Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. I

This paper deals with the spectral theory of the Laplace‐Beltrami operator Δ acting on automorphic functions in n ‐dimensional hyperbolic space H n . We study discrete subgroups Γ which have a fundamental polyhedron F with a finite number of sides and infinite volume. Concerning these we have shown previously that the spectrum of Δ contains at most a finite number of point eigenvalues in [‐(1/2( n ‐ 1)) 2 , 0], and none less than (1/2( n ‐1)) 2 . Here we prove that the spectrum of Δ is absolutely continuous and of infinite multiplicity in (‐∞, ‐(1/2( n ‐ 1)) 2 ). Our approach uses the non‐Euclidean wave equation introduced by Faddeev and Pavlov, Energy E F is defined as ( u t , u t )‐(u, Lu ), where the bracket is the L 2 scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric. Energy is conserved under the group of operator U(t) relating initial data to data at time t. We construct two isometric representations of the space of automorphic data by L 2 (R, N) which transmute the action of U(t) into translation. These representations are given explicitly in terms of integrals of the data over horospheres. In Part II we shall show the completeness of these representations. u tt ‐Lu = 0, L = Δ + (1/2( n ‐ 1)) 2 .