On the complex geometry of invariant domains in complexified symmetric spaces

Neeb, Karl-Hermann

doi:10.5802/aif.1671

Cited by 19 publications

(14 citation statements)

References 7 publications

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“…In the two-dimensional case, the domains S 1 (0) and S 2 (0) are related to the causal structure of the symmetric space G/H = SO 0 (2, 1)/SO(1, 1). Domains of this type were studied in [Neeb 1999]. …”

Section: Univalence On G-orbits In Gmentioning

confidence: 99%

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Univalence of equivariant Riemann domains over the complexifications of rank-one Riemannian symmetric spaces

Geatti¹,

Iannuzzi²

2008

Pacific J. Math.

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Let G/K be a noncompact, rank-one, Riemannian symmetric space, and let G ‫ރ‬ be the universal complexification of G. We prove that a holomorphically separable, G-equivariant Riemann domain over G ‫ރ‬ /K ‫ރ‬ is necessarily univalent, provided that G is not a covering of SL(2, ‫.)ޒ‬ As a consequence, one obtains a univalence result for holomorphically separable, G×K -equivariant Riemann domains over G ‫ރ‬ . Here G×K acts on G ‫ރ‬ by left and right translations. The proof of such results involves a detailed study of the G-invariant complex geometry of the quotient G ‫ރ‬ /K ‫ރ‬ , including a complete classification of all its Stein G-invariant subdomains.

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Section: Univalence On G-orbits In Gmentioning

confidence: 99%

“…As a result, the above map exchanges the singular orbits G · z 1 and G · z 3 and maps G · 1 (a) onto G · 3 (a), for 0 < a < 1. When G = SO 0 (2, 1), the domains S 1 (0) and S 2 (0) and their subdomains S 1 (b) and S 2 (b) for 0 < b < ∞ were shown to be Stein in [Neeb 1999]. …”

Section: Univalence On G-orbits In Gmentioning

confidence: 99%

Univalence of equivariant Riemann domains over the complexifications of rank-one Riemannian symmetric spaces

Geatti¹,

Iannuzzi²

2008

Pacific J. Math.

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“…Earlier examples of this result are due to Azad and Loeb [3] for compact symmetric spaces, and K.-H. Neeb [18] for certian non-degenerate semigroups.…”

Section: Steinmentioning

confidence: 99%

The geometry of Grauert tubes and complexification of symmetric spaces

Burns¹,

Halverscheid

Hind

2003

Duke Math. J.

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We study the canonical complexifications of non-compact Riemannian symmetric spaces by the Grauert tube construction. We determine the maximal such complexification, a domain already constructed by Akhiezer and Gindikin [1], and show that this domain is Stein. We also determine when invariant complexifications, including the maximal one, are Hermitian symmetric. This is expressed simply in terms of the ranks of the symmetric spaces involved.

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“…The domain S + is G-equivariantly biholomorphic to an invariant domain in the Lie group complexification of the symmetric space G/H and its Steiness follows from a result of Neeb [16]. Here we show that it is contained in + by proving the following identity (Proposition 7.5)…”

mentioning

confidence: 85%

“…This shows that inside thej th subdomain of r defined in (16), the element gj induces the reflection with respect to thej th-coordinate hyperplane.…”

Section: Remark 66mentioning

confidence: 99%

Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space

Geatti

Iannuzzi

2014

Math. Z.

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We carry out a detailed study of + , a distinguished G-invariant Stein domain in the complexification of an irreducible Hermitian symmetric space G/K . The domain + contains the crown domain and is naturally diffeomorphic to the anti-holomorphic tangent bundle of G/K . The unipotent parametrization of + introduced in Krötz and Opdam (GAFA Geom Funct Anal 18:1326-1421, 2008 and Krötz (Invent Math 172:277-288, 2008) suggests that + also admits the structure of a twisted bundle G × K N + , with fiber a nilpotent cone N + . Here we give a complete proof of this fact and use it to describe the G-orbit structure of + via the K -orbit structure of N + . In the tube case, we also single out a Stein, G-invariant domain contained in + \ which is relevant in the classification of envelopes of holomorphy of invariant subdomains of + .

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On the complex geometry of invariant domains in complexified symmetric spaces

Cited by 19 publications

References 7 publications

Univalence of equivariant Riemann domains over the complexifications of rank-one Riemannian symmetric spaces

Univalence of equivariant Riemann domains over the complexifications of rank-one Riemannian symmetric spaces

The geometry of Grauert tubes and complexification of symmetric spaces

Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space

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