Tic Group Research Articles

Eisenstein-series on real, complex, and quaternionic half-spaces

The real, complex, and quaternionic half-spaces are introduced in certain analogy with the Siegel half-space. The modified symplec- tic group acts on the attached half-space in the usual way. At first properties of these half-spaces considered as symmetric spaces are derived. Then fundamental domain with respect to the modified modular group, which consists of integral modified symplectic matri- ces, is constructed. The behavior of convergence of the corresponding Eisenstein-series is determined carefully. The Fourier-coefficients of the Eisenstein-series are calculated explicitly, whenever the degree is sufficiently small. Introduction. The present paper deals with half-spaces, which are built in analogy with the Siegel half-space, and the corresponding non- analytic Eisenstein-series. The roots can be traced back to C. L. SiegePs paper Die Modulgruppe in einer einfachen involutorischen Algebra (30). A special case of these investigations is considered and contin- ued by the examination of the Riemannian geometry as well as the attached Eisenstein-series. To be more precise, throughout this paper let F stand for R, C or H, where H is the skew-field of real Hamiltonian quaternions. Just as in (16) let r = r(F) = dim RF and denote the standard basis of F over R by 1 = β\,..., er. Given = £); =1 Ujβj e F, aj e R, put Re(ά) := a and let *-+ = 2 Re(#) - denote the canonical conjugation in F. Then A^n resp. A e Mat(w F), means that A is an n x n matrix with entries in F and A! denotes the transpose of A. The letter / is reserved for the identity matrix and 0 for the zero matrix of appropriate size. GL(fl F) stands for the group of units in the ring Mat(rc F). The half-space &(n; F) consists of all Z e Mat(π; F) such that Z+z' becomes positive definite Hermitian matrix. Thus i^(n C) equals the Hermitian half-space, which was investigated by H. Braun (3), But the remaining cases are related, because X(n H) can always be embedded into the Hermitian half-space of degree In. The attached modified symplectic group MSρ(w F) consists of the automorphs of the symmetric matrix Q = (^Q), / = /W, having the