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Let $G$ be an algebraic group over a field $k$ (say of characteristic $0$) and let $H,H'$ be two closed subgroups. I would like to understand the category $Rep_k(H \cap H')$ of finite dimensional representations of $H \cap H'$.

Here was my thought : by the tannakian formalism I think that, since $H \cap H' = H \times_G H'$ that $Rep_k(H\cap H')$ should be $Rep_k(H)\coprod_{Rep_k(G)} Rep_k(H')$ the pushout of $Rep_k(H)$ and $Rep_k(H')$ over $Rep_k(G)$ in the category of neutral tannakian categories with a fixed fiber functor.

And my first guess to describe this category would be that the objects would be elements of either $Rep_k(H)$ or $Rep_k(H')$ modulo the equivalence relation $V \sim V'$ if there exists $V_0 \in Rep_k(G)$ such that $V_{0|H} = V$ and $V_{0|H'} = V'$.

But that seems weird and probably false. Does anyone know the answer ?

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    $\begingroup$ I think that consideration of finite groups (which can be viewed as algebraic groups of course) might help you assess whether anything like this is at all plausible. $\endgroup$
    – Simon Wadsley
    Commented Oct 3, 2017 at 9:10
  • $\begingroup$ As indicated by @SimonWadsley finite groups is a good place to start to see what is going on, and why this is not the right answer. For example if you take $G = S_3$ and subgroups of order $2$ and $3$ then your description would give way too many representations. $\endgroup$
    – Tobias Kildetoft
    Commented Oct 3, 2017 at 11:25
  • $\begingroup$ Hey, thanks for your comments. Yeah I think that if $H \cap H'$ is trivial then my description gives way to many representations. So it's definitely not the right answer. But I was wondering if someone knows the right one :) $\endgroup$
    – bob
    Commented Oct 3, 2017 at 12:46

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