## No file available

To read the file of this research,

you can request a copy directly from the author.

Preprints and early-stage research may not have been peer reviewed yet.

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.

To read the file of this research,

you can request a copy directly from the author.

ResearchGate has not been able to resolve any citations for this publication.

We examine a q-analogue of Mahler expansions for continuous functions in p-adic analysis, replacing binomial coefficient polynomials (xn) with a q-analogue (xn)q for a p-adic variable q with |q−1|p

- D Mumford
- M Nori
- P Norman

D. Mumford, M. Nori, P. Norman: Tata lectures on Theta III, Progress in Mathematics 97, Birkhauser,
Boston (1991).

- A M Robert

A. M. Robert: A course in p-adic analysis, Graduate Texts in Mathematics 198 (2000).

- M Emerton

M. Emerton: p-adic L-functions and unitary completions of representations of p-adic reductive groups,
Duke Math. J. 130 (2005), no. 2, 353-392.

- G Van Dijk

G. Van Dijk: Smooth and admissible representations of p-adic unipotent groups, Compositio Mathematica 37 (1978), 77-101

- A Ophir
- E Shalit

A. Ophir, E. de Shalit: q-binomials and non-continuity of the p-adic Fourier transform, The Quarterly
Journal of Mathematics 67, 4, (2016), 653-668.