The little group method for a semidirect product with abelian kernel

Every element of $G$ has a unique expression $ah$ with $a\in A$, $h\in H$: existence is $G=AH$ by [F1], and if $ah=a^{\mathrm{\prime }}{h}^{\mathrm{\prime }}$ then $h^{\mathrm{\prime }}{h}^{-1}=(a^{\mathrm{\prime }}{)}^{-1}a\in A\cap H=\{1\}$, so $h=h^{\mathrm{\prime }}$ and $a=a^{\mathrm{\prime }}$. Also $H_{\theta }=\{h\in H:{}^h\theta =\theta \}$ is the stabilizer in $H$ of $\theta$ under the action of [F4], hence a subgroup of $H$; and $A\le I_G(\theta )$, because $A$ is abelian and therefore ${}^a\theta (b)=\theta (a^{-1}ba)=\theta (b)$ for all $a,b\in A$.

The inertia group is $I_{\theta }=I_G(\theta )=A⋊H_{\theta }$: an element $g=ah$ satisfies ${}^g\theta ={}^h\theta$ because $A$ acts trivially, so $g\in I_G(\theta )$ if and only if $h\in H_{\theta }$; thus $I_G(\theta )=A{H}_{\theta }=\{ah:a\in A,h\in H_{\theta }\}$, which is a subgroup of $G$ by steps 1.1 and the normality of $A$ in $G$. Moreover the assignment $h{H}_{\theta }\mapsto h{I}_{\theta }$ is a bijection $H\mathrm{/}{H}_{\theta }\to G\mathrm{/}{I}_{\theta }$, since $I_{\theta }\cap H=H_{\theta }$ and $G=I_{\theta }H$; hence $[G:I_{\theta }]=[H:H_{\theta }]$.

The formula $\tilde{\theta }(ah):=\theta (a)$ defines a linear character of $I_{\theta }$ with $\tilde{\theta }{\mathrm{\mid }}_A=\theta$: it is well defined by the uniqueness in step 1.1, takes values in ${\mathbb{C}}^{\times }$, and $\tilde{\theta }(1)=1$. For $a,b\in A$ and $h,k\in H_{\theta }$ one has $(ah)(bk)=a(hb{h}^{-1})(hk)$ with $hb{h}^{-1}\in A$, so $\tilde{\theta }((ah)(bk))=\theta (a)\theta (hb{h}^{-1})=\theta (a)\theta (b)=\tilde{\theta }(ah)\tilde{\theta }(bk)$, using $h\in H_{\theta }$; and $\tilde{\theta }(a\cdot 1)=\theta (a)$ for $a\in A$, so $\tilde{\theta }$ extends $\theta$.

Gallagher's correspondence applies to the extension $\tilde{\theta }$ of $\theta$ to $I_{\theta }$: by [F6] the map $\eta \mapsto {\chi }_{\tilde{\theta }}\mathrm{Inf}\eta$ is a bijection from $\mathrm{Irr}(I_{\theta }\mathrm{/}A)$ onto $\mathrm{Irr}(I_{\theta }\mid \theta )$. The quotient $I_{\theta }\mathrm{/}A$ is isomorphic to $H_{\theta }$ via $ah\mapsto h$, and by [F7] and [A1] inflation identifies $\mathrm{Irr}(H_{\theta })$ with $\mathrm{Irr}(I_{\theta }\mathrm{/}A)$ through the irreducible representations of $I_{\theta }$ having $A$ in their kernel; hence the irreducible characters of $I_{\theta }$ lying over $\theta$ are exactly the characters of the representations $\tilde{\theta }\otimes {\mathrm{Infl}}_{H_{\theta }}^{I_{\theta }}\sigma$ with $\sigma \in \mathrm{Irr}(H_{\theta })$, since $\tilde{\theta }$ is one-dimensional and ${\chi }_{\tilde{\theta }\otimes \mathrm{Infl}\sigma }=\tilde{\theta }\cdot (\mathrm{Infl}{\chi }_{\sigma })$ by [F8].

