Let Θπ be the character of
an irreducible supercuspidal representation π of the special linear group SLn(F),
where F is a p-adic field of characteristic zero and residual characteristic greater than
n. In this paper, we investigate the existence of a regular elliptic adjoint orbit 𝒪π
such that, up to a nonzero constant, Θπ (composed with the exponential map)
coincides on a neighbourhood of zero with the Fourier transform of the invariant
measure on 𝒪π. When such an orbit 𝒪π exists, the coefficients in the local expansion
of Θπ as a linear combination of Fourier transforms of nilpotent adjoint
orbits are given as multiples of values of the correponding Shalika germs at
𝒪π. Let q be the order of the residue class field of F. If n and q − 1 are
relatively prime, we show that there is an elliptic orbit 𝒪π as above attached
to every irreducible supercuspidal π. When n and q − 1 have a common
divisor, necessary and sufficient conditions for existence of an orbit 𝒪π are
given in terms of the number of representations in the Langlands L-packet of
π.
