Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space
[Enveloppes d’holomorphie invariantes dans la complexification d’un espace symétrique hermitien]
Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 143-174.

Cet article est consacré à l’étude des domaines invariants dans Ξ + , un domaine de Stein particulier dans la complexification d’un espace symétrique Hermitien irréductible G/K. Le domaine Ξ + , introduit récemment par Krötz et Opdam, contient la couronne Ξ et il est maximal en ce qui concerne la propreté de l’action de G. Dans le cas tubulaire, Ξ + contient aussi S + , un domaine de Stein invariant lié à la structure causale d’une orbite symétrique dans le bord de Ξ.

On demontre que l’enveloppe d’holomorphie d’un domaine invariant dans Ξ + , non contenu ni dans Ξ ni dans S + , est univalent et coincide avec Ξ + . Ce fait, en combination avec des résultats connus pour Ξ et S + , démontre l’univalence de l’enveloppe d’holomorphie d’un domaine arbitraire dans Ξ + et complète la classification des domains de Stein invariants dans Ξ + .

In this paper we investigate invariant domains in Ξ + , a distinguished G-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space G/K. The domain Ξ + , recently introduced by Krötz and Opdam, contains the crown domain Ξ and it is maximal with respect to properness of the G-action. In the tube case, it also contains S + , an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of Ξ. We prove that the envelope of holomorphy of an invariant domain in Ξ + , which is contained neither in Ξ nor in S + , is univalent and coincides with Ξ + . This fact, together with known results concerning Ξ and S + , proves the univalence of the envelope of holomorphy of an arbitrary invariant domain in Ξ + and completes the classification of invariant Stein domains therein.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3008
Classification : 32D10, 32M15, 32Q28
Keywords: Hermitian symmetric space, Lie group complexification, envelope of holomorphy, invariant Stein domain
Mot clés : Espace symétrique hermitien, complexification, enveloppe d’holomorphie, domaine de Stein invariant
Geatti, Laura 1 ; Iannuzzi, Andrea 1

1 Università di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133 Roma (Italy)
@article{AIF_2016__66_1_143_0,
     author = {Geatti, Laura and Iannuzzi, Andrea},
     title = {Invariant envelopes of holomorphy in the complexification of a {Hermitian} symmetric space},
     journal = {Annales de l'Institut Fourier},
     pages = {143--174},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     doi = {10.5802/aif.3008},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.3008/}
}
TY  - JOUR
AU  - Geatti, Laura
AU  - Iannuzzi, Andrea
TI  - Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 143
EP  - 174
VL  - 66
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.3008/
DO  - 10.5802/aif.3008
LA  - en
ID  - AIF_2016__66_1_143_0
ER  - 
%0 Journal Article
%A Geatti, Laura
%A Iannuzzi, Andrea
%T Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space
%J Annales de l'Institut Fourier
%D 2016
%P 143-174
%V 66
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.3008/
%R 10.5802/aif.3008
%G en
%F AIF_2016__66_1_143_0
Geatti, Laura; Iannuzzi, Andrea. Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space. Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 143-174. doi : 10.5802/aif.3008. http://www.numdam.org/articles/10.5802/aif.3008/

[1] Akhiezer, D. N.; Gindikin, S. G. On Stein extensions of real symmetric spaces, Math. Ann., Volume 286 (1990) no. 1-3, pp. 1-12 | DOI

[2] Bröcker, Theodor; tom Dieck, Tammo Representations of compact Lie groups, Graduate Texts in Mathematics, 98, Springer-Verlag, New York, 1985, x+313 pages | DOI

[3] Docquier, Ferdinand; Grauert, Hans Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., Volume 140 (1960), pp. 94-123

[4] Geatti, L.; Iannuzzi, A. Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space., Math. Z., Volume 278 (2014) no. 3-4, pp. 769-793 | DOI

[5] Geatti, Laura Invariant domains in the complexification of a noncompact Riemannian symmetric space, J. Algebra, Volume 251 (2002) no. 2, pp. 619-685 | DOI

[6] Geatti, Laura; Iannuzzi, Andrea Univalence of equivariant Riemann domains over the complexifications of rank-one Riemannian symmetric spaces, Pacific J. Math., Volume 238 (2008) no. 2, pp. 275-330 | DOI

[7] Gindikin, Simon; Krötz, Bernhard Invariant Stein domains in Stein symmetric spaces and a nonlinear complex convexity theorem, Int. Math. Res. Not. (2002) no. 18, pp. 959-971 | DOI

[8] Gunning, Robert C. Introduction to holomorphic functions of several variables. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990, xx+203 pages (Function theory)

[9] Hörmander, Lars An introduction to complex analysis in several variables, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973, x+213 pages (North-Holland Mathematical Library, Vol. 7)

[10] Knapp, Anthony W. Lie groups beyond an introduction, Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002, xviii+812 pages

[11] Krötz, Bernhard Domains of holomorphy for irreducible unitary representations of simple Lie groups, Invent. Math., Volume 172 (2008) no. 2, pp. 277-288 | DOI

[12] Krötz, Bernhard; Opdam, Eric Analysis on the crown domain, Geom. Funct. Anal., Volume 18 (2008) no. 4, pp. 1326-1421 | DOI

[13] Krötz, Bernhard; Stanton, Robert J. Holomorphic extensions of representations. I. Automorphic functions, Ann. of Math. (2), Volume 159 (2004) no. 2, pp. 641-724 | DOI

[14] Krötz, Bernhard; Stanton, Robert J. Holomorphic extensions of representations. II. Geometry and harmonic analysis, Geom. Funct. Anal., Volume 15 (2005) no. 1, pp. 190-245 | DOI

[15] Neeb, Karl-Hermann On the complex and convex geometry of Ol ' shanskiĭ semigroups, Ann. Inst. Fourier (Grenoble), Volume 48 (1998) no. 1, pp. 149-203

[16] Neeb, Karl-Hermann On the complex geometry of invariant domains in complexified symmetric spaces, Ann. Inst. Fourier (Grenoble), Volume 49 (1999) no. 1, pp. vi, x, 177-225

[17] Rosenlicht, Maxwell On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc., Volume 101 (1961), pp. 211-223

[18] Rossi, Hugo On envelopes of holomorphy, Comm. Pure Appl. Math., Volume 16 (1963), pp. 9-17

Cité par Sources :