Timeline for Mackey theory for semidirect products: equivalence between constructions for modules

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Jun 3, 2020 at 21:06	comment	added	LSpice		@AngeloLucia, you are right that it is a homomorphism; it corresponds to choosing the trivial action of $H_p$ on $\mathbb C_p$. Then we always have $\mathbb C_p \cong \mathbb C_p \otimes_{\mathbb C(H_p)} \mathcal H_\sigma$ as $\mathbb C(G_p)$-modules via any map $z \mapsto z \otimes_{\mathbb C(H_p)} v$ with $v \ne 0$, which doesn't seem to be what you want.
Jun 3, 2020 at 19:46	vote	accept	Angelo Lucia
Jun 3, 2020 at 19:46	comment	added	Angelo Lucia		Regarding your question about the $\mathbb{C}(H_p)$ structure of $\mathbb{C}_p$: I think that every character $p:N \to \mathbb{C}^{\times}$ (I am abusing notation and calling it $p$ again) could extended to a character of $G_p$ (and thus by restriction of $H_p$), as $\tilde{p}(nh) = p(n)$ for $n\in N$ and $h\in H_p$. This is an homomorphism due to the fact that $G_p$ leaves $p$ invariant, so if $n,m\in N$ and $h,k \in H_p$ then $\tilde{p}(nh)\tilde{p}(mk) = \tilde{p}(n) \tilde{p}(m)$ is the same as $\tilde{p}(nhmk) = \tilde{p}(n\phi_h(m)hk)=\tilde{p}(n)\tilde{\phi_h(m)}$.
Jun 3, 2020 at 0:05	history	answered	LSpice	CC BY-SA 4.0