Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Positive definite spherical functions
on Ol$'$shanskii domains

Authors: Joachim Hilgert and Karl-Hermann Neeb
Journal: Trans. Amer. Math. Soc. 352 (2000), 1345-1380
MSC (1991): Primary 22E46; Secondary 22A25
Published electronically: May 21, 1999
MathSciNet review: 1473443
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a simply connected complex Lie group with Lie algebra $\mathfrak{g}$, $\mathfrak{h}$ a real form of $\mathfrak{g}$, and $H$ the analytic subgroup of $G$ corresponding to $\mathfrak{h}$. The symmetric space ${\mathcal{M}}=H\backslash G$ together with a $G$-invariant partial order $\le $ is referred to as an Ol$'$shanskii space. In a previous paper we constructed a family of integral spherical functions $\phi _{\mu }$ on the positive domain ${\mathcal{M}}^{+} := \{Hx\colon Hx\ge H\}$ of ${\mathcal{M}}$. In this paper we determine all of those spherical functions on ${\mathcal{M}}^{+}$ which are positive definite in a certain sense.

References [Enhancements On Off] (What's this?)

  • [Ar50] N. Aronszajn, Theory of reproducing kernels, Transactions of the Amer. Math. Soc. 68 (1950), 337-404. MR 14:479c
  • [FHÓ94] J. Faraut, J. Hilgert, and G. Ólafsson, Spherical functions on ordered symmetric spaces, Ann. Inst. Fourier 44 (1994), 927-966. MR 96a:43012
  • [Hel84] S. Helgason, Groups and geometric analysis, Academic Press, Orlando (1984). MR 86c:22017
  • [HHL89] J. Hilgert, K. H. Hofmann, and J. D. Lawson, Lie Groups, Convex Cones, and Semigroups, Oxford University Press (1989). MR 91k:22020
  • [HiNe93] J. Hilgert and K. - H. Neeb, Lie semigroups and their applications, Springer Lecture Notes in Math. 1552 (1993). MR 96j:22002
  • [HiNe95] J. Hilgert and K. - H. Neeb, Compression semigroups of open orbits on complex manifolds, Arkiv för Mat. 33 (1995), 293-322. MR 97j:22013
  • [HiNe96] J. Hilgert and K. - H. Neeb, Spherical functions on Ol$'$shanski[??]i spaces, J. Funct. Anal. 142 (1996), 446-493. MR 98c:43015
  • [HÓ96] J. Hilgert and G. Ólafsson, G., Causal Symmetric Spaces, Geometry and Harmonic Analysis, Academic Press (1996). MR 97m:43006
  • [HÓØ91] J. Hilgert, G. Ólafsson, and B. Ørsted, Hardy Spaces on Affine Symmetric Spaces, J. reine angew. Math. 415 (1991), 189-218. MR 92h:22030
  • [KrNe96] B. Krötz and K.-H. Neeb, On Hyperbolic Cones and Mixed Symmetric Spaces, Journal of Lie Theory 6:1 (1996), 69-146. MR 97k:17007
  • [KNÓ96] B. Krötz, K.-H. Neeb, and G. Ólafsson, Spherical Representations and Mixed Symmetric Spaces, Representations Theory (1997), 424-461. (electronic) MR 99a:22031.
  • [Lo69] O. Loos, Symmetric Spaces I : General Theory, Benjamin, New York, Amsterdam (1969). MR 39:365a
  • [LM75] M. Lüscher and G. Mack, Global Conformal Invariance in Quantum Field Theory, Comm. Math. Phys. 41 (1975), 203-234. MR 51:7503
  • [Ne94a] K.-H. Neeb, The classification of Lie algebras with invariant cones, Journal of Lie Theory 4:2 (1994), 1-46. MR 96e:17013
  • [Ne94b] K.-H. Neeb, Holomorphic representation theory II, Acta Math. 173:1 (1994), 103-133. MR 96a:22025
  • [Ne94c] K.-H. Neeb, A convexity theorem for semisimple symmetric spaces, Pac. J. Math. 162:2 (1994), 305-349. MR 95b:22016
  • [Ne95a] K.-H. Neeb, Holomorphic representation theory I, Math. Ann. 301 (1995), 155-181. MR 96a:22024
  • [Ne95b] K.-H. Neeb, Square integrable highest weight representations, Glasgow Math. J. 39 (1987), 295-321. MR 99a:22021
  • [Ne96] K.-H. Neeb, Coherent states, holomorphic extensions, and highest weight representations, Pac. J. Math. 174:2 (1996), 497-542. MR 97k:22018
  • [Ne99] K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, de Gruyter, to appear.
  • [Nel70] E. Nelson, Analytic vectors, Ann. of Math. 70 (1959), 572-615. MR 21:5901
  • [Ols82] G. I. Ol$'$shanski[??]i, Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. and Appl. 15 (1982), 275-285. MR 83e:32032
  • [RS72] M. Reed and B. Simon, Methods of Mathematical Physics I: Functional Analysis, Academic Press, New York (1972). MR 58:12429a
  • [Wa72] G. Warner, Harmonic analysis on semi-simple Lie groups I, Springer, Berlin (1972). MR 58:16979

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 22E46, 22A25

Retrieve articles in all journals with MSC (1991): 22E46, 22A25

Additional Information

Joachim Hilgert
Affiliation: Mathematisches Institut, Technische Universität Clausthal, Erzstr. $1$, 38678 Claus- thal-Zellerfeld, Germany

Karl-Hermann Neeb
Affiliation: Mathematisches Institut, Universität Erlangen, Bismarckstr. $1\frac{1}2$, 91054 Erlangen, Germany
Address at time of publication: Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, Germany

Keywords: Positive definite function, ordered symmetric space, holomorphic representation, spherical function, involutive semigroup
Published electronically: May 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society