Clifford induction then gives the parametrization: by [F5] induction is a bijection $\mathrm{Irr}(I_{\theta }\mid \theta )\to \mathrm{Irr}(G\mid \theta )$, so composing with step 4.1, the assignment $\sigma \mapsto {\mathrm{Ind}}_{I_{\theta }}^G(\tilde{\theta }\otimes \mathrm{Infl}\sigma )$ is a bijection from $\mathrm{Irr}(H_{\theta })$ onto $\mathrm{Irr}(G\mid \theta )$, and every representation in its image is irreducible.

The degrees are $[H:H_{\theta }]\dim \sigma$: the tensor product $\tilde{\theta }\otimes \mathrm{Infl}\sigma$ has dimension $\dim \sigma$ because $\tilde{\theta }$ is one-dimensional, so [F9] with $G$, $I_{\theta }$ and this module gives $\dim {\mathrm{Ind}}_{I_{\theta }}^G(\tilde{\theta }\otimes \mathrm{Infl}\sigma )=[G:I_{\theta }]\dim \sigma =[H:H_{\theta }]\dim \sigma$ by step 2.1; the ramification index over $\theta$ is instead $\dim \sigma$ by [F6].

In this split situation the Clifford obstruction vanishes and no projective correction is needed: $\tilde{\theta }$ is a genuine extension of $\theta$ to $I_{\theta }$ by step 3.1, so the obstruction class of $\theta$ is zero by [F10]; correspondingly, in the general correspondence of [F11] the factor set can be taken to be $1$ and the irreducible projective representations of $I_{\theta }\mathrm{/}A$ are the ordinary irreducible representations of $H_{\theta }$, so the assignment of step 4.1 is exactly Gallagher's correspondence and the theorem above is its orbit-parametrized form. For a nonsplit invariant type the obstruction can be nonzero; when it is, [F11] replaces the ordinary quotient representations by the irreducible projective representations attached to the class. A nonsplit extension by itself does not force a nonzero obstruction (the trivial type always extends).

The list is exhaustive and repetition-free over the orbits: the irreducible characters of $A$ are exactly the homomorphisms $A\to {\mathbb{C}}^{\times }$, because $\mathbb{C}$ is a splitting field for the finite group $A$ by [F2] and every irreducible of a finite abelian group over a splitting field is one-dimensional by [F3]; since $A$ acts trivially on $\widehat{A}$, the $G$-orbits on these characters are exactly the $H$-orbits, and by [F5] the sets $\mathrm{Irr}(G\mid \theta )$ depend only on the orbit of $\theta$ and partition $\mathrm{Irr}(G)$. Taking one $\theta$ per $H$-orbit therefore lists every irreducible $G$-representation exactly once through step 5.1.

The degenerate cases are included: if $H_{\theta }=H$ then $I_{\theta }=G$, induction is the identity, and the list is $\{\tilde{\theta }\otimes \mathrm{Infl}\sigma :\sigma \in \mathrm{Irr}(H)\}$ with degrees $\dim \sigma$; if $H_{\theta }=1$ then $I_{\theta }=A$ and the list reduces to the single representation ${\mathrm{Ind}}_A^G\theta$ of degree $[H:1]=\dim {\mathrm{Ind}}_A^G\theta$; if $A=1$ then $G=H$, the dual $\widehat{A}$ is trivial, $H_{\theta }=H$, and the statement is the tautology $\mathrm{Irr}(H)=\mathrm{Irr}(H)$ with degrees $\dim \sigma$.

Steps 3.1, 5.1, 6.1, 5.2 and 6.2 prove the assertion: $\tilde{\theta }(ah)=\theta (a)$ is a linear character of $I_{\theta }=A⋊H_{\theta }$ extending $\theta$, and the representations ${\mathrm{Ind}}_{I_{\theta }}^G(\tilde{\theta }\otimes \mathrm{Infl}\sigma )$, for one $\theta$ from each $H$-orbit in $\widehat{A}$ and $\sigma \in \mathrm{Irr}(H_{\theta })$, are exactly the irreducible complex representations of $G$ up to isomorphism, with the stated degrees; step 5.3 records that the obstruction vanishes in this split case and that the projective correspondence is the correction required when it does not.