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 *I*_{M}, in terms of which the trace formula can be restated. The ordinary orbital integrals reappear unchanged as the terms *I*_{G}.

Arthur has announced a proof of the stabilization of the distributions *I*_{M} on real groups *G*. An important technical device are the differential equations satisfied by *I*_{M} as a function of the orbit. In fact, *I*_{M} 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 *I*_{M} can be explicitly calculated in terms of higher-dimensional hypergeometric series at least for groups of low real rank.

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