Symmetry breaking operators for real reductive groups of rank one

Abstract

For a pair of real reductive groups $G^{\prime }\subset G$ we consider the space ${\mathrm{Hom}}_{G^{\prime }}(\pi {|}_{G^{\prime }},\tau )$ of intertwining operators between spherical principal series representations $\pi$ of $G$ and $\tau$ of $G^{\prime }$, also called \emph{symmetry breaking operators}. Restricting to those pairs $(G,G^{\prime })$ where ${\dim \mathrm{Hom}}_{G^{\prime }}(\pi {|}_{G^{\prime }},\tau )<\infty$ and $G$ and $G^{\prime }$ are of real rank one, we classify all symmetry breaking operators explicitly in terms of their distribution kernels. This generalizes previous work by Kobayashi--Speh for $(G,G^{\prime })=(\mathrm{O}(1,n+1),\mathrm{O}(1,n))$ to the reductive pairs $$ (G,G') = ({\rm U}(1,n+1;\mathbb{F}),{\rm U}(1,m+1;\mathbb{F})\times F) \qquad \mbox{with $\mathbb{F}=\mathbb{C},\mathbb{H},\mathbb{O}$ and $F<{\rm U}(n-m;\mathbb{F})$.} $$ In most cases, all symmetry breaking operators can be constructed using one meromorphic family of distributions whose poles and residues we describe in detail. In addition to this family, there may occur some sporadic symmetry breaking operators which we determine explicitly.

56 pages