×

The non-semi-simple term in the trace formula for rank one lattices. (English) Zbl 0609.22005

A problem in the derivation of the Selberg trace formula for rank one lattices \(\Gamma\) in real reductive Lie groups \(G\) is solved. The representation \(L^{dis}\) of \(G\) by left translations in the maximal completely reducible subspace of \(L^ 2(G/\Gamma)\) extends to the algebra of smooth compactly supported functions \(\alpha\) on \(G\). The point is to express \(trace(L^{dis}(\alpha))\) by familiar distributions. It is shown how the contribution of the non-semi-simple elements of \(\Gamma\) to this trace can be expressed by weighted orbital integrals, which was previously known in general only for the semi-simple elements.

MSC:

22E40 Discrete subgroups of Lie groups
22E35 Analysis on \(p\)-adic Lie groups
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
43A85 Harmonic analysis on homogeneous spaces
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
PDFBibTeX XMLCite
Full Text: DOI Crelle EuDML