Gallagher correspondence for an extendible type

Write $\rho (i)$ for the fixed extension on $S$. For any $\theta$-isotypical $I$-module $U$, put $M={\mathrm{Hom}}_N(S,U)$ and define $(i\cdot f)(s)=i\cdot f(\rho (i{)}^{-1}s)$. For $n\in N$, one has $\rho (i{)}^{-1}\rho (n)=\rho (i^{-1}ni)\rho (i{)}^{-1}$, so $(i\cdot f)(\rho (n)s)=n\cdot (i\cdot f)(s)$. Thus the formula stays in $M$.

The action law holds because $i\cdot (j\cdot f)(s)=ij\cdot f(\rho (j{)}^{-1}\rho (i{)}^{-1}s)=((ij)\cdot f)(s)$, and the identity acts identically. For $n\in N$, $n\cdot f(\rho (n{)}^{-1}s)=f(s)$ by $N$-linearity. Hence $N$ acts trivially on $M$ and this is a representation of $I\mathrm{/}N$.

The evaluation isomorphism is $I$-equivariant for the diagonal action on $S\otimes M$: $E_U(\rho (i)s\otimes (i\cdot f))=i\cdot f(s)$. Every $N$-submodule is $E_U(S\otimes M_0)$ for a unique $M_0\subseteq M$. Since $\rho (i)$ is invertible, its translate is $E_U(S\otimes i{M}_0)$. Uniqueness shows that this submodule is $I$-stable exactly when $M_0$ is stable under $I\mathrm{/}N$. For nonzero $U$, the multiplicity space is nonzero, so $U$ is irreducible if and only if $M$ is irreducible.

Conversely start with a quotient module $M$ and form $\tilde{S}\otimes \mathrm{Inf}M$. The map $m\mapsto f_m$, where $f_m(s)=s\otimes m$, is an isomorphism $M\to {\mathrm{Hom}}_N(S,\tilde{S}\otimes M)$ by the evaluation lemma and scalar-coordinate identification. The action constructed above satisfies $i\cdot f_m=f_{im}$. Therefore this recovers the quotient module, and step 3.1 proves irreducibility for every irreducible parameter. An isomorphism of $I$-modules induces an isomorphism of their Hom spaces by composition, respecting the quotient action; thus distinct quotient parameters cannot give isomorphic $I$-modules.

Every irreducible $I$-module above $\theta$ is $\theta$-isotypical, so steps 1.1–3.1 apply and its evaluation isomorphism supplies the required tensor form. This proves exhaustivity as well as injectivity. Taking tensor-product characters yields the stated character map.

Clifford correspondence now supplies the bijection after induction. Its inverse identifies the $\theta$-component with the inducing tensor module, whose restriction to $N$ is $\dim M=\eta (1)$ copies of $S$. Thus its ramification index is $\eta (1)$. If $I=N$, the quotient is trivial and only $S$ occurs; if $I=G$, induction is identity. An extension was assumed throughout, not obtained merely from invariance.