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Given a topological group acting on a topological space, Bernstein and Lunts construct the equivariant derived category, which looks like the derived category of the quotient would, if action was free ("derived category of the quotient stack").

What are applications of this formalism to representation theory or other subjects?

In their book Bernstein and Lunts give a beautiful description of the equivariant derived category of a point (for a reasonable Lie group). Are there applications of this specific result?

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Aren't all Lie groups reasonable?! –  Mariano Suárez-Alvarez Mar 25 '12 at 9:08
BL do it for any connected Lie group. –  Jan Weidner Mar 25 '12 at 12:07
On the other hand the discrete Lie group $\mathbb Z$, or $\mathbb Z \times \text{connected Lie group}$ should also work. –  Jan Weidner Mar 25 '12 at 12:16
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