Abstract

Let \(K\) be a complete valued field extension of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. Let \(\mathcal{D}\) be a closed unitary subring of the valuation ring \(\Lambda_K\) of \(K\). Let \(\mathcal{H}(3 , \mathcal{D})\) be the \(3\)-dimensional Heisenberg group with entries in \(\mathcal{D}\). We shall give continuous linear representations of \(\mathcal{H}(3 , \mathcal{D})\) in spaces of functions analogous to Schrödinger representations of the classical Heisenberg group. First, we assume that the ring \(\mathcal{D}\) is compact, then we obtain continuous linear representations of the profinite group \(\mathcal{H}(3 , \mathcal{D})\) in the Banach space of continuous functions \(\mathcal{C}( \mathcal{D}, K)\). On the other hand, considering only that \(\mathcal{D}\) is closed, we obtain by the same way continuous linear representations of \(\mathcal{H}(3 , \mathcal{D})\) in the Banach space \(K< z >\) of restricted power series with coefficients in \(K\) (= the Tate algebra in one variable, i.e. the space of analytic functions on \(\Lambda_K\)) with values in \(K\) on which we consider the Gauss norm. These representations are topologically irreducible. From the second type of representations, one obtains position and momentum bounded operators satisfying Heisenberg commutation relation and the Weyl algebra \(A_1(K) \) as subalgebra of the algebra of bounded linear operators of \(K< z >\). The closure \(\widetilde A_1(K) \) of \(A_1(K) \) is described.

中文翻译：

---

关于 $$p$$ -Adic Heisenberg Groups 的线性表示

摘要

令\(K\)是\(p\)进数的域\(\mathbb{Q}_p\)的完整值域扩展。令\(\mathcal{D}\)是\(K\)的估值环\(\Lambda_K\)的闭合酉子环。令\(\mathcal{H}(3 , \mathcal{D})\)为\(\mathcal{D}\)中具有条目的\(3\)维海森堡群。我们将在函数空间中给出\(\mathcal{H}(3 , \mathcal{D})\)的连续线性表示，类似于经典海森堡群的薛定谔表示。首先，我们假设环\(\mathcal{D}\)是紧的，然后我们获得连续函数的 Banach 空间中的有限群\(\mathcal{H}(3 , \mathcal{D})\)的连续线性表示\(\mathcal{C}( \mathcal{D}, K)\)。另一方面，仅考虑\(\mathcal{D}\)是封闭的，我们通过相同的方式获得\(\mathcal{H}(3 , \mathcal{D})\)在Banach 空间\(K< z >\)的受限幂级数，系数在\(K\)（= 一个变量中的 Tate 代数，即\(\Lambda_K\)上的解析函数空间），值在\(克\)我们考虑高斯范数。这些表示在拓扑上是不可约的。从第二种类型的表示中，我们获得了满足海森堡交换关系的位置和动量有界算子以及作为\(K< z >\)的有界线性算子的代数的外尔代数\(A_1(K) \)的子代数。描述了 \(A_1(K) \)的闭包\(\widetilde A_1(K) \ )。