Hardy's theorem for simply connected nilpotent Lie groups (Q2758159)

Suppose that \({\mathfrak g}\) is a nilpotent Lie algebra, and \(G\) the corresponding connected simply connected nilpotent Lie group. It is possible to choose dual inner products on \({\mathfrak g}\) and \({\mathfrak g}^*\) and a subspace \(V\) of \({\mathfrak g}^*\) such that a dense open subset \(X\) of \(V\) parametrises the generic coadjoint orbits of \(G\) in \({\mathfrak g}^*\). Then the generic unitary representations \(\pi_\xi\) of \(G\) are parameterised by \(\xi\) in \(X\). Take exponential coordinates on \(G\). Suppose that \(f\) is an integrable function on \(G\) such that \(|f(x) |\leq C e^{-\pi\alpha|x|^2}\) for all \(x\) in \(G\) and \(\|\pi_\xi (f) \|_{\text{ HS}}\leq C e^{-\pi\beta|\xi|^2}\) for all \(\xi\) in \(X\), where \(\alpha \beta < 1\). Then \(f=0\). This generalises similar theorems on \({\mathbb R}^n\) (due to Hardy) and many other Lie groups.