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Geometric estimates for the trace formula. (English) Zbl 1165.11051

This paper is part of a program to extend Arthur’s trace formula to non-compactly supported test functions. Not only that, in the best of all worlds one would also like to have absolute convergence in the trace formlula. The spectral side of the trace formula has been dealt with satisfyingly by W. Lapid, B. Müller and E. M. Speh in [Geom. Funct. Anal. 14, No. 1, 58–93 (2004; Zbl 1083.11031)]. However, the geometric side remains open.
In the present paper the author presents a proof for the convergence of the geometric side of the trace formula for functions in the Harish-Chandra Schwartz space \({\mathcal C}^p\) for \(p>0\) sufficiently small. The precise statement is that each sum and integral on the geometric side converges absolutely. The author shows a bit more that that, as he proves absolute convergence of the so called coarse geometric expansion, i.e., the absolute value is located a bit further inside the iterated sums an integrals that make up the geometric side of the formula. This is not yet absolute convergence, but it is an important step that facilitates geometric applications of the trace formula.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Citations:

Zbl 1083.11031
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References:

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