A numerical criterion for admissibility of semi-simple elements
type: Item
instance of: scholarly article
title: A numerical criterion for admissibility of semi-simple elements
zbMATH Open document ID: 0791.32009
author: Sampei Usui
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1994-04-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14D07
zbMATH DE Number: 549618
zbMATH Keywords: semi-simple elements
zbMATH Keywords: classifying spaces for Hodge structures
zbMATH Keywords: horizontal SL(2)-representations
zbMATH Keywords: limiting split mixed Hodge structures
zbMATH Keywords: admissibility
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225841
OpenAlex ID: W2055044493
cites work: Extension of period mappings for Hodge structures of weight two
cites work: Q3941676
cites work: Variation of Hodge structure: The singularities of the period mapping
cites work: Q4018079
DOI: 10.2748/TMJ/1178225841
description: scientific article; zbMATH DE number 549618