The question is regarding the proof of Wigner-Mackey given as Proposition 25 of Linear Representation of finite groups by Serre. It is on page 23. The fifth line of the proof of $(b)$ on page 63 states that $W_i$ is stable under $H_i$, why?. For this if we can show that $\theta_{i,\rho}$ restricted to $A$ is $\chi_i$, then we will be done. But I am not getting that, i am getting a multiple of $\chi_i$. This should be pretty obvious but i am missing something. Can someone have a look? Thanks.

Edit 1: Basically one has to show that $\theta_{i,\rho}(ah)(w)=\theta_{i,\rho}(a)\theta_{i,\rho}(h)(w)=\chi_i(a)\theta_{i,\rho}(h)$ for all $h\in H_i$ and $w\in W_i$.

Edit 2: $A$ is an abelian normal subgroup of $G$ and $G$ is a semidirect product of $A$ and $H$. The group $G$ acts on $X=Hom(A,\mathbb{C}^*)$ by $(g\chi)(a)=\chi(g^{-1}ag)$, where $\chi\in X$. Let $\chi_i$ be the system of representatives for the orbits of $H$ in $X$. Let $H_i$ be the stabilier of $\chi_i$. Form the semi-direct product of $A$ and $H_i$ and call it $G_i$. Extend $\chi_i$ to $G_i$. Let $\rho$ be an irreducible representation of $H_i$, composing it with the natural projection, we get an irreducible representation $\overline{\rho}$ of $G_i$. Now $\theta_{i,\rho}$ is the induction to $G$ from $G_i$ of $\chi_i\otimes \overline{\rho}$

Edit 3: Let $W$ be the representation space of $\theta_{i,\rho}$ and let $W_i=\{x\in W|\theta_{i,\rho}(a)x=\chi_i(a)x, a\in A\}$.

  • $\begingroup$ Please include all necessary information in the question. This includes the meanings of the various symbols. $\endgroup$ – Tobias Kildetoft Mar 10 '15 at 12:29

You have $\theta_{i,\rho} (h^{-1}ah)(w) = \chi_i(h^{-1}ah)(w)$, since $A$ is normal. Also, beacuse $h\in H_i$ you get $\chi_i(h^{-1}ah)=\chi_i(a)$ and therefore $\theta_{i,\rho}(h^{-1}) \theta_{i,\rho}(a) \theta_{i,\rho}(h)(w) = \chi_i(a)(w)$. Multiply by $\theta_{i,\rho}(h)$ both sides from left and you get it.


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