Global representations of the conformal group and eigenspaces of the Yamabe operator on $S^1 \times S^n$

Mark R. Sepanski and Jose A. Franco

Vol. 275 (2015), No. 2, 463–480

DOI: 10.2140/pjm.2015.275.463

Abstract

Using parabolic induction, a global representation of a double cover of the conformal group $SO{\left(2,n+1\right)}_0$ is constructed. Its space of finite vectors is realized as a direct sum of eigenspaces of the Yamabe operator on $S^1\times S^n$. The explicit form of the corresponding eigenvalues is obtained. An explicit basis of $K$-finite eigenvectors is used to study its structure as a representation of the Lie algebra of the conformal group.

Keywords

globalizations, Yamabe operator, conformal Laplace operator, Lie group, conformal group, parabolic induction

Mathematical Subject Classification 2010

Primary: 22E46, 22E70

Milestones

Received: 8 July 2014

Revised: 10 November 2014

Accepted: 13 November 2014

Published: 15 May 2015