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
$$\text{Tr}(Frob:IC\_{\mathcal{N}}(\mathbb{Q}\_{\ell})|\_e)=\,?$$



counting points on nilpotent Springer fiber
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.


