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

2021/01/01 by Amit Ophir, Ophir, Amit
Computer Science · Mathematics · #05A30 #22D12 #46S10 #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2101.00342

openalex publication_date 2021/01/01 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Motivated by questions about \ℂ_p-valued Fourier transform on the\nlocally compact group (\ℚ_p^d,+), we study invariant norms on the\np-adic Schr "odinger representation of the Heisenberg group. Our main result\nis a minimality and rigidity property for norms in a family of invariant norms\nparameterized by a Grassmannian. This family is the orbit of the sup norm under\nthe action of the symplectic group, acting via intertwining operators. We also\nprove general fundamental properties of quotients of the universal unitary\ncompletion of cyclic algebraic representations. Combined with the rigidity\nproperty, we are able to show that the completion of the Schr "odinger\nrepresentation in any of the norms in that family satisfies a strong notion of\nirreducibility and a strong version of Schur's lemma. Norms that can be formed\nas the maximum of a finite number of norms from that family are also studied.\nWe conclude this paper with a list of open questions.\n

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