Description
Let $G$ be a semisimple Lie group and $K$ the maximal compact subgroup. Principal series representations of $G$ induced from a parabolic subgroup $P = MAN$ whose unipotent radical $N$ is abelian have been well studied. The analysis of these representations exploits the fact that $(K,L)$ has the structure a symmetric pair, where $L = K \cap M$. The next easiest case is when $N$ has the structure of a Heisenberg group. We study principal series representations induced from Heisenberg parabolic subgroups and we use the Peter-Weyl theorem for circle bundles over a symmetric space of $K$ to analyze these representations. We study reducibility, complementary series, and unitary subrepresentations, with a focus on $SL(n+2,\mathbb{R})$ and $G_{2(2)}$. This talk is based on joint work with Jan Frahm, Clemens Weiske and Genkai Zhang.