A new matrix representation for the Heisenberg group (Q6591711)

The full Heisenberg group \((\mathbb{H},\ast)\), where \(\mathbb{H}\!=\!\mathbb{R}\!\times\!\mathbb{R}^n\!\times\!\mathbb{R}^n\), \((r,t,\omega)\ast(\tilde r,\tilde t,\tilde \omega)\!=\!(r\!+\!\tilde r\!+\!\frac{1}{2}(t'\tilde \omega\!-\!\tilde t'\omega),t\!+\!\tilde t,\omega\!+\!\tilde \omega)\), \(t'\) is the transpose of a column vector \(t\), plays a fundamental role in quantum mechanics and Fourier analysis. The relation \(\pi (r,t,\omega)\!=\!\mathcal{R}_{r+\frac{1}{2}t'\omega}\mathcal{M}_\omega \mathcal{T}_{-t}\), where \((\mathcal{R}_r f)(x)\!=\!\mathrm{e}^{\mathrm{i}r}f(x)\), \((\mathcal{M}_\omega f)(x)\!=\!\mathrm{e}^{\mathrm{i}\omega'x}f(x)\), \((\mathcal{T}_tf)(x)\!=\!f(x\!-\!t)\), defines the Schrödinger representation of \((\mathbb{H},\ast)\) in \(L^2(\mathbb{R}^n)\). The authors investigate a more general case obtained using three non-singular real matrices \(B\), \(W\), \(A\) and the more general transformations \((\mathcal{D}_A f)(x)\!=\!|\det(A)|^{-\frac{1}{2}}f(A^{-1}x)\), \((\mathcal{M}_{W\omega} f)(x)\!=\!\mathrm{e}^{\mathrm{i}x'W\omega}f(x)\), \((\mathcal{T}_{Bt}f)(x)\!=\!f(x\!-\!Bt)\). If \(B\) and \(W\) are chosen such that \(B'W\!=\!\mathbb{I}\), then the considered relations also define a representation of \((\mathbb{H},\ast)\). But, in the case \(G\!=\!B'W\!\not=\!\mathbb{I}\), a more general version \((\mathbb{H},\ast_G)\) of the Heisenberg group is obtained by choosing the multiplication \((r,t,\omega)\ast_G(\tilde r,\tilde t,\tilde \omega)\!=\!(r\!+\!\tilde r\!+\!\frac{1}{2}(t'G\tilde \omega\!-\!\tilde t'G\omega),t\!+\!\tilde t,\omega\!+\!\tilde \omega)\). A unitary representation of \((\mathbb{H},\ast_G)\) is investigated in the second part of the article.