On restriction of admissible representations

Abstract

We give some results on restriction of admissible representations in local and global cases. In the local case, $\tilde G$ is a locally compact totally disconnected group, and G is its open normal subgroup such that $\tilde G/G$ is finite abelian. We describe explicitly the restriction of an irreducible admissible representation $\tilde \pi$ of $\tilde G$ to G. In the global case, $\tilde G$ is a connected reductive algebraic group and G is its connected reductive algebraic subgroup such that $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{G} = ZG$ with the central torus Z of $\tilde G$, all defined over an algebraic number field. For an irreducible cuspidal automorphic representation $\tilde \pi$ of $\tilde G\left( \mathbb{A} \right)$, we describe the structure as a $G(\mathbb{A})$-module of the space of the cusp forms on $G(\mathbb{A})$ obtained by restriction from the cusp forms on $\tilde G\left( \mathbb{A} \right)$ in the isotypic space of $\tilde \pi$.