Lecture notes for Introduction to Representation Theory

Proof. Proof that (ii) implies (i). Assume that g G does not belong to any of the subgroups
H X. Then, since X is conjugation invariant, it cannot be conjugated into such a subgroup.
Hence by the Mackey formula, ν Ind G (V )(g) = 0 for all H X and V . So by (ii), for any irreducible
H
representation W of G, ν W (g) = 0. But irreducible characters span the space of class functions, so
any class function vanishes on g, which is a contradiction.
Proof that (i) implies (ii). Let U be a virtual representation of G over C (i.e., a linear combination
of irreducible representations with nonzero integer coefficients) such that (ν U , ν Ind G V ) = 0 for
H
all H, V . So by Frobenius reciprocity, (ν U|H , ν V ) = 0. This means that ν U vanishes on H for any
H X. Hence by (i), ν U is identically zero. This implies (ii) (because of the above remark).
Corollary 4.74. Any irreducible character of a finite group is a rational linear combination of
induced characters from its cyclic subgroups.
4.26 Representations of semidirect products
Let G, A be groups and θ : G ⊃ Aut(A) be a homomorphism. For a A, denote θ(g)a by g(a).
The semidirect product G ∼ A is defined to be the product A × G with multiplication law
(a 1 , g 1 )(a 2 , g 2 ) = (a 1 g 1 (a 2 ), g 1 g 2 ).
Clearly, G and A are subgroups of G ∼ A in a natural way.
We would like to study irreducible complex representations of G ∼ A. For simplicity, let us do
it when A is abelian.
In this case, irreducible representations of A are 1-dimensional and form the character group
A ∗ , which carries an action of G. Let O be an orbit of this action, x O a chosen element,
and G x the stabilizer of x in G. Let U be an irreducible representation of G x . Then we define a
representation V (O,U) of G ∼ A as follows.
As a representation of G, we set
V (O,x,U) = Ind G U = {f : G ⊃ U|f(hg) = hf(g), h G x }.
G x
Next, we introduce an additional action of A on this space by (af)(g) = x(g(a))f(g). Then it’s
easy to check that these two actions combine into an action of G ∼ A. Also, it is clear that this
representation does not really depend on the choice of x, in the following sense. Let x, y O,
and g G be such that gx = y, and let g(U) be the representation of G y obtained from the
representation U of G x by the action of g. Then V (O,x,U) is (naturally) isomorphic to V (O,y,g(U)) .
Thus we will denote V (O,x,U) by V (O,U) (remembering, however, that x has been fixed).
Theorem 4.75. (i) The representations V (O,U) are irreducible.
(ii) They are pairwise nonisomorphic.
(iii) They form a complete set of irreducible representations of G ∼ A.
(iv) The character of V = V (O,U) is given by the Mackey-type formula
ν V (a, g) =
1
|G x |
hG:hgh −1 G x
x(h(a))ν U (hgh −1 ).
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