THÈSE DE DOCTORAT ELLOUMI Mounir Espaces duaux de ...

36 Dual topology of the motion groups SO(n) ⋉ R n
instead of ρ for the representation of SO(n − 1) with highest weight µ and
π(µ,r) instead of π(ρµ,ℓr). The representation π(µ,r) is realized on L 2 (SO(n)) as
follows ; for all (A, x) ∈ Mn and all B ∈ SO(n)
π(µ,r)(A, x)F (B) = e −i〈Bℓr,x〉 F (A −1 B), (F ∈ L 2 (SO(n))). (2.10)
In this way we obtain all the irreducible representations of Mn, which are
not trivial on its normal subgroup R n .
On the other hand, every irreducible unitary representation τλ of SO(n)
extends to an irreducible representation (also denoted by τλ) of the entire
group Mn, defined by
Now Mackey’s theory tells us that
Proposition 2.2.2.
Pn := � SO(n − 1) × R∗ +
τλ(A, x) := τλ(A), A ∈ SO(n), x ∈ R n .
� SO(n) ⋉ Rn is in bijection with the set of parameters
�
�SO(n).
2.2.3 Co-adjoint orbits attached to irreducible representations.
Let J =
�
0
−1
�
1
. We associate to the representation π(µ,r)
0
the linear
functional (Jµ, ℓr) in m∗ n where
⎛
µ1J
⎜
Jµ = ⎜
.
⎝ 0
. . .
. ..
. . .
0
.
µdJ
⎞
0
⎟
. ⎟
0 ⎠
0 . . . 0 0
,
if n = 2d + 1 is odd and if n = 2d is even, then
⎛
µ1J . . . 0 0
⎜
.
⎜
. .. . .
Jµ = ⎜
0 . . . µd−1J 0
⎝ 0 . . . 0 0
⎞
0
⎟
. ⎟
0 ⎟ .
⎟
0 ⎠
0 . . . 0 0 0
We see that the stabilizer Mn(ℓ) of ℓ = (Jµ, ℓr) in Mn is equal to Mn(ℓ) =
SO(n)(ℓ) ⋉ R n (ℓ). Indeed, by (2.3), we have that
Mn(ℓ) = {(A, a) ∈ Mn; (AJµA t + (Aℓra t − a(Aℓr) t ), Aℓr) = (Jµ, ℓr)}
= {(A, a) ∈ Mn; A ∈ SO(n − 1), AJµA t + (ℓra t − a(ℓr) t ) = Jµ}
= {(A, a) ∈ Mn; a ∈ Rℓr, A ∈ SO(n − 1), AJµA t = Jµ},