Harmonic Analysis

EPvan den Ban

Master Class Course MRI Fall

Intro duction

For an intro duction to the following material we refer to the rst lecture given by the author

One may also confer Sug which will b e the basic reference for the rest of this course

Haar measure

Let X b e a lo cally compact Hausdor space

Denition A p ositive Radon integral on X is a linear functional I C X C such that

c

f I f

Remark By the Riesz representation theorem I is actually the integral asso ciated with a

regular Borel measure We shall refer to this measure as the Radon measure asso ciated with

I and we shall use the notation

Z

I f f x dx

G

Lemma Let I b e a p ositive Radon measure on X Then

a f g I f I g f g C X

c

b f C X jI f j I jf j

c

c The map f I f is continuous on C G

c

Proof For a use that g f and the denition of I

b First assume that f is realvalued and dene continuous functions by f maxf

and f maxf Then f f f and jf j f f Hence

jI f j jI f I f jI f I f I jf j

Now assume that f is complex valued and that I f R Then I Im f Hence

jI f j jI Re f j I jRe f j I jf j

Sug M Sugiura Unitary Representations and Harmonic Analysis nd ed NorthHollandKo ddans ha

Amsterdam

by what we proved ab ove Let f now b e arbitrary Then there exists a z C with jz j and

zI f Then I zf R hence jI f j jI zf j I jzf j I jf j

It remains to establish c Let K X b e a compact subset Then wemust show that there

exists a C such that for all f C K wehave jI f j C kf k There exists a nonnegative

c

function g C X such that g on K Hence for all f C K wehave jf j jf jg kf k g

c c

and it follows that

jI f j I jf j kf k I g

which establishes the desired estimate Note that by taking the inmum over g one obtains the

estimate jI f j kf k volume K

By a Gspace we shall from now on mean a lo cally compact Hausdor space X together

with a continuous action of G on X

We agree to write L for the induced action of G on C X Thus if C X then L x

g

g x

The Gspace X is called homogeneous if there is only one Gorbit ie if x X is a p oint

then G x X One readily veries that in this setting the stabilizer

G fg G gx x g

x

is a closed subgroup of G Moreover the map G X g gx induces a homeomorphism

GG X Thus any homogeneous Gspace is of the form GH with H a closed subgroup of

x

G ConverselyifH is a closed subgroup then GH is a homogeneous space in a natural way

Example The unit sphere S in R is a homogeneous space for the sp ecial orthogonal group

SO Let e Then the stabilizer of e in SO is the group of rotations with axis

Re which is isomorphic to SO The sphere is thus identied with the quotientSOSO

This is the starting p oint for the spherical harmonics to b e discussed later

Lemma Let X be a Gspace and let C X Then g L is a continuous map

c g

G C X

c

Proof Let g G Then wemust show that L L is close to in C X for g close to

g g c

where L and we see that without loss of generality w L L L g No

g g g

gg

wemayaswell assume that g e Let C b e a compact neighb ourho o d of e in G Then the set

K C supp is compact in X Moreover for g C the function L has supp ort contained

g

in K Fix Then wemust show that there exists an op en neighb ourho o d U of e in C such

that

jL kL k sup x xj

g g K

xK

for all x U The map g x g xiscontinuous hence for every x K there exists an

op en neighb ourho o d N x in X and an op en neighb ourho o d U e in C G such that

x x

jg y y j for all y N g U By compactness of K there exists a nite collection

x x

x x of p oints in K such that the sets N cover K Let U be the intersection of the sets

n x

j

U Then U is an op en neighb ourho o d of e in G and one readily checks that holds for g U

j

Corollary Let X be a Gspace and let I b e a p ositive Radon integral on G Then for

C X wehavethatg I L denes a continuous function G C

c g

A p ositive Radon integral I on a Gspace X is called invariant if for all C X and

c

g G wehave

I L I

g

or equivalentlyif denotes the measure asso ciated with I

Z Z

g xdx x dx

X X

Lemma Let I be a Ginvariant p ositive nontrivial Radon measure on a Ghomogeneous

space X Let f C X and supp ose that f everywhere Then I f f

c

Proof Supp ose that f is as ab ove and I f but f Then there exists a p oint x X

such that f x Hence there exists a compact neighb ourho o d U x and a p ositve constant

Let C suppg b e the compact supp ort such that f on U Let now g C X g

c

of g Then every p ointofC is contained in Gx and by compactness there exists a nite subset

S G such that the sets sU s S cover C For every s S there exists a such that on

s

P

sU wehave g L f Hence g L f and it follows that

s s s s s

sS

X

I g I L f

s s

sS

tegral It follows from this the latter equality b eing a consequnece of the invariance of the in

that I contradiction

Corollary Let the hyp othesis of the ab ove lemma b e fullled Then

Z

hf jg i I f g f xg x dx

X

denes a p ositive denite Hermitian inner pro duct on C X

c

We denote the completion of C X with resp ect to the ab ove inner pro duct by L X Let

c

L denote the left represenation of G on X Then byinvariance of the measure the map L is

x

unitary for the inner pro duct on C x and therefore has a unique extension to a unitary map

c

L L X L X

x

The group G may b e viewed as a homogeneous space for its left right action A left

right invariantpositive nontrivial Radon integral on X is called a left right Haar integral

on G

Theorem Let G b e a lo cally compact group Then there exists a left right Haar integral

I on G If I is a second left right Haar integral on G then I cI for a p ositive constant

c

Proof For the existence part of this theorem we refer the reader to BI We will prove

uniqueness If f C X then we dene the function f C X by f xf x If J is a

c c

BI Bourbaki Integration Chapter

right Haar integral then we dene the left Haar integral J by J f J f Thus wehavea

bijection b etween left and rightHaarintegrals

Let now I and J b e a left and a right Haar integral resp ectively Then it suces to show that

J cI for some c Let and denote the measures asso ciated with I and J resp ectively

and let f C Gbesuch that I f Dene

c

Z

D x I f f y x d y

f

G

Then D is a continuous function on Gby Lemma

f

Let g C G Then

c

Z

J g g xy d x

G

for every y G Multiplying the ab ove identity with f y andintegrating with resp ect to dy

we obtain

Z Z Z Z

I f J g f y g xy d xdy f y g xy dy d x

G G G G

where the change of order of integration is allowed byFubinis therorem Note that the integrand

is compactly supp orted as a function on G G Using left invariance of and after that again

changing the order of integration wemay rewrite the integral as

Z Z

f x y d x g y dy f D g

f

G G

Hence J g I D g for every g C GLet f f C Gbetwo functions with I f

f c c j

ws from the ab ove that I D D g for all g hence D D Thus it then it follo

F f f f

follows that D D is a continuous function which is actually indep endent of the particular

f

choice of f Substituting x e in the denition of D wenow infer that

f

D eI f J f

for all f in the set S of functions C GwithI The set S is readily seen to b e dense

c

in C G so that is actually valid for all f C G Hence J D eI

c c

Remark The existence and the uniqueness of Haar measure is much easier to establish

when G is a Lie group ie G p ossesses the structure of C manifold compatible with its

top ologyandsuch that the map G G G x y xy is smo oth ie C Examples of

Lie groups are SUn SOn SLn C GLn C SLn R GLn R For simplicitywe assume

n

denotes that G is connected Let n dim G and x a nontrivial nform T G Here T G

e

e x

the cotangentspace of G at an element x G Let l G G g xg and dene for x G

x

n

the nform T G by

x

x

T l

x e x e

Then x is a nowhere vanishing smo oth dierential nform on G hence denes an

x

orientation on G We dene I C G C by the oriented integral

c

Z

I f f f C G

c

G

One readily veries that I is a p ositiveRadonintegral which is nontrivial Moreover one also

readily checks that l for all g G from which it follows that I is a left Haar integral

g

For further details we refer to BD or Wa

Let I b e a left Haar integral on G Then for every x G we dene the Radon integral R I

x

by R I I R Then one readily veres that R I isaleftHaarintegral use that R

x x x

x

and L commute Hence R I xI for a uniquely determined x The function

y x

G which is readily seen to b e indep endentoftyhe particular choice of I is called

the mo dular function of G

Exercise Show that is a continuous group homomorphism

A group G with is called unimo dular Thus a group is unimo dular if its left Haar

measure is a right Haar measure For obvious reasons the Haar measure is said to b e biinvariant

in this case

Exercise Show that a compact group G is unimo dular Hint use that the image G

is a compact group

Representations

From nowonG will always b e a lo cally compact top ological group

Exercise Let b e a representation of G in a Banach space V Show that the following

conditions are equivalent

a G V V is continuous

b For every x G the map x is coninuous and for every v V the map G V x x v

is continuous

Hint to show that a follows from b use the BanachSteinhaus theorem

Lemma Let X be a Ghomogeneous space and let I b e a nontrivial invariant p ositive

Radon integral on X Then the representation L of G in L X dened in the previous section

is continuous

Proof In view of the ab ove exercise it suces to show that for every L X the map

x L G L X iscontinuous Using that L is a representation we readily see that it

x

suces to establish the continuityofate Thus wemust estimate the L norm of the function

Let X such that k k L as x e Let Then there exists a C

c x

BD T Brocker T tom Dieck Representations of Compact Lie Groups Graduate Texts in Math

SpringerVerlag

Wa F Warner Foundations of dierentiable manifolds and Lie groups Scott Foresman and Co Glenview

Illinois

g C G b e a nonnegative function such that g on an op en neigb ourho o d of supp Then

c

for x suciently close to e wehave g on suppL Thus for such x wehave

x

kL k kL k

x x

kL g k

x

kL k kg k

x

as x e Using Lemma one sees that the last term b ecomes smaller than

Let b e a representation of G in a complex linear space V By an invariant subspace we

mean a linear subspace W V suchthat xW W for every x G

Acontinuous representation of G in a Banachspace V is called irreducible if and V are

the only closed invariant subspaces of V

Remark Note that for a nite dimensional representation an invariant subspace is auto

matically closed

By a unitary representation of G we will always mean a continuous representation of G

in a Hilb ert space H such that x is unitary for every x G

Prop osition Let G b e compact and supp ose that V is a continuous nite dimensional

represen tation of G Then there exists a p ositive denite Hermitian inner pro duct hji on V for

which the representation is unitary

Proof Let dx denote Haar measure on G and x any p ositive denite Hermitian inner

pro duct hji on V Then we dene a new Hermitian pairing on V by

Z

hv jw i h xv j xw i dx v w V

G

Notice that the integrand x h xv j xw i in the ab ove equation is a continuous func

vw

tion of x We claim that the pairing thus dened is p ositive denite Indeed if v V then the

R

x dx by p ositivityof is continuous and p ositiveonG Hence hv jv i function

vv vv

G

the measure Also if hv jv i then by Lemma and hence hv jv i e and

vv vv

p ositive deniteness follows

Finally we claim that is unitary for the inner pro duct thus dened Indeed this follows

from the invariance of the measure If y G and v w V then

Z Z

h y v j y w i xy dx x dx hv jw i

vw vw

G G

Lemma Let H be a continuous representation of G If H is an invariant subspace for

then its ortho complement H H is a closed invariant subspace for If H is closed then

wehave the direct sum H H H of closed invariant subspaces

Proof Left to the reader

Corollary Assume that G is compact and let V be a continuous nite dimensional

representation of G Then decomp oses as a nite direct sum of irreducibles ie there exists

a direct sum decomp osition V V of V into invariant subspaces suchthatforevery j

j n j

the representation dened by x xjV is irreducible

j j j

Proof Fix an inner pro duct for which is unitary and apply the ab ove lemma rep eatedly

Denition Let H b e a unitary representation of G Then by a matrix co ecientof

wemeanany function m G C of the form

mx m xh xv jw i

vw

with v w H

Notice that the matrix co ecent m denied ab oveisa continuous function on G

Let now V b e a nite dimensional unityary representation of G and x an orthonormal

basis u u of V Then for every x G we dene the matrix M x M xby

n u

M x m x

ij u u

i j

This is just the matrix of x with resp ect to the basis u Note that it is unitary Note also

that M xy M xM y Thus M is a continuous group homomorphism from G to the group

U n of unitary n n matrices

If V j are continuous representations of G in top ological linear spaces then

j j

a continuous linear map T V V is said to b e equivariant or called interwining if the

following diagram commutes for all x G

T

V V

x x

T

V V

The representations and are said to b e equivalent if there exists a top ological linear

isomorphism T from V onto V which is equivariant

Exercise Let V betwo nite dimensional representations of G Show that and

j j

are equivalent if and only if there exists choices of bases for V and V such that for the

asso cxiated matrices one has

mat x mat x

We will now discuss an imp ortant example of representations Let SU denote the group

ofby unitary matrices with determinant one Thus SU is the group of matrices of the

form

g

with C and jj j j Then SU acts on C in a natural way and wehavethe

asso ciated representation on the space P C of p olynomial functions p C C It is given

by the formula

g pz pg z pz z z z

The subspace P P C of homogeneous p olynomials of degree n is an invariant subspace for

n n

We write for the restriction of to P

n n

We will now discuss a result that will allow us to show that the representations of the

n

ab ove example are irreducible We rst need the following fundamental lemma from linear

algebra If V is a complex linear space we write End V for the space of linear maps from V

to itself and GLV for the group of invertible elements in End V If is a representation of

G in V then wemay dene a representation of G in EndV by

g A g A g

Note that if is nite dimensional and continuous then so is Note also that the space

G

End V fA V g A Ag

of Ginvariants in V is just the space of Gequivariant linear maps V V

Lemma Let V b e a nite dimensional complex linear space and let A B EndV be

such that AB BA Then A leaves ker B imB and all the eigenspaces of B invariant

Proof Elementary and left to the reader

From nowonwe assume that G is a lo cally compact group All representations are assumed

to b e continuous

Lemma Schurs lemma Let V b e a representation of G in a nite dimensional com

plex linear space V Then the following holds

G

a If is irreducible then EndV CI

V

G

b If G is compact and EndV CI then is irreducible

V

G

Proof aSupp ose that is irreducible and let A EndV Let C b e an eigenvalue

of A and let E kerA I b e the asso ciated eigenspace Note that for the existence of this

eigenspace we need that V is complex For every x G wehave that x commutes with A

hence leaves E invariant In view of the irreducibil ityof it now follows that E V hence

A I

bBy compactness of G there exists a Hermitean inner pro duct hji for which is unitary

Let W V be a Ginvariant subspace For the pro of that is irreducible it suces

ve W V Let P b e the orthogonal pro jection V W Since W to show that wemust ha

and W are b oth Ginvariant wehave for g G that g P g Pg on W and

G

g P Pg on W Hence P EndV and it follows that P I for some C

Now P hence Also P P hence and we see that Therefore P I

and W V

We will now apply the ab ove lemma to prove the following

Prop osition The representations P C n of SU are irreducible

n n

For the pro of we will need compactness of SU In fact wehave the following more general

result

Exercise For n let M R and M C denote the linear spaces of n n matrices with

n n

entries in R and C resp ectively ShowthatSUn is a closed and b ounded subset of M C Show

n

that SOnSUn M R and nally show that SOn and SUn are compact top ological

n

groups

Proof of Prop Let n b e xed and put and V P C Supp ose that

n n

A EndV is equivariant Then in view of Lemma b it suces to showthatA is a scalar

V by For k n we dene the p olynomial p

k

nk k

p z z z

k

Then fp k ng is a basis for V For R we put

k

i

e cos sin

t r

i

e sin cos

Then

T ft Rg and R fr Rg

are closed subgroups of SU One readily veries that for k n and R weha ve

ik n

t p e p

k k

ik n

Thus every p is a joint eigenvector for T Fix a such that the numb ers e are mutually

k

dierent Then for every k n the space Cp is eigenspace for t with eigenvalue

k

ik n

e Since A and t commute it follows that A leaves all the spaces Cp invariant

k

Hence there exist C such that

k

Ap p k n

k k k

Let E b e the eigenspace of A with eigenvalue Wewillshow that E V thereby completing

the pro of The space E is SUinvariant and contains p Hence it contains r p for every

R By a straightforward computation one sees that

n

X

n

nk k n

cos sin r p z z cosz sinz p

k

k

k

From this one sees that the r p which are contained in E span V Hence E V

We end this section with two useful consequences of Schurs lemma

Lemma Let V V be two irreducible nite dimensional representations of the

lo cally compact group G If and are not equivalent then every intertwining linear map

T V V is trivial

Proof Let T be intertwining and nontrivial Then ker T V is a prop er Ginvariant

subspace Hence ker T anditfollows that T is injective Therefore its image im T is a non

ariant subspace of V It follows that imT V hence T is a bijection contradicting trivial Ginv

the inequivalen ce

Let V b e a complex linear space Then by a sesquilinear form on V we mean a map

V V C which is linear in the rst variable additive in the second and which satises

v w v wvw V C Thus a Hermitean inner pro duct is a sesquilinear form

If V is a representation for a group G then a sesquilinear form on V is called equivariant

if g v g w v w for all v w V g G

Lemma Let V b e a nite dimensional unitary representation of a lo cally compact

group G Then the equivariant sesquilinear forms on V are precisely the maps V V C

of the form hji C Here hji denotes the equivariant inner pro duct of the Hilb ert

space V

Proof Let V V C b e sesquilinear Then for every w V the map v v wis

a linear functional on V Hence there exists a unique Aw V such that v w hv Aw ji

One readily veries that A V V is a linear map Moreover the equivariance of and hji

implies that A is equivariant Hence A I for some C and the result follows

Schur orthogonality

In this section G will b e a compact top ological group unless stated otherwise Let I b e a left

