No Access

Invariant trilinear forms on spherical principal series of real rank one semisimple Lie groups

Salem Ben Said

Université de Lorraine–Nancy, Institut Elie Cartan, B.P. 239, 54506 Vandoeuvre-Les-Nancy, Cedex, France

Search for more papers by this author

Khalid Koufany

Université de Lorraine–Nancy, Institut Elie Cartan, B.P. 239, 54506 Vandoeuvre-Les-Nancy, Cedex, France

Search for more papers by this author

, and

Genkai Zhang

Department of Mathematics, Chalmers University of Technology and Gothenburg University, 412 96 Gothenburg, Sweden

Search for more papers by this author

https://doi.org/10.1142/S0129167X14500177Cited by:5 (Source: Crossref)

Abstract

Let G be a connected semisimple real-rank one Lie group with finite center. We consider intertwining operators on tensor products of spherical principal series representations of G. This allows us to construct an invariant trilinear form indexed by a complex multiparameter and defined on the space of smooth functions on the product of three spheres in 𝔽ⁿ, where 𝔽 is either ℝ, ℂ, ℍ, or 𝕆 with n = 2. We then study the analytic continuation of the trilinear form with respect to (ν₁, ν₂, ν₃), where we locate the hyperplanes containing the poles. Using a result due to Johnson and Wallach on the so-called "partial intertwining operator", we obtain an expression for the generalized Bernstein–Reznikov integral in terms of hypergeometric functions.

Keywords:

AMSC: Primary: 43A85, Primary: 22E45, Secondary: 33A75, Secondary: 22E25

References

R. Askey and J. Fitch, J. Math. Anal. Appl. 26, 411 (1969), DOI: 10.1016/0022-247X(69)90165-6. Web of Science, Google Scholar
W. N. Bailey , Generalized Hypergeometric Series ( Cambridge University Press , Cambridge , 1935 ) . Google Scholar
R. Beckmann and J.-L. Clerc, J. Funct. Anal. 262, 4341 (2012), DOI: 10.1016/j.jfa.2012.02.021. Web of Science, Google Scholar
J. Berndt , F. Tricerri and L. Vanhecke , Generalized Heisenberg Groups and Damek–Ricci Harmonic Spaces , Lecture Notes in Mathematics 1598 ( Springer-Verlag , Berlin , 1995 ) . Google Scholar
J. Bernstein and A. Reznikov, Ann. Math. 150, 329 (1999), DOI: 10.2307/121105. Web of Science, Google Scholar
J. Bernstein and A. Reznikov, Mosc. Math. J. 4, 19 (2004). Web of Science, Google Scholar
N. Bourbaki , Groupes et Algèbres de Lie ( Hermann , Paris , 1968 ) . Google Scholar
J.-L. Clercet al., Math. Ann. 349, 395 (2011), DOI: 10.1007/s00208-010-0516-4. Web of Science, Google Scholar
J.-L. Clerc and B. Ørsted, Ann. Inst. Fourier 61, 1807 (2011), DOI: 10.5802/aif.2659. Web of Science, Google Scholar
M. Cowling and A. Korányi, Harmonic Analysis on Heisenberg Type Groups from a Geometric Viewpoint, Lecture Notes in Mathematics, Vol. 1077 (Springer, Berlin, 1984), pp. 60–100 . Google Scholar
A. Deitmar, Int. J. Math. Math. Sci. 2006, 22 (2006), DOI: 10.1155/IJMMS/2006/48274. Google Scholar
A. Erdélyi et al. , Higher Transcendental Functions I ( McGraw-Hill Book Company , New York , 1953 ) . Google Scholar
G. B. Folland and E. M. Stein , Hardy Spaces on Homogeneous Groups , Mathematical Notes 28 ( Princeton University Press , Princeton, NJ , 1982 ) . Google Scholar
I. M. Gel'fand and G. E. Shilov, Generalized Functions: Properties and Operations 1 (Academic Press, New York, 1964) p. xviii+423. Google Scholar
S. Helgason , The Groups and Geometric Analysis , Mathematical Surveys and Monographs 39 ( American Mathematical Society , Providence, RI , 1994 ) . Google Scholar
L. Hörmander , The Analysis of Linear Partial Differential Operators I ( Springer-Verlag , Berlin , 1990 ) . Google Scholar
K. D. Johnson, Trans. Amer. Math. Soc. 215, 269 (1976), DOI: 10.1090/S0002-9947-1976-0385012-X. Web of Science, Google Scholar
K. D. Johnson and N. R. Wallach, Trans. Amer. Math. Soc. 229, 137 (1977), DOI: 10.1090/S0002-9947-1977-0447483-0. Web of Science, Google Scholar
A. W. Knapp , Representation Theory of Semisimple Groups. An Overview Based on Examples , Princeton Mathematical Series 36 ( Princeton University Press , Princeton, NJ , 1986 ) . Google Scholar
A. W. Knapp and E. M. Stein, Ann. of Math. 93, 489 (1971), DOI: 10.2307/1970887. Web of Science, Google Scholar
T. Kobayashi, F-method for symmetry breaking operators, Differential Geom. Appl., Special issue in honor of M.Eastwood, doi:10.1016/j.difgeo.2013.10.003 , arXiv:1303.3545 . Google Scholar
T. Kobayashi and T. Oshima, Adv. Math. 248, 921 (2013), DOI: 10.1016/j.aim.2013.07.015. Web of Science, Google Scholar
A. Korányi, Adv. Math. 56, 28 (1985). Web of Science, Google Scholar
B. Kostant, Bull. Amer. Math. Soc. 75, 627 (1969), DOI: 10.1090/S0002-9904-1969-12235-4. Web of Science, Google Scholar
S. D. Miller and W. Schmid, Representation Theory and Automorphic Forms, Progress in Mathematics 255 (Birkhäuser, Boston, MA, 2008) pp. 111–150. Google Scholar
A. I. Osak, Comm. Math. Phys. 29, 189 (1973). Web of Science, Google Scholar
J. Repka, Amer. J. Math. 100, 747 (1978), DOI: 10.2307/2373909. Web of Science, Google Scholar
C. Sabbah, Polynômes de Bernstein-Sato à plusieurs variables, in Séminaire sur les Équations aux Dérivées Partielles, 1986–1987, Exp. No. XIX (École Polytechnique, Palaiseau 1987), 6 pp . Google Scholar
H. M. Srivastava and P. W. Karlsson , Multiple Gaussian Hypergeometric Series , Ellis Horwood Series in Mathematics and Its Applications ( Ellis Horwood , Chichester , 1985 ) . Google Scholar
R. Takahashi, Analyse Harmonique sur les Groupes de Lie. II, Lecture Notes in Mathematics 739 (Springer, Berlin, 1979) pp. 511–567. Google Scholar
A. Unterberger , Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi , Progress in Mathematics 209 ( Birkhäuser-Verlag , Basel , 2003 ) . Google Scholar
A. Unterberger , Quantization and Arithmetic, Pseudo-Differential Operators , Theory and Applications 1 ( Birkhäuser-Verlag , Basel , 2008 ) . Google Scholar