The Hausdorff-Young theorem for the matricial groups \(G_{nm} = ax + b\) (Q866815)

For a local field \(\mathbb K\) and for \(m\geq n\geq 1\), let \(G=G_{nm}=\text{GL}_n(\mathbb K)\rtimes M_{nm}(\mathbb K)\) be the group of pairs \((b,a)\), where \(b\in M_{nm}(\mathbb K)\), \(a\in \text{GL}_n(\mathbb K)\), and the multiplication be given by \((b,a)(b',a')=(b+ab',aa')\). Let \(H=L^2(\text{GL}_n(\mathbb K),\frac{du}{| \det(u)| ^n})\) and let \(\pi_\lambda\), \(\lambda\in M_{nm}(\mathbb K)\), be the unitary representation of \(G\) on \(H\) given by \([\pi_\lambda(b,a)\xi](u)=\tau(\text{Tr}(b\lambda u))\xi(ua)\), where \(\tau\) is a non-trivial additive unitary character on \(\mathbb K\). Let \({\mathcal L}_q(H)\), \(1\leq q<\infty\), be the Banach space of bounded linear operators on \(H\) with \(\text{Tr}(| A| ^q)<\infty\) and let \(\delta_q\) denote the unbounded operator defined by \(\delta_q\xi(u)=| \det(u)| ^{mq}\xi(u)\). Let \(1<p<2\), \(\frac{1}{p}+\frac{1}{q}=1\), \(f\in L^1\cap L^p(G_{nm})\), \(F_p(f)\lambda=\pi_\lambda(f)\circ\delta_{1/q}\). Let \(S\) denote the `essential' dual of \(G\), which is the canonical realization of the Grassmannian \(\text{Grass}(m,n)\). The main result of the paper is the following Hausdorff-Young theorem: \(F_p(f)\in L^q(S;{\mathcal L}_q(H))\) and the map \(f\mapsto F_p(f)\) extends uniquely to a linear contraction from \(L^p(G_{nm})\) into \(L^q(S;{\mathcal L}_q(H))\). Note that the case \(p=2\) is (a version of) the Plancherel theorem for \(G\).