation in example 1.24 we get
dΦ∗:k∗−→ k∗×k∗
ξ7−→ (Ad∗y(ξ), ξ).
Hence, the wave front set of Φ∗(u⊗δΩ) is contained in
{(z, ζ)| ∃x, y, ξ, η :z=xy = Φ(x, y),(x, y;ξ, η)∈WF(u⊗δΩ), ξ = Ad∗y(ζ), η =ζ}
In particular, WF(Φ∗(u⊗δΩ)) ⊂K×Ad∗K(−iΩ).
Because of the relation between Fourier transformation and convolution, for a distri-
bution u=Pλ∈L∩Cϕλon Kwe get
u∗δΩ=X
λ∈L∩Ω
ϕλ=uΩ
Proposition 2.14. Let Γ := K×Ad∗K(−iΩ) be a closed cone in T∗K. Then the
convolution
u∗δΩconverges in D0
Γ(K)
In particular
WF(uΩ) = WF(u∗δΩ)⊂K×Ad∗K(−iΩ)
Proof. Due to proposition 2.10 we get that δΩconverges in D0
Γ(K). Moreover, we get
from proposition 2.13 that WF(u∗δΩ)⊂K×Ad∗K(−iΩ). Finally, due the separately
continuity of the map ∗:D0(K)×D0
Γ(K)→D0
Γ(K), (u, δΩ)7→ δΩ∗u(by fixing the
first component) and the properties of the convolution we get that u∗δΩconverges in
D0
Γ(K).
2.3 Characterization of The Wave Front Set of Distributions
Finally we gather the results from the preceding discussion in the last two sections
to state the following characterization theorem. The next theorem was introduced by
[KV79] in the hyperfunction setting.
Theorem 2.15. Let u=Pλ∈L∩Cϕλbe a distribution on Kand Ωbe a closed cone
contained in C\0. Then the following assertions are equivalent:
(i) WF(u)⊂K×Ad∗K(−iΩ).
40 2.3 Characterization of The Wave Front Set of Distributions
(ii) For every closed cone e
Ωin Csuch that e
Ω∩Ω = ∅and every N∈N, there exists
a constant LN>0such that
kϕλkL2≤LN(1 + |λ|)−N,for λ∈e
Ω (2.17)
Proof. First we are going to prove that (ii) implies (i). Choose e
Ω as in (ii), then we
write
u=u∗δe
Ω+u∗δC\
e
Ω,
set ue
Ω=u∗δe
Ω=Pλ∈L∩
e
Ωϕλand uC\
e
Ω=u∗δC\
e
Ω=Pλ∈L∩C\
e
Ωϕλ. Due to the
characterization theorem 1.35 the assertion (ii) is equivalent to ue
Ω∈C∞(K) which
implies that WF(ue
Ω) = ∅. Moreover, we obtain from proposition 2.14 that the WF(u) =
WF(uC\
e
Ω)⊂K×Ad∗K(C\e
Ω). Since we can choose (C\e
Ω) as close to Ω as we like,
then we obtain assertion (i).
On the other hand, if assertion (i) is satisfied, then the WF(ue
Ω) = ∅, hence ue
Ω∈
C∞(K). Then assertion (ii) follows from Theorem 1.35.
3 Restriction of Characters
In the last chapter we presented the wave front set of a distribution defined on con-
nected, semisimple, and compact Lie groups K. In this chapter we are going to apply
the results from the last chapter to achieve the restriction of characters of an irreducible
unitary representation of Kto a closed subgroup H.
Let Kbe a connected, compact, and semisimple Lie group. We denote by C⊂it∗
the closure of the dominant Weyl chamber (see definition 1.12) and L⊂it∗the weight
lattice (see definition 1.15). Let λbe the highest weight of an irreducible representation
πof K.
Definition 3.1. Let Sbe subset of a real vector space Rn. We define a closed cone
S∞in Rn\{0}by
S∞:= ny∈Rn\{0}|∃(yn, tn)⊂S×R+such that lim
n→∞ tnyn=yand lim
n→∞ tn= 0o.
The following lemma is without proof in [Ko98], Lemma 2.5.
Lemma 3.2. Let Sbe a subset of a real vector space Rn, and Ya closed cone in Rn.
Then the following two condition are equivalent.
1. S∞∩Y=∅.
2. There exists an open cone Vcontaining Ysuch that S∩Vis relatively compact.
Proof. First we are going to prove that (1) implies (2). Let Y⊂V⊂Vand S∞∩V=∅.
