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The Ginibre point process is given by the eigenvalue distribution of anon-hermitian complex Gaussian matrix in the infinite matrix-size limit. Thisis a determinantal point process (DPP) on the complex plane ${\mathbb{C}}$ inthe sense that all correlation functions are given by determinants specified byan integral kernel called the correlation kernel. Shirai introduced theone-parameter ($m \in {\mathbb{N}}_0$) extensions of the Ginibre DPP and calledthem the Ginibre-type point processes. In the present paper we consider ageneralization of the Ginibre and the Ginibre-type point processes on${\mathbb{C}}$ to the DPPs in the higher-dimensional spaces, ${\mathbb{C}}^D,D=2,3, \dots$, in which they are parameterized by a multivariate level $m \in{\mathbb{N}}_0^D$. We call the obtained point processes the extended Heisenbergfamily of DPPs, since the correlation kernels are generally identified with thecorrelations of two points in the space of Heisenberg group expressed by theSchr\"{o}dinger representations. We prove that all DPPs in this large familyare in Class I of hyperuniformity.

Hyperuniformity of the determinantal point processes associated with the Heisenberg group