scholarly journals Composition series and intertwining operators for the spherical principal series

1972 ◽  
Vol 78 (6) ◽  
pp. 1053-1060 ◽  
Author(s):  
Kenneth Johnson ◽  
Nolan R. Wallach
Keyword(s):  
Principal Series ◽  
Intertwining Operators ◽  
Composition Series

scholarly journals On the composition series of principal series representations of a three-fold covering group of SL(2, K)

1980 ◽  
Vol 77 ◽  
pp. 177-196 ◽  
Author(s):  
Haluk Aritürk
Keyword(s):  
Local Field ◽  
Principal Series ◽  
Roots Of Unity ◽  
Composition Series ◽  
Covering Group ◽  
Series Representations ◽  
Principal Series Representations

In this paper, we study the composition series of certain principal series representations of the three-fold metaplectic covering group of SL(2, K), where K is a non-archimedean local field. These representations are parametrized by unramified characters μ(x) = |x|s of K× and characters ω of the group of third roots of unity.

Spherical Functions for the Semisimple Symmetric Pair (Sp(2, ℝ), SL(2, ℂ))

2002 ◽  
Vol 54 (4) ◽  
pp. 828-865 ◽  
Author(s):  
Tomonori Moriyama
Keyword(s):  
Spherical Function ◽  
Series Representation ◽  
Principal Series ◽  
Spherical Functions ◽  
Intertwining Operators ◽  
Symmetric Pair ◽  
Radial Part ◽  
Principal Series Representation ◽  
Integral Expression ◽  
One Dimensional

AbstractLet π be an irreducible generalized principal series representation of G = Sp(2, ℝ) induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from π to the representation induced from an irreducible admissible representation of SL(2, ℂ) in G is at most one dimensional. Spherical functions in the title are the images of K-finite vectors by this intertwining operator. We obtain an integral expression of Mellin-Barnes type for the radial part of our spherical function.

Derived intertwining norms for reducible spherical principal series

1996 ◽  
Vol 61 (2) ◽  
pp. 171-188
Author(s):  
J. E. Gilbert ◽  
R. A. Kunze ◽  
C. Meaney
Keyword(s):  
Differential Operators ◽  
Principal Series ◽  
Intertwining Operators ◽  
Second Derivative ◽  
First Order

AbstractWe use the second derivative of intertwining operators to realize a unitary structure for the irreducible subrepresentations in the reducible spherical principal series of U(1, n). These representations can also be realized as the kernels of certain invariant first-order differential operators acting on sections of homogeneous bundles over the hyperboloid (U(1) × U(n))/U(1, n).

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

2014 ◽  
Vol 25 (03) ◽  
pp. 1450017 ◽  
Author(s):  
Salem Ben Said ◽  
Khalid Koufany ◽  
Genkai Zhang
Keyword(s):  
Hypergeometric Functions ◽  
Integral Formula ◽  
Principal Series ◽  
Intertwining Operators ◽  
Real Rank ◽  
Semisimple Lie Groups ◽  
Smooth Functions ◽  
Trilinear Form ◽  
Finite Center ◽  
Rank One

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 [Formula: see text] indexed by a complex multiparameter [Formula: see text] 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 [Formula: see text] in terms of hypergeometric functions.

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