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