# counting points on nilpotent Springer fiber

Computing $p$-adic orbital integral I come to the following question. My ground field $k$ is the residue field of a non-arch local field, i.e. a finite field. I am happy to put any assumption on $\text{char}(k)$ so that Springer theory behaves just like over $\mathbb{C}$. Let $$\pi:\widetilde{\mathcal{N}}\rightarrow\mathcal{N}$$ be the Springer resolution, say for any simple (split) algebraic group $G$. Let $e\in\mathcal{N}(k)$ be an arbitrary non-regular nilpotent element.

Question 1 $\;$ Can we, maybe inductively, compute the number of rational points on $\pi^{-1}(e)$?

Question 2 $\;$ Let $\mathcal{N}^o\subset\mathcal{N}$ be the open subset of regular nilpotent element, and $IC_{\mathcal{N}}(\mathbb{Q}_{\ell})$ the intermediate extension of the constant sheaf on $\mathcal{N}^o$. Then $$\text{Tr}(Frob:IC_{\mathcal{N}}(\mathbb{Q}_{\ell})|_e)=\,?$$

• Minor edits. Have you looked at the work of Daniel Juteau on the modular Springer correspondence? Possibly that would help here, though I'm not sure. (Also, it complicates things to ask two separate questions here, since they might be answered independently.) Commented Dec 8, 2014 at 23:19
• @JimHumphreys I'm not sure what the modular Springer correspondence has to do with anything. That is when the base field of the sheaf is finite, not the underlying variety (which is irrevelant for the category of constructible sheaves). Commented Dec 9, 2014 at 2:39
• @Ben: Good point. I mentioned Juteau here because his work deals with prime characteristic, though as you say it's not relevant to this question. Commented Dec 9, 2014 at 18:16

The nilpotent cone is "rationally smooth" i.e. one has $IC(\mathcal{N}) = (\mathbb{Q}_\ell)_{\mathcal{N}}[d]$ (where $d$ denotes the dimension of the nilpotent cone). One can find this in "Partial resolutions of nilpotent varieties" by Borho-MacPherson. In particular the trace of Frobenius on any stalk agrees with the trace at any point on the regular locus, which answers Question 2.
It's proven in a paper of DeConcini, Lusztig and Procesi that in all types other than $E_7,E_8$, the Springer fibers all have polynomial point count (I'm probably assuming $p$ isn't too small here), which is of course determined by their compactly supported Betti numbers. This is proven (and the Betti numbers can be computed) by showing the Springer fibers have a paving by affine spaces; the total Betti numbers are thus just the number of fixed points of the torus in the Springer fiber, and the dimension of the affine space attached to each can be computed combinatorially.
• One should be a little careful here. De Concini, Lusztig and Procesi do not show the existence of an affine paving in exceptional type. Moreover I think that the argument with a torus having finitely many fixed points only works in type $A$. Commented Dec 9, 2014 at 9:22