Decomposition of $C^{\infty }$ intertwining operators for Lie groups

Abstract:

Let $U$ be a unitary representation of a Lie group $G$ in a Hilbert space $\mathcal {K}$ and let ${C^\infty }(U)$ denote the space of differentiable vectors for $U$ given its usual topology. A continuous operator on ${C^\infty }(U)$ is said to be a ${C^\infty }$ intertwining operator for $U$ if it commutes with $U$. It is shown that if one decomposes $U$ via a central decomposition into a direct integral of unitary representations, then every ${C^\infty }$ intertwining operator decomposes into a direct integral of unique ${C^\infty }$ intertwining operators. Furthermore, it is shown that if $U$ is type I and primary, then every ${C^\infty }$ intertwining operator extends to unique bounded (in the sense of $\mathcal {K}$) intertwining operator defined on all of $\mathcal {K}$.