A local trace formula. (English) Zbl 0741.22013

This paper develops an idea that was put forward about 10 years ago by D. A. Kazhdan. According to this for a locally compact group \(G\) the diagonal embedding of \(G\) into \(G\times G\) is analogous to that of a discrete group \(\Gamma\) into a locally compact group \(H\). In particular if \(F\) is a local field and \(G\) consists of the \(F\)-points of a linear reductive group defined over \(F\) then the decomposition of \(L^ 2(G\setminus G\times G)\) into irreducible representations is given by Harish-Chandra’s Plancherel theorem. The spectrum consists of a discrete part and a continuous part. One can therefore ask for a description of the character of the representation on the discrete part, analogous to the formula given by the Selberg trace formula. If \(G\) is compact then the formula follows from the Peter-Weyl theorem.
The author solves precisely this problem. His final formula is an identity between a sum of distributions parametrized by elliptic (and so semisimple) conjugacy classes in \(G\) and in the Levi components of its parabolic subgroups, and a sum of distributions parametrized by the discrete spectrum of \(G\) and also that of the Levi components of the parabolic subgroups. The author points out that the distributions are not invariant except in some special cases but asserts that it is not difficult to find an invariant form. The proof of the formula makes use of Harish-Chandra’s theory and techniques developed by the author in his work on the general trace formula.


22E50 Representations of Lie and linear algebraic groups over local fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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