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On the spectral side of Arthur’s trace formula – absolute convergence. (English) Zbl 1242.11036

Introduced by Selberg for hyperbolic surfaces, the trace formula has been systematically studied by J. Arthur [“An introduction to the trace formula”, Providence, RI: American Mathematical Society. Clay Math. Proc. 4, 1–263 (2005; Zbl 1152.11021)] for adelic quotients \(G(F)\backslash G(\mathbb A)\) for any reductive group \(G\) defined on a number field \(F\). The trace formula settles the identity of a sum of geometric terms (weighted orbital integrals) with a sum of spectral terms (weighted traces of representations). The aim of this paper is a better comprehension of some limits of intertwining operators, with specific expressions for these limits as sums of products of first-order derivatives of intertwining operators. This analysis is based on the companion paper by the first two authors [Ann. Math. (2) 174, No. 1, 197–223 (2011; Zbl 1241.52006)], which proves combinatorial identities for some formal series families indexed by the chambers of a fan and with values in a non abelian algebra.
Even if the exposition follows strictly Arthur’s original expansion, it offers a refined spectral expansion and, as a original corollary, a convergence in trace norm, which is of primary importance for some application of the trace formula, e.g. Weyl’s laws or limit multiplicities. It explains also the behaviour of Maass-Selberg relations in some degenerate cases.

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
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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