# Derived category of representations

Suppose we are given an algebraic group $G$ (linearly reductive). Let $D^b(Repr(G))$ be the bounded derived category of finite dimensional algebraic representations of $G$. I am intressted in tilting objects for this category. Is there something known...some reference or articles treating this problem?

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The category is semisimple --- just take the sum of all simple representations. – Sasha Dec 29 '13 at 12:40
Ok. But the number of these simple representations may be infinite right? Are there some nontrivial tilting objects known for certain groups? – Aleksa Dec 29 '13 at 12:58
If the number of simple objects is infinite, then there is no finite dimensional tilting generator (for any finite dimensional representation there is a simple object which is not its summand and so it is in the orthogonal). – Sasha Dec 29 '13 at 14:05
Do you assume the tilting object is a finite dimensional representation...I mean the tilting object could be a komplex in $D^b(Repr(G))$... So what is when $G$ is not assumed to be linearly reductive? – Aleksa Dec 29 '13 at 14:47
The book of Jantzen is a good starting point. – Sasha Dec 29 '13 at 15:25