comsegueucesofpeter-Weyl theorem-

Rennert . Uniform version

of Peter -Weyl is strongerthan the E- version (asdiscussed last time)

.

The only place in the proofof uniform version where

we used L'- version is to

show that RCA) differspoints in G .

If G is a lined groupline . a group of matriceslike so in) ) the natural

representation differs pointsof G- and Peter -Weyl theorem

follows immediately fromStone -Weierstrass theorem .

We shall discuss this later

in more detail .-

since RIG) differs pointsof G , for way of E G, g t 1

,

there exists f e R (G) suchthat fCg) t f Ci) .Since f is a matrix coefficientof a finite -dimensionalrepresentation , we get

3

Lemmy .

Let g c- G- , g Fl.

Then there exists a finite -

dimensional representation(it,V) of G- such thatit(g) t I .Now we prove a variant

of a result we proved forfinite groups .

Let (it,V) be a finite-dimensionalrepresentation of G- . LetL . I . > be an arbitrary inner

product on V . Since itis continuous for any u ,o EV

the function

4

g '→⇐Ig)u fatg) o>is continuous . Hence

(ulu ) = fastCg) u ht CgJv> dreg)is well -defined functionon V x V

.

As in the ease of finite

groups we show directlythat C . I . ) is linear in

first and antilinear in

second vanlable .

Moreover, Cutie = (ulu)

and Culm) Z O for n,v E V

Moreover,we have

5

(ulu) = Sg Stig) u IFLg) a>dug) .CTCg)u INg) m> 30 forall g c- G- , and

gi-ktlglultcgso.us iscontinuous

.

Assume that u t O .

Then Lulu> > O . Hence

there existsan open meiglooshookU of 1 such that

Ctlg) ulitlgn> > cou U for some E 70

.

Hence

(ulu) = Sqsntgdnlitcgn>dreg) =

= Su CTIg) u Itcgsu> dy Cg) t6

af,u

etcg) a ITCg) u> dieCg)

Z E Tulu) .Since uw) so it followsthat Lulu) > o .

It follows that ( ulu ) =Oimplies u=0 andC - I . ) in an inner productonV .

Moreover

(it (g)u I tlglv ) == Sqft tht ITIg)u ITChtt (g) re> que h)

7

{sting) ulHugh> dych) == SaitChul Hh) v> dmCah) = Lulu)by the invariance ofHaar measure .

Hence,it Cg) are unitary

operators with respect toC. I .)

,and it is a

unitary representation .

This proves the followingresult .

Proposition Any finite -dimensional representationof G- is unitary (withrespect to an appropriate

8

inner product).Let (it,V) be a finite.dimensionedrepresentation of a-(unitarywith respect toC.I.)) .

Let U be a

G- invariant subspaceof V . Then U is invariantfor all it(g) , g E G -

Hence Ut is invariant

for all it IgE , g E G -

Since it is unitary ,it (g)*= IT(g5

'= it ( g-

'),so

Ut is G - invariant .Hence

,V = U ⑤ Ut is

qa direct sum of representationsBy induction in dimension

of it we prove the followingtheorem .

theorem . Every finite --dimensional representationof Ct is a direct sum

of irreducible representation.

A direct consequence isthe following result .theorem .

Let g E G, get 1.

Then there exists an irreduciblefinite-dimensional representation

Cail of G- such that it (g) ¥ In

10As we remarked

,the

irreducible representationin above theorem is unitary(with respect to appropriate innerproduct) .

Remark .

.

- unitaryLet Ct,V) be a representation^

of a locally compact groupG- on a Hibbert space V.We say that it is irreducibleif the only closed a- - invariant

subspaces of V are to} andV .

If V is finite-dimensional

'l

this agrees with old

definition ( since all finite-dim.

subspaces are closed ) .Wehave the followinggeneralization of the aboveresult

.

theorem (Gelfand -RaiKoo)Let G- be a locally compactgroup . Let g c- G , get 1

.

Thenthere exists are

irreducible unitary representation(it, V) of G- such that Alg) *I .

12The proof of Gelfand -RaiKootheorem is quite differentfrom our discussion .

The essential point ofthe theorem is that we

have to consider infinitedimensional irreducibleunitary representationsof G . Finite-dimensionalirreducible unitary representationsdo not suffice !This can be seen fromthe following theorem .

13

theorem(Segal - von Neumann)Let G- be a noncompactsimple Lie group .

Let

fit,V) be a finite -dimensional

irreducible unitary representativeof G . There (it,V) istrivial .(i. e , V is one

-chin. and

itCgl = I, for all g E G) .

M7,2

Remedy : S L Cn,IR) areK

examples of noncompactsimple Lie groups .