 Solution

We use complex representations. By Maschke's theorem, restriction to any finite subgroup is a semisimple module, so it has a canonical isotypic decomposition. In particular, the $\theta$-isotypic component $W_{\theta }$ is the sum of all copies of the irreducible $N$-module with character $\theta$. The central character idempotentprojects onto this component. These projections preserve every $N$-submodule, which consequently decomposes as the direct sum of its intersections with the isotypic components.

Let $W$ afford $\xi \in \mathrm{Irr}(T\mid \theta )$. The component $W_{\theta }$ is nonzero. Since $T=I_G(\theta )$ fixes $\theta$, it preserves $W_{\theta }$. Irreducibility of $W$ as a $T$-module therefore gives $W=W_{\theta }$: its restriction to $N$ consists entirely of copies of $\theta$.

Form the induced representationwhere $\mathcal{R}$ represents the left cosets $G/T$. Because $N◃G$, for $n\in N$ we haveThus $g\otimes W$ is $N$-isotypic of type ${\theta }^g$, where ${\theta }^g(n)=\theta (g^{-1}ng)$. These types are distinct for distinct cosets $gT$, precisely by the definition of the inertia group of a character $T$.

Let $Y$ be a nonzero $G$-submodule of $V$. Its isotypic decomposition as an $N$-module shows that it meets some $g\otimes W$ nontrivially. Acting by $g^{-1}$ gives $Y\cap (1\otimes W)\ne 0$. This intersection is a $T$-submodule of the irreducible $W$, hence equals $1\otimes W$. The $G$-translates of that component span $V$, so $Y=V$. The induced representation is therefore irreducible, and its restriction contains $\theta$. Its induced character satisfiesThis proves the irreducibility assertion directly from isotypic components; no form of the correspondence being proved has been assumed.