Module categories over $\mathrm{Rep}(G)$

Question

$\DeclareMathOperator\Rep{Rep}$Related to this question I also had some troubles to understand the classification of module categories over $\Rep(G)$. Specifically, on page 12 of Ostrik's paper what is the category $\mathrm{\Rep}^1(\tilde{H})$? $k^*$ acting as "identity character on V" means $a.v=av$ for all $a \in k^*$ and $v \in V$? Then what is the structure of module category over $\Rep(G)$? Tensor product should be after restricting representations of $G$ to $H$ and then inducing back to $\tilde{H}$?

Concretely, I was thinking about the following example. Let $H$ be a subgroup of $G$. Then $\Rep(H)$ is a module category over $\Rep(G)$ via tensor product as $H$-modules. What is the decomposition of $\Rep(H)$ in indecomposable module categories and what are the corresponding subgroups $H$ and cocyles $\omega \in H^2(H,\;k^*)$ for each indecomposable subcategory?

Answer 1

Sebastian: your definition of $Rep^1(\tilde H)$ is absolutely correct. If you have a representation of $G$ you can restrict it to $H$ and consider it as a representation of $\tilde H$ (this operation is called inflation). Now you can tensor it with any representation of $\tilde H$; this tensoring preserves $Rep^1(\tilde H)$; this is a module category structure (same thing was explained above by t3suji).

The category $Rep(H)$ considered as a module category over $Rep(G)$ is indecomposable. It corresponds to subgroup $H$ and trivial cocycle $\omega$ (so $\tilde H$ is a direct product of $H$ and multiplicative group $G_m$).