## Module categories over $Rep(G)$.

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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?

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Why is this post community wiki? – Leonid Positselski Jan 5 2010 at 17:03
I'd think tensor product is by pulling back a representation from $G$ to $\tilde H$ and then tensoring over $\tilde H$. How is $\mathcal M$ a module category? – t3suji Jan 5 2010 at 17:09
One rstricts a rep. of $G$ to $H$ and then it tensors over $H$ with some rep. from $\mathcal{O}$. The irreducible constituents of the tensor product are in the same orbit, $\mathcal{O}$. – Sebastian Burciu Jan 5 2010 at 17:13
@ Leonid: It gives a classification of all modules categories over $Rep(G)$. Maybe I should have said this in the post. – Sebastian Burciu Jan 5 2010 at 17:17

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@Sebastian Burciu: Are they? Take $G=H$ and $\mathcal O$ to be the orbit of the one-point orbit of the trivial representation of $G$, for instance. Aren't you saying that the tensor product of the trivial representation and any representation is trivial? – t3suji Jan 5 2010 at 17:19
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## 1 Answer

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).

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Thank you very much! – Sebastian Burciu Jan 5 2010 at 18:20
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