Invariant harmonic analysis and Selberg zeta functions
In this project we study canonical invariant distributions on reductive groups, which appear in the Arthur-Selberg trace formula. Among them are orbital integrals, which are undergoing intensive study in the framework of endoscopy in order to facilitate progress in the Langlands program. The project concentrates on weighted orbital integrals, weighted characters and the associated invariant distributions, which appear in the case of non-compact quotients. The Fourier transforms of these distributions, which are known for real groups of rank one, are to be determined for groups of higher rank as well.
With the aid of these results, the trace formula will be applied to zeta functions of Selberg's type, also in the case of bundles on non-compact locally-symmetric spaces of higher rank, in order to continue them meromorphically, to interpret them as regularised determinants of differential operators and to relate their special values to spectral invariants.