It is sufficient to show that S∩Vis bounded. Assume that S∩Vis unbounded,
then exits a sequence (zn)⊂S∩Vsuch that |zn| → ∞. Consider the sequence
(zn/|zn|)⊂V, (znj/|znj|)→y∈V. Since S∞∩V=∅, then y /∈S∞. On the other
hand y= limj→∞ tnjznjwhere tnj= 1/|znj|, then y∈S∞. Hence, S∩Vis bounded.
Second we are going to prove that (2) implies (1). Due to the boundedness of S∩V
we get that S∞∩Y= (S\(S∩V))∞∩Y= (S\V)∞∩Y. Since (S\V)∩Y=∅,
it remains to show that (S\V)∞∩Y=∅. Assume that y∈(S\V)∞∩Y, it follows
that ∃(yn, tn)⊂(S\V)×R+such that limn→∞ tnyn=y∈Yand limn→∞ tn= 0.
Moreover, we get that tnyn∈Vfor n∈Nlarge enough. On the other hand, we have
that (yn)⊂S\V⇒(yn)⊂| V, then (tnyn)⊂| Vbecause Vis an open cone. Hence,
(tnyn)⊂S\Vbut this contradicts to tnyn∈Vwhen n∈Nis large enough.
Definition 3.3. Given u=Pλ∈L∩Cϕλ∈D0(K) where ϕλ∈Eλ(Eλ=Eπλsee
theorem 1.19 ), we define
fsupp(u) := λ∈L∩C|ϕλ6= 0⊂it∗.(3.1)
Analogous to Lemma 2.6 in [Ko98], we will prove the following lemma using wave
front set of distribution instead of singular spectrum of hyperfunction.
Lemma 3.4. Using definition 3.3, we have
42 3.1 K-Characters
1. WF(u)⊂K×Ad∗K(−ifsupp(u)∞)⊂T∗K\0
2. Assume ∃M > 0∀λ∈fsupp(u):kϕλkL2(K)≥(1 + |λ|)−M/M. Assume there
exists a closed cone W⊂C\0such that WF(u)⊂K×Ad∗K(−iW). Then
fsupp(u)∞⊂W.
Proof. 1) We define a closed cone S:= fsupp(u)∞⊂it∗. Let Ω be an arbitrary closed
cone in it∗such that Ω ∩S=∅. By lemma 3.2 1⇒2 we can find an open cone
Vcontaining Ω such that V∩fsupp(u) is relatively compact in it∗. In particular,
Ω∩fsupp(u) is a finite set because fsupp(u)⊂L∩Cis discrete. Hence, for every
N∈N, there exists a constant LN>0 such that
kϕλkL2(K)≤LN(1 + |λ|)−N∀λ∈Ω∩(L∩C),(3.2)
here LN:= maxλ∈Ω(1 + |λ|)NkϕλkL2(K)<+∞. It follows from theorem 2.15 (ii)⇒(i)
that WF(u)⊂K×Ad∗K(−ifsupp(u)∞).
2) Suppose fsupp(u)∞⊂| W, then we can find a closed cones S0and S00 and an open
cone Vsuch that
∅ 6=S00 ⊂V⊂S0⊂C,
S00 ⊂fsupp(u)∞,(3.3)
S0∩W=∅.(3.4)
By assumption,
WF(u)⊂K×Ad∗K(−iW).(3.5)
From (3.4) and (3.5) we obtain using theorem 2.15 (i)⇒(ii) that; for every N∈N,
there exists a constant LNsuch that
kϕλkL2(K)≤LN(1 + |λ|)−Nfor any λ∈S0∩(L∩C).(3.6)
Because ∃M > 0∀λ∈fsupp(u) such that kϕλkL2(K)≥(1 + |λ|)−M/M, we have that
fsupp(u)∩S0is bounded. In particular,
#(fsupp(u)∩S0)<∞.(3.7)
Hence, fsupp(u)∩Vis relatively compact. It follows from lemma 3.2 (2) ⇒(1) that
S00 ∩fsupp(u)∞=∅. This contradicts to (3.3) and S00 6=∅. Hence, we get that
fsupp(u)∞⊂W.
3.1 K-Characters
Let τbe a representation of Kon the Hilbert space H. The K-multiplicity is defined
by
mK(·:τ) : b
K→N0∪{∞}, π 7→ mK(π:τ) := dim HomK(π, τ).
