Intuition behind the representation theory of semi-direct products (Mackey's theorem)

Question

I've been attempting to understand the irreducible representations of semi-direct products involving abelian groups; this is primarily motivated by a desire to understand Wigner's classification of elementary particles. It seems that these representations are well-understood for finite groups, according to a result sometimes called Mackey's theorem. This theorem states that an irrep of a finite group $G = H \ltimes A$ with $A$ abelian can always be expressed as the representation induced by the representation $\hat{\chi} \otimes \hat{\rho}$ of $G_\chi$, where:

• $\chi: A \to \mathbb{C}^\times$ is an irreducible representation of $A$;
• $H_\chi \leq H$ is the isotropy subgroup of $\chi$ under the adjoint action;
• $G_\chi = H_\chi \ltimes A$ is a subgroup of $G$;
• $\rho: H_\chi \to \mathrm{GL}(V)$ is an irrep of $H_\chi$ and $\hat{\rho}$ is the representation of $G_\chi$ formed by precomposing with the projection $G_\chi \to H_\chi$, and likewise for $\hat{\chi}: G_\chi \to \mathrm{GL}(V)$ (with the projection $G_\chi \to A$);
• the induced representation of $\hat{\chi} \otimes \hat{\rho}$ is the collection of all $G_\chi$-equivariant maps $\phi: G \to V$, with action by $G$ given by right multiplication with $g^{-1}$.

A proof of this fact is given in Serre's book on representations of finite groups (p.69), but it's all done in a sort of "abstract-nonsense" way (or perhaps I only see it this way because I don't have a deep enough understanding of character theory). The point of issue for me is that the result is genuinely surprising, in that I would expect we would need more than just the representation theory of isotropy subgroups to get all of the representations of the entire group. Unfortunately, none of the explanations I have found give me any more insight into this relationship other than "it happens to work."

My question, therefore, is the following: is it possible to "see" how we are able to build up an entire irrep of $G$ just from an irrep of $A$ and some $H_\chi$? By this I do not necessarily mean an explanation of the anatomy of the induced representation, but something more like a decomposition of an arbitrary irrep of $G$ into these two pieces. I am hoping to find a proof which is more illuminating in this regard, but if you are able to explain the deeper meaning of Serre's argument, or even perhaps give a motivating example, then that would also go a long way. Thank you!

Answer 1

I don't know how Serre does it, but I think the basic idea is pretty intuitive and not too hard to explain. Let $V$ be an irreducible representation of $G = A \rtimes H$ (I prefer to write semidirect products in this order but it doesn't really matter) over $\mathbb{C}$. Consider its restriction to $A$: since $A$ is abelian, $V$ decomposes into a direct sum of $1$-dimensional representations of $A$, corresponding to characters $\chi : A \to \mathbb{C}^{\times}$.

Let $v_{\chi}$ be a vector lying in the isotypic component

$$V_{\chi} = \{ v \in V : av = \chi(a) v \}.$$

Equivalently, this means $v_{\chi}$ is a simultaneous eigenvector for the elements of $A$, with eigenvalues $\chi(a)$. What happens if we act on $v_{\chi}$ by an element of $H$? Writing the action of $H$ on $A$ as $a^h = hah^{-1}$, we have

$$a^{h^{-1}} v_{\chi} = \chi(a^{h^{-1}}) v_{\chi} = h^{-1}ah v_{\chi}$$

which gives

$$a (hv_{\chi}) = h \chi(a^{h^{-1}}) v_{\chi} = \chi(a^{h^{-1}}) h v_{\chi}$$

so we get that $h v_{\chi}$ remains a simultaneous eigenvector for the elements of $A$, but with different eigenvalues: in other words, $h v_{\chi}$ ends up in a different isotypic subspace, namely the one determined by the character $\chi(a^{h^{-1}})$. This is the natural action of $H$ on the character group $A^{\vee} = \text{Hom}(A, \mathbb{C}^{\times})$ given by dualizing the action of $H$ on $A$, so let's notate it by $\chi^h$. Then this computation gives

$$\boxed{ hv_{\chi} \in V_{\chi^h} }.$$

So the action of $H$ permutes the isotypic components of $V$ according to the action of $H$ on the characters of $A!$ This is the key computation. This means if we take the direct sum $\bigoplus V_{\chi^h}$ of the isotypic components over the orbit of $\chi$ under the action of $H$, the result is invariant under the action of both $A$ and $H$, hence $G$, and so by irreducibility this must already be all of $V$; in other words, only a single orbit appears.

Moreover, $V_{\chi}$ is now acted on by the stabilizer $H_{\chi} = \{ h \in H : \chi^h = \chi \}$ of $\chi$ under the action of $H,$ and together with $A$ acting by the character $\chi$ it is a representation of $A \rtimes H_{\chi}$. From here it's not too hard to guess by comparing dimensions that $V$ itself is the result of inducing this representation up to $A \rtimes H$, and you can check that. It also shouldn't be too hard to guess that in order for $V$ to be irreducible, $V_{\chi}$ needs to be an irreducible representation of $H_{\chi}$, and you can check that too.