I am trying to define representations of semidirect products of non-abelian groups following the example of representations of semidirect products of abelian groups by Etingof et al. Here is what I have.
Am I doing it write?
Let $G, A$ be two non-abelian groups and let $\phi: G \to Aut(A)$ be a homomorphism. For $a \in A$ denote $\phi(g)a$ by $g(a)$. The semidirect product $G \ltimes A$ is defined to be the product $A \times G$ with multiplication law $$ (a_1, g_1) (a_2, g_2) = (a_1 g_1 (a_2), g_1 g_2)$$.
The irreducible representations of $A$ can be more than one dimensional and, together, they form the character group $A^\vee$ (direct product as the group operation), which carries an action of $G$. Let $O$ be an orbit of this action, $x \in O$ 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 $G \ltimes A$ as follows.
As a representation of $G$, we set $$ V_{(O, x, U)} = Ind^{G}_{G_x} U= \{f: G \to U | f(hg) = hf(g), h \in G_x\} $$.
Next we introduce an additional action of $A$ on this space by $(a f)(g) = x(g(a))f(g) $. Then it is easy to check that these two actions combine into an action of $G \ltimes A$. Also, it is clear that this representation does not really depend on the choice of $x$, in the following sense. Let $x, y \in O$ and $g \in G$ be such that $g x = y$, 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 $V_{(O, U)}$.