43 3.1 K-Characters
The asymptotic K-support of τ, denoted by ASK(τ), is defined by
suppK(τ) := λ∈L∩C|mK(πλ:τ)6= 0,(3.8)
ASK(τ) := suppK(τ)∞.(3.9)
ASK(τ) is a closed cone contained in C⊂it∗because suppK(τ)⊂(L∩C).
We say that mK(·:τ) is of polynomial growth, if there exist constants Cand N
such that
mK(πλ:τ)≤C(1 + |λ|)Nfor λ∈L∩C. (3.10)
We denote by χπthe (trace) character of the irreducible representation πof K.
Lemma 3.5. Suppose that mK(·:τ)is of polynomial growth. Then the K-character
ΘK
τof τ,
ΘK
τ=X
λ∈L∩C
mK(πλ:τ)χπλ,(3.11)
is well-defined as a distribution on K.
Proof. Due to characterization theorem 1.35, ΘK
τis a distribution if and only if there
exists a positive numbers Mand a constant Lsuch that
kmK(πλ:τ)χπλkL2≤L(1 + |λ|)M.(3.12)
Since characters form an orthonormal basis for L2(K) (see [DK00], Theorem 4.3.4) and
because of the polynomial growth of mK(·:τ) we get that
kmK(πλ:τ)χπλk2
L2= (mK(πλ:τ)χπλ, mK(πλ:τ)χπλ)L2
= (mK(πλ:τ))2(χπλ, χπλ)L2
(3.10)
≤C2(1 + |λ|)2N.
This proves inequality (3.12) which proves that ΘK
τis a distribution on K. Since ΘK
τ
depends only on the equivalent class of τ, then ΘK
τis well-defined.
Analogous to Proposition 2.7. in [Ko98], we will prove the following proposition using
wave front set of distribution instead of singular spectrum of hyperfunction.
Proposition 3.6. Suppose that mK(·:τ)is of polynomial growth. Then
1. The wave front set of the K-character ΘK
τof τsatisfies
WF(ΘK
τ)⊂K×Ad∗K(−iASK(τ)) ⊂T∗K\0≃K×k∗.(3.13)
2. Conversely, if W⊂C\0is a closed cone such that
WF(ΘK
τ)⊂K×Ad∗K(−iW).
Then
ASK(τ)⊂W. (3.14)
44 3.2 Restriction of Characters to a Closed Subgroup
Proof. By definition ΘK
τ=Pλ∈L∩Cϕλ,ϕλ=mK(πλ:τ)χπλ. It follows from (3.1) and
(3.8), that
fsupp(ΘK
τ) = suppK(τ).(3.15)
In particular, fsupp(ΘK
τ)∞= suppK(τ)∞= ASK(τ). We apply now lemma 3.4 (1) with
u= ΘK
τ
WF(ΘK
τ)⊂K×Ad∗K(−ifsupp(ΘK
τ)∞) = K×Ad∗K(−iASK(τ)).(3.16)
2) Observe that kmK(πλ:τ)χπλkL2(K)≥1 if λ∈suppK(τ). Using lemma 3.4 (2)
and the assumption WF(ΘK
τ)⊂K×Ad∗K(−iW) we obtain for u= ΘK
τthat
ASK(τ) = fsupp(u)∞⊂W. (3.17)
This complete the proof of the proposition.
3.2 Restriction of Characters to a Closed Subgroup
Let Hbe a closed subgroup of K. We write prk→h:k∗→h∗for the projection dual to
the inclusion of the Lie algebras h,→k. Let h⊥:= ker prk→h.
Let µ∈b
H, define the set
4(µ) := λ∈L∩C|[πλ|H:µ]6= 0(3.18)
where [πλ|H:µ] is the multiplicity of µin πλ|H(see the following remark 3.7).
Remark 3.7. Let σbe a representation of a compact group Hon a finite dimensional
Hilbert space V.Then σis completely reducible, i.e., Vcan be written as a direct sum
of σ(H)-invariant subspaces Vj, such that σ|Vjis irreducible for each j(see [DK00],
Corollary 4.2.2). Furthermore,
eχσ=X
µ∈
b
H
[σ:µ]eχµ,
is a finite sum, where eχµis the trace of µ,eχσis the trace of σ, and [σ:µ] is a number
which denote the multiplicity of µin σ(see [DK00], Corollary 4.3.5).