Haar integral on G Then I is determined up to multiplication by a p ositive constan t Thus

I is uniquely determined by the additional requirement that I We denote the measure

R

dx asso ciated with I by dx Then

G

If is a nite dimensional irreducible unitary representation of G we write C G for the

linear span of the space of matrix co ecients of Notice that the space C G do es not dep end

on the chosen unitary inner pro duct on V Thus by Prop osition we can dene C G for

any irreducible nite dimensional representation of G

There is a nice way to express sums of matrix co ecients of a nite dimensional unitary

representation V of a lo cally compact group G Let v w V Then we shall write L for

vw

the linear map V V given by

v L u hujw i

vw

One readily sees that

tr L hv jw i v w V

vw

Indeed b oth sides of the ab ove equation are sesquilinear forms in v w so it suces to check

the equation for v w memb ers of an orthonormal basis which is easily done

It follows from the ab ove equation that

m x tr xL

vw vw

Hence every sum m of matrix co ecients is of the form mg tr g A with A EndV

Conversely if fe k ng is an orthonormal basis for V then one readily sees that

k

X

L A hAe je i

e e j i

i j

ij n

Using this one may express every function of the form g tr g A as a sum of matrix

co ecients Wenow dene the linear map T EndV C Gby

T Ax tr xA x G

for every A End V Let b e irreducible then it follows from the ab ove discussion that T

maps V onto C G Dene the representation of G G on EndV by

x y A xA y

for A End V xy V

We dene the representation R L of G G on C Gby

R Lx y L R

y x

Then wehave the following

Lemma Let V b e a nite dimensional irreducible representation of G Then C G

is invariant under R L and the map T V C G is surjective and intertwines the

representations and R L of G G

Proof W e rst prove the equivariance of T End V C G Let A EndV and

x y G then

T x y Ag tr g xA y tr y gxA R L T Ax

x y

Note that it follows from this equivariance that the image of T is R Linvariant In a previous

dicussion wesaw already that im T C G

Corollary If and are equivalent nite dimensional irreducible representations of G



then C G C G

Proof Let V V b e the asso ciated representation spaces Then by equivalence there exists

a linear isomorphism T V V such that T x x T for all x G Hence for

A EndV x G wehave



TAT x tr xTAT tr T x TA tr xA T Ax T

Now use Lemma

Wenowhave the following

Theorem Schur orthogonality Let V and V be two irreducible nite dimen

sional representations of G Then wehave the following



with resp ect to the Hilb ert structure a If and are not equivalent then C G C G

of L G

b Let V b e equipp ed with an inner pro duct for which is unitaryIfv w v w V then

 

the L inner pro duct of the matrix co ecien is given by ts m and m

vw

v w

Z

 

x dx dim hv jv i m x m hw jw i

vw

v w

G

Remark The relations are known as the Schur orthogonality relations

 

Proof For w V and w V we dene the linear map L V V by L u hujw iw

w w w w

Consider the map

Z

 

I x L x dx

w w w w

G

Then one readily veries that

  

hI v jv i hm jm i

vw

w w v w



Moreover using invariance of the measure one readily sees that I is intertwining

w w



a If and are inequivalent then the interwtwining map I is trivial by Lemma

w w

Now apply to prove a

G

 

b Now assume V V Then for all w w V wehave I End V hence I is a

w w w w

scalar It follows that there exists a sesquilinear form on V such that



I w wI

w w



Applying the trace to b oth sides of the ab oveequationwe nd that d w w tr I Here

w w

w ehave abbreviated d dim On the other hand since tr is linear wehave that

Z Z

   

tr I tr x L x dx tr L dx trL hw jw i

w w w w w w w w

G G

Hence



hw jw i I d

w w

Now apply to prove b

Another waytoformulate the orthogonality relations is the following V is assumed to b e

denote the equipp ed with an inner pro duct for which is unitary If A End V let A

Hermitean adjointofA Then one readily veries that

A B hAjB i tr B A

denes a Hermitean inner pro duct on EndV Moreover the representation is readily seen

to b e unitary for this inner pro duct

p

Corollary The map d T is a unitary isomorphism from End V onto C G

Notice that it follows from the ab ove result that C G has a unique conjugation invariant

function whose L norm is one In fact one has

x T I x tr x

V

The function is called the character of

Characters

For the moment assume that G is a lo cally compact group If is any nite dimensional

representation of G we dene its character C Gby x tr x

Exercise Let V b e a nite dimensional representation of G Show that

a xy x y

x x b

t

For x G dene x End V by x x

c Show that is a continuous representation of G in V This representation is called

the contragredientof



d Show that

Exercise Let b e nite dimensional representations of G Show that

If V and V aretwocontinuous representations of G then we dene the direct

sum representation in the direct sum V V V of top ological linear spaces by

xv v xv xv v V v V x G

Exercise Let b e nite dimensional representations of G Show that

Exercise Recall the denition of the irreducible representations n N of SU Show

n

that the character of is completely determined by its restriction to T ft Rg

n n

Hint use that every matrix in SU is conjugate to a matrix of T

Show that

sin n

t

n

sin

From nowonwe assume that G is a compact group

Lemma Let b e nite dimensional irreducible representations of G

a If then h j i

b If then h j i

Proof This follows easily from Theorem

Let b e a nite dimensional representation of the compact group G Then is unitarizable

n

and therefore is equivalent to a direct sum of irreducible representations It follows that

i

i

P

n

Using the lemma ab ovewe see that for every irreducible representation of G

i

i

wehave

fi g h j i

i

In particular this numb er is indep endent of the particular decomp osition of into irreducibles

For obvious reasons the numb er is called the multiplicityof in We shall also denote it

by m

b

Let G denote the set of equivalence classes of nite dimensional irreducible representations

b

of G Then by abuse of language we shall write G to indicate that is a representative for

b b b

an elementofG A b etter notation would p erhaps b e GIf G and m N then we

write m for the equivalence class of the direct sum of m copies of

Wehaveproved the folllowing lemma

Lemma Let b e a nite dimensional representation of the compact group G Then

M

m

b

G

Moreover any decomp osition of into irreducibles is equivalent to the ab ove one

Exercise This exercise is meant to illustrate that a decomp osition of a representation into

irreducibles is not unique Let b e irreducible representations in V V resp ectively Assume

that are equivalent and let T V V be an intertwining isomorphism

Equip V V V with the direct sum representation and show that W fv T v v

V g is an invariant subspace of V Show that the restriction of to W is irreducible and

equivalentto Find a complementary invariant subspace W and show that the restriction of

to this space is also equivalentto

The following result expresses that the character is a p owerful invariant

Corollary Let be two nite dimensional representations of G Then

Proof The only if part is obvious Toprove the if part assume that Then for

b

every G wehave m h j i h j i m Now use the ab ove lemma

Integration with values in a lo cally convex space

Let V b e a complex linear space Then by a seminorm on V wemeanamapp V R

satisfying

a pv w pv pw v w V

b pv jjpv v V C

If p is a seminorm on V v V and wewriteB v p fw V pv w g

A set P of seminorms on V will b e called fundamental if for every p p P there exists a

p P such that p p p and moreover if pv p P v To a set P of seminorms

one may asso ciate the set F fB p p P g If P is fundamental then there

P

exists a unique structure of top ological linear space on V such that the set F is a fundamental

P

system of neighb ourho o ds ie for every op en neighb ourho o d in V there exist p P

and such that B p Note that fg is closed for this top ology so the top ology is

Hausdor By a lo cally convex space wewillalways mean a complex top ological linear space

whose top ology is given by a fundamental system of seminorms in the fashion describ ed ab ove

The name convex refers to the fact that the fundamental neigb ourho o ds B p are convex in

V viewed as a real vector space One mayalsouseconvexity prop erties of neighb ourho o ds to

characterize lo cally convex spaces

Note that a normed linear space V kk is lo cally convex b ecause P fk kg is fundamental

We recall that a lter in a set S is any collection F of subsets of S satisfying the following

conditions

a A B F A B F

b A F and A B S B F

c F

A lter F in a top ological linear space V is said to b e convergent with limit v V if for every

neighb ourho o d v there exists a A F such that A If V is Hausdor then every lter in

V has at most one limit b ecause of conditions b and c The notion of convergence of a lter

extends the notion of convergence of a sequence Indeed let v b e a sequence To this sequence

k

one may asso ciate the lter G consisting of all of sets A V such that fv k NgnA is

k

nite Then fv g is convergent with limit v if and only if G is convergent with limit v

k

A lter F in a top ological linear space V is called a Cauchy lter if for every neighb ourho o d

ofin V there exists a A F such that A A This notion generalizes the notion of

Cauchy sequence in the same fashion as ab ove

A top ological linear space is said to b e complete if every Cauchy lter is convergent This

denition obviously extends the old denition of completeness for a normed space V k k

In the following we shall need the following prop erty of complete top ological linear Hausdor

spaces

Lemma Let V b e a dense linear subspace of a top ological linear space V equipp ed with the

restriction toplogy Moreover let A be a continuous linear map of V to a complete top ological

linear Hausdor space Then A has a unique continuous linear extension to a map V W

Proof Uniqueness is obvious To establish existence let v V let N b e the lter of all

F b e the lter of sets B V B F Let G b e the lter of all neighb ourho o ds of v in V and let

sets S W such that A S F Then from the continuity and the linearityofA it follows that

G is Cauchy hence has a unique limit Av One readily checks that A is the desired extension

see eg Bourbaki Espaces Vectoriels Top ologiques Chap par

A lo cally convex space whose top ology is determined by a countable fundamental set of

seminorms fp j Ng is complete if and only if a sequence whichisCauchy with resp ect to

j

every p is convergent Such a space is called a Frechet space

j

Example Let X b e a lo cally compact Hausdor space and V a lo cally convex space with a

fundamental set P of seminorms Then C X V the space of continuous functions X V may

b e equipp ed with a lo cally convex top ology in the following natural wayFor every compact

subset K X and every p P we dene the seminorm

s f sup pf x

Kp

xK

The set of these seminorms is easily seen to b e fundamental

Exercise Show that the lo cally convex space C X V dened ab ove is complete if V is

complete Show that if X is compact ie X is a countable union of compact sets and V is

Frechet then C X V isaFrechet space

In the following we assume that X is a lo cally compact Hausdor space and V a complete

lo cally convex space We also assume that I is a p ositive Radon integral on X Our purp ose is

to extend the denition of I to the space C X V of compactly supp orted continuous functions

c

with values in V We will rst do this under the assumption that X is compact whichwe assume

to b e fullled until asserted otherwise Let V denote the space of continuous linear functionals

on V

Prop osition There exists a unique continuous linear map I C X V V such that for

every V wehave

I f I f f C X V

Moreover if p is a continuous seminorm on V then for all f C X V wehave

pI f I pf

We will prove this prop osition in a numb er of steps If V is nite dimensional then one

readily veries the existence of I For the estimate we need the following

Lemma Let W b e a nite dimensional real linear space and p a seminorm on W Then for

every w W nfg and every there exists a linear hyp erplane H W such that for all

e pw h pw h H wehav

Proof If pw there is nothing to prove Thus assume that pw Let L be the

linear subspace of W consisting of v W with pv Then the imagew of w in W WL

is nonzero and p induces a normp on W It suces to prove the assertion for W p w Thus

without loss of generalitywemay assume from the start that p is a norm Then the set

w g S fv W pv p

is compact and convex and do es not contain w Hence there exists a linear hyp erplane H W

such that w H S This implies the desired estimate

Proof of Prop when V is nite dimensional It remains to prove the estimate Let p

b e a seminorm let f C X V and put w I f Let b e arbitrary Then by the ab ove

lemma there exists a real linear subspace H V of real co dimension such that for all h H

wehave pw h pw Obviously this must imply that V Rw H as a real linear space

Write f f w f where f C X Rand f C X H Then I f w implies I f

and I f Now I jf j jI f j hence

pI f pw I jf jpw I pw jf j I pf w I pf

X

where denotes the constant function with value Hence

X

pI f I pf I

X

for arbitrary from which the desired estimate follows

In general if W V is a nite dimensional linear subspace then wemay view C X W asa

subspace of C X V and we obtain a continuous linear functional I C X W W satisfying

with W instead of V One now readily sees that I extends to the algebraic direct sum of such

subspaces C X W with W nite dimensional we denote this direct sum by C X V If p is

a continuous seminorm on V then for every f C X V wehave pI f I p f Hence I

on C X V is continuous for the top ology on C X V Thus in view of Lemma the pro of

of Prop osition is completed by the following lemma

Lemma The space C X V is dense in C X V

Proof Let f C X V let p be a continuous seminorm on V and let Then it suces

to show that there exists a g C X V such that

pf y g y

h that for all y N we for all y X For every x X there exists a neighb ourho o d N suc

x x

have pf x f y By compactness there exists a nite set of x X such that the sets

j

N N cover X Let b e a partition of unity sub ordinate to the covering fN g Then we

j x j j

j

P

claim that g f x satises our requirements First of all it is clear that g has its values

j

j

in the linear span of the f x hence b elongs to C X V Secondlylety X and let J

j y

denote the set of j such that y N Then j J implies pf y f x hence

j y j

p y f y y f x y

j j j j

If j J then y Thus summing the ab ove inequalityover the j we obtain the desired

y j

inequality

We will now describ e the extension of the Radon integral to C X V when X is lo cally

c

compact Let Y X b e a compact subset then it suces to dene I on the space C X V of

Y

continuous functions X V with supp ort con tained in Y Let Z b e a compact neighb ourho o d

of Y and select a function C X with onY and such that Z is a

c

neighb ourho o d of supp If f C Z then f may b e viewed as an elementofC X with

c

supp ort contained in Z Hence wemay dene the p ositiveRadonintegral I C Z C by

f I f Wenow dene I to b e restriction to C X V ofI s extension to C Z V One

Y

readily veries that this denition of I do es not dep end on the choices of Z involved

We end this section with a useful result assertying that continuous linear maps commute

with integration

Lemma Let X b e a lo cally compact Hausdor space and I a p ositive Radon integral on

X Let A V W be a continous linear map b etween complete lo cally convex spaces Then

for all f C X V wehave A f C X W Moreover

c c

AI f I A f

Proof As in the previous discussion wemay reduce to the situation that X is compact Let

t

be anycontinuous linear functional on W Then A A isacontinous linear functional

on V Hence

t t

AI f A I f I A f I A f I A f

Since was arbitrary this completes the pro of

We shall now apply the material of this section to representation theoryLetG b e a lo cally

compact group and let dx be a choice of left Haar measure on G Let V b e a continous

representation of G in a complete lo cally convex space If f C G then we dene the linear

c

op erator f V V by

Z

f v f x xvdx

G

Lemma If f C G then

c

a the linear op erator f is continuous

b for all y G wehave

L f y f

y

Proof Exercise for the reader Toprove the last equality one needs the previous lemma

Remark Note that if dx is a right Haar measure then the ab ove lemma is also valid but

with eqn replaced by R f f y

y

The following lemma asserts that the identity op erator I can b e approximated by f

V

Lemma For every v V every continuous seminorm p on V and every there exists

R

an op en neighb ourho o d U of e in G such that for every C U with and xdx

c

G

wehave

p v v

Proof There exists an op en neighb ourho o d U e such that x U p xv v

Let C U satisfy the ab ovehyp otheses Then

c

Z

p v v p x xv xv dx

G

Z

xp xv v dx

G

Z

xdx

G

The algebra of representative functions

Let V b e a continuous representation of a lo cally compact group G Then a vector v V is

called Gnite if the linear span h xv x Gi is nite dimensional

Let RG denote the space of left and right Gnite functions in C G This space is called

the algebra of representative functions

Exercise Show that RG is indeed a subalgebra of C G

Prop osition Let G b e compact Then

M

RG C G

b

G

b

Proof For every G the space C G is nite dimensional and left and righ t Ginvariant

hence contained in RG Therefore it remains to b e shown that every elementofRG is a nite

linear combination of matrix co ecients of nite dimensional representations Let RG

and let V b e the span of all translates of the form L R x y G Moreover let V V V

x y

the bar denoting complex conjugation Then V is a nite dimensional left and right Ginvariant

subspace of RG hence decomp oses as a nite direct sum of irreducible G Gmo dules We

must show that every direct summand V in this decomp osition is contained in the righthand

side of Let f V nfg Then

W fL f V g

is a left and right Ginvariant subspace of V Moreover since Lf f e hf jf i we see that

f f W is nontrivial Therefore W V In particular there exists a V such that L

But this means that

Z

R f x y f xy dy hL jf i

x

G

Hence f is a matrix co ecient of the left regular representation restricted to the nite dimen

sional space V

Let H b e a collection of Hilb ert spaces indexed by a set Then the algebraic direct sum

