On the continuity of the geometric side of the trace formula

Abstract

We extend the geometric side of Arthur's non-invariant trace formula for a reductive group $G$ defined over $\mathbb{Q}$ continuously to a natural space $\mathcal{C}(G({\mathbb{A}}^1))$ of test functions which are not necessarily compactly supported. The analogous result for the spectral side was obtained in [MR2811597]. The geometric side is decomposed according to the following equivalence relation on $G(\mathbb{Q})$: ${\gamma }_1\sim {\gamma }_2$ if ${\gamma }_1$ and ${\gamma }_2$ are conjugate in $G(\bar{\mathbb{Q}})$ and their semisimple parts are conjugate in $G(\mathbb{Q})$. All terms in the resulting decomposition are continuous linear forms on the space $\mathcal{C}(G(\mathbb{A}{)}^1)$, and can be approximated (with continuous error terms) by naively truncated integrals.

Fixed a mistake found by Werner Hoffmann Explicated dependence on level

References in corpus (6)

• On the formal arc space of a reductive monoid
• Sur le comptage des fibrés de Hitchin nilpotents
• On the geometric side of the Arthur trace formula for the symplectic group of rank 2
• Induced conjugacy classes, prehomogeneous varieties, and canonical parabolic subgroups
• Zeta Functions for the Adjoint Action of GL(n) and density of residues of Dedekind zeta functions
• The trace formula and prehomogeneous vector spaces

Cited by in corpus (9)

• On the remainder term of the Weyl law for congruence subgroups of Chevalley groups
• Analytic torsion of arithmetic quotients of the symmetric space SL(n,R)/SO(n)
• Absolute convergence of the twisted Arthur-Selberg trace formula
• The dimensions of spaces of Siegel cusp forms of general degree
• Développement fin de la contribution unipotente à la formule des traces sur un corps global de caractéristique p>0, I
• On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties
• On the convergence of zeta functions of prehomogeneous vector spaces
• Refinements of the trace formula for GL(2)
• Approximation of $L^2$-analytic torsion for arithmetic quotients of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$