Lecture notes for Introduction to Representation Theory

Proof. (i) Let us decompose V = V (O,U) as an A-module. Then we get
V = yO V y ,
where V y = {v V (O,U) |av = (y, a)v, a A}. (Equivalently, V y = {v V (O,U) |v(g) = 0 unless gy =
x}). So if W → V is a subrepresentation, then W = yO W y , where W y → V y . Now, V y is a
representation of G y , which goes to U under any isomorphism G y ⊃ G x determined by g G
mapping x to y. Hence, V y is irreducible over G y , so W y = 0 or W y = V y for each y. Also, if hy = z
then hW y = W z , so either W y = 0 for all y or W y = V y for all y, as desired.
(ii) The orbit O is determined by the A-module structure of V , and the representation U by
the structure of V x as a G x -module.
(iii) We have
dim
2 V( U,O) = |O| 2 (dim U) 2 =
U,O
U,O
|O|
2 |Gx | = |O||G/G x ||G x | = |G| |O| = |G||A ∗ | = |G ∼ A|.
O O O
(iv) The proof is essentially the same as that of the Mackey formula.
Exercise. Redo Problems 3.17(a), 3.18, 3.22 using Theorem 4.75.
Exercise. Deduce parts (i)-(iii) of Theorem 4.75 from part (iv).
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