Lemma 7

We have

for every $f\in \mathscr{F}_{\chi}(G)$ and all $f_1$ and $f_2$ in $\mathscr{L}_{\chi}^2(G)$.

These formulas are consequences of the definitions.

For every $g\in G$ and every $f\in \mathscr{F}_{\chi}$(G), the functions $x\mapsto f(g^{-1}x)$ and $x\mapsto f(xg)$ belong to $\mathscr{F}_{\chi}(G)$. They are denoted respectively by $\boldsymbol{\gamma }_{G,\chi}(g)f$ and $\boldsymbol{\delta }_{G,\chi}(g)f$. The mappings $\boldsymbol{\gamma }_{G,\chi}$ and $\boldsymbol{\delta }_{G,\chi}$ are linear representations of G in $\mathscr{F}_{\chi}(G)$. For every $z\in Z$, one has

Let $f\in \mathscr{F}_{\chi}(G)$ and $g\in G$; one has

where all the functions occurring in these equalities are identified with functions on $G/Z$.

The subspace $\mathscr{K}_{\chi}(G)$ of $\mathscr{F}_{\chi}(G)$ is stable under $\boldsymbol{\gamma }_{G,\chi}$ and $\boldsymbol{\delta }_{G,\chi}$. Let $p$ be a real number $\geqslant 1$. Formulae (8) imply that the representations $\boldsymbol{\gamma }_{G,\chi}$ and $\boldsymbol{\delta }_{G,\chi}$, restricted to $\mathscr{K}_{\chi}$(G), extend to continuous isometric linear representations of G in $\mathscr{L}_{\chi}^p$(G), which will be denoted by $\boldsymbol{\gamma }^{(p)}_{G,\chi}$ and $\boldsymbol{\delta }^{(p)}_{G,\chi}$. By passing to quotients, these representations also define isometric representations of G in $L^p_{\chi}$(G), denoted in the same way.

The representations $\boldsymbol{\gamma }^{(2)}_{G,\chi}$ and $\boldsymbol{\delta }^{(2)}_{G,\chi}$ in $L_{\chi}^2(G)$ are unitary, and will be denoted simply by $\boldsymbol{\gamma }_{G,\chi}$ and $\boldsymbol{\delta }_{G,\chi}$, respectively, whenever no confusion with the representations in $\mathscr{F}_{\chi}(G)$ is possible. One also denotes by $\boldsymbol{\varrho }_{G,\chi}$ the continuous representation of $G\times G$ in $\mathscr{L}_{\chi}^2(G)$ or $L^2_{\chi}(G)$ defined by

for every $(g, h)\in G\times G ($cf. Lemma 1 of V, p. 377).

The representation $\boldsymbol{\gamma }_{G,\chi}$ on $L^2_{\chi}(G)$ is none other than the induced representation Ind$^G_Z(\overline{\chi})$.

When $Z =\{e\}$, the following lemma results from INT, VIII, p. 166, § 4, n$^o5$, Prop. 12, since G is unimodular.