Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating object for $D^b(\mathrm{Repr}(G))$? I mean for finite groups this is clear, since one can take all irreducible representations. For reductive groups this is not possible. Are there examples of groups where this can happen?
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1$\begingroup$ For any unipotent group,the trivial representation does the job. $\endgroup$– abxCommented Feb 7, 2014 at 16:42
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2$\begingroup$ Any representation of a unipotent group contains the trivial representation, hence, by induction, is a successive extension of trivial representations. No restriction on the characteristic. $\endgroup$– abxCommented Feb 7, 2014 at 17:13
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1$\begingroup$ There's more than one meaning of "generating" for derived categories. Could you clarify exactly what you mean by "generating object"? $\endgroup$– Jeremy RickardCommented Feb 7, 2014 at 17:24
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1$\begingroup$ @Aleksa: With that definition of generating, then there's only a generating object if the only irreducible representation is the trivial one, so even finite groups won't have them (except $p$-groups in characteristic $p$, and the trivial group). $\endgroup$– Jeremy RickardCommented Feb 8, 2014 at 11:43
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2$\begingroup$ @Aleksa: For "if": the direct sum of irreducibles works, since any representation can be constructed from the irreducibles by iterated extensions. For "only if": only finitely many irreducibles occur as composition factors of the homology of an object $X$, and the thick subcategory generated by $X$ contains no objects that have other irreducibles involved. $\endgroup$– Jeremy RickardCommented Feb 8, 2014 at 12:41
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