P P

is a preHilb ert space when equipp ed with the direct sum inner pro duct h v aj w i

H g

P

hv jw i Its completion is called the Hilb ert direct sum of the spaces H and denoted by

M

H

A

If is a unitary representation of G in H for every A then the direct sum of the

extends to a unitary representation of G in We call this representation the Hilb ert sum of

the

Theorem The PeterWeyl Theorem The space L G decomp oses as the Hilb ert sum

M

L G C G

b

G

each of the summands b eing an irreducible invariant subspace for the representation R L of

G G

It remains to establish densityofRGin L G This will b e achieved in the next section

cf Lemma after some necessary preparations

Exercise Fix for every equivalence class of an ireducible unitary representation V

an orthonormal basis e e Denote the matrix co ecient asso ciated to e and e

i j

dim

by m Use Schur orthogonality and the PeterWeyl theorem to show that the functions

ij

q

b

dim m G i j dim

ij

constitute a complete orthonormal system for L G

Compact op erators

Let X Y b e lo cally compact Hausdor spaces If C X and C Y then we write

for the continuous function on X Y dened by

x y x y

The linear span of such functions in C X Y is denoted by C X C Y If C X and

c

C Y then is compactly supp orted Hence the span C X C Y ofsuch functions

c c c

is a subspace of C X Y

c

Lemma Let X Y b e lo cally compact Hausdor spaces Then the space C X C Y is

c c

dense in C X Y

c

Proof Fix C X Y and let K supp Then K K K for compact subsets

c X X

K X K Y Fix an op en neighb ourho o d U K with a compact closure Let

X Y X X

Then by compactness there exists a nite op en covering fV g of K such that for every j and

j X

all x x V y K one has

j Y

x y x y

Without loss of generalitywemay assume that V U for all j Select a partition of unity

j X

f g which is sub ordinate to the covering fV g and x for every j a p oint V Let x

j j j j

K y K If j is such that x V then jx y x y j It follows from this that

X Y j j

X X

j xx y x y j j xx y xx y j

j j j j j

j j

X

xjx y x y j

j j

j

X

x

j

j

Hence if we put y x y then

j j

X

k k

j j

j

Moreover supp U K and we see that wehave supp ort control which is uniform

j j X Y

in

Let now G b e a lo cally compact group Let J b e a left Haar integral on G G Then one

readily veries that for a xed C G with one has that J is a left Haar

c

integral on G hence equal to a constant times the Haar integral I on G Applying the same

reasoning with interchanged variables one sees that there exists a constant c such that

J cI I Without loss of generalitywemay assume that c Of course in the

sense of measure theory this means that the Haar measure of G G is the square of the Haar

measure of G More precisely wehave

Lemma Let f C G G Then

c

Z Z

J f f x y dx dy

G G

Proof By what we said ab ove the identityisvalid for f C G C G Now use a density

c c

argument

If K C G G then we dene the linear op erator T C G C Gby

c K c c

Z

T x K x y y dy

K

G

For obvious reasons this is called an integral op erator with kernel K

Lemma Let K C G G Then the op erator T extends uniquely to a b ounded linear

c K

endomorphism of L G with op erator norm kT k kK k Moreover this extension is compact

K

Proof Let C G Then

c

hT j i hK j i kK k k k kK k kk k k

K

Hence kT k kK k kk This implies the rst assertion since C G is dense in L G

K c

For the second assertion note that there exists a sequence K in C G C G which

j c c

verges to K It follows that con

kT T kkK K k

K K j

j

has a nite dimensional image hence is compact The subspace of compact Every op erator T

K

j

op erators is closed for the op erator norm hence T is compact

K

Corollary Assume that G is compact and let f C G Then the op erator Rf

L G L G is compact

Proof If C G then

Z Z

Rf x f y xy dy f x y y dy

G G

Hence Rf T with K x y f x y Note that for this argument it is crucial that G be

K

compact For if not and f C G then the asso ciated K need not b e compactly supp orted

c

Exercise Let H b e a unitary representation of G Let f C G and put f x

f x Show that

f f

Show also if f is conjugation invariant ie f xy x f y for all x y G then f is

intertwining

Corollary Assume that G is compact and let f C G be such that f f Then Rf

and Lf as well is a compact selfadjoint op erator

Wenow recall the imp ortant sp ectral theorem for compact selfadjoint op erators in Hilb ert

space

Theorem Let T b e a compact selfadjoint op erator in the complex Hilb ert space H Then

wing holds there exists a discrete subset R nfg such that the follo

a For every the asso ciated eigenspace H of T in H is nite dimensional

b If then H H

c For every let P denote the orthogonal pro jection HH Then

X

T P

the convergence b eing absolute with resp ect to the op erator norm

Pro of of the PeterWeyl Theorem

In this section we assume that G is a compact group We will establish the PeterWeyl theorem

by proving the following lemma

Lemma Let G b e compact Then the space RG is dense in L G

We rst need some preparation

Lemma Let U b e a neighb ourho o d of e in G Then there exists a C G such that

c

R

a and x dx

G

b

c is conjugation invariant

Proof From the continuity of the map x x one sees that there exists a compact neigh

b ourho o d V of e such that V U and V U For every x G there exist an op en

neighb ourho o d N of x and a compact neighb ourho o d V of e in V such that zyz V for all

x x

z N y V By compactness of G nitely many of the N cover G Let b e the intersection

x x x

of the corresp onding V Then is a compact neighb ourho o d of e and for all x G and y

x

wehave xy x V

R

Now select C such that and x dx Dene

c

G

Z

x yxy dy

G

Using the result of the exercise b elowwe see that is a continuous function Clearly

Moreover by application of Fubinis theorem and biinvariance of the Haar measure it follows

R

that x dx If x then yxy supp for some y G hence x y y

y G

G

satises V It follows that supp V One now readily veries that the function

all our requirements

Exercise Let G b e a compact group I the normalized Haar integral and assume that

f G G C is a continuous function Dene the function F G C Gby F y x f x y

a Show that F is a continuous function with values in the Frechet space C G

R

f x y dy b Show that for all x G wehave I F x

G

Corollary Let f L Gf Then there exists a left and right Gequiv ariant b ounded

linear op erator T L G L G with

a Tf

b T is selfadjoint and compact

c T maps every right Ginvariant closed subspace of L G into itself

Proof Let kf k and x an op en neighb ourho o d U of e in G that satises the assertion

of Lemma with V L Gv f Let C U b e as in Lemma and dene T R

c

Then T satises a and c T is left Gequivariant since L and R commute It is right

Gequivariant b ecause is conjugation invariant cf Exercise Finally b follows from

Corollary

Proof of Lemma The space RGisleftandright Ginvariant and by unitarity so is its

ortho complement V Supp ose that V contains a nontrivial element f Let T b e as in the ab ove

lemma Then T jV V V is a nontrivial compact selfadjoint op erator whichisbothleftand

right Gequivariant By the sp ectral theorem for compact selfadjoint op erators there exists

a R such that the eigenspace V kerT I is nontrivial By compactness

V

of T the eigenspace V is nite dimensional and by equivariance of T it is b oth left and right

Ginvariant Hence V RG contradiction Therefore V must b e trivial

Densityof Gnite vectors

For the momentwe assume that G is a lo cally compact group and that dx is a choice of left

Haar measure on G If f g C G we dene the convolution pro duct of f and g to b e the

c

function f g G C given by

Z

f g x f y g y x dy

G

If x G then ev f f x is a continuous linear functional on C G thus applying

x c

Prop osition we see that f g x ev Lf g Hence f g C G One readily checks that

x c

supp f g supp f suppg

Exercise Show that C G equipp ed with addition and the conv olution pro duct is an

c

asso ciative algebra Show that in general this algebra has no unit element

An imp ortant motivation for the denition of the convolution pro duct is the following

Exercise Let be a continuous representation of G in a complete lo cally convex space V

Show that for all f g C G one has

c

f g f g

The following fact will b e needed in the sequel

Exercise Let f g C G and let x G Then

c

L f g L f g and R f g f R g

x x x x

We will also need the following

Lemma Let f g C G Then

c

kf g k kf k kg k

Proof Deneg G C byg x g x Then from the denition of the convolution

pro duct it follows that for x G wehave

f g x hf jL gi

x

By the CauchySchwartz inequality this implies that jf g xj kf k kL gk From the bi

x

invariance of the Haar integral it follows that kL gk kg k Hence jf g xj kf k kg k for

x

every x G

From nowonwe assume that the group G is compact and that dx is normalized Haar

measure Moreover we assume that is a continuous representation of G in a complete lo cally

convex space V

Let V denote the space of Gnite vectors in V Then V is a Ginvariant linear subspace of

G G

V Avector v V is called isotypical if there exists an irreducible nite dimensional represen

G

Gv i is equivalent to a necessarily nite multiple of tation of G such that the linear span h

The representation or rather its equivalence class is then called the typ e of the isotypical

b

vector v If G then by V we denote the set of isotypical Gnite vectors of typ e in V

Obviously V is a Ginvariant linear subspace of V Moreover in view of Corollary the space

V is the algebraic direct sum of the spaces V

G

M

V V

G

b

G

Notice that the ab ove decomp osition of V is canonical The decomp osition of each V into

G

irreducibles is not canonical in general cf Exercise

The notation intro duced ab ove is consistent with the notation C G intro duced b efore

b

Lemma Let G Then C G equals the space of isotypical right Gnite functions of

typ e

Proof Let V denote the space of right Gnite functions of typ e in C G Then obviously

C G V It remains to prove the reversed inclusion

b

If G let P L G C G denote the orthogonal pro jection By the invariance of the

subspace C G it follows that P is equivariant The restriction T of P to V is an intertwining

op erator V C G for the right regular representation If then using Lemma we

see that T By the PeterWeyl theorem this implies that V is contained in the closure of

C G This closure equals C G by nite dimensionality

b

Exercise Let G Show that C G equals the space of isotypical left Gnite vectors

of typ e

b



G Then f maps V into the space V Lemma Let f C G

Proof Let v V and consider the map T C G V v Thenitfollows from

that T intertwines L with Therefore it maps Lisotypical vectors of typ e to isotypical

vectors of typ e Now use Exercise

Prop osition The space RG is dense in C G with resp ect to the supnorm

Proof Fix f C G and let Then by Lemma there exists a C Gsuch

that kf f k By Lemma there exists a RGsuch that k k hence

k f f k kf k by Lemma Combining these estimates we infer that

kf f k kf k

Using Exercise we see that f L f RG Since was arbitrary this establishes

densityofRGin C G

Corollary The space V of Gnite vectors is dense in V

G

Proof Let v V and let p be a continuous seminorm on V Let Then by Lemma

there exists a C Gsuch that pv v By Lemma there exists a left Gnite

C G such that k k One now readily veries that p v v pv

so that pv v pv But v is Gnite by Lemma and we see that

pv V This establishes density

G

Corollary If is irreducible then V is nite dimensional

Proof By the previous corollary wemay select a nontrivial element v V The linear span

G

W of Gv is a nontrivial invariant subspace whcih is nite dimensional hence closed Since

is irreducible wemust have W V

V Corollary V cl

b

G

Proof This follows from combining with Corollary

We will end this section bycharacterizing the pro jections onto comp onents of the ab ove

b

decomp osition If G we denote its representation space byH

b

Lemma Let G Then the following holds

a d

b d I

H

Proof Let H b e the representation space of and put T Then from the conjugation

invariance of and the biinvariance of dx it follows that T H H is equivariant Hence

T I for some C bySchurs lemma From this we nd that

H

Z

d trT x tr x dx h j i

G

Now apply Lemma

b

Exercise Let G Show that the following holds

a If then d

b If then d

b

Theorem Let G Then V is a closed subspace of V Moreover

V V cl V

b

Gnfg

and P d is the asso ciated pro jection op erator V V Finally P is equivariant

Proof From the ab o ve lemma it follows that P is the identityonV and annihilates every V

By continuityofP it follows that P I on cl V On the other hand P maps into V by

Lemma It follows that cl V V hence V is closed Moreover P is a pro jection op erator

with image V Hence V W V with W ker P Since is conjugation invariant it follows

that P is equivariant and therefore W is a Ginvariant complete lo cally convex space It follows

from Corollary that W is dense in W On the other hand obviously W W V

G

Hence W equals the closure of V

b

G nf g

Wecannow completely describ e the structure of every unitary representation of the compact

group G in terms of irreducibles

b

Corollary Let H b e a unitary representation of G Then for every G the space

H is closed and d is the orthogonal pro jection HH Moreover wehave the following

Hilb ert decomp osition

M

H H

b

G

Proof The only thing that remains to b e veried is the orthogonality of the summands If

b

G let P denote the orthogonal pro jection HH Then P is equivariantsince is unitary

b

and H is invariant Let G Then the restriction of P to H is an intertwining

op erator H H Using Lemma we see that P jH whence H H

Exercise Let G b e a compact group and dx the normalized Haar measure on G Show

that for every f L Gwehave

X

d f f

b

G

norm with convergence in the L

We complete the description of unitary representations by describing the isotypical ones

b

Let G V its representation space and let S b e a set provided with the counting measure

Then we dene the representation of G on L S V by

L S

xf s x f s s S x G

L S

for f L S V

Prop osition Let H b e an isotypical unitary representation of G ie there exists a

b

G such that H H Then there exists a set S such that

L S

Proof If v Hwe put Hv for the linear span of the xv x G Note that Hv is

nite dimensional by the assumption and the rst assumption of Theorem Let S be the

collection of subsets S Hnfg satisfying

a if v S then Hv is irreducible

b if v w S v w then Hv Hw

We order S by inclusion Let CS b e a completely ordered subset Then C b elongs to S and

dominates C By Zorns lemma the set S has a maximal element S For every s S wemayx

an intertwining op erator T V H with im T Hs Wenow dene T L S V H by

s s

X

Tf T f s

s

sS

Then one readily veries that T is an equivariant isometry It remains to prove that T is

surjective Assume not Then im T is a nontrivial Ginvariant subspace of H hence contains

an element s such that Hs is irreducible Obviously Hs imT so Hs Hs for every

s S This contradicts the maximalityofS

Class functions

By a class function on a lo cally compact group G we mean a function f G C whichis

conjugation invariant ie L R f f for all x G The space C G classofcontinuous class

x x

functions is a closed subspace of C G

b

Now assume that G is compact Then the pro jections P C G C G Gmap

class functions to class functions by equivariance Hence

P C G class C G C G classC

It follows from this that the space RG class C G class RG of biGnite class functions

b

is the linear span of the characters G

Lemma Let f C G class Then

X

f hf j i

b

G

with convergence in the L norm Moreover the space RG class is dense in C G class

equipp ed with the supnorm

P

By the PeterWeyl theorem wehave f Proof P f in L sense Now P f C

b

G

and hence

P f hP f j i hf j i

The second result is proved as follows Let f RG class and Then there exists a

function g RGsuch that kf g k The function g is contained in a nite direct sum W

R

of spaces of the form C G The same is true for the functiong L R gdy By invariance

y y

G

of the Haar measureg is a continuous class function thus g RG class Finallyifx G

then

Z

jf x gxj j f x g y xy dy j

G

Z

f yxy g y xy dy j j

G

Z

kf g k dy

G

This shows that kf gk and establishes density

Ab elian groups

Wenow consider the case that the compact group G is ab elian ie xy yx for all x y G By

amultiplicativecharacter of G wemeanacontinuous group homomorphism G C where

C C nfg is equipp ed with complex multiplication Notice that every compact subgroup of

C must b e contained in the unit circle jz j Therefore if is a multiplicativecharacter then

j xj x G

Lemma Let G b e a compact ab elian group If V is a nite dimensional irreducible

representation of G then dim V Moreover x xI The map induces a

V

b

bijection from G onto the set of multiplicativecharacters of G

Proof If x G then y x yx xy x y for all x y G hence xis

equivariant and it follows that

x xI

for some x C bySchurs lemma It follows from this that every linear subspace of V is

invariant Therefore the dimension of V must b e one From the fact that is a representation

it follows immediately that x xisa character Applying the trace to we see that

b

the character of Thus induces a map from the space G of equivalence classes

of nite dimensional irreducible representationstothesetofmultiplicativecharacters of G

If isamultiplicativecharacter then denes an irreducible representation of G in C and

Therefore the map is onto the multiplicativecharacters

Corollary Assume that G is compact and ab elian Then the set of multiplicativechar

b

acters G is a complete orthonormal system for L G

Proof This follows immediately from the previous lemma combined with the theorem of

Peter and Weyl Theorem

Corollary Assume that G is compact and ab elian Then the linear span of the set of

multiplicativecharacters is dense in C G for the supnorm

Proof This follows from Prop osition

b

In the present setting we dene the Fourier transform f G C of a function f L Gby

f hf j i

b b

Let G b e equipp ed with the counting measure Then the asso ciated L space is l G the space

