Geometric cycles in compact Riemannian locally symmetric spaces of type IV and automorphic representations of complex simple Lie groups

Abstract

Let G be a connected complex simple Lie group with maximal compact subgroup U. Let g be the Lie algebra of G, and X = G/U be the associated Riemannian globally symmetric space of type IV. We have constructed three types of arithmetic uniform lattices in G, say of type 1, type 2, and type 3 respectively. If g is not equal to b_n, n>0, then for each 0<i<4, there is an arithmetic uniform torsion-free lattice \Gamma in G which is commensurable with a lattice of type i such that the corresponding locally symmetric space \Gamma \ X has some non-vanishing (in the cohomology level) geometric cycles, and the Poincare duals of fundamental classes of such cycles are not represented by G-invariant differential forms on X. As a consequence, we are able to detect some automorphic representations of G when g = \delta_n (n >4), c_n (n > 5), or f_4. To prove these, we have simplified Kac's description of finite order automorphisms of g with respect to a Chevalley basis of g. Also we have determined some orientation preserving group action on some subsymmetric spaces of X.