On the differential equations satisfied by weighted orbital integrals (Q5960964)

Weighted orbital integrals are distributions on reductive groups over local fields appearing both in the local and global trace formulae. Associated are analogous invariant distributions which play the same role in the invariant trace formulae. In the Archimedean case the Fourier transforms of these distributions satisfy a system of differential equations induced by the center of the universal enveloping algebra. It is highly desirable to compute these Fourier transforms. The paper under review gives an important step towards this computation by showing that the system of differential equations is holonomic and has a simple singularity at infinity. Thus any solution has a series expansion and is a linear combination of certain canonical solutions. For some groups of small rank the recursion formula for the coefficients is solved explicitly.