Square integrable automorphic forms and cohomology of arithmetic quotients of \(SU(p,q)\) (Q789430)

Let G be a semi-simple algebraic group defined over \({\mathbb{Q}}\). Let \(\Gamma \subset G_{{\mathbb{Q}}}\) be an arithmetic subgroup, \(G=G_{{\mathbb{R}}}\) (the real points of G). In addition let \(A(\Gamma\),G) denote the space of automorphic forms on \(\Gamma\backslash G\). Then \(A(\Gamma\),G) is a (\({\mathfrak g},K)\)-module under right translation. Here \(K\subset G\) is a maximal compact subgroup, \({\mathfrak g}\) is the complexified Lie algebra of G. In the paper under review the author shows how the internal structure of the (\({\mathfrak g},K)\)-module generated by \(f\in A(\Gamma,G)\) gives information on the growth of f on Siegel sets. For a particular class of automorphic forms and for \(G=SU(p,q)\) the author receives the following results. 1) If \(p+q\geq 3\), then there exist subgroups \(\Gamma\) of SU(p,q)(\({\mathbb{Z}}[i])\) of finite index such that dim \(H^ q(\Gamma,{\mathbb{C}})\) can be made arbitrarily large. 2) If \(q=1\), \(p\geq 3\) there are non-tempered representations that occur in the space of cusp forms on \(\Gamma\backslash G\) for \(\Gamma\) a congruence subgroup of SU(P,1)(\({\mathbb{Z}}[i])\).