La formule des traces pour les rev\^etements de groupes r\'eductifs connexes. IV. Distributions invariantes

TL;DR

This paper proves the invariant trace formula for adelic covers of connected reductive groups over number fields, assuming the trace Paley-Wiener theorem holds for all Levi subgroups at real places, with applications to metaplectic covers.

Contribution

It establishes the invariant trace formula for adelic covers of reductive groups under a key hypothesis, extending previous results to new classes of covers like metaplectic groups.

Findings

Proves the invariant trace formula under the Paley-Wiener hypothesis.

Provides simplified trace formulas for cuspidal test functions.

Applies to metaplectic covers of GL(n) and Sp(2n).

Abstract

We establish the invariant trace formula (\`a la Arthur) for the ad\'elic covers of connected reductive groups over a number field, under the hypothesis that the trace Paley-Wiener theorem is verified for all Levi subgroups at the real archimedean places. For instance, this hypothesis can be verified for the metaplectic covers of GL(n), or the twofold metaplectic cover of Sp(2n). We also give simple trace formulae when the test function satisfies certain cuspidality properties.