Journal of Lie Theory

\def\tr{{\rm tr}} \def\GL{{\rm GL}} The character $\tr\pi$ of an irreducible admissible representation $\pi$ of the group $G(F)$ of $F$-points of a reductive connected linear algebraic group $G$ over a local non-Archimedean field $F$ has been shown by Harish-Chandra to be locally constant on the regular set and {\it locally integrable}, that is, representable by a function $\chi$ with such properties, when the characteristic of $F$ is $0$. His method was extended to $G = \GL(n)$ and its inner forms for all characteristics. Earlier this result had been proven for $G = \GL(2)$ and $F$ of any characteristic, characteristic two being the difficult case, in Jacquet-Langlands, by a direct and relatively elementary approach. We give here another proof by explicit computation, in this case of $\GL(2)$ and $F$ of any characteristic, especially two, which we believe extends to other low rank groups. Our computation gives an explicit evaluation of the orbital integral of the characteristic function $\chi_K$ of the maximal compact subgroup $K$. We use this to compute the coefficients in the germ expansion of the orbital integrals on $G$, and observe that the germ expansion of the orbital integral of $\chi_K$ extends to all of $K$.