Local integrability of characters on GL(2), orbital integrals, germs

The character tr p of an irreducible admissible representation p 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 locally integrable, that is, representable by a function 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 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 K extends to all of K.