Definition 3.8. The embedding ι:H ,→Kdefines the co-normal bundle N∗(H) :=
ker ρ≃H×h⊥, here ρ:T∗K|H≃H×k∗→T∗H≃H×h∗is the natural projection
(h, α)7→ (h, prk→h(α)). The projection (ι0)tin remark 1.59 agrees with ρin this case.
Lemma 3.9. Let Hbe a closed subgroup of Kand τa representation of Kin H.
Suppose that mK(·:τ)is of polynomial growth. Define the closed cone Γ := K×
Ad∗K(−iASK(τ)) in T∗K\0. Then we obtain the following:
1. The K-character of τ
ΘK
τ=X
λ∈L∩C
mK(πλ:τ)χπλconverges in D0
Γ(K) (3.19)
45 3.2 Restriction of Characters to a Closed Subgroup
2. Assume
Γ∩N∗(H) = ∅.(3.20)
Then ΘK
τ|His a well-defined distribution on H. Moreover,
ΘK
τ|H=X
λ∈L∩C
mK(πλ:τ)χπλ|Hconverges in D0(H),(3.21)
and the wave front set of ΘK
τ|Hsatisfies,
WF(ΘK
τ|H)⊂H×prk→h(Ad∗K(−iASK(τ))) (3.22)
Proof. Let Ybe an arbitrary closed cone in it∗such that ASK(τ)∩Y=∅. By lemma
3.2 1⇒2, there exists an open cone Vcontaining Ysuch that suppK(τ)∩Vis relative
compact in it∗. In particular, suppK(τ)∩Vis finite set because suppK(τ) is discrete.
This implies that, for every closed cone W⊂C\0 such that ASK(τ)⊂◦
W, the series
ΘK
τ∗δC\Wconverges in C∞(K). Using proposition 2.14 we deduce that the series
ΘK
τ∗δWconverges in D0
Γ1(K) where Γ1:= K×Ad∗K(−iW). It follows that the series
ΘK
τ= ΘK
τ∗δC\W+ ΘK
τ∗δWconverges in D0
Γ1(K). The assertion (3.19) follows from
remark 1.54.
From assumption (3.20) and proposition 1.43 we obtain that ΘK
τ|His a well-defined
distribution on H. Moreover, ΘK
τ|His the pull-back of ΘK
τby the inclusion ι:H ,→K.
Using theorem 1.56 and remark 1.59 we obtain that ι∗:D0
Γ(K)→D0
ι∗Γ(H) is contin-
uous. Hence, (3.21) holds and the wave front set of ΘK
τ|His contained in the image of
the projection ρ(see definition 3.8), i.e., WF(ΘK
τ|H)⊂H×prk→h(Ad∗K(−iASK(τ))),
where prk→h:k∗→h∗.
Remark 3.10. The following theorem is introduced in [Ko98], Theorem 2.8 using
hyperfunctions. It was remarked in [Ko98] (remark following Theorem 2.8) that for a
convergent sequence of analytic functionals (hyperfunctions) restricted to a submanifold
may not converge in general. This limitation can be avoided when working, as in this
thesis, with the distribution spaces D0
Γ(K).
Theorem 3.11. Let τbe a representation of Kand Hbe a closed subgroup of Kwith
the inclusion H ,→K. We assume
ASK(τ)∩iAd∗K(h⊥) = ∅.(3.23)
Then we obtain the following:
1.
# ( suppK(τ)∩ 4(µ)) <∞(3.24)
for each µ∈b
H. If the K-multiplicity function mK(π:τ)<∞for any π∈b
K.
Then the H-multiplicity function mH(µ:τ|H) := HomH(µ, τ|H)<∞for any
µ∈b
H.
46 3.2 Restriction of Characters to a Closed Subgroup
2. If the K-multiplicity function mK(·:τ) : b
K→N0is of polynomial growth,
then so is the H-multiplicity function mH(·:τ|H) : b
H→N0. Furthermore, the
restriction of ΘK
τto the submanifold His well-defined as a distribution and the
resulting distribution ΘK
τ|Hcoincides with
ΘH
τ|H:= X
µ∈
b
H
mH(µ:τ|H)eχµ.