P

b

of functions G C such that j j equipp ed with the inner pro duct

b

G

X

hj i

b

G

Corollary The Plancherel theorem Let G b e a compact ab elian group Then the

b

G then G Moreover if f L G onto l Fourier transform f f is an isometry from L

X

f f

b

G

Proof Exercise for the reader

The purp ose of the following exercise is to view the classical theory of Fourier series as a

sp ecial case of the PeterWeyl theory

n n n

Exercise Let G R Z If m Z show that

im x

x e

m

denes a multiplicativecharacter of G Here m x m x m x Show that every

n n

n

b

multiplicativecharacter is of this form Thus G Z Accordingly for f L Gwe view the

n

Fourier transform f as a map Z C

Show that the normalized Haar integral of G is given by

Z Z

f x x dx dx I f

n n

n

n

Show that for f L Gm Z wehave

Z Z

im x m x

n n

f m f x x e dx dx

n n

n

Moreover show that wehave the inversion formula

X

imx n n

f x f m e x R Z

n

mZ

in the L sense

The group SU

Recall the denition of the representation of SU in the space V P C of homogeneous

n n n

p olynomials of degree n from Section In Prop osition it was shown that is irreducible

n

Moreover the asso ciated character is determined by the formula

sin n

t R

n

sin

see Exercise The purp ose of this section is to prove the following result

Prop osition Every nite dimensional irreducible representation of SU is equivalentto

one of the

n

We recall that every element of SU is conjugate to an elementofT Therefore a class

function on SU is completely determined by its restriction to T This restriction to T is obvi

ously invariant under the substitution t t Thus if C T denotes the space of continuous

ev

functions f T C satisfying f t f t for all t T then restriction to T denes an

injective linear map r C G class C T

ev

Lemma The map r C G class C T is a bijective isometry for the supnorms

ev

Proof That r is isometric follows from the observation that the set of values of a function

f C G class is equal to the set of values of its restriction r f Thus it remains to establish

ix

the surjectivityofr Let g C T An element x SU has two eigenvalues e and

ev

ix

e whichmay lo cally b e chosen such that x x dep ends continuously on x Dene

f x g t Then f is well dened and indep endent of the particular choice of x

x

Moreover f C G class and r f g

Corollary The linear span of the characters n N is dense in C G class

n

Proof By Lemma it suces to show that the linear span S of the functions jT is

n

P

n

ink

e Hence S equals the dense in C T From formulawe see that t

ev n

k

in in

linear span of the functions t e e n N The latter span is dense in C T

n ev

by classical Fourier theory see Section

Corollary Every nite dimensional irreducible representation of SU is equivalentto

one of the n N

n

Proof The representations are irreducible and mutually inequivalent Therefore their

n

characters constitute an orthonormal system in L G class If f C G class and hf j i

n n

y the fact that dy for all n N then hf jg i for all g C G class Let C G Then b

is normalized and f is a class function one obtains

Z Z Z

x dx x dy dx f yxy hf ji f x

G G G

Applying Fubinis theorem to interchange the order of integrations and biinvariance of the

measure dx one sees that the ab oveintegral equals

Z Z

f x y xy dxdy f x y xy dy dx hf j i

G G

R

L R dy is a continuous class function Here Fubinis theorem has b een applied where

y y

G

once more The inner pro duct at the extreme right of the latter equation vanishes and therefore

f C G Hence f It follows that the orthonormal system f n Ng is complete

n

b

But this must imply that the exhaust the irreducible representations of G by the PeterWeyl

n

theorem

From the fact that every element of SU is conjugate to an elementofT it follows that

there should exist a Jacobian J T such that for every continuous class function f on

SU wehave

Z Z

f x dx f t J t d

SU

It is p ossible to compute this Jacobian by a substitution of variables We shall obtain it by other

means

Lemma For every continuous class function f SU C wehave

Z Z

sin

f x dx f t d

SU

Proof Consider the linear map L which assigns to f C G class the expression on the left

hand side minus the expression opn the right hand side of the ab ove equation Then wemust

show that L is zero

Obviously L is continuous linear so that it suces to show that L forevery n N

n

The function is identically one therefore left and right hand side of b oth equal if one

substitutes f Hence L On the other hand if n and f then the left hand

n

side of equals h j i The right hand side of also equals hence L for all

n n

n

Manifolds

Notations and preliminaries

Let V V b e nite dimensional real linear spaces and let b e an op en subset of V We recall

that a map V is called dierentiable at a p oint a in the direction v V if

d

a a tv

v t

dt

exists

The map f is called dierentiable at a if there exists a linear map Df aV V such

that

f a h f a Df ah o h h

The linear map Df a which is unique when it exists is called the derivativeoff at a If f is

dierentiable in a then f a exists for every v V and wehave

v

f a Df av

v

n m

When V R V R then the ab oveformula may b e used to express the matrix of Df a

in terms of the partial derivatives f f the Jacobi matrix

j i e i

j

If f is dierentiable in anypointof then Df is a map from to the space HomV V

of linear maps V V If this map is dierentiable then f is called twice dierentiable The

derivativeofDf is denoted by D f It is now clear how to dene the notion of an ptimes

p

dierentiable function and its pth derivative D f A function f is called ptimes continuously

p p

dierentiable or briey C if it is ptimes dierentiable and D f is continuous We recall that

p

f is C on if all mixed partial derivatives of f order at most p exist and are continuous on

p p

Let C V denote the linear space of C maps V Then the eect of any sequence of

p

at most ppartial derivatives applied to C V is indep endent of the order of the sequence

p

C for every p We put A map f V is called smo oth or C ifitis

p

C V C V

p

for the space of smo oth maps V

Let e e b e a basis of V and abbreviate Then is a linear op erator on the

n j e j

j

space C V By the ab ovementioned result on the order of mixed partial derivatives we

have that and commute i j n Hence as an endomorphism of C V every

i j

mixed partial derivatives of order at most p is of the form

n

n

with jj p

n

p p

We briey write C for C C p By a linear partial dierential op erator

with C co ecients on we mean a linear endomorpism P of C of the form

X

P c

with nitely many nontrivial functions c C The number k maxfjj c g is

called the order of P

Manifolds

Let b e a bijection b etween op en subsets of nite dimensional real linear spaces Then

p p

is called a C dieomorphism p if and are C Note that by the inverse

p

function theorem this is equivalent to the requirement that C and Da is bijective for

every a

We shall now develop the theory of C manifolds weleaveittoreadertokeep trackof

p

what can b e done in a C context for further reading we suggest the references La Wa

La S Lang Dierential Manifolds Addison Wesley Reading Massachusetts

Wa F Warner Foundations of dierentiable manifolds and Lie groups Scott Foresman and Co Glenview

Illinois

Let X b e a Hausdor top ological space A pair U consisting of an op en subset U X

n

and a homeomorphism from U onto an op en subset of R is called an ndimensional chart

of X If U is a second ndimensional chart of X such that U U then the map

is a homeomorphism from U U onto U U This homeomorphism is called

the transition map from the chart to the chart

A set fU Ag of ndimensional charts is called a C or smo oth ndimensional

atlas of X if

a U X

A

b all transition maps are smo oth ie C

Remark Note that since is the inverse of it actually follows that all transition

maps are dieomorphisms

An ndimensional smo oth or C manifold is a Hausdor top ological space X equipp ed

with a smo oth ndimensional atlas fU Ag An ndimensional chart U ofthe

manifold X is called smo oth if all the transition maps are dieomorphisms The com

p onents will then b e called a system of lo cal co ordinates of X The collection of all

n

smo oth charts of X is an atlas by its own right called the maximal atlas of the smo oth manifold

X

Remark Any op en subset of a nite dimensional linear space is a smo oth manifold in a

natural way its dimension b eing the dimension of the linear space More generally anyopen

subset of an ndimensional smo oth manifold X is a smo oth manifold of dimension n in a natural

way

p

A map f X Y of smo oth manifolds of p ossibly dierent dimensions is called C at a

point x X if there exist smo oth charts U and V of X and Y resp ectivelysuch that

p

x U f U V and f is a C map from U to V Similarly one may dene the

concept of a k times dierentiable map b etween smo oth manifolds

p p

One readily checks that the comp osition of C maps b etween smo oth manifolds is C etc

verse The map f X Y is called a C dieomorphism if it is bijective and if f and its in

f are smo oth ie C Note that dieomorphic manifolds have the same dimension The

present notion generalizes that of a dieomorphism of op en subsets of nite dimensional real

linear spaces

Our next ob jective is to generalize the notion of derivative of a dierentiable smo oth map

between manifolds The key to this is the concept of a tangentvector Since our manifold is not

contained in an ambient linear space it may seem strange that tangentvectors can b e dened

at all The basic idea is that it makes sense to say that two curves are tangent at a p oint A

tangentvector of a manifold is then dened as an equivalence class of tangential curves More

preciselyletX b e smo oth manifold of dimension n Then by a dierentiable curveinX we

mean a dierentiable map c I X where I R is some op en interval containing The

point c is called the initial p ointofc Let x X b e a xed p oint Then two dierentiable

curves c d with initial p oint x are said to b e tangentatx if there exists a smo oth chart U

containing x such that

d d

ctj dtj

t t

dt dt

Supp ose now that V is another smo oth chart and let is the asso ciated transition

map Then by the chain rule wehave

d d

ctj D x ctj

t t

dt dt

From this we see that if holds in one chart containing x then it holds in any other chart

containing x Let C denote the set of all dierentiable curves in X with initial p oint x Dene

x

the equivalence relation on C by c d if and only if c and d are tangential at xWe dene

x

T X C

x x

The class of an element c C is denoted byc The elements of T X are called the tangent

x x

vectors of X at x

Let U bea chart containing x Then for every c C the vector ddt c only

x

dep ends on the equivalence classc We denote it by T c this notation will b e justied

x

at a later stage

n

Lemma The map T T X R is bijective

x x

Proof The injectivityofT is an immediate consequence of the denitions To establish

x

n

its surjectivityletv R andxany dierentiable curve c in U with initial p oint x and

with ddtc v Let c c Then c C and by denition wehave T c v

x x

Let nowV b e another chart containing x Then bywehave that

T D x T on T X

x x x

This implies the following

Corollary The set T X has a unique structure of real linear space such that for every

x

n

chart U containing x the map T T X R is linear

x x

The set T X equipp ed with the structure of linear space describ ed in the ab ove corollary

x

is called the tangent space of X at x

The tangent map

W e can now generalize the concept of derivative to manifolds Let f X Y be a map between

smo oth manifolds and supp ose that f is dierentiable at the p oint x X If c d C then

x

f c f d C Let U and V becharts of X and Y such that x U f U V Then

f x

the map F f is dierentiable Moreover ifc d then by denition wehave

d d

c d

dt dt

If we apply DF x to this expression we obtain

d d

f c f d

dt dt

by the chain rule It follows from this that f c and f d are equivalent elements of C This

f x

shows that the map C C c f c induces a map T X T Y whichwe denote by

x x

f x f x

T f

x

Note that it is immediate from the ab ove discussion that the following diagram commutes

T f

x

T X T Y

x

f x

T T

x

f x

D f x

n m

R R

Hence it follows from Lemma and Cor that the map T f T T is linear it

x x

f x

is called the tangentmapof f at x

Theorem The chain rule Let f X Y and g Y Z b e maps such that f is

dierntaible at x X and g is dierentiable at f x Then g f is dierentiable at x and

T g f T g T f

x x

f x

Proof This follows from the ordinary chain rule by using the commutative diagram

three times once for f X Y at x once for g Y Z at f x and once for g f X Z

at x

Let U b e an op en subset of X Then if x U one readily checks that the tangentmap

T i of the inclusion map i U X is an isomorphism T U T X Via this isomorphism we

x x x

n n

shall identify T U T X In particular if U is an op en subset of R then T U T R The

x x x x

n n n

latter space is identied with R as follows For v R dene the curve c R by

v

n n

c tx tv Then we identify R T R via the map v c Weleave it to the reader

v x v

n m

to check the following If f U V is a map b etween op en subsets V R and V R

then via the identications discussed ab ove the tangent map T f T U T V x U

x x

f x

n m

corresp onds to the derivative Df x R R Also if U isachart of a smo oth manifold

n

X containing the p oint x X then the map T T X R of Lemma corresp onds to

x x

n

the map T T X T R Finally observe that when c C then the elementc dened

x x x x

as the class of c equals T c

Remark In the literature one also nds the notations df xandDf xforT f

x

Submanifolds

Let X be a ndimensional smo oth manifold A subset Y X is called a smo oth submanifold

of dimension k if for every y Y there exists a chart U containing y such that U Y

k k n

U R Here we agree to identify R with the subspace fx R x j k g

j

Supp ose that Y is a submanifold of X and let i Y X denote the inclusion map Then

Y

one readily veries that for every y Y the map T i is an injective linear map Via this map

y Y

we shall identify T Y with a linear subspace of T X

y y

Notice that the notion of submanifold as dened ab ove generalizes the notion of smo oth

n

submanifold of R Notice also that a subset Y X is a submanifold of X if it lo oks likea

n

submanifold of R in any set of lo cal co ordinates More precisely Y is a submanifold if for every

n

smo oth chart U of X the set U Y is a smo oth submanifold of R whichmay b e empty

Let X Y b e smo oth manifolds A map f X Y is called an immersion at a p oint x X

if the tangent map T f T X T Y is injective It is called a submersion at x X if for all

x x

f x

the tangent map T f is surjective

x

One now has the following useful result which is a consequence of the implicit function

n

theorem for R

Lemma Immersion Lemma Let f X Y b e a smo oth map and let x X Then f

is an immersion at x if and only if dim X dim Y p and there exist op en neigb ourho o ds

X U x and Y V f x and a dieomorphism of V onto a pro duct U with

p

an op en subset of R such that the following diagram commutes

f

U V

I

U

i

U U

Here i denotes the inclusion x x

Lemma Submersion Lemma Let f X Y b e a smo oth map b etween smo oth

manifolds Then f is submersiveatx X if and only if dim X dim Y p and there exist

op en neigb ourho o ds X U x and Y V f x and a dieomorphism U V with

p

an op en subset of R such that the following diagram commutes

f

U V

I

V

pr

V V

Here pr denotes the pro jection on the rst comp onent

In particular it follows from the ab ove lemmas that an immersion is lo cally injective and

that a submersion always has an op en image From the denitions given b efore combined with

the two lemmas ab ovewenow obtain

Theorem Let X b e a smo oth manifold and let Y X b e a subset Then the following

conditions are equivalent

a Y is a smo oth submanifold

b Y is lo cally closed in X and for every y Y there exists an op en neighb ourho o d U y

such that U Y is the image of an injective immersion

c for every y Y there exists an op en neigb ourho o d X U y and a submersion of U

onto a smo oth manifold Z such that Y U z for some z Z

Vector elds

If X is a smo oth manifold wewriteTX for the disjoint union of the tangent spaces T x X

x

The set TX is called the tangent bundle of X We dene the map TX X bythe

requirement that T X fxg for every x X

x

A map v X TX with v x T X for every x X is called a vector eld on X The

x

set of vector elds on X is denoted byTX By dening addition and scalar multiplication of

vector elds p ointwise we turn TXinto a linear space

n

Recall that if U R is op en and x U then wehav e a natural identication

x

n n

T U R We identify TU with U R via the map given by x for x

x x

n

U T U Via this identication a vector eld on U maybeviewed as a map v U U R

x

n

with v x fxg R for all x U Thus v x x f x for a uniquely dened function

n n

U R In this waywemay identify TU with the linear space of maps U R Wenow

p p n

agree to call a vector eld v U ofclass C if it is C as a map U R

If f X Y is a dierentiable map then we dene the map Tf TX TY by Tf T f

x

on T X

x

A smo oth map f X Y is a dieomorphism if and only if Tf TX TY is bijective

Moreover if this is the case then wehave an induced bijectivemapf TX TY

dened by the formula

f v f x T fvx

x

p

ecanintro duce the notion of a C vector eld On op en subsets After these preliminaries w

n

of R this has b een done alreadyIfX is a smo oth manifold then a vector eld v TX

p

is said to b e C if for every x X there exists a chart U containing x so that v jU is

smo oth By what we said ab ove the latter assertion can also b e rephrased as the map

v j x T v x

U x

p p

is C By the chain rule it follows that if v is a C vector eld on X then for any smo oth chart

p p p

U the map is C The set TXof C vector elds on X is ob viously a linear subspace

of TX

p

From nowonwe assume that p that X is a smo oth manifold and that v X

Let x X Then byanintegral curveforv with initial p oint x we mean a dierentiable map

c I X with I an op en interval containing such that

c x

c t v ct t I

d

Here wehave writtenc tfor ct T c Wenow come to a nice reformulation of the

t

dt

existence and uniqueness theorem for systems of rst order ordinary dierential equations use

lo cal co ordinates to see this

p

Theorem Let v TXx X Then there exists an op en interval I such that

a there exists an integral curve c I X for v with initial p oint x

b if d J X is a second integral curve for v with initial p oint x then d c on I J

usually one reserves this name for TX equipp ed with a canonical structure of vector bundle

