This comment thread is ridiculous. I'm about as far from dynamical systems mathematically as possible and this result looks amazing and fundamental to me. Thank you for posting this.

A correction to my previous comment: This only holds if you consider the Chow motive rationally.

Having a paving by affine spaces certainly suffices, but there are lots of examples of varieties with motives of this form which do not have affine pavings (in fact, they may even be of general type.) I believe that your converse is true assuming the Tate conjecture, but I may be misremembering. Another possibly mis-remembered statement: assuming the Hodge conjecture, a smooth projective variety over $\mathbb{C}$ has motive of this form if and only if all off-diagonal hodge numbers are zero.

I can't immediately rule out that there may be some exotic category equivalence, but if there is it can't preserve $N$-integrality (for the same reasons why it doesn't exist on category O.) And any equivalence gotten via Beilinson-Bernstein should preserve $N$-integrality.

Isn't this false even for category O if e.g. $\lambda=-2\rho?$

I'm not sure that this question is appropriate for MO, but here is a quick answer: This is not elliptic cohomology, actually, this is a sum of two copies of ordinary singular cohomology with coefficients in $\mathbb{Z}$ (with an index shift by $1$). The reason is that the homotopy type of an elliptic curve is $S^1\times S^1$, and $S^1$ is the Eilenberg-Maclane space $K(\mathbb{Z},1)$.

Crossposted from Math.SE: math.stackexchange.com/questions/4374022/…. (In the future, please note MSE crossposts in the original question to avoid duplication of effort.)

P.S. I think reading the nLab is a terrible way to actually learn any of these concepts. The nLab is useful as a reference and occasionally useful for gaining a new perspective on things one already knows well, but in this case I think you would be better served by reading any article/book on derived categories.

By "homotopical derived functor" I mean the same thing as the "nPOV derived functor. The classical derived functors arise in the following situation: Let $D(A)$ and $D(B)$ be the derived category of two abelian categories, and let $f:A\rightarrow B$ be a functor which is either left- or right- t-exact. Then there is a homotopical derived functor $Df:D(A)\rightarrow D(B).$ The classical derived functors are obtained by taking the composition $A\rightarrow D(A)\rightarrow D(B)\rightarrow B$, where the last functor is taking $i$th cohomology.

For better or for worse, this notion of derived functor is NOT the same (or even morally the same) as the classical notion of derived functor. The classical derived functors are given by the cohomology of the homotopical derived functor. So the answer is: Yes, homotopical derived functors and cohomology are somewhat orthogonal concepts, and you have to combine them to get the classical derived functors.

I think the answer to mathoverflow.net/questions/117098/… applies here as well.

@FZaldivar I believe that the bijection is between all positive roots and indecomposable representations (for $A_{n}$, think about the indecomposable representation assigning a $1$-dimensional representation to each vertex.) The mistake in the anecdote is that the $n+1$ should be replaced with $n(n+1)/2.$

@DavidBen-Zvi I don't think Kashiwara's is a problem - the category $D(\mathbb{A}^1)_0$ of D-modules supported at $0$ is not closed under limits, no? (I.e. it admits limits but they don't agree with the limits in $D(\mathbb{A}^1).$)