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Journal of Lie Theory 30 (2020), No. 3, 851--908
Copyright Heldermann Verlag 2020

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

Pampa Paul
Dept. of Mathematics, Presidency University, Kolkata 700073, India


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 $X$.

Keywords: Arithmetic lattice, automorphism of finite order of Lie algebra, orientation preserving isometry, geometric cycle, automorphic representation.

MSC: 22E40, 22E46, 22E15, 17B10, 17B40, 57S15.

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