A Poisson formula for Heisenberg manifolds
The author obtains a Poisson type formula for the \((2n + 1)\)-dimensional Heisenberg manifolds \((\Gamma \setminus H_ n,{\mathbf m})\) where \(H_ n\) is the \((2n + 1)\)-dimensional Heisenberg group, \(\Gamma\) a cocompact discrete group in \(H_ n\) and \(\mathbf m\) a metric which can be lifted to a left invariant metric on \(H_ n\). Proposition II.2. Let \((\Gamma_ r \setminus H_ n,{\mathbf m})\) be a normalized Heisenberg manifold and \(Z(t) = \sum_{\lambda \in \Sigma(r,{\mathbf m})} e^{-\lambda t}\) \((t > 0)\); then \(Z = L_ H + L_ V + L_ T\) where: 1. \(L_ H(t) = (\text{Vol}({\mathcal L}_ r\setminus {\mathbf R}^{2n},h)/(4\pi t)^ n)\cdot (\sum_{X \in {\mathcal L}_ r - \{0\}} \exp(-\| X\|^ 2 / 4t))\), 2. \(L_ V(t) = (\text{Vol}(\Gamma_ r \setminus H_ n,{\mathbf m})/(4 \pi t)^{(2n + 1)/2})\cdot (\sum_{c \in \mathbb{Z}} \theta_ c(t) \exp(- c^ 2 g/4t))\), 3. \(L_ T(t) = (\text{Vol}(\Gamma_ r \setminus H_ n,{\mathbf m})/(4 \pi t)^{(2n + 1)/2})\cdot (\sum_{c \in \mathbb{Z}^*} \sum_{\alpha \in {\mathcal A}_ c} P_{c,\alpha}(1/\sqrt{t}) \times \exp(-(4\pi | c| \alpha - (4 \pi^ 2 \alpha^ 2/g)) / 4t))\) where \(\theta_ c(t) = \lim_{\varepsilon \to 0} {1\over \sqrt{x}} \int_{\Delta_ \varepsilon} e^{-z^ 2} \times \prod^ n_{j = 1} {d_ j \sqrt{gt} z + icd_ j g/2\over sh(d_ j \sqrt{gt} z + icd_ j g/2)} dz\) and \(P_{c,\alpha}\) is a polynomial of degree \(2 \times \#({\mathcal B}_{c,\alpha}) - 1\), which can be expressed through Hermite polynomials.
scientific article; zbMATH DE number 572052