Let $k=\Bbb F\_q$ be a finite field, $G$ be a connected reductive group over $k$, $\mathcal{N}$ be the nilpotent cone of $G$ which consists of nilpotent elements in the lie algebra of $G$. 


Motivation & example: If $G=GL\_n$, then $\#\mathcal{N}(k)=q^{n^2-n}$ ([here](https://projecteuclid.org/euclid.ijm/1255629831) is a direct proof) i.e there are exactly $q^{\text{dim}\mathcal{N}}$ nilpotent matrices over $k$. 


The direct proof uses no geometry while the result is simple, so I am interested in an explanation by Grothendieck trace formula. However, $\mathcal{N}$ is not smooth ( $\mathcal{N} \cong \{xy=z^2\}$ if $G={GL}\_2$ ) in general, so interesection cohomology might be better than etale cohomology (The nilpotent cone is rationally smooth when using intersection cohomology at least in characterestic zero).


**Q1**. What is $\#\mathcal{N}(k)$ for general $G$?


**Q2** What is the cohomology (etale/intersection cohomology) of $\mathcal{N}\_{\bar k}$ and the trace of geometric Frobenius action on it?


**Q3** Do we have Grothendieck trace formula for intersection cohomology? 


My vague idea for **Q1** is to consider Springer resolution and reduce to counting size of the flag variety and Springer fiber.



Number of points of the nilpotent cone over a finite field and its cohomology
When formulating a question about the cohomology of a variety it's important to determine which cohomology group you want to ask about. One rule of thumb is:



> 
> If you want to understand the number of $\mathbb F\_q$-rational points, or you already understand the number of $\mathbb F\_q$-rational points and want to extend that understanding to cohomology, you almost always want to study the etale cohomology with compact supports of the constant sheaf.
> 
> 
> 


This is for the simple reason that the Grothendieck-Lefschetz fixed point formula relates the etale cohomology with compact supports to a sum over points, where the summand is $1$ in the case of the constant sheaf.


With regards to your question 3, in the etale setting we very rarely think of intersection cohomology as a different cohomology theory, but rather as a choice of coefficient sheaf (the intersection cohomology complex). In particular, there exists a Grothendieck-Lefschetz fixed point formula for the intersection cohomology with compact supports, but it relates to the sum over points of the trace function of the intersection cohomology complex. (This is often a very interesting function to sum, more interesting than the constant function 1, e.g. in the case of a Schubert cell in a Grassmanian or affine Grassmanian.)


Anyways, with regards to your question 2 we have the general result that if $X$ is any closed, scaling-invariant subset of $\mathbb A^n$ containing the origin, then $H^\*(X, \mathbb Q\_\ell) = \mathbb Q\_\ell$ in degree zero.


Indeed, let $j: X - \{0 \} \to X$ be the open immersion, then we have an exact sequence $0 \to j\_! \mathbb Q\_\ell \to \mathbb Q\_\ell \to \delta\_0\to0$, where $\delta\_0$ is the skyscraper sheaf, hence a long exact sequence $H^\*(X, j\_! \mathbb Q\_\ell) \to H^\*(X,\mathbb Q\_\ell) \to H^\*( 0, \mathbb Q\_\ell)$, so it is sufficient to prove that the second map is an isomorphism by proving that $H^\*(X, j\_! \mathbb Q\_\ell)=0$.


Then if we let $Y$ be the blowup of $X$ at $0$, $u: X-\{0\} \to Y$ the open immersion, $\pi: Y \to X$ the projection map, then one can check that $\pi\_\* u\_! \mathbb Q\_\ell= j\_! \mathbb Q\_\ell$, so by the Leray spectral sequence it suffices to show that $H^\* (Y, j\_! \mathbb Q\_\ell)=0$. Applying Leray again, it suffices to show that the derived pushforward from $Y$ to the projective variety on which $X$ is an affine cone of $u\_! \mathbb Q\_\ell$ vanishes. This is a locally trivial fibration by $\mathbb A^1$, on which $j\_! \mathbb Q\_\ell$ is locally the extension by zero of the constant sheaf on $\mathbb A^1-\{0\}$, so by smooth base chance we can reduce to $H^\* (\mathbb A^1, j\_! \mathbb Q\_\ell)=0$ for $j : \mathbb A^1 - \{0\} \to \mathbb A^1$ the open immersion, where it is straightforward.


It follows that if $X$ is a rational homology manifold of dimension $d$, in the sense that its dualizing complex is a constant sheaf in degree $-2d$, then $H^\*\_c(X,\mathbb Q\_\ell) = \mathbb Q\_\ell(-d)$ in degree $2d$, and so the trace of Frobenius on $H^\*\_c(X,\mathbb Q\_\ell)$ is $q^d$, and thus $X$ has $q^d$ rational points.


As you note, this is only possible if $X$ is the affine cone on a projective variety with $\frac{q^d-1}{q-1}$ points. So the condition of being a rational homology manifold is quite restrictive, already at the level of counting rational points!


However, the vanishing for ordinary cohomology is no restriction on $X$ because there is no Grothendieck-Lefschetz formula for ordinary cohomology.


My understanding (which I didn't check) is that the foundational papers on the Springer resolution and its cohomology do prove that the nilpotent cone is a rational homology manifold in characteristic $p$ in general. If this is the case, then indeed the number of points must be $q^{ \dim G - \dim T}$ with $T$ the maximal torus of $G$.


