The Cos\u3csup\u3eλ\u3c/sup\u3e and Sin\u3csup\u3eλ\u3c/sup\u3e transforms as intertwining operators between generalized principal series representations of SL(n+1,K)

In this article we connect topics from convex and integral geometry with well-known topics in representation theory of semisimple Lie groups by showing that the Cosλ and Sinλ transforms on the Grassmann manifolds Grp(K)=SU(n+1,K)/S(U(p,K)×U(n+1-p,K)) are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup Pp of SL(n+1,K). The index p indicates the dependence of the parabolic on p. The general results of Knapp and Stein (1971, 1980) [20,21] and Vogan and Wallach (1990) [44] then show that both transforms have meromorphic extension to C and are invertible for generic λ∈C. Furthermore, known methods from representation theory combined with a Selberg type integral allow us to determine the K-spectrum of those operators. © 2011 Elsevier Inc.

A Paley-Wiener theorem for the Θ-hypergeometric transform: The even multiplicity case

The Θ-hypergeometric functions generalize the spherical functions on Riemannian symmetric spaces and the spherical functions on non-compactly causal symmetric spaces. In this paper we consider the case of even multiplicity functions. We construct a differential shift operator Dm with smooth coefficients which generates the Θ-hypergeometric functions from finite sums of exponential functions. We then use this fact to prove a Paley-Wiener theorem for the Θ-hypergeometric transform. © 2004 Elsevier SAS. All rights reserved

Support properties and Holmgren\u27s uniqueness theorem for differential operators with hyperplane singularities

Let W be a finite Coxeter group acting linearly on Rn. In this article we study the support properties of a W-invariant partial differential operator D on Rn with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv (supp D f) = conv (supp f) holds for all compactly supported smooth functions f so that conv (supp f) is W-invariant. The main tools in the proof are Holmgren\u27s uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented. © 2005

Ramanujan\u27s Master Theorem for Riemannian symmetric spaces

Ramanujan\u27s Master Theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of compact and noncompact reductive Riemannian symmetric spaces inside a common complexification, we prove an analogue of Ramanujan\u27s Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extends the results proven by Bertram for Riemannian symmetric spaces of rank-one. © 2012 Elsevier Inc.