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

TL;DR

This paper investigates invariant norms on the $p$-adic Schr"odinger representation of the Heisenberg group, establishing minimality, rigidity, and irreducibility properties, with implications for Fourier analysis over $C_p$.

Contribution

It introduces a family of invariant norms parameterized by a Grassmannian, proves their minimality and rigidity, and explores the structure of their completions and quotients.

Findings

Invariant norms form a family parameterized by a Grassmannian.

The norms exhibit a minimality and rigidity property.

Completing the Schr"odinger representation yields irreducible modules.

Abstract

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