The Cos^λ and Sin^λ transforms as intertwining operators between generalized principal series representations of SL(n+1,K)

Authors

Gestur Ólafsson, Louisiana State University
Angela Pasquale, Université de Lorraine

Document Type

Article

Publication Date

1-15-2012

Abstract

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..

Publication Source (Journal or Book title)

Advances in Mathematics

First Page

267

Last Page

293

Recommended Citation

Ólafsson, G., & Pasquale, A. (2012). The Cos^λ and Sin^λ transforms as intertwining operators between generalized principal series representations of SL(n+1,K). Advances in Mathematics, 229 (1), 267-293. https://doi.org/10.1016/j.aim.2011.08.015