Corollary

Let $E\subset L^2(G)$ be a finite-dimensional vector subspace defining a subrepresentation of $\boldsymbol{\gamma }_G($resp. of $\boldsymbol{\delta }_G$, of $\boldsymbol{\varrho }_G)$. Then E is contained in $\mathscr{C}(G)$.

In fact, every element of E is a G-finite vector of the representation $\boldsymbol{\gamma }_G$.