Lemma Let c I X be an integral curveforv with initial p oint x Fix t I let

x

I I t b e the translated interval and let c I X b e dened by c t ct t Then

c is an integral curveforv with initial p oint x

Proof By an easy application of the chain rule it follows that

c tc t t v ct t v c t

Moreover c x by denition

Corollary Let c d I X be integral curves for v with initial p oint x Then c d

Proof Let J b e the set of t I for which ctdt Then J is a closed subset of I

by continuityofc and d On the other hand if t J then ct t dt t fort in a

neighb ourho o d of in view of Lemma and Theorem This implies that J is op en in

I as well Hence J is an op en and closed subset of I containing and we see that J I

From this corollary it follows that there exists a maximal op en interval I for which

x

there exists an integral curve c I X for v with initial p oint x Indeed I is the union of all

x x

the intervals which are domain for an integral curve with initial p oint x

The asso ciated unique integral curve I X is called the maximal integral curve with initial

x

point x

Exercise Let v be a C vector eld on a compact manifold X and let x X Show that

I R Hint assume that I is b ounded from ab ove and let s b e its sup Let I X

x x x

b e the maximal integral curve Show that there exists a sequence s I with s s so that

n x n

s x Now apply the existence and uniqueness theorem to v and the starting p oint x

n

The following results will b e of crucial imp ortance in the theory of Lie groups

p

Corollary Let v be a C vector eld on a smo oth manifold X Let x X and let

I X b e the asso ciated maximal integral curve Let t I and let b e the maximal

x x

integral curve with initial p oint x t Then I equals the translated interval I t

x x

Moreover for t I wehave

x

t t t

Proof It follows from Lemma that c I t X t t t denes an integral curve

x

with initial p oint x Hence I t I Moreover c on I t In particular it follows

x x x

that t I Applying the same argumentto and x t we see that I t I

x x x

Hence I I t The desired equalitynow follows from c on I t

x x x

The following result which is stated without pro of expresses that the integral curve dep ends

smo othly on the initial value Let b e the union of the subsets I fxg of R X For x X

x

let I X b e the maximal integral curveofv with initial p oint x Then we dene the ow

x x

of the vector eld v to b e the map X given byt x x t

t x

p

Theorem Let v be a C vectoreld on X Then is an op en subset of R X and the

p

ow X is a C map

Lie groups

A Lie group is a smo oth manifold G equipp ed with a group structure so that the maps m

x y xy G G G and x x G G are smo oth

Example An imp ortant example of a Lie group is the group GLV ofinvertible linear

endomorphisms of a nite dimensional real linear space V Indeed GLV is the subset of elements

in EndV with determinant nonzero hence an op en subset of a linear space and therefore

a smo oth manifold Moreover the group op erations are obviously smo oth for this manifold

structure

If x G then the map l y xy G G is a smo oth bijective map whose inverse

x

l is also smo oth Therefore l is a dieomorphism from G onto itself Likewise the right

x

x

multiplication map r y yx is a dieomorphism of G onto itself and therefore so is the

x

conjugation map Ad x l r y xy x The latter map xes the neutral element e

x

x

therefore its tangentmap

Ad x T Ad x

e

is smo oth it follows that is a linear automorphism of T G From the fact that x y xy x

e

x Ad x is a smo oth map from G to GLT G From Ad eI it follows that Ad e

e G

I Moreover dierentiating the relation Ad xy Ad xAd y at e we nd Ad xy

T G

e

Ad xAd y for all x y G A smo oth map from a Lie group G to a Lie group G whichisa

homomorphism of groups is called a Lie group homomorphism Thus wehave proved

Lemma Ad G GLT G is a Lie group homomorphism

e

Example We return to the example of GLV with V a nite dimensional real linear space

Since GLV is an op en subset of the linear space EndV wemay identify its tangent space

at I with End V If x GL V then Ad x is the restriction of the linear map C A

x

xAx EndV End V Hence Ad x C is conjugation by x

x

G or equivalently Avector eld v TG is called left invariant if l v v for all x

x

if

v xy T l v y x y G

y x

From the ab ove equation with y e we see that a left invariantvector eld is completely

determined by its value v e T G at e ConverselyifX T G then wemaydeneavector

e e

eld v on G by v x T l X for x G Dierentiating the relation l l l and

X X e x xy x y

applying the chain rule w e see that T l T l T l Applying this to the denition of v

e xy y x e y X

we see that v satises hence is left invariant Thus we see that X v is a bijection

X X

from T G onto the left invariantvector elds on G Notice that v is smo oth by smo othness of

e X

the group structure

If X T G we dene to b e the maximal integral curveofv with initial p oint e

e X X

Lemma Let X T G Then the integral curve has domain R Moreover wehave

e X

s t for all s t R Finally the map t X t R T G is smo oth s t

X X X e X

Proof Let be anyintegral curveforv let y G and put tyt Dierentiating

X

this relation with resp ect to t we obtain

d d

t T l tT l v t v t

y X y X X X

t t

dt dt

by left invariance of v Hence is an integral curveforv as well

X X

Let now I b e the domain of x t I and put x t Then t x tis

X X X

an integral curve for v with starting p oint x hence has domain contained in I t in view

X

of Corollary On the other hand from its denition we see that the domain of is I so

that I I t Since this holds for any t I this implies that s t I for all s t I Hence

I R

Fix s R then by what wesawabove c t s t is the maximal integral curve

X X

for v with initial p ont s On the other hand the same holds for d t s t cf

X X X

Corollary Hence c d

The nal assertion follows from Theorem

Wenow dene the exp onential map exp T G G by

e

expX

X

Lemma For all s t RX T G wehave

e

a expsX s

X

b exps tX expsX exp tX

Moreover the map exp T G G is a lo cal dieomorphism at Its tangent map at the origin

e

is given by T exp I

T G

e

Proof Consider the curve ct st Then c e and

X

d

ct s stsv st v ct

X X X sX

dt

Hence c is the maximal integral curveofv with initial p oint e and we conclude that ct

sX

t Nowevaluate at t to obtain the equality

sX

Formula b is an immediate consequence of a and Lemma Finally we note that

d

exptX j v e X T expX

t X X

dt

Hence T exp I and from the inverse function theorem it follows that exp is a lo cal

T X

e

dieomorphism at ie there exists an op en neighb ourho o d U of in T G such that exp maps

e

U dieomorphically onto an op en neighb ourho o d of e in G

Example We return to the example of the group GLV with V a nite dimensional real

linear space If x GLV then l is the restriction of the linear map L A xA End V

x x

End V the invariantvectoreld EndV to GLV hence T l L and we see that for X

e x x

v is given by v x xX Hence the integral curve saties the equation

X X X

d

ttX

dt

tX

Since t e is a solution to this equation with the same initial value wemust have that

tX X

t e Thus in this case exp is the ordinary exp onential map X e EndV GLV

X

A smo oth group homomorphism R G is called a one parameter subgroup of G

Lemma If X T G then t exp tX is a one parameter subgroup of G Moreover all

e

one parameter subgroups are obtained in this way More preciselylet b e a one parameter

subgroup in G and put X Then texptX t R

The rst assertion follows from Lemma Let R G b e a one parameter subgroup

Then e and

d d d

t t sj tsj T l v t

s s e X

t

dt ds ds

hence is an integral curvefor v with initial p oint e Hence by Corollary Now

X X

apply Lemma

Wenowcometoavery imp ortant corollary

Corollary Let G H b e a homomorphism of Lie groups Then the following diagram

commutes

G H

exp exp

G H

T

e

T G T H

e e

Proof Let X T G Then texp tX is a one parameter subgroup of H Dieren

e

G

tiating at t we obtain T T exp X T X Now apply the ab ove lemma to

e e

G

conclude that t exp T X

e

H

Exercise Let V b e a nite dimensional real linear space Show that det GLV R is

a Lie group homomorphism Show that T det tr Show that for all A EndV wehave

I

A tr A

dete e

Applying the ab ove corollary to the Lie group homomorphism Ad x G G for x G

we obtain the following

Corollary Let x G then for every X T G wehave

e

x exp Xx expAd x X

Dierentiating Ad at e we obtain a linear map

ad T Ad T G EndT G

e e e

Moreover applying Corollary to the Lie group homomorphism Ad G GLT G we

e

obtain

Corollary For all X T G wehave

e

ad X

Ad exp X e

Example Let V b e nite dimensional real linear space Then for x GLV the linear

tX

map Ad x End V EndV isgiven byAdxY xY x Substituting x e and

dierentiating this with resp ect to t at t we obtain

d

tX tX

ad X Y e Ye XY YX

t

dt

Hence in this case ad X Y is the commutator bracket of X and Y

In general we put

X Y adX Y for X Y T G

e

Then X Y X Y denes a bilinear map T G T G T G This map is antisymmetric

e e e

Lemma For all X Y T G wehave X Y Y X

e

Proof Let Z T G Then for all s t R wehave

e

exptZ exp sZ exp tAd expsZ Z exptZ expsZ

by Lemma and Corollary Dierentiating this relation with resp ect to t at t we

obtain

Z Ad expsZ Z s R

Dierentiating this with resp ect to s at s we obtain

adZ Z Z Z

Now substitute Z X Y and use the bilinaritytoarrive at the desired conclusion

Let G H b e a homomorphism of Lie groups Then one readily veries that Ad x

G

Ad x Taking the tangent map of b oth sides of this equation at e we obtain that the

H

following diagram commutes

T

e

T G T H

e e

Ad x Ad x

G H

T

e

T G T H

e e

Dierentiating once more at x e in the direction of X T G we obtain that the following

e

diagram commutes

T

e

T G T H

e e

ad X ad T X

e

G H

T

e

T H T G

e e

Wenow agree to write X Y ad X Y Then by applying T ad X to Y T G the

e G e

commutativity of the ab ove diagram yields

T X Y T XT Y

e G e e H

Applying the ab ove relation to Ad G GLT Gwe obtain

e

ad X Y ad X ad Y ad Y ad X

Applying the latter relation to Z T G we obtain

e

X Y Z X Y Z Y X Z

for all X Y Z T G

e

A real Lie algebra is a real linear space a equipp ed with a bilinear map a a a such

that for all X Y Z a wehave

a X Y Y X antisymmetry

b X Y Z Y Z X Z X Y Jacobi identity

Remark Note that condition a may b e replaced by the equivalent condition a X X

for all X a

Lemma Let G b e a Lie group Then T G equipp ed with the bilinear map X Y

e

X Y ad X Y is a Lie algebra

Proof The antilinearitywas established b efore The Jacobi identity follows from com

bined with the antilinearity

A homomorphism of Lie algebras a b is a linear map a b such that X Y

a

X Y for all X Y a

b

Exercise Let A b e an asso ciative real algebra Show that x y xy yx x y A

denes a Lie algebra structure on A

From nowonwe will adopt the convention that Roman capitals denote Lie groups The

corresp onding Gothic lower case letters will denote the asso ciated Lie algebras If G H

is a Lie group homomorphism then the asso ciated tangentmapT will b e denoted by We

e

nowhave the following

Lemma Let H G b e a homomorphism of Lie groups Then the asso ciated tangent

map g h is a homomorphism of Lie algebras Moreover the following diagram commutes

G H

exp exp

G H



h g

Proof The rst assertion follows from the second is Corollary

Intermezzo partial dierentiations

Let X Y b e smo oth manifolds of dimensions m and n resp ectively Then X Y is a manifold in

a natural way for the charts one maytake pro ducts of X and Y charts Fix x X y Y

and consider the maps X X Y x x y and Y X Y y x y Then

X Y

the asso ciated tangent maps are injective linear maps

T T T X Y and T T T X Y

x X x y Y y

x y x y

Using lo cal co ordinates one sees that their images have zero intersection Via the ab ove tangent

Y with linear subspaces of T X Y thus we see that X and T maps we identify T

y x

x y

T X Y T X T Y

x y

x y

X

If f X Y Z is a dierentiable map to a smo oth manifold Z wewrite T f for the

x y

Y

f for the tangentmapof y f x yat tangent map of x f x y atx x and T

x y

y y

Lemma Let T T Then

x y

X Y

T f T f T f

x y

x y x y

X

Proof It is immediate from the denitions that T f T f Applying the chain

x X

x y

X

f T f T The identication of T X with a subspace of rule we see that T

x X x

x y

x y

T X Y means that

x y

X

T f T f

x y

x y

Now apply a similar reasoning but with interchanged roles of X and Y to nish the pro of

Corollary Let m G G G b e the multiplication map x y xy Then for all

X Y T G wehave

e

m X Y X Y T

ee

Another useful application is the following

Corollary Let f R X b e dierentiable at Then

f t tj f t j f tj

t t t

t t t

Proof Dene t t t and apply the chain rule to f t Then one sees that

f t tj T f T f T f T f

t

t t

The right side of this equation equals the right side of the desired equation

Note that in the ab oveproofwehave made use of the following lemma in a sp ecial case

Lemma Let f f f Z X Y b e dierentiable at the p oint z Z and put

x y f z Then

T f T f T f

z z z

Proof Left to the reader

Exercise Let G b e a Lie group g its Lie algebra Showthatfor X Y g wehave

X Y expsX exptY expsX exptY j

st

s t

Commutativity of a Lie group

A Lie algebra l is said to b e ab elian or commutative if X Y forall X Y l Obviously

the Lie algebra of an ab elian Lie group is ab elian The converse is true if the group is connected

We will nish this section by establishing this result

Lemma Let G b e a Lie group and assume that X Y g Then

X Y exp X exp Y expX Y

Remark If X Y g satify the hyp othesis of the ab ove lemma wesay that X and Y

commute

Proof Assume that X Y g commute It follows from the ab ove assertion that exp X and

exp Y commute We will b egin byproving this result and then we will deduce the lemma from

it

By Corollaries and wehave

ad X

exp X exp Y exp X exp Ad exp X Y exp e Y exp Y

Hence exp X and exp Y commute Applying the same reasoning to tX and tY we see that exp tX

and exp tY commute for all t R Hence texptX exp tY is a parameter subgroup of G

It follows that t exp tZ where

d d d

Z exp tX exp tY exp tX exp tY X Y

t t t

dt dt dt

see also Corollary

If G is a Lie group then the connected comp onentofG containing e is denoted by G This

e

comp onent is called the identity comp onentofG Obviously e G Moreover if x G then

e e

xG is connected and op en and closed in G and it contains x xe Hence xG G We see

e e e

that G is a subgroup of G

e

Exercise Show that G is a normal subgroup ie xG x G for all x G

e e e

Lemma The identity comp onent G of a Lie group G equals the subgroup generated by

e

the elements exp X X g

Remark Thus G is the subgroup consisting of elemen ts of the form

e

exp X exp X exp X X X g

k k

Proof Let H b e the subgroup of G generated by the elements exp X X g Since obviously

exp RX G for all X g we see that H G To prove that this inclusion is an equalityit

e e

suces to show that H is op en and closed in G Let g b e an op en neighbourhood of in g

such that exp is a dieomorphism of onto an op en subset U G Let h H Then h exp is

an op en neighbourhood of h contained in H Hence H is an op en subgroup of G Now use the

lemma b elow to conclude that H is closed as well

Lemma Let G b e a top ological group Then any op en subgroup of G is closed as well

Proof Let H b e an op en subgroup of G Select for every coset x GH a representative

g G Then G is the disjoint union of the sets g H x GH Left translation b eing a

x x

homeomorphism these cosets are all op en The complementofH in G equals the union of the

cosets g H x GH x eH Thus we see that this complementisopensothatH is closed

x

Prop osition Let G b e a connected Lie group Then G is commutative if and only if its

Lie algebra g is commutative

Proof If G is commutative then Ad x I for all x G Dierentiation yields Ad x I

G g

for all x G and dieren tiating this at x e we see that ad Hence g is commutative

Conversely assume that g is commutative Then exp X and exp Y commute for all X Y g

Hence the subgroup G generated byexpg is commutative But this subgroup equals G since

e

G is connected

Subgroups

Let G b e a Lie group Then a subgroup H which is a submanifold is called a Lie subgroup of G

Obviously a Lie subgroup H is a Lie group in a natural way and the inclusion map H G

is a homomorphism of Lie groups The asso ciated tangent map h g is an injective

homomorphism of Lie algebras Via this map we view h as a subalgebra g The commutative

diagram of Lemma now means that the exp onential map of H is the restriction of the

exp onential map of G to h

Remark If H is a Lie subgroup of GLV then h is a linear subspace of EndV such that

X

X Y h XY YX h and for all X h wehaveexpX e Many Lie groups maybe

realized as Lie subgroups of a general linear group such Lie groups are called linear

Lemma Let H b e a Lie subgroup of a Lie group G Then its Lie algebra h is given by

h f X g j exp RX H g

Proof This is immediate from the ab ove remarks

Lemma Let H b e a subgroup of a Lie group G Then H is a Lie subgroup if and only if

