On the Compactness of Arithmetically Defined Homogeneous Spaces

Abstract

Let G be an algebraic matric group defined over the field Q of rational numbers. Let GR denote the subgroup of elements in G with real coefficients, and let G, denote the subgroup of elements in G with rational integer coefficients and determinant + 1. The main theorem of Chapter I gives a necessary and sufficient condition for GRIGZ to be compact: that G have no non-constant rational characters over Q and no unipotent elements outside of its radical. This theorem generalizes results of C. L. Siegel and A. Weil on the compactness of certain arithmetically defined homogeneous spaces related to quadratic forms. In Chapter II, we generalize this theorem to the adelization GA of G, thereby obtaining a strengthened version of the conjecture of R. Godement: if G is a semi-simple group defined over Q, then GAIGQ is compact if and only if GQ has no unipotent elements other than 1. We have been informed that A. Borel and Harish-Chandra have also proved the main theorem of Chapter I, and that A. Borel has extended their joint result to the adele case, thereby proving the Godement conjecture also. The proof that we present in Chapter I has the merit of carrying over almost verbatim to the adele case in Chapter II, when one interprets a finite dimensional vector space VQ over Q as a lattice in VA, that is, a discrete subgroup of VA such that the quotient is compact. The proofs in Chapters I and II are thus parallel, but essentially independent. In both Chapter I and Chapter II, one proves the result first for the adjoint group G* of a fully reducible group G, and then one lifts the result to G. The proofs for the compactness of G*IG* and GI/G' are identical. However, the lifting argument is entirely different in the two cases (Chapter I, ? 3 and II, ? 3). The main result in Chapter II asserts: For any algebraic matric group G defined over Q, G' /GQ is compact if and only if GQ has no unipotent elements outside of its radical. This result implies that if moreover G has no nonconstant rational characters over Q, then GR/GZ is compact and the