july 2023 p-Adic Schrödinger representations of the higher p-adic Heisenberg groups
Bertin Diarra, Tongobé Mounkoro
Bull. Belg. Math. Soc. Simon Stevin 30(1): 91-129 (july 2023). DOI: 10.36045/j.bbms.220912

Abstract

Let $p$ be a prime number, $\mathbb{Z}_p$ the ring of $p$-adic integers, $\mathbb{Q}_p$ the field of $p$-adic numbers. Let $K$ be a complete valued field extension of $\mathbb{Q}_p$ and $\Lambda_K$ the valuation ring of $K$. Let $\mathcal{D}$ be a closed unitary subring of the ring $\Lambda_K$. Let $n\geg 2$ be an integer and let $\mathcal{H}(2n+1,\mathcal{D})$ be the $(2n+1)$-dimensional Heisenberg group with entries in $\mathcal{D}$. We have studied ultrametric Schrödinger representations of the Heisenberg group $\mathcal{H}(3, \mathcal{D})$ in two kinds of function spaces. In this paper we extend the study to the $(2n + 1)$-dimensional Heisenberg group with entries in $\mathcal{D}$. We study these representations in the case when the ring $\mathcal{D}$ is compact and in the general case their restriction to the algebra of analytic functions on $\mathcal{D}^n$ identified with the Tate algebra $T_n(K)$ in $n$ variables. These representations are topologically irreducible. From the last representations one obtains bounded linear operators satisfying Heisenberg commutation relations and the Weyl algebra $A_n(K)$ as subalgebra of the Banach algebra $\mathcal{L}(T_n(K))$ of bounded linear operators of $T_n(K)$. Its closure $\widetilde{A_n}(K)$ in $\mathcal{L}(T_n(K))$ is described and is seen to be a central simple algebra. Notice that this algebra is different from those considered by Pangalos.

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Bertin Diarra. Tongobé Mounkoro. "p-Adic Schrödinger representations of the higher p-adic Heisenberg groups." Bull. Belg. Math. Soc. Simon Stevin 30 (1) 91 - 129, july 2023. https://doi.org/10.36045/j.bbms.220912

Information

Published: july 2023
First available in Project Euclid: 6 August 2023

Digital Object Identifier: 10.36045/j.bbms.220912

Subjects:
Primary: 20C99
Secondary: 16D30 , 16S32 , 20E50

Keywords: complete Weyl algebras , Higher p-adic Heisenberg groups , Schrödinger linear representations

Rights: Copyright © 2023 The Belgian Mathematical Society

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Vol.30 • No. 1 • july 2023
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