For a linearly reductive Group $G$ over a field $k$ one has that the category of finite dimensional representations of $G$ is semisimple. What can one say about the number of simple representations? For what kind of groups the number is finite?
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2$\begingroup$ The number is finite only for finite groups (at least among split groups). $\endgroup$– abzCommented Jan 24, 2014 at 19:58
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$\begingroup$ Hence the derived category of finite dimensional representations $D^b(\mathrm{Repr}(G))$ has no generating object... $\endgroup$– AleksaCommented Jan 24, 2014 at 20:06
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Let $G$ be an affine linearly reductive group scheme over a field. Suppose that there are only finitely many simple representations (up to isomorphism) and let $X$ be the direct sum of them. Then every representation of $G$ is isomorphic to a subquotient (in fact, direct factor) of $X^n$ for some $n$. This implies that $G$ is finite (see, for example, Deligne and Milne, Tannakian Categories, 2.20).
