I am wondering if there is a theorem of Shalika germ (as below) for local function field, for both the group version or the Lie algebra version, probably under assumption on the characteristic to be very good. Thank you!
Theorem (Shalika) Let $G$ be a connected reductive group over $F$. $u_1,...,u_n\in G(F)$ (resp. $\in\mathfrak{g}(F)$) be a list of the finite set of unipotent (resp. nilpotent) $G(F)$-conjugacy orbits. There exists locally constant functions $\Gamma_1,...,\Gamma_n$ on $G(F)^{rs}$ (resp. $\mathfrak{g}(F)^{rs}$), the regular semisimple locus, such that for any smooth compactly supported function $f\in C_c^{\infty}(G(F))$ (resp. $C_c^{\infty}(\mathfrak{g}(F))$, one has $\mu_{\gamma}(f)=\sum\Gamma_i(\gamma)\mu_{u_i}(f)$ for $\gamma$ close enough to $1\in G(F)$ (resp. $0\in\mathfrak{g}(F)$), where $\mu_{\gamma}$ denotes the orbital integral.
