Poincaré series for \(\mathrm{SO}(n,1)\) (Q1112094)

Generalized Kloosterman sums are introduced for \(G=\mathrm{SO}(r+1,1)(\mathbb R)\) with \(r\geq 2\) and a discrete subgroup \(\Gamma\) for which \(\Gamma\setminus G\) has finite volume, but is not compact. The corresponding Kloosterman-Selberg zeta function \(Z(s)\) is defined. For the case of certain arithmetically given \(\Gamma\) an adelic treatment is indicated which shows on the one hand that this zeta function is holomorphic for \(\Re s>(r-1)/2\), and on the other hand that exceptional eigenvalues correspond to poles of \(Z\). This implies \(\lambda \geq r/2-1/4\) for all positive eigenvalues \(\lambda\). (The generalized Ramanujan conjecture would be \(\lambda \geq r^ 2/4\) in this case.) Proofs of the various results are indicated.