Introduction xix

ρF,H – the 1-dimensional unirrep of H given by ρF,H (exp X) =

e2πi F, X

;

PA – the G-invariant polynomial on

g∗

related to A ∈ Z(g), the center

of U(g).

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

Appendices.

USER’S GUIDE

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 ∈

g∗.

a subalgebra h of maximal

dimension subordinate to F ,

and put πΩ = Ind

G

H

ρF,H .

3. Describe the spectrum Take the projection p(Ω) and

of Res

G

H

πΩ. split it into H-orbits.

4. Describe the spectrum Take the G-saturation of

p−1(ω)

of Ind

G

H

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) =

Ω

e2πi F,X +σ

or

character of πΩ. χΩ , ϕ =

Ω

ϕ (F )

eσ.

7. Compute the infinitesimal For A ∈ Z(g) take the value of

character of πΩ. PA ∈

Pol(g∗)G

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

1

2

dim Ω.

dimension of πΩ.