A trace formula for symmetric spaces
The trace formula of the title is of the following type. We have a symmetric space \(G/H\) and a semisimple group \(G'\) such that there is a certain correspondence between closed double cosets for \(H\times H\) in \(G\) and closed conjugacy classes in \(G'\). The formula in question is an identity between integrals of the form \(\int K(h_ 1,h_ 2) dh_ 1 dh_ 2\) and \(\int K'(x,x) dx\). Here \(K\) and \(K'\) are the kernels belonging to the discrete spectrum of \(G_ \mathbb{A}\) and \(G_ \mathbb{A}'\) and defined by ``matching'' functions on \(G_ \mathbb{A}\) and \(G_ \mathbb{A}'\). The integrals are taken over \(H_ F\setminus H_ \mathbb{A}\times H_ F\setminus H_ \mathbb{A}\) resp. \(G_ F' \setminus G_ \mathbb{A}'\). In fact, on at least one of the two sides of the formula one has to insert an automorphic function as a factor in the integrand. All this is conjectural. Only \(H\)-distinguished representations should contribute to the left hand side. In the present paper this idea is worked out for \(G= R_{E/F} GL(2)\), where \(E/F\) is a quadratic extension, \(G'= GL(2)\) and \(H\) is the group of similitudes of a hermitian form in two variables (all these groups \(H\) enter into the formula).
scientific article; zbMATH DE number 469208
scientific article