On Compactifications of the Quotient Spaces for Arithmetically Defined Discontinuous Groups

Abstract

In my previous papers [2], [4], I have given a method of compactifying the quotient spaces of the generalized upper half plane with respect to Siegel's modular and, more generally, to any commensurable with Siegel's modular group. Now a similar problem can be considered in a more general situation as follows. Let G be a semi-simple linear algebraic defined over Q, and let G4, GR be the groups consisting of points in G rational over Q, R, respectively. GR is a semisimple Lie with a finite number of connected components. Let K be a maximal compact subgroup of GR and let S = K\GR be the associated (not necessarily connected) symmetric space. Let furthermore Gz be the group of units in G, i.e., the consisting of all elements in G whose coefficients (together with those of its inverse) are rational integers. Then Gz is a discrete subgroup of GR, whose commensurable class is uniquely determined, independently of the choice of the matrix expression of G. Let F be any subgroup of Gq commensurable with Gz. Then one may ask the possibility of constructing a reasonable compactification of the quotient space S/r. To approach this problem, it will be convenient to use a for F, which has been constructed by Weil [7], by means of the reduction theory, in the case where G is a of automorphisms of a semi-simple associative algebra over Q with or without involution. Moreover he has also shown in [6] that these cases cover all semi-simple linear algebraic groups without center, of classical type, over Q, with few exceptions. The purpose of this paper is to give some results on the above problem in the case treated by Weil [7]. The outline of the paper is as follows. We shall recall in ? 1 the main result of our previous paper [3], on which our whole construction will be based, concerning a general method of compactification of a symmetric space S by means of an irreducible, faithful projective representation p of the corresponding Lie group; we give also a trivial generalization of it to the non-connected case. In ? 2 we shall give a general condition (the condition (D)) for a discontinuous r operating on S and for a fundamental set f2 for F, which enables us to construct a suitable compacti-