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

TL;DR

This paper constructs specific arithmetic lattices in complex simple Lie groups of type IV, demonstrating the existence of non-trivial geometric cycles in associated symmetric spaces and linking these to automorphic representations.

Contribution

It introduces new arithmetic lattices in complex simple Lie groups and connects geometric cycles in symmetric spaces to automorphic representations, simplifying automorphism descriptions.

Findings

Existence of non-vanishing geometric cycles in certain locally symmetric spaces.

Detection of specific automorphic representations based on geometric and cohomological properties.

Simplification of Kac's automorphism classification for complex simple Lie algebras.

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.