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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
.