We recall Chat the action of a group in a function is given

b9Ogfcx) = fcgtx )

⇒ ( Or, f. Or

,

h ) = |d2x f*CF' x ) h ( 9-'

x )

In this case the representation for the inverse of a

rotation in the coordinate space is given by

Doe's -

.

fs;sn¥±EMI)

Therefore,

we can do a transformation of coordinates in

the previous integralI

' =

Dcri'

) e

⇒ ( Or, f. Or ,h > =) dx

'

wet JI f*CE ' )htx ' )

but in tn 's case kktJl=ldetEssn¥¥IIs¥¥H=L

⇒ ( Or .f,

Or ,h)=)d2x'

f*tx' )htx

'

) = ( fin )

Since the inner product is invariant under representationsof the group

we have

( fin'

, to'

)=tµ§ loofah,

ogfi"

) -

to.

§ with sacks# Doha

= §.

C fin'

.fi'

)took

( Diana ) )*DYits)= saw , too'

) daddabdm

= § ( fin'

, fin'

) dabdnu - Ccmdmdab

where on'

'

-

{ ( fin'

, fin'

>

By looking at the slides,

we see thatany function can

be decomposed in a basis determined by the differentrepresentations and the rows of each representation .

In this

sense,

we can think of the fin'

as the projection ofsaid function f in this basis

.

Since in this group we know that the inner product is

invariant under the group operations we can see that

( fin'

,Hfbi

"

t.nl.

§ last 'a"

,9Hfi"

) =tµ§( osfi'

, Hosts" )

Where we used the commutation of H withg

in the last

inequality .Now

, using the results from b) we see that

Ogfan'

- fin'

DTACS ) Osf 's'

= fed"

Dmdb (9)

⇒ ( fam,

ntfi"

) =

t.gs#',9tfoiYEalDYatsD*Dmdbc9) )

= E ( fin, Hfoi

"

) dad Sabdm = § hf in

, Hfi'

) dabdm:' n'

dabdnv

where we expresshen '

-

§ 1 fin'

, ntfin'

).