H is closed

Proof If H is a Lie subgroup then it is a submanifold hence lo cally closed Hence there exists

a compact neighb ourho o d U of e in G such that U H is closed Select an op en neighb ourho o d

V of e in G such that VV U

Let h b e a sequence in H converging to an element g G Then there exists a Nsuch

n

that p N h g V Hence p q N h h VV U In particular this implies that

p p

q

h h U H for all n N and taking the limit for n we see that h g U H This

N N

n

implies that g H Hence H is closed

The pro of of the converse implitcation is deep er Wedonotgive its pro of here but refer to

BD Theorem instead

Corollary Let H H b e Lie subgroups of a Lie group G Then H H H is a Lie

subgroup of G with Lie algebra h h h

Proof This is an easy consequence of the previous results

We will now illustrate howtousetheabove to ols to determine Lie groups and there Lie

algebras Let V b e a complex linear space Then V viewed as a real linear space is denoted by

V Let J EndV b e the linear map v iv multiplication by i Then EndV the space of

R R

complex linear maps V V may b e viewed as the space of A End V such that AJ JA

R

Jg From this it is obvious that Similarly GLV is the group of g GLV such that gJ

R

GLV is a closed subgroup of GLV hence a Lie subgroup We claim that its Lie algebra

R

X JXJ X

equals EndV To see this let X End V Then J exp XJ Je J e e

Hence exp maps EndV into GLV On the other hand if X EndV and exp RX GLV

R

then

tJXJ

exp tX J exp tX J e

Dierentiating this expression at t we nd that X commutes with J hence b elongs to

EndV In view of Lemma the claim has now b een established Use similar tec hniques to

make the following exercises

BD T Brocker T tom Dieck Representations of Compact Lie Groups Graduate Texts in Math

SpringerVerlag

Exercise Let V b e a real linear space and dene SL V to b e the group of g GLV

with determinant ShowthatSLV is a Lie subgroup of GLV with Lie algebra

sl V fX EndV j tr X g

Let V b e equipp ed with an inner pro duct and for X EndV let X denote the adjointofX

with resp ect to the given inner pro duct Let OV b e the asso ciated group of orthogonal maps

ie

OV fx GLV j x x g

Show that OV is a Lie subgroup of GLV with Lie algebra

oV fX End V j X X g

Show that SO V OV SL V is a Lie subgroup with Lie algebra soV oV

Exercise Let V b e a complex linear space and dene SLV to b e the group of g GLV

with determinant ShowthatSLV is a Lie subgroup of GLV with Lie algebra

sl V fX EndV j tr X g

Let V b e equipp ed with a Hermitean inner pro duct and for X EndV let X denote the

Hermiten adjointofX with resp ect to the given inner pro duct Let UV b e the asso ciated

group of unitary maps ie

UV fx GLV j x x g

Show that UV is a Lie subgroup of GLV with Lie algebra

uV fX End V j X X g

Show that SU V UV SL V is a Lie subgroup with Lie algebra

suV fX EndV j X X and tr X g

The groups SU and SO

In the previous section wesaw that SU is a Lie subgroup of GL C The Lie algebra of

the latter group is the algebra M C of complex matrices The Lie algebra su is the

algebra of X M C with

X X tr X

From this one sees that as a real linear space su is generated by the elements

i i

i i

One readily veries that and Hence the commutator

brackets are given by

From this it follows that the endomorphisms ad Endsu have the following matrices

j

with resp ect to the basis

B C B C B C

mat ad mat ad mat ad

A A A

The ab ove elements b elong to

so fX M R j X X g

the Lie algebra of the group SO

If a R then the exterior pro duct map X a X R R has matrix

a a

B C

a R a

A

a

a a

with resp ect to the standard basis e e e of R Clearly R so

a

Lemma Let t R Then exp tR is the rotation with axis a and angle tjaj

a

Proof Let r SO Then one readily veries that R r R r and hence

a

r a

exp tR r exp tR r

a

r a

a jaje we see that wemay reduce to the case that a jaje In that Selecting r such that r

case one readily computes that

B C

exp tR cos tjaj sin tjaj

A

a

sin tjaj cos tjaj

Write R R for j Then by the ab oveformulas for mat ad wehave

j e j

j

mat ad R j

j j

Wenow dene the map SU GL Rby xmatAdx the matrix b eing taken

with resp ect to the basis Then is a homomorphism of Lie groups Moreover from

ad X mat ad X

exp X mat e e

we see that maps SU into SO Since SU is obviously connected wehave SU

e

SU so that is a Lie group homomorphism from SU to SO The tangent map of

e

is given by X mat ad X It maps the basis f g of su onto the basis fR g of

j j

so hence is a linear isomorphism It follows that is a lo cal dieomorphism at I hence

its image im contains an op en neighb ourho o d of I in SO Hence im is an op en connected

subgroup of SO and we see that im SO The latter group equals SO by the lemma

e

b elow From this we conclude that SU SO is a surjective group homomorphism

Hence SO SU ker The kernel of may b e computed as follows If x ker then

Ad x I Hence x x for j From this one sees that x fI Ig Hence

j j

ker fI Ig

Prop osition The map SU GLx mat Ad x is a surjective group homo

morphism onto SO and induces an isomorphism

SUfI g SO

For k N the representation of SU factorizes to a representation of SO The

k k

d

representations are mutually inequivalent and exaust SO

k

Proof By the preceding discussion it suces to prove the assertions ab out the representa

tions One readily veries that x I for x fI g Hence factorizes to a representation

k k

of SO Every invariant subspace of the representation space V of is SU in

k k k k

variant if and only if it is SO invariant A nontrivial SO equivariantmapV V

k k l

would also b e SUequivariant Hence the are mutually inequivalent Finally to see

k

d

that they exhaust SO assume that V is an irreducible representation of SO Then

is an irreducible representation of SU hence equivalenttosome n N From

n

it follows that I on fI g hence n is even I on ker

n

Lemma The group SO is connected

Proof Let x SO There exists an orthogonal matrix m such that mxm exp tR

The curve ct m exp tR m t lies in SO and connects I with x

Exercise Let n ShowthatSOn On

e

Invariant dierential op erators

Let X b e a smo oth manifold and v a smo oth vector eld on X If f C G then we dene

the function vf C Gby

vf x T fvx

x

If c is an integral curveforv then using the chain rule one sees that

d

vf ct f ct

dt

If U x is a ndimensional chart for X then the comp onent functions x are called lo cal

j

co ordinates for X Wedenethevector elds x on U by

j

T e

p p j

x

j

Here e denotes the j th standard basis vector of f Wenow see that

j

f p f p

j

x

j

thus the left side of the ab ove equality corresp onds indeed to a partial derivativeoff in the

n

lo cal co ordinates x From the fact that partial dierentiations in R commute when applied

j

to smo oth functions wenow conclude that for i j n the endomorphisms and of

x x

i j

C U commute

n

For a multiindex N we dene the following endomorphism of C U

n

x x x

n

A linear dierential op erator with C co ecients of order at most m N is a linear

map P C X C X such that for every chart U xof X there exist nitely many

n

c C U N jj m such that

X

Pfj c f j

U U

x

for all f C X

In particular it follows that if v TX then f vf is a rst order dierential

v

op erator on X which annihilates the constant functions Using lo cal co ordinates one sees that

the linear map v is injective Using lo cal co ordinates one also sees that any rst order

v

dierential op erator on X which annihilates the constant functions arises from a unique vector

eld in this way

Using lo cal co ordinates one sees that for v w TX the dierential op erator

v w

is rst order and annihilates constants Hence for a unique smo oth

v w w v v w u

vector eld u This vector eld denoted byv w is called the Lie brack et of the vector elds v

and w From the denitions one readily sees that TX equipp ed with the Lie bracket is an

innite dimensional Lie algebra

Let G b e a Lie group Then one readily sees that a vector eld v TG is left invariant

i

v L f L vf for all f C Gx G

x x

ie i commutes with every left translation L x G From this one sees that the bracket

v x

of left invariantvector elds is left invariant again Thus the space L of left invariantvector

elds is a Lie subalgebra of TG

Lemma The map X v is a Lie algebra isomorphism from g onto L

X

Pro of Let X Y g Then v v is left invariant hence equals v for some Z g Thus if

X Y Z

f C G then

T fZv v f e v v f e v v f e

e X Y X Y Y X

Now

v f exp sX j v v f e

Y s X Y

s

f exp sX exp tY j

st

s t

tad X

f exp se Y exptX j

st

s t

tad X

f expse Y f exp sY exptX j

st

s t

v f ej v f exp sY j

tad X

t X s

e Y

t t

tad X

T f e Y j v v f e

e t Y X

t

T f X Y v v f e

e Y X

Hence T fZ T f X Y for every f C G and it follows that Z X Y

e e

The algebra of invariant dierential op erators

A dierential op erator P on a Lie group G is called left invariant if for all x G wehave

L P P L on C G

x x

Here L denotes the left regular action of G on C G L f y f x y The space U Gof

x

left invariant dierential op erators is an algebra From nowonwe shall use the notation for

X

X g Then the map X is a linear emb edding of g into U G W ehave seen that

v X

X

for all X Y g wehave

X Y Y X

XY

n

From now on let X X b e a xed linear basis for g If N we shall use the multiindex

n

notation

n

X X

n

Here it should b e noted that the order of the op erators is imp ortant since they do not

X

j

commute

n

Lemma Let N Then for all f C G wehave

n

f x f x exp t X exp t X j

n n t t

n

t t

n

Proof Let f C Gand X g Then we recall that for every x G wehave

f x exp sX f x

s X

s

Hence

k k

f x exp tX f x expt sX j

t s t

t t s

k

f x exp tX

X t

t

By induction wethus see that

k k

f x exp tX f x

t X

t

The lemma follows from rep eatedly applying the latter equation with X k equal to X X

n n

resp ectively

n

Prop osition The elements N constitute a basis for the complex linear space

U G

Proof The smo oth map

n

t t exp t X exp t X R G

n n n

n

has tangent map T R g given by

X X

n n n

n

Hence there exists an op en neighb ourho o d of in R such that maps dieomorphically

onto an op en neighb ourho o d U of e in G Any mth order dierential op erator P on U is given

by

Pft Qf t f C U

P

c t with Q a dierential op erator on Write p c Then in particular

jjm

t

we see that

X

Pfe p f exp t X exp t X

n n

t

jjm

X

p f e

jjm

by rep eated application of If we assume in addition that P is left invariant then

Pfx L PfeP L f e

x x

X X

p L f e p f x

x

jjm jjm

P

Hence P p and we see that the span U G To see that they are linearly

jjm

indep endent let m and assume that p C are such that

X

p

jjm

Then in particular it follows that for all f C Gwehave

X X

p f t t p f e

n

t

jjm jjm

From the fact that is a dieomorphism it follows that f can b e found such that f has any

prescrib ed Taylor expansion at Therefore all p jj m must b e zero

From nowonitwillbeconvenient to view g as a subspace of the algebra U G via the

injective map X Thus if X Y g then wehaveX Y XY YX in the algebra

X

ehave U G Moreover w

M

n

CX U G X

n

n

N

as a linear space Since the X X do not commute in general the pro duct in U Gdoes

n

P

j

not well b ehave with resp ect to the ab ove decomp ostion Write X X c X Then we

k l j

j

kl

have for instance

n

X

j

X X X X X X X X X c X X

j

j

X

X X X c X X X c X c X X

j

j

The terms of the last sum can b e rewritten as

n

X

k

X X X X X X X X X X X X c X X

j j j j k

j

k

This indicates how to rewrite a general pro duct as a linear combination of the by

using commutation relations The decomp osition and the commutation relations XY

YX X Y completely determine the structure of the algebra U G This is b est formulated

by means of the following universal prop erty

Prop osition Let G b e a Lie group and A an asso ciative algebra over C and with unit

element Assume that g A is a linear map such that

X Y X Y Y X for all X Y g

Then there exists a unique algebra homomorphism U G A such that on g

Uniqueness follows from the fact that g generates U Gasanalgebraby In this pro of

it will b e convenient to use the notation

n

X X X

n

Wemay dene a linear extension of of bysetting

n

X X X

n

n

for all N The problem is then to show that is a homomorphism of algebras This can

b e done by a double induction argument involving commutator relations The rst induction

o ccurs in the pro of of the following lemma If m N let U G denote the nite dimensional

m

subspace of op erators of order at most m in U G

Lemma Let X gP U G Then XP X P

m

Proof We prove this by induction on m For m the result is obvious So let m and

assume the result has b een proved already for smaller values of m By linearity of it suces

to prove the result for X X with j n By linearity of and the induction hyp othesis

j

n

N jj m Let k b e the smallest index for wemayaswell assume that P X where

k

n

which Then XP X X X If j k then the result follows from the denition

k j

n

k

of So assume that jkLet be the multiindex e where e is the ntuple with zeros

k k



in all co ordinates except for the k th whichis Then X X X Hence

k

  

XP X X X X X X X X X

j k k j j k

From the denition and the linearityof and using the induction hyp othesis wenow nd

 

XP X X X X X X

k j j k

 

X X X X X X

k j j k

Using the rule X Y X Y Y X wemay rewrite the ab oveas



X XP X X

j k

The expression on the right side of this equation equals X X in view of the denition

j

of

To complete the pro of of Prop osition it suces to show

PQP Q P U GQ U G

m

We proveby induction on m For m the result is obvious from the denition of

Thus assume that m and that the result has b een proved already for smaller values of m

By linearity of and the induction hyp othesis wemayaswell assume that P P X with

P U GX g Then by the induction hyp othesis wehave

m

PQ P XQ

From the previous lemma wehaveXQ X Q X Q and using the induction

hyp othesis once more wenownd

PQP X Q P X Q P Q

The universal prop erty describ ed in the ab ove prop osition determines the algebra U Gup

to isomorphism Moreover if G and G are Lie groups with isomorphic Lie algebras then

U G and U G are isomorphic In fact wehave

Lemma Let G G b e Lie groups and assume that g g is a homomorphism of

their Lie algebras Then has a unique extension to an algebra homomorphism U G

U G If is an isomorphism of Lie algebras then is an isomorphism of asso ciative algebras

Proof Asamapg U G the map satises X Y X Y X Y

Y X By the universal prop erty has a unique extension to an algebra homomorphism

U G U G This proves the rst assertion Now assume that is an isomorphism and

let g g be its inverse Then by the rst part of the lemma has a unique extension

to an algebra homomorphism U G U G The comp osition extends I

g

hence must equal the identityofU G by uniqueness Likewise I and we see that

is an isomorphism with inverse

Because of the universal prop erty of Prop osition the algebra U G is called a universal

enveloping algebra for g In view of the ab ove remark its isomorphism class only dep ends on

the isomorphism class of the Lie algebra g and therefore it is natural to construct the universal

enveloping algebra in terms of g without reference to a group The construction is as follows

Let g b e the complexication of g and

c

k

T g g

c k N c

its tensor algebra Let I b e the twosided ideal in T g generated by the elements X Y Y

c

X X Y XY g Then U g is dened as the quotient algebra of T g by I It is not hard

c c

to show that the map g U g fullls the universal prop erty of Prop osition For details

c

see eg Hum Section In particular if g is the Lie algebra of a Lie group G then we

map g U g induces a natural isomorphism U G U g

c c

Hum JE Humphreys Intro duction to Lie Algebras and Representation Theory SpringerVerlag New York

Heidelb erg Berlin

Biinvariant dierential op erators

Let x G Then the map Ad xg g is a Lie algebra isomorphism hence by Lemma

extends to an algebra automorphism of U G whichwe also denote byAdx By the uniqueness

part of that lemma it follows that Ad denes a linear representation of G in U G

Lemma For x G let R denote the right translation C G C G dened by

x

R f y f yx for all f C Gy G Then for all D U G wehave

x

Ad x D R D R

x

x

Proof Since g generates the algebra U G it suces to prove this for D X g Let

f C Gy G Then

R X R f y R f yx

x X

x x

R f yx exp tX j

t

x

t

f yxexp tX x j

t

t

f y exp tAd xX j

t

t

f y Ad xX f y

Ad xX

An op erator P U G is called biGinvariantifforallx G wehave R P R P

x

x

The algebra of such op erators is denoted by DG By the ab ove lemma it equals the algebra

