Weighted orbital integrals
Werner Hoffmann (Bielefeld)
The derivation of the trace formula requires truncation of an integral kernel to make it integrable over a non-compact quotient. On the geometric side, this produces weighted orbital integrals, which are integrals of a test function on a reductive group G with respect to a certain non-invariant measure supported on a conjugacy class. On the spectral side, truncation gives rise to weighted characters, which are traces of induced representations of G twisted by certain logarithmic derivatives of intertwining operators.
In this way one gets two families, each indexed by the Levi subgroups M of G, of distributions which are non-invariant under inner automorphisms in a parallel pattern. Arthur constructed from those two families a new family of invariant distributions IM, in terms of which the trace formula can be restated. The ordinary orbital integrals reappear unchanged as the terms IG.
Arthur has announced a proof of the stabilization of the distributions IM on real groups G. An important technical device are the differential equations satisfied by IM as a function of the orbit. In fact, IM is fully characterized by those differential equations together with the behaviour at infinity and at the singular orbits.
The differential equations form a holonomic system with a regular singularity at infinity similar to the system satisfied by the matrix coefficients of an admissible representation. Thereby the Fourier transform of IM can be explicitly calculated in terms of higher-dimensional hypergeometric series at least for groups of low real rank.
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