As should be evident from the title this question has a similar flavor to:


[Formality of classifying spaces](https://mathoverflow.net/questions/46521/formality-of-classifying-spaces)


However, unlike Geordie's question, I will be working with torsion free coefficients (say the complex numbers). The `torsion' in the current question is going to be from a different source.


Let $G$ be a reductive linear algebraic group (over $\mathbb{C}$) say. Relatively general formal considerations show that the $G$-equivariant derived category of a point is equivalent to the dg-derived category of a dg-algebra (I am going to ignore finiteness issues).


For instance, if $G$ is connected (so that $BG$ is simply connected), the dg-algebra is the algebra associated to the De Rham complex of $BG$ (let's be friendly and ignore finiteness issues again). In this case the $G$-equivariant derived category of a point is the subcategory of $D(BG)$ generated by the constant sheaf. The latter is equivalent to the derived category of the De Rham complex. Further, since $H^\*(BG)$ is a polynomial ring (I am still in the $G$ is connected case) generated in even degree, it is easy to see that this dg-algebra is formal (the point is that the De Rham complex is (super)commutative before even passing to cohomology, so you can very naively construct your quasi-isomorphism by hand).


Ok, that's pretty nice. Now let's look at $G$ not connected. Then the $G$-equivariant derived category of a point is still governed by a dg-algebra, but it's a bit nastier. Essentially we want the subcategory of $D(BG)$ generated by local systems, and since $BG$ isn't simply connected anymore we are going to have non-trivial local systems. The dg-algebra we are after is the $Ext$-algebra of the sum of all the irreducible local systems. Now we can certainly compute the cohomology of this dg-algebra: it is the $H^\*(BG^0)$-twisted group algebra of $G/G^0 = \pi\_1(BG)$. **Added later:** to see this, loop the fibration $BG^0\hookrightarrow BG \twoheadrightarrow B(G/G^0)$.


**The question:** is the dg-algebra governing $D(BG)$ (for $G$ not necessarily connected) formal? If so, is it possible to deduce this formality from the connected case?


I am quite happy assuming that $G/G^0$ is abelian. 



Formality of classifying spaces (for not necessarily connected groups)
The answer is yes $D^b\_G(X)$ is equivariantly formal. The result has been proved in a diploma thesis written under the supervision of Wolfgang Soergel. (Unfortunately it is not available electronically). 


The proof goes roughly as follows. Let $\pi:BG\_0 \to BG$ the quotient map. Let $\mathcal{L}$ be the sum of simple perverse sheaves on $D\_G(pt)$ and $\mathcal{L}\_0$ the sum of simple perverse sheaves on $D\_{G\_0}(pt)$. Let $I\_{\mathcal{L}}$ and $I\_{\mathcal{L}\_0}$ (note that $I\_{\mathcal{L}\_0}$ is a direct summand of $\pi^\*(I\_{\mathcal{L}})$). The corresponding injective resolutions i.e. $D^b\_G(pt)\cong D^b(End(I\_{\mathcal{L}}))$ ... 


You want to show that the $dg$-algebra $End(I\_{\mathcal{L}})$ is formal. Of course it would also be fine to show that $End(\pi\_\*\pi^\*I\_{\mathcal{L}})$ is formal. Now by [Soe01] Theorem 2.4.2 there exists an isomorphism 
$$End(\pi\_\*\pi^\*I\_{\mathcal{L}})\cong End( I\_{\mathcal{L}\_0})\otimes \mathbb{C}[G/G\_0] $$
 **Be carefull: The multiplication of $End( I\_{\mathcal{L}\_0})\otimes \mathbb{C}[G/G\_0]$ has some twist corresponding to the action of $G/G\_0$ on $End( I\_{\mathcal{L}\_0})$**  Anyway, one can show that the "classical" quasi-isomorphism of $End(I\_{\mathcal{L}\_0})$ to $H^\*(...)$ is equivariant with respect to the $G/G\_0$ action on so induces the quasi-isomorphism you are looking for.


[Soe01] Langlands' Philosophy and Koszul Duality <http://home.mathematik.uni-freiburg.de/soergel/PReprints/langlands.ps>


