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

Pampa Paul¹

¹Dept. of Mathematics, Presidency University, Kolkata 700073, India

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