Proposition 6

The following subspaces of $L^2(G)$ are equal:

a) The space Θ(G) ;

b) The algebraic direct sum of the subspaces $\boldsymbol{\varrho }_G(\pi )$ of $L^2(G)$;

c) The space of G-finite vectors (cf. V, p. 376) of $\boldsymbol{\gamma }_G$;

d) The space of G-finite vectors of $\boldsymbol{\delta }_G$;

e) The space of $(G\times G)$-finite vectors of $\boldsymbol{\varrho }_G$.

In particular, every G-finite vector of $\boldsymbol{\gamma }_G,\boldsymbol{\delta }_G$ or $\boldsymbol{\varrho }_G$ belongs to $\mathscr{C}(G)$.

Let us denote by $F_a$ (resp. $F_b, F_c, F_d, F_e$) the space defined by condition a) (resp. b), c), d), e)). We have already observed that $F_a= F_b$.

We have $F_b\subset F_c$ because $\boldsymbol{\varrho }_G(\pi )$ is a finite-dimensional subrepresentation of $\boldsymbol{\gamma }_G$ for every $\pi \in \widehat{G}$. Conversely, let $f$ be a G-finite vector of $\boldsymbol{\gamma }_G$. The subspace $E_f$ generated by the functions $\boldsymbol{\gamma }_G(g)f$ for $g\in G$ is a finite-dimensional subrepresentation of $\boldsymbol{\gamma }_G$. It is equal to the direct sum of its $\pi$-isotypic components for $\pi \in \widehat{G}$ (Prop. 1 of V, p. 457). By Cor. 4 of V, p. 463, this implies that $E_f$ is contained in the sum of the spaces $\boldsymbol{\varrho }_G(\pi )$, hence $F_c\subset F_b$.

By analogous reasoning, one obtains $F_b= F_d$, and since $\boldsymbol{\varrho }_G(\pi )$ is a subrepresentation of $\boldsymbol{\varrho }_G$, one establishes in the same way that $F_b= F_e$.