ρ′(l 1 )f=d
dtf
et
0 0 0
0 0 1
0 −1 0
x 1
x 2
x 3
|t=0=
d
dtf
1 0 0
0 cost sint
0 −sint cost
x 1
x 2
x 3
|t=0=
d
dtf
x 1
x 2 cost+x 3 sint
−x 2 sint+x 3 cost
|t=0=
(
∂f
∂x 1,
∂f
∂x 2,
∂f
∂x 3)
·
0
x 3
−x 2
=x 3∂f
∂x 2−x 2∂f
∂x 3so
ρ′(l 1 ) =x 3∂
∂x 2−x 2∂
∂x 3and similar calculations give
ρ′(l 2 ) =x 1∂
∂x 3−x 3∂
∂x 1, ρ′(l 3 ) =x 2∂
∂x 1−x 1∂
∂x 2The space of all functions onR^3 is much too big: it will give us an infinity of
copies of each finite dimensional representation that we want. Notice that when
SO(3) acts onR^3 , it leaves the distance to the origin invariant. If we work in
spherical coordinates (r,θ,φ) (see picture)