Index of Hecke operators (Q1060713)

Let M be a complete Riemannian manifold. Suppose the discrete group \(\Gamma\) acts isometrically and properly discontinuously on M with compact quotient \(\bar M=\Gamma \setminus M\). Assume that S is a set of isometries of M satisfying (i) \(S=\Gamma S=S\Gamma\) and (ii) \(\Gamma\) \(\setminus S\) is a finite set. The associated Hecke operator \(T_ S\) acts on \(L^ 2\bar M\). We study the asymptotic behavior of its trace on the eigenspaces of the Laplacian. This leads to a signature theorem for the Hecke operator. If M is locally symmetric, related results were obtained by \textit{H. Moscovici} [Lefschetz formulae for Hecke operators, Lect. Notes in Math. 1077, 321-358 (1984; Zbl 0561.58046)].