Geometric Cycles in Compact Riemannian Locally Symmetric Spaces of Type IV and Automorphic Representations of Complex Simple Lie Groups

P. Paul

Journal of Lie theory, Tome 30 (2020) no. 3, pp. 851-908

Résumé

Let $G$ be a connected complex simple Lie group with maximal compact subgroup $U$. Let $\frak{g}$ 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 $\frak{g} \neq \frak{b}_n$ $(n \ge 1)$, then for each $1 \le i \le 3$, 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 $\Gamma \backslash X$ has some non-vanishing (in the cohomology level) geometric cycles, and the Poincar\'{e} 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 $\frak{g} = \frak{\delta}_n$ $(n >4)$, $\frak{c}_n$ $(n \ge 6)$, or $\frak{f}_4$. To prove these, we have simplified Ka\v{c}'s description of finite order automorphisms of $\frak{g}$ with respect to a Chevalley basis of $\frak{g}$. Also we have determined some orientation preserving group action on some subsymmetric spaces of $X$.

Détail

Classification : 22E40, 22E46, 22E15, 17B10, 17B40, 57S15
Mots-clés : Arithmetic lattice, automorphism of finite order of Lie algebra, orientation preserving isometry, geometric cycle, automorphic representation