TITLE: Formality of classifying spaces
QUESTION [21 upvotes]: Let $G$ be a compact Lie group (or reductive algebraic group over $\mathbb{C}$), and let $BG$ be its classifying space. Fix a prime $p$. Let $\mathcal{A}$ denote the dg algebra of singular cochains on $BG$ with coefficients in a field or characteristic $p$ (or if you prefer the dg algebra of endomorphisms of the constant sheaf). My question is:

Is it known for which primes $p$ the dg algebra $\mathcal{A}$ is formal, that is, quasi-isomorphic to a dg algebra with trivial differential?

I assume / hope that the answer is that this is true if $p$ is not a torsion prime for $G$ (i.e. $p$ arbitrary in types $A$ and $C$, $p \ne 2$ in types $B$, $D$ and $G_2$, $p \ne 2, 3$ in types $F_4$, $E_6$ and $E_7$, and $p \ne 2,3,5$ in type $E_8$.)
Note that we know* that $\mathcal{A}$ is formal in characteristic 0.

Can one then conclude that it is formal in any characteristic in which the cohomology of $\mathcal{A}$ is torsion free?

If so I think this would give the above list of primes.
*) for example because $H(BG, \mathbb{Q})$ is a poynomial algebra, and $\mathcal{A}$ admits a graded commutative model using the de Rham complex -- see Bernstein-Lunts "Equivariant sheaves and functors".

REPLY [10 votes]: This is an old question. But sometimes old questions get answered!
Benson, Greenlees, Formality of cochains on BG
Here is the abstract:
Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(T)/T|$ is invertible in the field $k$ then the algebra of cochains $C_*(BG;k)$ is formal as an $A_\infty$ algebra, or equivalently as a DG algebra.
(Note that this is not the same condition as I postulate in the question. I still wonder if it is true under the "$p$ not a torsion prime" hypothesis. However in the meantime, I should understand the above paper...)
Update: latest version of paper completely answers question! From Section 5:
"...if $G$ is a connected, simply connected compact Lie group, $k$ is a field of characteristic $p$, and $p$ is not a torsion prime then $H^∗(BG; k)$ is a polynomial ring on even classes, and $C^∗(BG; k)$ is formal."