ρF,H – the 1-dimensional unirrep of H given by ρF,H (exp X) =
e2πi F, X
PA – the G-invariant polynomial on
related to A ∈ Z(g), the center
For other notation, when it is not self-explanatory, the reader must con-
sult the Index and look for definitions given in the main text or in the
What you want What you have to do
1. Describe the unitary dual G Take the space O(G) of coadjoint
as a topological space. orbits with the quotient topology.
2. Construct the unirrep πΩ Choose a point F ∈ Ω, take
associated to the orbit Ω ∈
a subalgebra h of maximal
dimension subordinate to F ,
and put πΩ = Ind
3. Describe the spectrum Take the projection p(Ω) and
πΩ. split it into H-orbits.
4. Describe the spectrum Take the G-saturation of
πω. and split it into G-orbits.
5. Describe the spectrum of Take the arithmetic sum Ω1 + Ω2
the tensor product πΩ1 ⊗ πΩ2 . and split it into orbits.
6. Compute the generalized tr πΩ(exp X) =
e2πi F,X +σ
character of πΩ. χΩ , ϕ =
ϕ (F )
7. Compute the infinitesimal For A ∈ Z(g) take the value of
character of πΩ. PA ∈
on the orbit Ω.
8. What is the relation between They are contragredient (dual)
πΩ and π−Ω? representations.
9. Find the functional It is equal to
dimension of πΩ.