G

U G of Ad Ginvariants in U G

uX extends the endomorphism ad X of g Therefore we If X g then the map u Xu

write ad X for this linear endomorphism of U G

Lemma Let X g Then ad X is a derivation of U G ie it is a linear endomorphism

such that

ad X uv ad Xu v u ad Xv for all u v U G

Proof This is immediate from the denitions

Remark Notice that any derivation of U G is completely determined by what it do es on

the generating space g

Lemma The map ad g End U G is a Lie algebra homomorphism

X Now this is an immediate Proof Wemust show that ad X Y adX ad Y ad Y ad

consequence of the denition

In the following sense the map ad is the tangent map of Ad Let m N Then obviously for

all x G the map Ad xleaves the nite dimensional subspace U Gof U Ginvariant Let

m

Ad xAdxjU G Then Ad G GLU G is readily seen to b e a homomorphism

m m m m

of Lie groups

Lemma The tangentmapofAd G GLU G at e is given by X ad X jU G

m m m

Proof Let Y Y g k m Then it suces to show that for all X g wehave

k

Ad X Y Y adX Y Y

m k k

Now this is seen as follows

d

Ad exp tX Y Y j Ad X Y Y

k t m k

dt

d

Ad exp tX Y Ad exp tX Y j

k t

dt

k

X

d

Y Y AdexptX Y Y Y j

j j j k t

dt

j

l

X

Y Y adX Y Y Y

j j j k

j

Since ad X is a derivation the last memb er of this equation equals ad X Y Y

k

Lemma Let u U G Then the following statements are equivalent

a u is Ad G invariant

e

b u is ad ginvariant

c u b elongs to the center of the algebra U G

Proof a b supp ose that u U GisAdG invariant Then it follows that

m e

Ad X u for all X g Now use that ad X u Ad X u

m m

b c supp ose that u is ad X invariant Then Xu uX adXu for all X g

Hence u commutes with a set of generators of the algebra U G so that it must b elong to the

center

c a Supp ose that u U Gisa central element Fix m N such that u U G

m

Ad X

m

Then Ad X adXu forall X g Hence Ad exp X u e u u for X g here

m

wehave used Lemma It nowfollows that Ad xu u for all x G

e

We shall now use the ab ove to determine a particular biinvariant dierential op erator of

SU

Lemma The element

b elongs to DSU

Proof Since SU is connected it suces to showthatadsu annihilates For this it

suces to show that commutes with

Since ad is a derivation it follows that

X

j j j j

j

Likewise one sees that commutes with and

The element

C DSU

is called the Casimir op erator of SU

The algebra isomorphism mat ad from su onto so induces an algebra isomor

phism U SU U SO whichwe also denote by The image of the Casimir op erator

under this isomorphism is also denoted by C From it follows that in U SO wehave

C R R R

This element b elongs to the center of U SO In view of Lemma it therefore is a bi

invariant dierential op erator on SO

Smo oth functions with values in a lo cally convex space

n

Let R b e an op en subset and V a complete lo cally convex space A map f V is said

n

to b e dierentiable at a p oint a if there exists a linear map Df aR V suchthat

f a h f a Df ah oh h

The linear map Df a which is uniquely determined by the ab ove prop erty is called the deriva

tiveoff at a Here the small oh notation is dened as follows If is a V valued function

n

dened on an op en neighb ourho o d of in R then

h oh h

h

means that lim Equivalently this means that for every continuous seminorm p on

h

khk

V wehavelim khk ph

h

The function f V is said to have a directional derivativeata in the direction

n

v R nfg if

f a tv f a

f lim

v

t

t

tR

exists The function is said to b e partially dierentiable in a with resp ect to the j th variable if

n

f a exists here e is the j th standard basis vector of R We write f for

e j j e

j j

n n

In a natural way HomR V the space of linear maps R V may b e equipp ed with

the structure of a lo cally convex space If p is a continuous seminorm of V then wehavean

n

asso ciated seminorm kk on HomR V dened by

p

n

kT k suppTx T HomR V

p

kxk

n n

Thus kTxk kT k kxk for all x R The space HomR V equipp ed with the seminorms

p p

kk is a complete lo cally convex space of its own right

p

n

A dierentiable map f V is said to b e C if Df HomR V is continuous It

p

is said to b e twice dierentiable if Df is dierentiable Using recursion we dene f to b e C if

p p p

its derivative Df is C Wewrite C V for the space of C maps V and put

p

C V C V

pN

Lemma Let I R b e an op en interval and f C I V Then for all a h R with

a a h I wehave

pf a h f a f ah jhj sup pf a th f a

t

Proof We claim that

Z

f a h f a f ah h f a th f adt

It suces to prove the equation which arises if one applies a continuous linear functional V

to the ab ove equation The functional passes through the dierentiation and the integration

use Prop osition Therefore the claim follows from elementary calculus

The pro of is completed by applying the estimate to

Using the ab ove estimate one can prove the following result in the same fasion as in the nte

dimensional case

p

Lemma Let p N fg Then the function f V is of class C if and only if for

all sequences j j fng of length k p the partial derivative

k

f

j j

k

exists and denes a continuous function V

p

It is now straightforward to dene the notion of a dierentiable map and of a C map X V

for X a smo oth nite dimensional manifold use lo cal co ordinates Moreover if f X V

is dierentiable at a p oint a X then we dene an asso ciated tangentmap T f T X V by

a a

d

T f c f ctj

a t

dt

for c a dierentiable curveinX with initial p oint a Notice that this tangent map is linear and

uniquely characterized by the prop erty that

T f T f

a a

for every continuous linear functional on V By using lo cal co ordinates one maynow provethe

following versions of the chain rule

Lemma Let X Y b e smo oth manifolds and V W complete lo cally convex spaces Assume

that the map f X Y is dierentiable at a X and that the map g Y V is dierentiable

at b f a Then the comp osition g f is dierentiable at a and its tangent map is given by

T f T g f T g

a a b

Assume that f X V is dierentiable at a X and that A V W is a continuous linear

map Then A f is dierentiable at a X and

T A f A T f

a a

Proof Left to the reader

Let PDOX denote the algebra of all smo oth dierential op erators on X The elements of

PDOX are linear endomorphisms of C X Let P PDOX Then by using lo cal co ordi

nates we see that wemay dene an endomorphism P of C X V by

PfP f

for all f C X V and every continuous linear functional on V From nowonwe shall

suppress the tilde in the notations

Smo oth vectors in a representation

In this section we assume that G is a Lie group and that is a continuous representation of G

in a complete lo cally convex space V Avector v V is called smo oth if x xv is a smo oth

ie C map from G to V The space V of smo oth vectors is a linear subspace of V which

is readily seen to b e Ginvariant

We shall rst discuss some examples of smo oth vectors

Example Let X b e a smo oth manifold equipp ed with a continuous action of the Lie group

G We assume the action to b e smo oth ie the map g x gx G X X is C This

action naturally induces the following representation L in the space C G If C X

g G then L x g x

g

The space C X can b e equipp ed with the structure of a lo cally convex space If K X

is a compact subset and PPDOX a nite set of dierential op erators then we dene the

seminorm by

P K

max sup jPxj

P K

P P

xK

The set of seminorms dened in this way is complete and thus C X b ecomes a lo cally convex

space

n

Let us consider the particular case that X is an op en subset of R For K X compact and

m N we dene the seminorm on C X by

Km

f xj maxsup j

Km

x jjm

xK

The set of seminorms thus obtained is already complete so that it determines the top ology

Km

of C X It is well known and not hard to prove that the space C X is complete

One may limit the set of seminorms to a countable set by restricting K to a sequence K of

j

compact subsets with K K and K X Thus one sees that C X isa Frechet space

j j j j

Wenow return to the case of an arbitrary smo oth manifold X Then by using lo cal co

ordinates one can show that the lo cally convex space C X is complete Moreover if X is

compact ie X allows a countable covering by compact sets then one can show that C X

is Fr echet

Lemma Let G act smo othly on the smo oth manifold X Then the natural representation

L of G in the complete lo cally convex space C X is continuous Moreover every function of

the space C G is a smo oth vector for L

Proof See app endix

Another example of smo oth vectors in a representation is provided by the following Let dx

b e left Haar measure on G and L G the asso ciated Banach space of integrable functions The

left regular representation L of G in L Giscontinuous

Lemma Let C G Then is a smo oth vector for the left regular representation

c

L L G

Proof See app endix

tees the existence of many smo oth vectors in a representation The following result garan

Lemma Let f C G Then f maps V to V

c

Proof Fix v V and consider the continuous linear map A v L G V Let

f C G Then F f L f is a smo oth map G L G by the previous lemma The

x

c

comp osition A F G V is smo oth by Lemma But

A F x AL f L f v x f v

x x

by Lemma and it follows that x x f v is smo oth Hence f v V

Remark A smo oth vector of the form f v with f C Gv V is called a Garding

c

vector in V A famous result of Dixmier and Malliavin cf DM asserts that if V is a Frechet

space then every smo oth vector is a Garding vector

DM J Dixmier P Malliavi n Factorisations de fonctions et de vecteurs indenimentdierentiables Bull

Sci Math

Corollary The subspace V of smo oth vectors is dense in V

Proof This follows from combining the previous lemma with Lemma

Corollary If is a nite dimensional representation then all of its vectors are smo oth

V V Moreover is a Lie group homomorphism as a map G GLV

Proof The rst assertion is an immediate consequence of the previous corollary As for the

second one if v V V then x xv is a smo oth function G C This implies that

the co ecients of x with resp ect to a xed basis of V dep end smo othly on x G Hence

is smo oth as a map G End V Since GLV is an op en subset of End V this implies that

is smo oth as a map G GLV

The algebra U G acts in a natural way on the space of smo oth V valued functions on G

This allows us to dene for P U G and v V

P v P x xj

xe

One readily veries that P b elongs to EndV the space of linear endomorphisms of V

Moreover the map P P is a homomorphism of algebras U G EndV The space

V is a U Gmo dule in this way

If X gv C then retracing the denitions we see that

d

X v exp tX v j

t

dt

From and the fact that U g EndV is an algebra homomorphism it follows that

for all X Y g wehave

X Y X Y Y X on V

Example Let R b e the right regular representation of G on C G All vectors of this represen

tation are smo oth so wehave an asso ciated algebra homomorphism R U G End C G

g then from we see that Let f C GX

d

RX f x f x exp tX j f x Xf x

t X

dt

Thus we see that R I on g Since g generates the algebra U g this implies that for all

D U Gwehave

RD D on C G

Lie algebra representations

Let W b e a complex linear space Then by a representation of g in W we mean a Lie alge

bra homomorphism g End W ie is a linear map such that for all X Y g wehave

X Y X Y Y X By the universal prop erty extends to an algebra homomor

phism U g EndW turning W into a U g mo dule ConverselyifW is a U g mo dule

c c c

then wemay dene a representation of g in W by X w Xw W W for X g

Thus we see that g representations are in one to one corresp ondence with U g mo dules In

c

the literature it also customary to sp eak of gmo dules instead of U gmo dules Likewise if

V isacontinuous representation of G then the top ological linear space V together with

the map G V V x v xv is called a Gmo dule

Finite dimensional representations

If is a nite dimensional representation then by Corollary is a Lie group homomorphism

G GL V Let g End V b e the asso ciated homomorphism of Lie algebras Then from

we readily deduce that X X Hence in a setting like this wewillfromnowon

suppress the star in the notations

Thus we see that a nite dimensional Gmo dule V is automatically a gmo dule Let denote

the asso ciated representations of G and g in V Since g EndV is the tangent map of the

Lie homomorphism G GLV it follows from Lemma that for all X g wehave

X

exp X e

When G is connected this equation allows us to compare the G and the gmo dule structures

on V

Lemma Assume that G is connected and let V V be two nite dimensional Gmo dules

a Let W b e a linear subspace of V Then W is Ginvariant if and only if W is ginvariant

b The Gmo dule V is irreducible if and only V is irreducible as a gmo dule

c Let T V V b e a linear map Then T is Gequivariant if and only if T is gequivariant

Proof Write and for the representations of G in V and V resp ectively

t under the group G a If W is ginvariant then it follows from that W is invarian

e

which is generated by exp g But G G since G is connected The converse implication is

e

proved by dierentiating exptX at t

b This is now an immediate consequence of a

c Supp ose that T is gequivariant Then for all X g wehave X T X T

n n

hence X T X T for all n N and it follows that



X X

e T T e

From this it follows that x T T x for all x exp g and hence for x G G The

e

reverse implication follows by a straightforward dierentiation argument as in part a of this

pro of

Lemma Let G b e a connected compact Lie group and let b e a representation of G in

a nite dimensional Hilb ert space V Then is unitary if and only if

X X

for all X g

Proof We recall that G GLV is a Lie group homomorphism Hence for all X

gt R wehave

t X

exp tX e

If is unitary then exp tX exptX hence



t X t X

e e

Dierentiating this relation at t we nd Conversely if holds then holds for

all X t and it follows that x is unitary for x exp g This implies that x is unitary for

x G G

e

The action of biinvariant dierential op erators

Wenow come to an imp ortant asp ect of harmonic analysis on a Lie group G involving the

action of DG on the space V of smo oth vectors in the complete lo cally convex space V

Lemma Let D U G Then for all x G wehave

x D x Ad xD on V

In particular if D DG then D commutes with the action of G on V

Proof It suces to prove the rst assertion for the generating elements D X g On V

wehave

d

x X x x exp tX j x

t

dt

d

x exp tX x

t

dt

d

exptAd xX j Ad xX

t

dt

Corollary Let V b e nite dimensional and irreducible Then the algebra DG acts

by scalars on V

Proof This follows from the previous lemma in combination with Schurs lemma Lemma

Let b e nite dimensional and irreducible Then it follows from the ab ove lemma that there

exists an algebra homomorphism G C such that D D I The homomorphism

V

is called the innitesimal character of

Relation to the PeterWeyl theorem

For the rest of this section we assume that G is a compact Lie group Our goal is to relate the

action of DGon C G to the PeterWeyl decomp osition

b

L G C G

b

G

b

If G then the elements of C G are nite linear combinations of matrix co ecents of

hence smo oth functions cf Corollary The C G are nite dimensional and right

invariant hence if C G then x Rx is a smo oth map G C G use Corollary

once more Since C G is nite dimensional this implies that is smo oth as a map

G L G Thus we see that the spaces C G consist of smo oth vectors for R L G

Let D U G Then from the discussion in the nal example of Section wehave that

R D D on C G In particular we see that C G is invariant under the left invariant

dierential op erators

b

Prop osition Let G Then C G is a nite dimensional subspace of C G consisting

of smo oth vectors for R L G Moreover C G is invariant under the action of all left

invariant dierential op erators Finally if D DG then D acts by the scalar D on C G

ve discussion it remains to establish the last assertion The right regular Proof By the ab o

representation restricted to C G splits as a nite direct sum of copies of From this it follows

that RD actsby the scalar D on C G Now use that RD D on C G by

Thus we see that the algebra DG admits a simultaneous diagonalization over the Peter

b

Weyl decomp osition The eigenvalues are given by the innitesimal characters G

Representations of su

Let C b e the Casimir op erator of SU Then we shall compute the scalar C C by

n

n

which C acts on the represen tation space of the irreducible representation of SU intro duced

n

in Section

We recall that the representation space of is the space P C of p olynomial functions

n n

C C which are homogeneous of degree n The action of SU on this space is given by

xpz px z

n

Let X X b e a matrix in su Then the asso ciated innitesimal representation of su

ij n

is given by

d

tX

X pz pe z j

n t

dt

Applying the chain rule to the right side of the ab ove equation we obtain

d p d p

tX tX

X pz e j z z e j z z

n t t

dt z dt z

p p

z X z X z z X z X z

z z

This equation allows us to compute the action of the generators of the algebra U SU

However for reasons that will b ecome apparent in a moment it will b e more convenienttouse

the generators and X i ie

X X

Recall that the p olynomials

nk k

p z z z z k n

k

constitute a basis for P C Using one sees that the action of on this basis is given by

n

p in k p

k k

Here wehave suppressed the in the notation wewillcontinutodoso

n

The action of the other two basis elements is describ ed by

p p

X p z X p z

z z

Hence

X p k n p X p kp

k k k k

These equations hold for all k n if we agree to write p for kand kn

k

We are now ready to compute the scalar C

n

Lemma Let n N Then

C nn

n

Proof It suces to compute the action of C on p One readily veries that in U SU we

have

X X X X

In view of the ab ove element maps p to np Using this in combination with we

P

acts by the scalar n n nn on p Hence C acts by see that

j

j

the scalar

The invariant inner pro duct

We end this section with the computation of a SUinvarian t inner pro duct on P C In

n

view of Lemma this inner pro duct is determined up to a p ositive factor Thus it is uniquely

determined if we require that kp k Applying Lemma to the representation of SU

n

we see that the action of is antiHermitian with resp ect to the inner pro duct This implies

that its eigenspaces are mutually orthogonal Hence the p constitute an orthogonal basis for

k

P C and it remains to compute their lengths Put c kp k Then c by assumption

n k k

The element X X acts antisymmetrically hence for all k wehave

hX X p p i hp X X p i

k k k k

Using that X X p kp n k p and X X p k np k p

k k k k k k

we see that this leads to

n k c k c k n

k k

This enables us to solvethe c recursively from c and we nd

k

n

c

k

k

Wehaveproved

Lemma Let P C b e equipp ed with an inner pro duct for which the representation

n n

of SU is unitary Then the p olynomials

nk

k

p z z z k n

k

are mutually orthogonal and

kp k n

k n

k kp k

k

The following corollary will b e used at a later stage

k

Corollary The p olynomials X p constitute an orthogonal basis for P C Moreover

n n

k

kX p k n

n

k

kp k k

n

Proof From rep eated application of it follows that

n n

k k k

p p k X p

nk nk n

k n k

Now use the ab ove lemma to complete the pro of

Harmonic analysis on a homogeneous space

In this section we assume that G is a compact top ological group and that H is a closed subgroup

