Cycle space constructions for exhaustions of flag domains
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 573-580.

A real semisimple group has only finitely many orbits on every flag manifold of its complexification. To each of these orbits there is a naturally associated space of algebraic cycles, and that cycle space is known to be a Stein manifold. In the past, properties of the cycle space have been proved by transforming functions or cohomology from, e.g., an open orbit in the flag manifold to its cycle space. Here the opposite is done: given an irreducible representation of a maximal compact subgroup of the real semisimple group, a canonical strictly plurisubharmonic exhaustion of the cycle space is constructed. This is then transformed to a (continuous) q-pseudoconvex exhaustion of the associated open orbit, where q is the complex dimension of the cycles under consideration.

Classification : 32M05, 32F10, 32M10, 22E46
Huckleberry, Alan 1 ; Wolf, Joseph 2

1 Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
2 Department of Mathematics, University of California, Berkeley, California, 94720-3840, U.S.A.
@article{ASNSP_2010_5_9_3_573_0,
     author = {Huckleberry, Alan and Wolf, Joseph},
     title = {Cycle space constructions for exhaustions of flag domains},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {573--580},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {3},
     year = {2010},
     mrnumber = {2722656},
     zbl = {1209.32019},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2010_5_9_3_573_0/}
}
TY  - JOUR
AU  - Huckleberry, Alan
AU  - Wolf, Joseph
TI  - Cycle space constructions for exhaustions of flag domains
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2010
SP  - 573
EP  - 580
VL  - 9
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2010_5_9_3_573_0/
LA  - en
ID  - ASNSP_2010_5_9_3_573_0
ER  - 
%0 Journal Article
%A Huckleberry, Alan
%A Wolf, Joseph
%T Cycle space constructions for exhaustions of flag domains
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2010
%P 573-580
%V 9
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2010_5_9_3_573_0/
%G en
%F ASNSP_2010_5_9_3_573_0
Huckleberry, Alan; Wolf, Joseph. Cycle space constructions for exhaustions of flag domains. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 573-580. http://archive.numdam.org/item/ASNSP_2010_5_9_3_573_0/

[1] A. Andreotti and H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math France 90 (1962), 193–259. | EuDML | Numdam | MR | Zbl

[2] D. Barlet, Convexité de l’espace des cycles, Bull. Soc. Math. France 106 (1978), 373–397. | EuDML | Numdam | MR | Zbl

[3] M. G. Eastwood and G. V. Souria, Cohomologically complete and pseudoconvex domains, Comment. Math. Helv. 55 (1980), 413–426. | EuDML | MR | Zbl

[4] G. Fels, A. T. Huckleberry and J. A. Wolf, “Cycle Spaces of Flag Domains: A Complex Geometric Viewpoint”, Progress in Mathematics, Vol. 245, Birkhäuser, Boston, 2006. | MR | Zbl

[5] J. Hong and A. T. Huckleberry, On closures of cycle spaces of flag domains, Manuscripta Math. 121 (2006), 317–327. | MR | Zbl

[6] B. Krötz and R. Stanton, Holomorphic extension of representations, I. Automorphic functions, Ann. of Math. 159 (2004), 641–724. | MR | Zbl

[7] B. Krötz and R. Stanton, Holomorphic extension of representations, II. Geometry and harmonic analysis, Geom. Funct. Anal. 15 (2005), 190–245. | MR | Zbl

[8] W. Schmid and J. A. Wolf, A vanishing theorem for open orbits on complex flag manifolds, Proc. Amer. Math. Soc. 92 (1984), 461–464. | MR | Zbl

[9] R. O. Wells, Jr., and J. A. Wolf, Poincaré series and automorphic cohomology on flag domains, Ann. of Math. 105 (1977), 397–448. | MR | Zbl

[10] J. A. Wolf, The action of a real semisimple Lie group on a complex manifold, I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. | MR | Zbl

[11] J. A. Wolf, The Stein condition for cycle spaces of open orbits on complex flag manifolds, Ann. of Math. 136 (1992), 541–555. | MR | Zbl

[12] J. A. Wolf, Exhaustion functions and cohomology vanishing theorems for open orbits on complex flag manifolds, Math. Res. Lett. 2 (1995), 179–191. | MR | Zbl