## Discrete group actions on Stein domains in complex Lie groups.(English)Zbl 1044.22007

The authors study certain non-compact Stein manifolds $$S$$ on which a real Lie group $$G$$ acts freely but not transitively and for a discrete subgroup $$\Gamma$$ of $$G$$ address the question, whether $$\Gamma\setminus S$$ is a Stein manifold. They succeed to answer the question for a special case, where $$\Gamma$$ is the modular group. On the other hand they show that for cocompact $$\Gamma$$ the bounded holomorphic functions separate the points of $$\Gamma\setminus S$$.
A key fact in their argument is that $$S$$ is a semigroup with a polar decomposition of the type $$S= G\times M$$ and that a large part of the holomorphic discrete series of $$G$$ admits a holomorphic extension to $$S$$. This allows them to construct Poincaré series which are $$\Gamma$$-invariant (hyperfunction) vector for certain holomorphic representations.

### MSC:

 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 22E40 Discrete subgroups of Lie groups 11F70 Representation-theoretic methods; automorphic representations over local and global fields 32D05 Domains of holomorphy
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### References:

 [1] Ach D., Math. Z. 230 pp 21– (1999) [2] BaOt W., Math. Ann. 201 pp 97– (1973) [3] Bo A., Topology 2 pp 111– (1963) [4] Bo, Proc. Sympos. Pure Math. pp 9– (1966) [5] [Bo69] : Introduction aux groupes arithmetiques. Hermann, Paris 1969 [6] [Fr83] Freitag, E.: Siegelsche Modulfunktionen. Grundlehren der Mathematischen Wissenschaften 254. Springer, 1983 [7] GiHu B., Math. Ann. 238 pp 39– (1978) [8] [Hel78] Helgason S.: Lie Groups, Di erential Geometry and Symmetric Spaces. Academic Press, London 1978 [9] HiKr J., Math. Z. 37 pp 31– (2001) [10] HiKr J., J. Funct. Anal. 169 pp 357– (1999) [11] HiKr, Math. Z. 37 pp 31– (2001) [12] [HiNe93] Hilgert J., and Neeb K.H.: Lie semigroups and their applications. Lecture Notes in Math. 1552. Springer, 1993 · Zbl 0807.22001 [13] [H 73] H rmander L.: An introduction to complex analysis in several variables. North-Holland, 1973 [14] HoMo R., J. Funct. Anal. 32 pp 72– (1979) [15] [Ko98] Kobayashi S.: Hyperbolic Complex Spaces. Grundlehren der Mathematischen Wissenschaften 318. Springer, 1998 [16] Kr B., Math. Ann. 312 pp 13– (1998) [17] Kr, Math. Z. 237 pp 505– (2001) [18] Kr, Publ. RIM 35 pp 91– (1999) [19] Kr, Comp. Math. 125 pp 155– (2001) [20] Oa B., Represent. Theory 1 pp 424– (1997) [21] Oa, Represent. Theory 5 pp 43– (2001) [22] Krtz B., J. Lie Theory 12 pp 409– (2002) [23] [Lo84] Loeb J.J.: Fonctions plurisousharmoniques sur un groupe de Lie complexe invariantes par une forme reelle. C. R. Acad. Sci. Paris Ser. I Math. 299 (1984), no. 14, 663-666 · Zbl 0616.31006 [24] MaMo Y., Bull. Soc. Math. France 88 pp 137– (1960) [25] Ma M., J. Funct. Anal. 143 pp 42– (1997) [26] Ne K.-H., Forum Math. 9 pp 613– (1997) [27] Ne, Ann. Inst. Fourier 48 pp 149– (1998) [28] [Ne99a] : Holomorphy and Convexity in Lie Theory. Expositions in Mathematics 28. de Gruyter, 1999 · Zbl 0936.22001 [29] Ne, Ann. Inst. Fourier 49 pp 177– (1999) [30] Ols G. I., Funct. Anal. and Appl. 15 pp 275– (1982) [31] [Ra72] Raghunathan M. S.: Discrete Subgroups of Lie Groups. Ergebnisse der Mathematik 68. Springer, 1972 [32] [Sh92] Shabat B. V.: Introduction to Complex Analysis, Part II: Functions of Several Variables. Amer. Math. Soc., Providence, Rhode Island 1992 [33] Sta R. J., Amer. J. Math. 108 pp 1411– (1986) [34] [WaWo93] Wallach N., and Wolf J. A.: Completeness of Poincare series for automorphic forms associated to the integrable discrete series. Representation theory of reductive groups (Park City, Utah, 1982), 265-281. Progr. Math. 40. Birkh user Boston, Boston, Mass. 1983 [35] [WeWo77] Wells R. O., and Wolf J. A.: Poincare series and automorphic cohomology on flag domains. Ann. of Math. (2) 105 (1977), no. 3, 397-448
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