In particular, ΘH
τ|Hhas the wave front set given by:
WF ΘH
τ|H⊂H×prk→h(Ad∗K(−iASK(τ)))
Proof. (1) Let µ∈b
H. We fix a representation space Hµ(finite dimension). Let
σ= indK
H(µ) be the unitary representation of Kinduced by µ. We consider the repre-
sentation space Hσof σas a subrepresentation of L2(K, Hµ) defined by,
u(kh) = µ(h−1)u(k),(3.25)
where h∈Hand k∈K.
Using (3.25) we obtain, for Y∈h,
d
dt|t=0 u(ketY )=d
dt|t=0 µ(e−tY )u(k)=−µ∗(Y)u(k),
where µ∗:h→End(Hµ) is the Lie algebra representation induced by µ. Here,
d
dt|t=0 u(ketY )= (k·Y)u(k) =: e
Y u(k).
Hence, u∈L2(K, Hµ) belongs to Hσ, if and only if it satisfies, in the sense of
distributions, the differential equations of first order,
e
Y u +µ∗(Y)u= 0.
This implies, using theorem 1.42,
WF(u)⊂Char(e
Y+µ∗(Y)) = Char(e
Y),for Y∈h
where Char(e
Y) denotes the characteristic variety of an operator e
Y(see (1.28)). More-
over, using remark 2.1, we get that
Char(e
Y)⊂K×Y⊥.
Hence,
WF(u)⊂\
Y∈h
Char(e
Y)⊂K×Ad∗K(h⊥).(3.26)
Claim 3.12. ASK(σ)⊂iAd∗K(h⊥).
47 3.2 Restriction of Characters to a Closed Subgroup
Proof. Assume, to the contrary, ASK(σ)⊂| iAd∗K(h⊥). Then there exists a closed
cone Ω ⊂C\0 such that Ω ∩iAd∗K(h⊥) = ∅and ASK(σ)∩◦
Ω6=∅. Choose ϕλ∈Eλ,
λ∈L∩Ω, such that Pλ∈L∩Ωkϕλk2
L2(K,Hµ)<+∞. Then u:= Pλ∈L∩Ωϕλconverges
in L2(K, Hµ), and since ϕλ∈Eλthen u∈Hσ. From (3.26) we obtain that
WF(u)⊂K×Ad∗K(h⊥).(3.27)
In addition, we can choose ϕλ’s such that for some M > 0,
∀λ∈L∩Ω : kϕλkL2(K)≥1
M(1 + |λ|)−M.(3.28)
Then L∩Ω⊂fsupp(u). It follows that ◦
Ω⊂fsupp(u)∞. Therefore,
fsupp(u)∞⊂| iAd∗K(h⊥).
On the other hand (3.27), (3.28), and lemma 3.4 (2) lead to the contradiction fsupp(u)∞⊂
iAd∗K(h⊥).
Since ASK(σ)⊂iAd∗K(h⊥) then we obtain, using assumption (3.23), that ASK(σ)∩
ASK(τ) = ∅. Therefore, using lemma 3.2 (1 ⇒2), suppK(σ)∩suppK(τ) is relatively
compact, hence finite. Using Frobenius reciprocity Theorem (see [Fo95a], Theorem
6.10) we obtain that suppK(σ) = 4(µ). This implies that #(4(µ)∩suppK(τ)) <∞.
Moreover, we obtain
mH(µ:τ|H) = X
λ∈L∩C
mK(πλ:τ) [πλ|H:µ]
=X
λ∈suppK(τ)∩ 4(µ)
mK(πλ:τ) [πλ|H:µ],(3.29)
is a finite sum. Since the K-multiplicity function mK(π:τ)<∞for every π∈b
K, then
the H-multiplicity function mH(µ:τ|H)<∞for every µ∈b
Hwhich proves assertion
(1).
(2) Because of proposition 3.6 and (3.23) we get
WF(ΘK
τ)∩N∗(H)⊂K×Ad∗K(−iASK(τ)) ∩H×h⊥=∅.
Therefore, it follows from lemma 3.9 that the restriction ΘK
τ|His well-defined as a
distribution on Hand its wave front set satisfies
WF(ΘK
τ|H)⊂H×prk→h(Ad∗K(−iASK(τ))) ⊂T∗H\0.
Set Γ := K×Ad∗K(−iASK(τ)) which is a closed cone in T∗K\0. It follows from
lemma 3.9 (1) that
ΘK
τ=X
λ∈L∩C
mK(πλ:τ)χπλconverges in D0
Γ(K).
