Some aspects of invariant harmonic analysis on the Lie algebra of a reductivep-adic group

Harish-Chandra developed invariant harmonic analysis on the Lie algebra in order to solve difficult problems on reductive groups. In all cases, real, p-adic, and finite, the "Fourier transforms of nilpotent orbits" play a central role. This thesis considers the case of p-adic reductive groups. We prove two main results. The first result concerns Harish-Chandra's formula for the character of an irreducible smooth representation near the identity. Via the exponential map, or some substitute, we can consider the character on the Lie algebra near 0. Harish-Chandra showed that the character can be expressed as a linear combination of the Fourier transforms of nilpotent orbits. For the discrete series, the formal degree shows up as the leading term (i.e., the constant term) in this expansion. Our first result shows that for tempered representations which do not belong to the discrete series, the coefficient of the leading term vanishes. Our second result is an integral formula for the Fourier transforms of nilpotent orbits. Following a suggestion of R. Kottwitz, we derive it as a special case of a formula for the Fourier transform of an invariant distribution with compactly-generated support.