Let a- be a compact group .

To determine € it is enoughto find all irreducible charactersch ft) , it C- I .Assume that G is connected .

As we remarked, any finite -

dimensional representation#is actually a representationof the group Ghent whichis a connected compactLie group .

We shall use some structuralresults about connectedcompact Lie groups proved

2

in Lie Groups class .Let G- be a connected compactLie group . A torus T in G

is a closed subgroup isomorphicto a product of circle groups .Let T be the family oftori in G- ordered by inclusion .Then T contains maximal

elements. Theyareall

conjugate by elements ofG- . Moreover

, airy torus iscontained in a maximal

torus . Any element g c- Ctis contained in a maximal

3

Fix a maximal toneas T

in G . Thenthe conjugationmap

G-at → G-

Cg ,t ) -7 gt g-l

is surjective . Since characters

are constant on conjugacyclasses of G, they are

completely determined

by their restriction toT -Eixample : Let G --SU(2) -2×2 unitary matrices ofdeterminant equal to 1 .

4

Therefore , the element

of SUCH are of the form

FEE)with laft IpTEI . ThereforeSU(2) is diffeomorphic to

a 3 -dimensional sphereand simply connected .

The eigenvalues X suchmatrix are complex numbers

of modulus 1. If I , and 72

are its eigenvalues , X . . Az=L,i. e.the eigenvalues are

oil and e-it.

5

Let t = { (foie-ie) ; y ETH} .There T is atorus in G .

One can show that T is

a maximal torus in G-,

Let ge G with eigenvaluese"and e-

ill.Then these

exists an orthonormal basis

Ni,Na of K

-

consisting ofeigenvectors of g . Thematrix harring N

,and Nz as

columns is unitary matrixwhich conjugates g with

( g""

f-ice) . By dividing6

it by a square root of thedeterminant

,we can assume

that this matrix is h c- SDK) .This proves that a- xT→ a-

is surjective inthis case .-

Since itIt is a finite -dimensioned

ofT, it is a direct sum ofcharacters . Therefore

ch ft) IT = I mo -V

vEF

where F is a group ofcharacters ofT (which is

isomorphic to Z'

,as we

7discussed before? .

The

number r is the dimension ofthe torusT - the stank of Ct-

mo e ZI's the multiplicityof u c-F in it .UEF is a weight ofit if mu t O .

Therefore , to find characters

of irreducible representationsof G- we need to determine

the multiplicities of their

weights .

We shall explain a method

8to do this

,due to Hermann

Weyl, based on orthogonalityrelations .

We shall just state variousresults in general .Thedetailed proofs for SUCH

are on thetexed note"Weyl character formula"

on the class web page .

The first step is the Weylintegral formula . It givesthe formula for the integral

qover G- as the integralover conjugacy classes followedby integration overT.This is the Weyl integral

.

-

First,since T is abelian

we can factor the mapG- xT → a-

atIt tIn this way ,dim (GA xT) = dim (GH t dimT= dim G - dimT t dimT = dimG

.

10Let NCT) = {me G-InTri '-53be the normalized ofT .Then NCT) is a closed subgroupof G-, T is its identitycomponentand

W = NE) ITis the VVeylgronp-of.fr,T) .There exists an open dense

subset Greg of a- ( mehthat G-Greg is of Haarmeasure zero) such that

p- ' (Greg) = GIT × Trey(Treg

=Tn Greg)is a 2W] - fold cover ofGreg .

"G- acts on a-IT byleft multiplication . Thereexists a unique positivea- - invariantmeasure

U ou a-It which isnormalized

,i. e,vCGA) = I .

Then we have

fgfcg) dyadg) =

Ey ! ( faffgets- 'I dug))

Dft) dm, It)where D is the absolute

value of Jacobian of p :a-G- xT → G

.

12In the taxed note

,

the functionD is calculatedexplicitly for Suk) .General case requiresrequires details on thestructure of lie alglare

of G.

inthe next installmentwe shall describe the

function D in generaland discuss Weyl characterformula .