A numerical criterion for admissibility of semi-simple elements (Q1319188)

We generalize a theorem of \textit{E. H. Cattani} and \textit{A. Kaplan} [Duke Math. J. 44, 1-43 (1977; Zbl 0363.32018)] on horizontal representation of SL(2). This theorem plays an important role in the construction of partial compactification of the classifying spaces \(D\) modulo an arithmetic subgroup of Hodge structures of weight 2. A horizontal SL(2)-representation is a generalization of the notion of ``\((H_ 1)\)-homomorphism'' of SL(2) in the case of the classical theory of Hermitian symmetric domains. More precisely, let \(G\) be the automorphism group of the classifying space \(D\) of Hodge structures of weight \(w\). A representation \(\rho:\text{SL} (2,\mathbb{R}) \to G\) is said to be horizontal at \(r \in D\) if the corresponding morphism of their Lie algebras is a morphism of Hodge structures of type (0,0) with respect to the Hodge structures on the Lie algebras induced by \(i \in\) (upper-half plane) and \(r \in D\) respectively. In this case, the pair \((\rho,r)\) is uniquely determined by the pair \((Y,r) \in \text{Lie} G \times D\) with \(Y:=\rho_ *(y)\), where \(y:=\Bigl( {1\atop 0} {0 \atop -1} \Bigr)\). Conversely, a pair \((Y,r) \in \text{Lie} G \times D\) is said to be admissible if there exists a representation \(\rho:\text{SL} (2,\mathbb{R}) \to G\) horizontal at \(r\) and satisfying \(Y=\rho_ *(y)\). The main result in the present article is a numerical criterion for admissibility of a pair \((Y,r)\) in the case of general weight.