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
SN - 0129-167X
VL - 25
JO - International Journal of Mathematics
JF - International Journal of Mathematics
IS - 3
M1 - 1450017
ER -