Then GH is a compact top ological space Let p G GH b e the canonical pro jection Then

the pullback map p f f p maps C GH injectively into C G The image of this map

H

equals the space C G of right H invariantcontinuous functions on G Via the top ological

H

linear isomorphism p we shall identify C GH with C G

Let I b e the normalized Haar integral on G Then one readily veries that the restriction

G

I of I to C GH is a p ositive Radon integral on GH whichisinvariant for the left regular

G

representation L of G on C GH Moreover I

Let L GH b e the completion of C GH with resp ect to the L inner pro duct asso ciated

with I Then the left regular representation extends to a unitary representation of G on L GH

Thus the theory of Section may b e applied and we see that wehave the left invariant Hilb ert

decomp osition

b

L GH L GH

b

G

Here L GH denotes the space of isotypical left Gnite vectors of typ e in L GH

Notice that the L structure is dened in sucha way that the map p extends to an isometric

and equivariantemb edding of L GH into L G the image b eing the subspace of right H

invariant functions Let L G denote the space of isotypical left Gnite vectors of typ e in



cf Exercise By equivariance wehave that L G Then L G C G

L GH L G L GH

b

If G dene the map S EndH C Gby

S Ax tr x A

Then one readily veries that S intertwines the representations and L R of G G In

analogy with Corollary wenowhave

p

Lemma The map d S is an equivariant isometry from End H onto L G

p

Proof It remains to verify that d S is an isometryFor this we compare with the map

t

T of Corollary The map A A is an isometry from EndH onto End H Moreover

one readily veries that S T The result now follows from Corollary

maps the space bijectively onto the space V of endomorphisms By equivariance S

A EndH withA x A for all x H By unitarityof this condition on A is equivalent

H H

to the condition that A onH Therefore restriction to H induces a bijective linear

H

map V HomH H Wethus view the latter space as a subspace of End H It follows that

S restricts to a linear isomorphism

H

H L GH S HomH

H

Let b e the representation of G on HomH H dened by xA xA Then the

ab ove map S intertwines and the left action L of G on L GH

In particular it follows that L GH is nontrivial if and only if has nontrivial H xed

H

b b

G of G for whichH vectors Hence the decomp osition ranges over the set

H

H

b

For G let n denote the dimension of the space H Then n Let u u be a

H n

H

basis of H Then from the natural equivariant isomorphisms

n n

H

HomH H HomCu H H

j

j j

H

maybeinterpreted one sees that isamultiple of infact n Thus n dim H

ultiplicity of the representation in L GH as the m

H

b

forall G Wesay that the decomp o Of particular interest is the case that dim H

H

H

b

sition is multiplicity free in this case For every G x a nontrivial element u H

H

p

of unit length and dene the map s H L GH by s v x d hv j xu i Then it

follows from the ab ove discussion that s is an equivariant isometry from H onto L GH

Wenow consider the particular case that G is a compact Lie group Then by Lemma the

group H is a Lie subgroup One can show that GH carries the structure of a smo oth manifold

such that the action map G GH GH is smo oth and such that p maps C GH

H

bijectively onto C G for details we refer the reader to BD Section I Thus bythe

ab ove discussion and the results of Section wehave that

L GH C GH C G

Moreover the algebra g acts from the left by rst order dierential op erators on L GH

This action lifts to an action L of U G If D DG then LGisaleftinvariant dierential

op erator on C GH which diagonalizes over the decomp osition On L GH the

action is given by the scalar D

Harmonic analysis on the sphere

In this section we study harmonic analysis on the two dimensional sphere from the viewp ointof

group theoryWe shall compare our results with the classical theory of spherical harmonics

The relation to group theory is ro oted in the fact that S the unit sphere in R is an orbit

for the natural action of SO on R Let e b e the rst standard basis vector in R Then the

stabilizer of e in SO is the image of SO under the injective group homomorphism

SO SO A

A

Via this homomorphism we view SO as a closed subgroup of SO Thus we see that the

map a SO S x xe factorizes to an isomorphism

a SOSO S

Equip S with the rotation invariant Radon measure of total measure and let L S bethe

L space Then via the left action L the group SO acts unitarily on L S corresp onding

Moreover the map a f f a is an equivariant isometry from L S onto L SOSO

Applying the theory of Section wethus obtain the decomp osition

b

L S L S

mN m

where L S denotes the space of functions of left typ e The latter space consists of

m m

SO

smo oth functions and is equivariantly isomorphic to HomP C P C Note that

m m

SO exp RR Hence the preimage of SO under the epimorphism SU SO of

Prop osition equals T It follows that

SO T

P C P C

m m

mk

k

We recall that the action of T diagonalizes over the p olynomials p z z which constitute

k

a basis of P C The action of T on p is trivial if and only if k m Hence the dimensions

m k

of the spaces in equal Thus we see that L S is isomorphic to the SO mo dule

m

P C forevery m N In particular it follows that the decomp osition is multiplicity

m

free and that its mth comp onen tis m dimensional

The left action induces an algebra homomorphism from U SO to the dierential op erators

on S The image LC of the Casimir under this map is a second order dierential op erator which

diagonalizes over the decomp osition in particular it acts by the scalar C mm

m

on the comp onent L S

m

BD T Brocker T tom Dieck Representations of compact Lie groups SpringerVerlag

Spherical harmonics

In this section we compare the results of the previous section with classical spherical harmonics

The natural action of SO on R induces a representation L of U SO in C R by means

of dierential op erators Let a R and let R so b e the matrix of the map x a x

Then the rst order dierential op erator LRisgiven by

d

tR

LRf x f e xj Df xRx

t

dt

gradf x x a detgradf xxa

Substituting a e e e resp ectively one thus nds

x LR x

x x

x LR x

x x

LR x x

x x

Wenow recall that the Casimir is given by C R R R Hence

X

LR x E E LC

j

j

P

and where where is the ordinary Laplacian where x denotes multiplication by x x

j

j

E denotes the Euler op erator

E x x x

x x x

For m a nonnegativeinteger let P R denote the nite dimensional space of homogeneous

m

p olynomial functions R C of degree m Then E acts by the scalar m on P R Let

m

H fp P R p g

m m

the space of harmonic p olynomials in P R This space is nontrivial since it obviously con

m

m

tains the function Y x x ix

Moreover the space H is invariant under the action L of SO In particular LC leaves

m

this space invariant and by using we see that LC actsby the scalar mm on it From

general theory we know that the nite dimensional SO mo dule H splits as a direct sum of

m

irreducibles If SO o ccurs in H then wemust have C mm This implies

k m k

that k m Thus H is equivalent toamultiple of the irreducible mo dule P C In fact we

m m

have

Lemma Restriction to S induces an equivariant isomorphism of H onto L S In

m m

particular the SO mo dule H is equivalent to the irreducible mo dule P C

m m

Proof Consider the restriction map

H C S f f jS

m

It maps H equivariantly and injectively into C S By the ab ove discussion its image is a

m

nontrivial subspace of L S and since the latter mo dule is irreducible it follows that the

m

image of equals L S

m

We shall now use our insight in the structure of the representations to compute the

m

elements of L S explicitly in terms of spherical co ordinates on S These co ordinates

m

are determined by the relations

i

x cos sin e

Here we identify the x x plane with the complex plane via x x x ix

In spherical co ordinates the function Y Y jS is given by

m im

Y sin e

im

It follows that the rotation expR actsby the scalar e on Y Via the epimorphism

SU SO dened in Prop osition wemay view C R as a SUmo dule Recall

im

that t expR Hence t acts by the scalar e on Y

By the results of Section there exists an equivariant isomorphism

T L S P C

m m

of SUmo dules The image T Y of Y is a nontrivial elementofP C on which t T

m

im m

acts by the scalar e This implies that T Y is a nontrivial multiple of p z Each

m

k

p olynomial iX p k m is a nontrivial multiple of p These p olynomials

m m mk

constitute a basis for P C The corresp onding functions

m

k

Y iX Y k m

mk

therefore constitute a basis for L S

m

ik m

We recall that t acts by the scalar e on p Hence the rotation expR acts

mk

ik m

by the scalar e on Y This implies that in spherical co ordinates Y must b e of the

mk k

form

ik

Y e f m k m

k k

Notice that the function Y is invariant under rotations around the x axis Its level curves

thus divide the sphere into spherical zones The function Y is therefore called a zonal spherical

function

i in so equals R iR Hence we obtain The image of X

k

Y iR R Y k m

mk

Using and a substitution of variables one nds that in terms of spherical co ordinates the

op erator iR R isgiven by

i

iR R e i cot

Usually a cyclic p ermutation of these co ordinates is used but the presentchoice is more convenient for our

purp oses

Combining this with and we see that

m

f sin

m

and the f are related by the formula

k

df

k

k cot f f jk j m

k k

d

Wenow substitute s cos This substitution is allowed since cos is a monotonically

decreasing function on If wewritep sforf viewed as a function of s the ab ove

k k

relation b ecomes

dp ks

k

s p p

k k

ds s

k

This relation may b e simplied by putting u s s p s It then b ecomes

k k

du

k

u

k

ds

From we see that

m m

u s s p s s

m m

Hence

mk

d

m

u s s jk j m

k

mk

ds

and it follows that

mk

d

k m

p s s s jk j m

k

mk

ds

One normally uses a dierent normalization for the p Put

k

mk k

d s

m k

s jk j m P s

m

m mk

m ds

k

Then P P is called the mth Legendre p olynomial and the P jk j m are called

m

m m

its asso ciated Legendre functions Notice that the chosen normalization has the eect that

P

m

It follows from the ab ove discussion that the functions

k ik k

Y e P cos

m m

constitute a basis for L S This basis is not yet orthonormal Put

m

m m k

k k k

Y Y

m m

m k

k

Prop osition The functions Y m N jk j m constitute a complete orthonormal sys

m

tem for L S Here S is equipp ed with the rotation invariant measure for which S has surface

k

Proof The spaces L S are orthogonal Thus it suces to show that for a xed m the Y

m

constitute an orthonormal basis for L S

m

m k mk

Y We recall that the isomorphism From the ab ove discussion it follows that Y iX

m m

m

T of maps Y onto a multiple of p After comp osing with a suitable scalar multiplication

m

m

m k mk

wema yaswell assume that T Y p Then T Y iX p by equivariance

m m

m m

The image of the inner pro duct on L S under T is an inner pro duct on P C for

m m

k

which is unitaryThus by Corollary we see that the Y are mutually orthogonal and

m

m

their lengths are given by

k mk

kY k kX p k m

m

m

m k

m

kp k m k

kY k

m

m

m im

The function Y sin e has L norm

Z Z Z

m

m

m

kY k jY j sin dd sin d

m

Hence

m

kY k

m

m

k

By combining the last equality with it nowfollows that the Y have length

m

Final comments

The Casimir of SO is closely related to the spherical Laplacian If f C S let f

denote its extension to R dened by setting f tx f xx S t Then the spherical

Laplacian is dened by

f f jS

It is a second order dierential op erator on S which annihilates the constants and is formally

selfadjoint In addition it is rotation invariant By these prop erties it is in fact characterized up

S to a constant factor From we immediately see that LC A function f C

which is an eigenfunction for the spherical Laplacian is called a surface spherical harmonic

According to the theory develop ed ab ove the Hilb ert space L S has a complete orthonormal

system of surface spherical harmonics Explicitly such a system is given by The eigenvalues

of the spherical Laplacian are mm and the eigenvalue o ccurs with multiplicity

m m

m For more details background material and information ab out applications we refer the

reader to BD Section I I T Chapter I I and HC

BD T Brocker T tom Dieck Representations of compact Lie groups SpringerVerlag

T A Terras Harmonic Analysis on symmetric spaces and application s I SpringerVerlag

CH R Courant D Hilb ert metho den der Mathematischen Physik I Edition SpringerVerlag

App endix

In this app endix wegive the pro ofs of Lemmas and We start with the following

lemma

Lemma Let X Y b e smo oth manifolds and let f C X Y Then the map F X

C Y dened by F xy f x y is smo oth

Proof The assertion is lo cal in the xvariable Hence wemayaswell assume that X is an

m

op en susbset of R

By using smo oth partitions of unityover compact subsets of Y we see that it suces to

prove the assertion for functions f with suppf X U where U is a chart in Y Thus wemay

n

as well assume that Y is an op en subset of R A complete set of seminorms for C Y is then

given by

k maxk

K Kk

y

j jk

Here k NK is a compact subset of Y and kk denotes the supnorm over K

K

We will rst show that F is continuous Fix a X By uniform continuity of the partial

derivatives f over compact subsets of X Y wehave

y

lim k F x F ak

K

xa

y y

for K Y compact From this it follows that lim F x F a whence the

xa Kk

continuityofF in a

m

Next we show that F has rst order partial derivatives Let a X and v R Then by

the mean value theorem wehave

X

t f a tv y f a f a t y v y

v

y y y

with t y between and t If y is restricted to a compact subset of Y then the ab ove expression

X

f a y as t From this it follows that tends uniformly to

v

y

F a tv F a

X

lim f a

v

t

t

X

in C Y Hence F has the directional derivative F a y f x y

v

v

The directional derivatives F are continuous by the rst part of the pro of Hence F is C

v

By using an obvious induction one maynow complete the pro of

Proof of Lemma Let C X Then f g x L x g x is a smo oth

g

g L dep ends function on G X Applying the ab ovelemmawe obtain that g F

g

smo othly on g G

It remains to prove Lemma

Proof of Lemma Let C G and dene the map F G L Gby F xL

x

c

Replacing by a left translate if necessary we see that it suces to establish the smo othness

of F in a neighb ourho o d of e This can b e done as follows Fix a function C Gsuch

c

that on an op en neighb ourho o d of supp Select an op en neighb ourho o d U of e such that

on supp L for every x U

x

The map T C G L G is continuous linear Using Lemma and Lemma

wenow see that T F is a smo oth map G L G In particular this map is smo oth on

U But for x U wehave T F xT L L F x

x x

Erratum A

In Section p it is asserted that every homogeneous Gspace is a coset space ie of the form

GH with H a closed subgroup of G This is true if G is compact but in general one needs an

additional requirement on the action

All the top ological spaces o ccurring in this section are assumed to b e Hausdor

A continuous map f X Y between top ological spaces is called prop er if the preimage

f C ofevery compact subset C Y is compact

Lemma Let f X Y b e prop er and assume that Y is lo cally compact Then f is

closed ie the image of any closed subset of X is a closed subset of Y

Proof Let S X b e closed Let t be a point in the closure of f S Then there exists

a compact neighb ourho o d C of t in Y Its preimage f C is closed in X Hence f C S is

compact in X The image of this set under f is compact in Y But this image equals C f S

Hence C f S is closed This implies that t f S hence f S is closed

Let X b e a top ological space and G a lo cally compact top ological group acting on X The

action is called prop er if the action map G X X is prop er

Lemma Supp ose that the action of G on X is prop er and transitive Let x X and let

G b e the stabilizer of x in G Then the natural bijection GG X is a homeomorphism

x x

Proof Let p G GG denote the canonical pro jection The map a g gx G X

x

is surjective by transitivity of the action Hence X is lo cally compact The induced map

b GG X is a bijection If U X is op en then a U is op en and right G invariant

x x

hence b U pa U is op en Thus we see that b is continuous To provethatitisa

homeomorphism it suces to show that b is closed For this it suces to show that a is closed

Now this is a consequence of the prop erness of a

Remark If a compact group G acts continuously on a top ological space then the action is

is homeomorphic to a quotient automatically prop er Hence if the action is transitive then X

of G by a compact subgroup

Erratum B

The nal argument in the pro of of Lemma is not complete The purp ose of this section is

to complete it

Let X Y b e smo oth manifolds and v a smo oth vectoreld on X dep ending smo othly on a

y

parameter y Y We shall investigate how the owofv dep ends on y

y

We rst explain what smo othness of the parameter dep endence means For every x

X y Y wehave v x T X Recall that wehave a canonical identication T X Y

y x

xy

T X T Y Via this identication we view w x y v x as an elementofT X Y

x y y

xy

Thus w X Y T X Y isavectoreld on X Y The smo othness requirementonv now

means that w is a smo oth vector eld

Let I X Y b e an integral curveofw with initial p ointx y Then from

tv t

t

we see that t hence t y for all t I Moreover wenow see that is an integral

curveofv with initial p oint x

y

Y b e the op en domain of the ow ofw Then it follows that for every Let R X

y Y the ow of v is given by t x t x y Moreover its domain is given by

y y y y

fy g R X fy g Thus we see that dep ends smo othly on y Y

y y

Wenow consider the particular case of a Lie group G Then the left invariantvector eld v

X

on G dep ends linearly hence smo othlyonX g Its owisgiven byt x x t R G G

X

From the ab ove discussion we see that the map t x X x t R G g G is smo oth

X