SATAKE-FURSTENBERG COMPACTIFICATIONS, THE MOMENT MAP AND λ1

Let G be a complex semisimple Lie group, K a maximal compact subgroup and τ an irreducible representation of K on V. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure γ on M we construct a map Ψ γ from the Satake compactification of G/K (associat...

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Veröffentlicht in:American journal of mathematics 2013-02, Vol.135 (1), p.237-274
Hauptverfasser: Biliotti, Leonardo, Ghigi, Alessandro
Format: Artikel
Sprache:eng
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Zusammenfassung:Let G be a complex semisimple Lie group, K a maximal compact subgroup and τ an irreducible representation of K on V. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure γ on M we construct a map Ψ γ from the Satake compactification of G/K (associated to V) to the Lie algebra of K. If γ is the K-invariant measure, then Ψ γ is a homeomorphism of the Satake compactification onto the convex envelope of O. For a large class of measures the image of Ψ γ is the convex envelope. As an application we get sharp upper bounds for the first eigenvalue of the Laplacian on functions for an arbitrary Kähler metric on a Hermitian symmetric space.
ISSN:0002-9327
1080-6377
DOI:10.1353/ajm.2013.0006