A square integrability criterion for the cohomology of arithmetic groups.

(1) Introduction.-For arithmetic subgroups r of algebraic semisimple groups defined over the rational numbers Q, and of Q-rank one, M. S. Raghunathan has given a criterion for all classes in certain cohomology groups associated to r to be representable by square integrable forms (cf. ref. 11). In the present note we announce a generalization of Raghunathan's criterion for arbitrary Q-rank (cf. Theorem 3.1, below). Using results of Andreotti and Vesentini, Raghunathan, and Weil (cf. refs. 1, 12, and 13) Theorem 3.1 can be used to obtain vanishing theorems for cohomology. In particular, one obtains the previously known result stated in Theorem 3.3. Theorem 3.1 follows from Theorem 4.8, which is our main tool. The complex EtQ&I which plays a role in Theorem 4.8 consists of forms which are well-behaved at a.* Our hope is that one can define a similar complex for certain holomorphic cohomology, prove an analogue of Theorem 4.8 in the holomorphic case, and using the special nature of the forms of these complexes at a, develop a Hodge theory. The goal would be to extend the results of Matsushima, and Matsushima and Murakami, to certain noncompact manifolds associated with arithmetic groups (cf. refs. 6-9). (2) Notation.-We will denote linear algebraic groups over the complex numbers C by capital boldface Latin letters. R will denote the real numbers. We then use the following notation scheme: If H denotes a linear algebraic group, then H will denote the R-rational points of H, and for F C R a subring, HF will denote the F-rational points of H. H' will denote the Zariski identity component of H. The Lie algebra of a real group will be denoted by the corresponding capital German letter, while the Lie algebra of an algebraic group will be denoted by the corresponding German letter with a bar over it. Thus, ! will denote the Lie algebra of H, and 6 will denote the Lie algebra of H. We also make the convention that the Lie algebra is the tangent space of the group, at the identity. We assume that G is Zariski connected, semisimple, and defined over Q, and we let r c GQ denote an arithmetic subgroup. We then fix a Cartan involution i,6 a maximal Q-split torus QS, a maximal R-split torus RS, and a maximal torus T, so that if Q3 = {X e @|I+*(X) = -X} (where AP* is the differential of y6 at the identity e), then