Lemma 6

Tag 03AL, permanent. Cite it as [Bourbaki, Tag 03AL].

Théories spectrales, chapter V, § 2. Représentations des groupes localement compacts

Let $f\in \mathscr{K}_{\overline{\pi}}(G)$ . For every $g\in G$ , the function

\widetilde{f}:x\mapsto \eta (g^{-1}, xH)^{1/p}f(g^{-1}x)

of G into E belongs to $\mathscr{K}_{\overline{\pi}}(G)$ and satisfies $N_p(\widetilde{f}) = N_p(f)$ .

It is verified without difficulty that $\widetilde{f}\in \mathscr{K}_{\overline{\pi}}(G)$ . Since

N_p(\widetilde{f})^p=\int_{G/H}^*\|\widetilde{f}\|^pd\nu =\int_{G/H}^*\gamma_{G/H}(g)(\|f\|^p)(y)\eta (g^{-1}, y)d\nu (y)

and since $(y\mapsto \eta (g^{-1}, y))\cdot \nu =\boldsymbol{\gamma }_{G/H}(g)\nu$ , one obtains $N_p(\widetilde{f}) = N_p(f)$ .

It follows from this lemma that there exists a continuous isometric representation $\widetilde{\pi}$ of G in $L^p_{\overline{\pi}}(G)$ such that for $f\in \mathscr{K}_{\pi}(G)$ and $g\in G$ , the element $\widetilde{\pi}(g)f$ is the class of the function $\widetilde{f}$ defined above. If $p= 2$ , then this representation is unitary.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

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Cites

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