48 3.2 Restriction of Characters to a Closed Subgroup
Furthermore, from (3.23) and lemma 3.9 (2) we obtain
ΘK
τ|H=X
λ∈L∩C
mK(πλ:τ)(χπλ)|H∈D0(H).
On the other hand we obtain
ΘK
τ|H=X
ν∈
b
H
cνeχν∈D0(H),
where eχνdenotes the trace of ν∈b
H. Observe that cνis of polynomial growth because
ΘK
τ|His a distribution on H. We want to show: cµ=mH(µ:τ|H) for each µ∈b
H.
Then the H-characters,
ΘH
τ|H=X
µ∈
b
H
mH(µ:τ|H)eχµ∈D0(H).
and ΘK
τ|H= ΘH
τ|H. This means we have to show
(ΘK
τ|H,eχµ) = mH(µ:τ|H) (3.30)
where (·,·) denotes the anti-duality brackets, anti-linear in the second variable, between
D0(H)×C∞
0(H) which is induced by the L2scalar product. Now
(ΘK
τ|H,eχµ) = X
πλ∈
b
K
mK(πλ:τ) ((χπλ)|H,eχµ)
remark 3.7
=X
πλ∈
b
KX
ν∈
b
H
mK(πλ:τ) [πλ|H:ν] (eχν,eχµ).
Since the characters are orthonormal basis for L2(H), then (eχν,eχµ) = δν
µ. Furthermore,
(ΘK
τ|H,eχµ) = X
πλ∈
b
KX
ν∈
b
H
mK(πλ:τ) [πλ|H:ν] (eχν,eχµ)
=X
πλ∈
b
KX
ν∈
b
H
mK(πλ:τ) [πλ|H:ν]δν
µ
=X
πλ∈
b
K
mK(πλ:τ) [πλ|H:µ].
This and (3.29) imply (3.30). Therefore cµ=mH(µ:τ|H). Moreover we conclude that
the H-multiplicity function mH(·:τ|H) in (3.29) is of polynomial growth. Using lemma
3.9 we obtain
WF ΘH
τ|H= WF ΘK
τ|H⊂H×prk→h(Ad∗K(−iASK(τ)))
this proves (2).
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51
Index
B(X, Y ), 10
Dα,19
Eλ,41
Eπ,13
K-multiplicity, 43
L,13
LT,13
S∞,41
Vλ,12
Ad,9
Ad, 9
Ad∗,16
Λ+(c), 9
Λ+,9
ΘK
τ,43
ad, 10
ASK(τ), 43
ASK,43
χα,11
χπλ,35
χπ∗f,14
fsupp(u), 41
λ(α)∨,12
≤,12
A0(B), 19
B(X), 19
D0(X), 18
D0
Γ(X), 21
E0(X) , 19
H,43
Hµ,46
Hσ,46
K,23
S,23
S0,23
c(Λ+), 9
k,9
kα,9
t,9
treg,9
π,14
φu, v,13
πij,13
σ(e
X), 29
sing supp, 18
suppK(τ), 43
WF(u), 21
WF(A), 24
WF(K), 23
b
K,12
b
f(π), 15
eχµ,44
mK(·:τ), 43
abstract root system, 11
adjoint representation, 10
algebraically integral, 12
analytic functionals, 19
analytically integral, 11
characteristic variety, 22
co-adjoint representation, 16
conic, 20
contragredient, 14
Dirac-distribution, 35
distribution, 18
distribution on a manifold, 20
distribution with compact support, 19
dominant, 11
Fourier inversion formula, 15
Fourier-Laplace transformation, 19
group Fourier transformation, 15
hyperfunctions, 19
Killing form, 10
matrix elements, 13
multiplicative character, 11
Peter-Weyl Theorem, 14
polynomial growth, 43
52 Index
positive roots, 9
principal symbol, 22
pseudo-differential operator, 23
pull-back, 20
pull-back of distribution, 26
reduced element, 11
reduced system, 11
reductive Lie algebra, 10
regular, 9
root space, 9
root-space decomposition, 9
roots, 9
Schwartz kernel, 23
semisimple Lie algebra, 10
simple Lie algebra, 10
singular support, 18
symbol, 23
system of simple roots, 12
Theorem of The Highest Weight, 13
truncated Dirac-Distribution, 36
unitary representation, 13
wave front set, 21
weight lattice, 13
Weyl chamber, 12
Weyl dimension formula, 36
