Picard-Lefschetz theory and characters of a semisimple Lie group
type: Item
instance of: scholarly article
title: Picard-Lefschetz theory and characters of a semisimple Lie group
zbMATH Open document ID: 0851.22013
DOI: 10.1007/BF01884312
published in: Inventiones Mathematicae
publication date: 1996-11-11T00:00:00Z
full work available at URL: https://eudml.org/doc/144311
review text: Let \(\mathfrak g\) be a complex semisimple Lie algebra, \(\mathfrak h\) a Cartan subalgebra and \(W\) the Weyl group. Let \(q : {\mathfrak g}^* \mapsto {\mathfrak h}^*/W\) be the quotient map onto the coadjoint orbits. For a regular \(\lambda \in {\mathfrak h}^*\) let \(p_\lambda : \Omega \to \Omega_\lambda\) be a homeomorphism from a standard fibre of \(q\) onto the fibre over \(\lambda\) and \(p_0 : \Omega \mapsto {\mathcal N}\) its limit map into the nilpotent variety \(\mathcal N\). Given a real subalgebra \({\mathfrak g}_0\) and its Lie group \(G_0\) denote by \(\mathcal S\) the inverse image of \(\mathcal N \cap {\mathfrak g}^*_0\) under \(p_0\). Let \(s = s_\alpha\) in \(W\) be a simple reflection and let \(\lambda \in {\mathfrak h}^*\) be orthonormal to \(\alpha\) and no other simple roots. The author proves that the monodromy representation of \(s\) in the top homology \(H_{2n}({\mathcal S})\) is a reflection along the subspace \(H_{2n} ({\mathcal S}_0)\), where \({\mathcal S}_0\) is the inverse image under \(p_0\) of the orbit \(G_0 \cdot \lambda\). The author also gives a graded filtration of \(H_{2n} ({\mathcal S})\) in terms of the homologies of inverse images under \(p_0\) of nilpotent orbits of \(G_0\). In the second part the author studies the character of representations of the Lie group \(G_0\) with semisimple Lie algebra \({\mathfrak g}_0\). Let \(\pi\) be an admissible representation of \(G_0\). It has been proved by the author [In: A. Connes et al. (eds), Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honor of J. Dixmier, Paris 1989, Prog. Math. 92, 263-287 (1990; Zbl 0744.22012)] that the character of \(\pi\) on \({\mathfrak g}_0\) is an integration \(\Theta(\Gamma, \lambda)\) of the exponential function over \(p_\lambda \Gamma\) for some \(\lambda \in {\mathfrak h}^*\) and a cycle \(\Gamma\) in \(H_{2n}({\mathcal S})\). The author proves that the wave front set of the representation \(\…
reviewed by: Genkai Zhang
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 14D05
zbMATH DE Number: 871785
zbMATH Keywords: Picard-Lefschetz theory
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: Cartan subalgebra
zbMATH Keywords: Weyl group
zbMATH Keywords: coadjoint orbits
zbMATH Keywords: Lie group
zbMATH Keywords: monodromy representation
zbMATH Keywords: homology
zbMATH Keywords: wave front set
zbMATH Keywords: Fourier transform
Wikidata QID: Q115393097
author: Wulf Rossmann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 871785