Local character expansions and Shalika germs for GL(n)
We study the values of characters of irreducible supercuspidal representations of \(p\)-adic general linear groups. Under the assumption that the residual characteristic of the field is greater than the rank of the group, we prove that on some neighbourhood of the identity element the character of such a representation coincides with a positive multiple of the Fourier transform of a regular elliptic adjoint orbit (composed with the inverse of the exponential map). As a consequence, the coefficients of the Fourier transforms of nilpotent adjoint orbits in Harish-Chandra's local character expansion are shown to be positive multiples of the Shalika germs evaluated at the elliptic orbit.
irreducible supercuspidal representations
Harish-Chandra's local character expansion
scientific article; zbMATH DE number 861919
scientific article