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Motivated by questions about $\mathbb{C}_p$-valued Fourier transform on thelocally compact group $(\mathbb{Q}_p^d,+)$, we study invariant norms on the$p$-adic Schr\"odinger representation of the Heisenberg group. Our main resultis a minimality and rigidity property for norms in a family of invariant normsparameterized by a Grassmannian. This family is the orbit of the sup norm underthe action of the symplectic group, acting via intertwining operators. We alsoprove general fundamental properties of quotients of the universal unitarycompletion of cyclic algebraic representations. Combined with the rigidityproperty, we are able to show that the completion of the Schr\"odingerrepresentation in any of the norms in that family satisfies a strong notion ofirreducibility and a strong version of Schur's lemma. Norms that can be formedas the maximum of a finite number of norms from that family are also studied.We conclude this paper with a list of open questions.

On the rigidity of invariant norms on the $p$-adic Schr\"odinger representation