TY - JOUR

T1 - Invariant trilinear forms on spherical principal series of real rank one semisimple lie groups

AU - Said, Salem Ben

AU - Koufany, Khalid

AU - Zhang, Genkai

N1 - Funding Information:
Research by G. Zhang partially supported by Swedish Research Council. This work started during the visit of G. Zhang to the Institute Élie Cartan at Université de Lorraine-Nancy as an invited professor. He would like to express his gratitude to the university for the support.

PY - 2014

Y1 - 2014

N2 - 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 K ν indexed by a complex multiparameter ν = (ν1, ν2, ν3) 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 K ν (1⊗1⊗1) in terms of hypergeometric functions.

AB - 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 K ν indexed by a complex multiparameter ν = (ν1, ν2, ν3) 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 K ν (1⊗1⊗1) in terms of hypergeometric functions.

KW - Generalized Bernstein-Reznikov integrals

KW - H-type groups

KW - Intertwining operators

KW - Invariant trilinear forms

KW - Meromorphic continuation

KW - Spherical principal series

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U2 - 10.1142/S0129167X14500177

DO - 10.1142/S0129167X14500177

M3 - Article

AN - SCOPUS:84898540234

VL - 25

JO - International Journal of Mathematics

JF - International Journal of Mathematics

SN - 0129-167X

IS - 3

M1 - 1450017

ER -